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DefineOptimization(decisions, maximize/minimize, constraints, type,...)
Defines an optimization problem to find a solution for the decision variables that maximizes (or minimizes) an objective and satisfies some constraints. Any optimization must have at least one
decision variable -- otherwise it has nothing to optimize. If it has no objective, it tries to find any solution that satisfies the constraints. If it has an objective, the constraints are optional.
Analytica automatically discovers the type of problem and selects the appropriate solver engine, according to whether the problem is a Linear Program (LP), Quadratic Program (QP), Quadratically
Constrained Program (QCP), or Non-Linear Program (NLP).
The key parameters for DefineOptimization are «decisions», «minimize» or «maximize», and «constraints». For example, if your has two decision variables, X and Y, this formulates a linear program to
find values of X and Y that maximize the expression subject to two linear constraints:
Decisions: X, Y,
Maximize: 3*X + 2*Y,
Constraints: 5*X + 2*Y <= 900,
8*X + 10*Y <= 2800)
It is best to use named-parameter syntax with DefineOptimization, spelling out the name of each parameter followed by a colon, and the value(s) for that parameter.
A problem definition always specifies one or more decision variables (i.e., there must be at least one thing to solve for or optimize). Decisions define the search space for the optimization. These
are typically decision nodes in your model (the green rectangle nodes), although local variables may also be used as decision variables. Each decision variable may be scalar or array-valued. For
example, when solving for an optimal portfolio allocation, the allocation decision would typically be dimensioned by an investment index.
Many problems include an objective, which is an expression passed to either the «minimize» or «maximize» parameter, such that the expression depends directly or indirectly on the decision variables.
To simply solve a system of equations and inequalities to find a feasible solution, the objective is omitted. When specified, the objective is a standard Analytica expression, typically one that
computes a scalar result when the decision variables are set to a candidate solution.
Constraints are expressions that specify an equality or inequality, which must be satisfied by any solution. A problem with no constraints is referred to as an unconstrained optimization problem and
must have an objective function. Each constraint may be an array, which specifies a collection of constraints to satisfy.
The Optimizer analyzes your model to discover automatically what type of optimization Program it is, and so which solver engine to use. But, you can also specify the «type» parameter as one of: 'LP',
'QP', 'QCP', 'CQCP', 'NCQCP', 'NLP', or 'NSP'. (Linear , Quadratic, Quadratically constrained, Convex Quadratically Constrained, Non-convex Quadratically Constrained, Non-Linear, or Non-Smooth
Program). If you specify the type, it will flag an error if it finds that the actual problem is outside that class -- for example, if you have specified it is an "LP" or linear program and it finds a
non-linear relationship.
All problems types (linear, quadratic, or non-linear) are formulated in the same way, using the structure of an Analytica model, and all formulated using DefineOptimization.
Suppose your model has two decision node variables, X and Y. The following specifies a linear program:
Decisions: X, Y,
Maximize: 3*X + 2*Y,
Constraints: 5*X + 2*Y <= 900,
8*X + 10*Y <= 2800)
Local variables may also be used for the decision variables:
Var X := 0;
Var Y := 0;
Decisions: X, Y,
Maximize: X + 4*Y,
4*X >= Y-2
2*X <= Y+1)
Obtaining the Solution
DefineOptimization defines a problem to be solved or optimized, and analyzes the structure of the problem -- for example to determine it is linear, quadratic, or nonlinear, etc. It does not actually
solve for the solution. The first evaluation of OptSolution will trigger finding a solution, as will any related function for accessing an aspect of the solution: OptStatusNum, OptStatusText,
OptObjective, OptSlack, OptShadow, OptReducedCost, OptObjectiveSa, OptRhsSa.
When DefineOptimization is evaluated, it returns a special object, which displays as: «LP», «QP», «QCP», «CQCP», «NCQCP», «NLP», or «NSP», depending on the type of problem. This object contains an
encoding of the problem definition, and is passed to any of the solution functions mentioned above. You can double-click on the result cell containing «LP» (etc) or evaluate the OptInfo function to
explore the internals of the problem specification. For example, you can use OptInfo to see the tables of linear or quadratic coefficients extracted from your model.
DefineOptimization will typically evaluate your model one to three times to discover the problem type and the dimensionalities of decisions and constraints. The bulk of the solving time is incurred
when it actually solves the problem, triggered by evaluating OptSolution, OptStatusText, or another solution function.
Variable/Node Classes
In most cases, constraints and objectives are encoded within Analytica variables as part of a larger model. When so doing, the variable (or node) class becomes relevant, where the following
distinguished variable classes are used for the respective components of the optimization problem:
For example, the expression to be maximized (or minimized) is placed in the definition of an objective node, and any equality or inequality constraints are placed directly in definitions of
constraint nodes.
The use of variable nodes in a model allows you to structure your optimization model in the form of an influence diagram, break up the computation into many steps, splitting among many variable nodes
and organizing into submodules. It also allows you to use your model outside of an optimization context, for example, when exploring the solution space manually or via parametric analysis. And it
allows further specification of each component within the model nodes themselves. So, for example, if a certain decision variable is integer-valued with special bounds, those can be indicated
directly in the decision variable. If certain decisions or constraints incorporate (assimilate) indexes as part of the optimization problem, these can be indicated directly in the decision or
constraint node. Thus, most of the problem specification exists within the Analytica model structure, rather than in the parameters to DefineOptimization. Once a model is defined (including decision,
constraint and objective nodes), the specific optimization to be solved is defined using DefineOptimization.
Decision Variables
Decision variables define the search space for the optimization, and the set of decision variables to be used is specified in the «decisions» parameter of DefineOptimization. For example:
DefineOptimization(decisions: X, Y, Z, ...)
The variables, X, Y, and Z, may be either identifiers of variables in your model, or may be local variable identifiers.
You can also specify that every decision node in the model should be included by specifying:
DefineOptimization(decisions: All, ...)
Or that all decisions in one or more selected modules using the syntax:
DefineOptimization(decisions: All in Module_A, All in Module_B, ...)
When using All in «module», any decision nodes, aliases or input/output nodes for a decision node that is contained in the indicated module, or in any of its submodules, is included in the
optimization except for those decisions that are descendants of the variable being defined with the DefineOptimization function.
An advantage of using the All or All in «module» syntax is that your optimization formulation automatically adapts as decisions are added or subtracted from your model. However, when using this, you
must take care not to utilize a decision node for a different purpose.
The key "components" of each decision variable are:
• Its domain, which includes:
□ Its integer type: Continuous, Integer, Binary, Grouped Integer, or an explicit set of Discrete possible values.
□ Its bounds
• Its dimensionality (the indexes that a solution will be indexed by)
• An initial guess
You have full control over these aspects through the attributes of a decision variable node.
The Domain Attribute allows you to select the integer-type and bounds. In most cases, you can use the pulldown and enter simple scalar values for bounds. For more complex domains, where some aspect
of the domain is computed, or where the integer type or bounds vary along some index, a computed expression can be specified for the domain. If you use a computed expression for the domain, it should
not depend on other decision variables -- the domain remains constant while any given problem space is being solved. When bounds depend on other decision variables, constraints are used (see below).
If you don't set the domain of a decision, the variable is treated as continuous. You can also override the domain specified in the decision using the optional «domain» parameter of
DefineOptimization. You must set both upper and lower bounds for variables if you plan to use the Evolutionary solver engine.
The decision variable's domain can be computed, but should never depend upon another decision variable in the optimization. If the domain is influenced by another decision variable, the actual domain
(integer type and variable bounds) are determined statically from the definitions of the other decision variables and do not change during the optimization search. For example, suppose you have:
Definition of X := 5;
Domain of Y := Continuous(lb: X)
optDef := DefineOptimization(decisions: X, Y,...)
This formulation encodes the bound that Y >= 5, and not a constraint that Y >= X. To enforce Y >= X, use a constraint.
Decision variables are often array-valued -- that is, they may contain one, two or more dimensions even when there is only a single optimization to be solved. The Intrinsic Indexes attribute in the
Object Window provides you full control when specifying the dimensionality of a decision variable. If you leave Intrinsic Indexes unspecified, Analytica will attempt to infer the intended
dimensionality of each decision variable. Even though it may be able to guess the dimensionality correctly, it is preferred style to explicitly indicate the dimensionality, even if there are no
dimensions (in which case we say it has a scalar dimensionality). To specify the dimensionality, press the Indexes button in the Intrinsic Indexes attribute from the object window of a decision node.
When using a local variable as a decision variable, the intrinsic indexes are specified by including parentheses after the variable name, e.g.:
Var x [Segment, Time] := ...
Finally, in certainly non-linear problems, the initial guess may influence both the time required to find the solution, and the actual solution found. Because non-linear problems may contain local
optima, the starting point for the search may influence which local optima is discovered, or even whether any feasible solution is found at all. The Initial Guess attribute on a decision node can be
used to specify an initial guess, or even an array of initial guesses (occasionally you may want to try solving the same problem from a set of different initial guesses). If you don't specify the
Initial Guess attribute, DefineOptimization will attempt to utilize values contained in the Definition for the initial guess, with some limitations. In general, you will probably want to utilize the
Definition of decision variables for uses of your model outside of optimization, such as simple evaluation, Parametric Analysis, sensitivity analysis, etc. It is for this reason that
DefineOptimization prefers you to use the Initial Guess attribute for specifying initial guesses. Note that when your problem is linear (or even convex quadratic), the initial guess is irrelevant.
You can also override the particular initial guess encoded in a decision node by specifying the optional «guess» parameter to DefineOptimization.
Constraints specify relationships between quantities in your model that must be satisfied in any feasible solution. Constraints should depend directly or indirectly on decision variables. A
constraint specification consists of two items of information:
• An equality or inequality expression.
• A list of intrinsic (assimilated) indexes
A constraint can be specified directly to the «constraints» parameter as an equality or inequality expression, or it can be placed within a constraint node with the definition consisting of an
equality or inequality expression. The equality or inequality expression must be the top-level operation in the definition. When using constraint nodes, list the constraint identifiers in the
«constraints» parameter of DefineOptimization.
You can also specify All or All in «module» as a constraint to indicate that all constraint nodes located in (or with aliases located in) the given module or any of its submodules are to be included
in the optimization.
The following illustrates the various ways that constraints can be specified:
decisions: All,
X + Y <= Z,
X <= Y,
All in Module_A,
All in Module_B,
A single constraint expression will generally encode a collection of constraints when inputs to the constraint expression are array-valued. These dimensions may either be intrinsic to your problem,
or abstracted. Intrinsic dimensions result in multiple constraints within a single optimization problem. Abstracted dimensions result in multiple separate optimization problems. In a constraint node,
use the OptDimensions attribute to list intrinsic dimensions of your problem. By pressing the Indexes button, you can select the intrinsic indexes, or select none when no extra dimensions should be
assimilated into the optimization problem. If you leave the Intrinsic Indexes unspecified, Analytica will attempt to infer which extra dimensions should be assimilated or abstracted.
See also Display of constraint results.
When you wish simply to find a solution to a set of equations and inequalities, the objective need not be specified. Otherwise, the expression is specified using either the «maximize» or «minimize»
parameter of DefineOptimization. The expression parameter accepts an Analytica expression that evaluates to a single number within a single optimization problem. E.g.:
decisions: All,
minimize: Cost)
It is a stylistic convention to place objective expressions within an objective node, so that the objective is visually distinguished when viewing an influence diagram depiction of your model. With
an objective node, simply place the identifier in the «maximize» or «minimize» parameter of DefineOptimization. If you are maximizing or minimizing a statistical quality such as the Mean, then it is
nicer to set the «maximize» parameter to the expression Mean(MyObjective), rather than including the call to Mean within the node's definition.
Problem Type
Most solver engines can only solve some problem types. So you, the modeler, need to figure out ahead of time what type of problem you are trying to solve, according to whether the objective and
constraints are linear, quadratic, or other nonlinear functions of the decision variables, and so select the appropriate engine. The Analytica Optimizer can usually figure out the problem type
itself. It then automatically chooses the appropriate solver engine. These are the problem types it can handle:
• LP: Linear objective and linear constraints
• QP: Quadratic objective and linear constraints
• QCP: Linear or quadratic objective and quadratic constraints. This is further sub-categorized as:
□ CQCP: Convex quadratic
□ NCQCP: Non-convex quadratic
• NLP: (Smooth) Non-linear objective and constraints
• NSP: Non-smooth (hence non-linear) objective and constraints
Parts of your model that do not depend on decision variables can be of any functional form without impacting the problem type.
If DefineOptimization finds that all constraints and the objective are linear functions of the decision variables, it automatically sets the problem type to «LP», infers all the linear coefficients,
and selects the "LP/Quadratic" solver engine. Or if it finds that some relationships are quadratic, it automatically selects that problem type, computes the quadratic coefficient matrices, and
selects the appropriate engine . It may not discover the convexity or non-convexity of a set of quadratic constraints until it starts to solve the problem; but if a non-convexity is detected, it
changes the problem type to NCQCP and and uses the "GRG Nonlinear" solver engine.
You always have the option of setting the problem type yourself using the «type» parameter, rather than rely on Analytica to infer it. This may eliminate the need for some aspects model analysis by
DefineOptimization, which could reduce computation time slightly. Also, if you specify the «type», DefineOptimization issue an error if the relationships in the model are not consistent with that
problem type. This is often useful when you expect the model to be linear (or quadratic), and want to be notified if you have accidentally introduced some relation that is not linear (or not
You can also specify any problem type as:
• continuous - every decision variables is continuous
• integer -- every decision variable is boolean, integer, or discrete with explicit values, or
• mixed-integer some decisions are continuous and some are integer.
The integer-ness or otherwise of each decision variable is taken directly from its Domain.
Solver Engines
Analytica Optimizer ships with four solver engines: "LP/Quadratic", "SOCP Barrier", "GRG Nonlinear", and "Evolutionary". Normally, DefineOptimization automatically chooses the most appropriate solver
based on how the problem is formulated. But, sometimes you may want to tell it which engine to use, specify the «engine» parameter, e.g.:
DefineOptimization(.., Engine: "GRG Nonlinear")
You can also purchase a range of more powerful add-on solver engines for challenging applications with large numbers of decision variables and constraints. See [1] to purchase one. See information
Examining Engine Capabilities to learn the numbers of decisions and constraints handled by each engine type.
If you have installed an add-on engine, you may specify its use by setting the «engine» parameter.
Engine settings
Each solver engine has settings that control its Termination Controls, algorithm methods, and so on. You can see the full list of settings relevant to a specified engine using this function:
OptEngineInfo(«Engine», "SettingNames")
You can see the default settings with:
OptEngineInfo(«Engine», "Defaults")
You can change one engine setting using the «settingName» and «settingValue» parameters, like this:
DefineOptimization(..., settingName: "Multistart", settingValue: 1)
If you want to specify several engine settings, it is easier to create a 1-D array where the index labels each contain a setting name and the array cells contain the corresponding values. You then
pass the array to the «settingValue» parameter.
Debugging Problem Formulations
Debugging an optimization model can be very difficult. Many nasty complications appear in non-linear models in particular, such as local minima and maxima, and unusual surfaces with gradients
pointing off in unexpected directions, and singularities that result in INF or NaN values, which may thwart successful convergence to a solution. Identifying which constraints to blame when no
feasible solution exists is seldom obvious. And mistakes with dimensionality may create a problem formation different from that which you expected.
No feasible solution
A big advantage of linear over nonlinear optimization is that you can use special tools to diagnose which constraint(s) are to responsible for the absence of any feasible solution. The OptFindIIS
function identifies the minimal set of constraints in your problem that form a contradiction -- i.e. cannot be satisfied simultaneously.
For quadratic and non-linear problems, debugging the no feasible solution problem requires more ingenuity. The best strategy is to relax constraints that you expect might be the hardest to satisfy.
For example, maybe it would be easy to find a feasible solution when you are not limited by budget; so, you might start by removing the budget constraint from the «constraints» parameter, or simply
increase the available budget amount. Once you obtain a formulation with a feasible solution, you may gain further insight into the nature of your problem.
For non-linear problems, it may help to study the trace file. For additional hints on diagnosing non-feasible formulations, see no feasible solution warning.
Trace File (NLPs)
When solving an NLP (or when using the "GRG Nonlinear" or "Evolutionary" engine on an LP or QP), you can generate a trace file log of the points explored during the solution process. To do this,
specify a file name for the trace file in the «traceFile» parameter, thus:
DefineOptimization(..,engine: "GRG Nonlinear", traceFile: "C:\Temp\trace.log")
Once the solve is attempted, load the indicated file into a text editor such as NotePad or TextPad. Each point searched is logged, along with the values of the objective and constraints, with
violated constraints flagged with a (!). Many mysteries are solved quickly when viewing the trace file -- it often becomes quite obvious why the solver engine did what it did.
Examining internals of problem specification
DefineOptimization returns an object that displays as «LP», «NLP», etc. When this displays in a result table, you can double click on it to view the internals. Double clicking on it brings up the
same result obtained when you evaluate OptInfo(def, "All").
You can access individual parts of the formulation more directly using other options of the OptInfo function. The OptInfo function returns information about the internals of an optimization problem
instance. For example, to see exactly what decision variables and constraints are present and how intrinsic dimensions were "flattened", view the result of each of these expressions:
OptInfo(probDef, "Variables")
OptInfo(probDef, "Constraints")
Numerous other components can be viewed using OptInfo, including bounds and linear/quadratic coefficients, which may help establish whether the problem formation was interpreted as you intended.
Array Abstraction Control
When an extra dimension is identified in the objective, a constraint, initial guess, a variable's domain, or an optional parameter of DefineOptimization, which is otherwise not determined to be
intrinsic to the optimization problem itself, Analytica's standard intelligent array mechanism will array abstract over that index to produce multiple problem instances, where each problem instance
shall be solved separately, and where each instance varies in its formation accordingly. Hence, a single DefineOptimization call may define an array of optimization problems, each to be solved
When you want to force array abstraction over a particular index or indexes, you can list these indexes in the «over» parameter of DefineOptimization. For example:
DefineOptimization(..., over: Plant)
indicates that a separate optimization should be solved for each Plant.
Linear and quadratic optimization problems array abstract pretty seamlessly, such that if you introduce new indexes into your model in the future, the principles of array abstraction kick in and
automatically generalize to an array of independent optimizations. The same can also be said of non-linear optimizations defined via DefineOptimization, except that certain substantial inefficiencies
may arise when non-linear models are array abstracted. In most cases, you can circumvent these with some model modification and appropriate use of the «SetContext» parameter. Without circumventing
these inefficiencies, the non-linear optimizations will array abstract in a functionally correct fashion, so that you can evaluate and view the results, but you may find excessive computation times,
especially as your abstracted indexes grow in length.
The potential inefficiency with abstracted NLPs results because your constraints, objective, and intermediate variables within your model may be computing array-valued results along the abstracted
index, even when the optimization problem actively being solved corresponds to only a single position along that index. To eliminate this inefficiency, you must refine your model so that all children
of the index can continue to successfully evaluate when the list definition of the index is replaced by a single value (for example, Tables must be replaced by DetermTables). Once this is the case,
you can list the index in the «SetContext» parameter. As any single NLP is being solved, the indicated context variable will be replaced with the single value. See Using SetContext to efficiently
solve NLPs.
Optional Parameters
Overriding Integer type and bounds
DefineOptimization has a large number of additional optional parameters, mostly esoteric.
The «Domain» parameter can be used to override the domain appearing in decision variable nodes, or to provide an integer type or bounds for local decision variables. The «Domain» parameter accepts
expressions composed of Domain Specification Functions, and is a repeated parameter where the expressions are placed in positional correspondence with the list of decision variables in the
«decisions» parameter. For example:
decisions: X, Y1, Y2, U, Z,
domain: Continuous(-1, 1), GroupedInteger("G1"), GroupedInteger("G1"), Null, Boolean(),
... )
In the above, X is treated as continuous, bounded by -1 ≤ X ≤ 1, Y1 and Y2 are grouped integer variables belonging to the same group, and Z is a boolean integer. U's domain is not overriden.
Overriding Initial Guess
The initial guess for each variable can likewise be specified as a parameter, overriding the Initial Guess attribute or other inferred initial value. Like «Domain», the «Guess» attribute is a
repeated parameter in positional correspondence with the variables listed in the «decisions» parameter.
decisions: X, Y1, Y2, U, Z,
guess: 0.5, null, null, 0, 1,
In the example, initial values for X, U, and Z are provided. The initial guess for Y1 and Y2 are not overridden.
Preserving values
DefineOptimization, and the solution process for an NLP, conduct a series of WhatIf type evaluations on the model. Previously computed values are preserved by default, meaning that they are returned
once the solution process completes.
You can set the optional parameter «preserve» to false to turn this off. Doing so eliminates the memory overhead required to save all the previously computed values, but will require all those values
to be recomputed if they are viewed again later. If you are having trouble completing your solution within available memory, this switch may help a little; otherwise, there is little reason to use
Indexes of local decision variables and constraint expressions
Whereas structured optimization models include explicit objects in the model for decisions and constraints, some code (often in User-Defined Functions) defines and solves an optimization in an
expression that stands on its own, without external decision and constraint objects. Without the objects, you don't have the attributes such as Intrinsic Indexes and Domain for specifying information
about the decisions and constraints.
In stand-alone problems, local variables are used for decision variables, and expressions for constraints and objective are specified as expressions. For example:
Local x := 0;
Local opt := DefineOptimization( Decisions: x, Constraints: x^2 = Exp(x) );
If you need to explicitly specify intrinsic indexes for decision variables, you do so by specifying the indexes in the local declaration. If you need to explicitly specify intrinsic indexes for any
of the constraints, use the optional «InstrinsicIndexes» parameter (which requires Analytica 5.0). For example:
Local pt[Dim] := target_point; { Uses the local as the decision variable }
Decisions: pt,
Minimize: Sum( (pt-target_point)^2, Dim ) ,
Constraints: Sum( (pt-circle_center)^2, Dim)) <= Circle_radii^2,
IntrinsicIndexes: Circle_index
The above example is found in the "NLP with Jacobian" model (with partial derivative specification not shown here), and the closest point target_point that is also inside all specified circles (or
spheres or hyper-spheres when the Dim index has more than 2 elements. The declaration of x specifies that x has the Dim index (i.e., it is a point in IndexLength(Dim)-dimensional space).
DefineOptimization uses is usual heuristics to infer the dimensionality of decisions and constraints from all the items in the problem definition, but in this case, without specifying Circle_index in
the «IntrinsicIndexes», it will infer that you want to solve a separate optimization problem for each circle (i.e., that Circle_index is extrinsic.
When one or more indexes are listed in «IntrinsicIndexes», these apply to any constraints whose evaluation has those indexes when the decisions are evaluated with their dimensionality. It applies to
all constraints where those indexes appear, including global constraint objects, even if they also specify the Intrinsic Indexes parameter.
DefineOptimization was introduced in Analytica 4.3 to replace the older LpDefine, QpDefine and NlpDefine functions. These older functions are still available for backward compatibility.
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2022 CMS Summer Meeting
Time series analysis: Inference and prediction
Masoud Nasari
(Bank of Canada and Carleton University) and
Mohamedou Ould-Haye
(Carleton University)
The rapid growth in the size and scope of data sets in a variety of disciplines have naturally led to the usage of the term, Big Data. The analysis of such data is important in multiple research
fields such as digital marketing, gene expression arrays, social network modeling, clinical, genetics and phenotypic data, bioinformatics, personalized medicine, environmental, neuroscience,
astronomy, nanoscience, among others. In high-dimensional models where number of predicting variables is greater than observations, many penalized regularization strategies were studied for
simultaneous submodel selection and post-estimation. Generally speaking, submodel are subject to inherited bias, and the prediction based on a selected submodel may not be preferable. For this
reason, we propose a high-dimensional shrinkage strategy to improve the prediction performance of a submodel. Such a high-dimensional shrinkage estimator (HDSE) is constructed by shrinking a
overfitted model estimator in the direction of a candidate submodel. We demonstrate that the proposed HDSE performs uniformly better than the overfitted model estimator. Interestingly, it
improves the prediction performance of a given candidate submodel. The relative performance of the proposed HDSE strategy is appraised by both simulation studies and the real data analysis.
Graphical models offer a powerful framework to capture intertemporal and contemporaneous relationships among the components of a multivariate time series. For stationary time series, these
relationships are encoded in the multivariate spectral density matrix and its inverse. We will present adaptive thresholding and penalization methods for estimation of these objects under
suitable sparsity assumptions. We will discuss new optimization algorithms and investigate consistency of estimation under a double-asymptotic regime where the dimension of the time series
increases with sample size. If time permits, we will introduce a frequency-domain graphical modeling framework for multivariate nonstationary time series that captures a new property called
conditional stationarity.
Cluster indices describe extremal behaviour of stationary time series. We consider their sliding blocks estimators. Using a modern theory of multivariate, regularly varying time series, we obtain
central limit theorems under conditions that can be easily verified for a large class of models. In particular, we show that in the Peaks-Over-Threshold framework, sliding and disjoint blocks
estimators have the same limiting variance.
The long-run variance matrix and its inverse, the so-called precision matrix, give respectively information about correlations and partial correlations between dependent component series of
multivariate time series around zero frequency. This talk will present non-asymptotic theory for estimation of the long-run variance and precision matrices for high-dimensional time series under
general assumptions on the dependence structure including long-range dependence. The presented results for thresholding and penalizing versions of the classical local Whittle estimator ensure
consistent estimation in a possibly high-dimensional regime. The highlight of this talk is a concentration inequality of the local Whittle estimator for the long-run variance matrix around the
true model parameters. In particular, it handles simultaneously the estimation of the memory parameters which enter the underlying model. Finally, we study the temporal and spatial dependence of
multiple realized volatilities of global stock indices.
Tests of serial independence are presented for a fixed number of consecutive observations from a stationary time series, first in the univariate case, and then in the multivariate case, where
even vectors of large dimensions can be used. The common distribution function of the time series is not assumed to be continuous, and the tests statistics are based on the multilinear copula
process. A case study using a time series of images is used to illustrate the usefulness of the methodologies presented.
We investigate some aspects of time series data spectral analysis. We consider random sampling of continuous processes and stationarity tests.
In this talk, it is shown that under weak assumptions, the change-point tests designed for independent random vectors can also be used with pseudo-observations for testing change-point in the
joint distribution of non-observable random vectors, the associated copula, or the margins, without modifying the limiting distributions. In particular, change-point tests can be applied to the
residuals of stochastic volatility models or conditional distribution functions applied to the observations, which are prime examples of pseudo-observations. Since the limiting distribution of
test statistics depends on the unknown joint distribution function or its associated unknown copula when the dimension is greater than one, we also show that iid multipliers and traditional
bootstrap can be used with pseudo-observations to approximate P-values for the test statistics. Numerical experiments are performed in order to compare the different statistics and bootstrapping
methods. Examples of applications to change-point problems are given. This is joint work with Bouchra R. Nasri and Tarik Bahraoui.
Epidemic models provide an insight on how to react to an epidemic outbreak. For this matter, we investigate several aspects of stochastic dynamical systems according to different incidence rates
and perturbations. We carry out a thorough analysis to show the existence of the global and positive solutions. We explore the extinction and the persistence of the disease regarding a derived
stochastic threshold of the model. Moreover, we use suitable Lyapunov functions in order to explore the impact of the perturbation on the stability around the equilibrium points. Finally, we give
some numerical illustrations to support our analytical results.
This work presents powerful goodness-of-fit procedures for general Markov regime-switching models with covariates when the outcomes are continuous, discrete, or zero-inflated. The EM algorithm is
used for the estimation method and a randomized Rosenblatt’s transform is applied to obtain formal goodness-of-fit tests. The latter then served for selecting the number of regimes. Numerical
experiments are used to assess the finite sample performance of the proposed methodologies and to compare with other criteria for the selection of models, including Bayesian methods. Finally, the
proposed methodologies are implemented in an R package available on CRAN.
We shall present an anomaly-detection method when systematic anomalies, possibly statistically very similar to genuine inputs, are affecting control systems. The method allows anomaly-free inputs
to originate from a wide class of random sequences. To illustrate how the method works on data, we shall provide a controlled experiment with anomaly-free inputs following an ARMA time series
model under various contamination scenarios. | {"url":"https://www2.cms.math.ca/Events/summer22/abs/ti","timestamp":"2024-11-07T23:24:31Z","content_type":"text/html","content_length":"20303","record_id":"<urn:uuid:bb99fba1-2b47-4f84-8145-9a6ef8937ce6>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00639.warc.gz"} |
Convert fahrenheit to rankine
How do you convert Fahrenheit to Rankine?
To convert a temperature from Fahrenheit to Rankine, you can use a simple formula: Rankine = Fahrenheit + 459.67. Just add 459.67 to the Fahrenheit temperature and you'll have the equivalent value in
Rankine. It's a straightforward calculation that helps you understand the relationship between these two temperature scales.
Why is it important to convert Fahrenheit to Rankine?
Converting from Fahrenheit to Rankine is important when you need to work with absolute temperature measurements. Rankine is an absolute temperature scale, similar to Kelvin. By converting Fahrenheit
to Rankine, you can accurately measure temperatures in scientific and engineering applications where absolute temperature values are required.
What are the similarities and differences between Fahrenheit and Rankine?
Fahrenheit and Rankine are both temperature scales used to measure temperature. However, the main difference lies in their starting points. Fahrenheit starts at 32°F while Rankine starts at absolute
zero, which is 0°R. This means that Rankine is an absolute temperature scale, while Fahrenheit is a relative temperature scale.
Can I convert temperatures from Rankine to Fahrenheit?
Certainly! Converting temperatures from Rankine to Fahrenheit is also possible. You can use the formula: Fahrenheit = Rankine - 459.67. By subtracting 459.67 from the Rankine temperature, you can
obtain the corresponding Fahrenheit value. This conversion is useful when you need to switch between these two temperature scales.
Where are Fahrenheit and Rankine commonly used?
Fahrenheit is most commonly used in the United States for everyday temperature measurements, such as weather forecasts and household thermostats. On the other hand, Rankine is primarily used in
scientific and engineering fields, especially when dealing with absolute temperature calculations and thermodynamics. | {"url":"https://calculatebox.com/fahrenheit-to-rankine","timestamp":"2024-11-10T21:01:29Z","content_type":"text/html","content_length":"22047","record_id":"<urn:uuid:6fe4086c-2dd0-4ced-9039-f8ce7667e8c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00222.warc.gz"} |
Math Question - Did anyone have to memorize squares to 30
Math Question - Did anyone have to memorize squares to 30
[ 16 posts ]
Joined: 19 Aug 2024
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Hi All. For all the math lovers out there I was just curious but did any of you have to memorize your squares up to a certain number in high school? My math teacher had us memorize the squares up to
30 and we could do higher for extra credit. I still have them all memorized up to 30 today. It was useful I think in Trig and Calc at times. I was just curious how many other teachers did the same
thing. I'm glad ours had us do this.
"In this galaxy, there’s a mathematical probability of three million Earth-type planets. And in all the universe, three million million galaxies like this. And in all of that, and perhaps more...only
one of each of us. Don’t destroy the one named Kirk." - Dr. Leonard McCoy, "Balance of Terror", Star Trek: The Original Series.
Joined: 19 Aug 2024
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MatchboxVagabond wrote:
No, we just did the times table to 10x10 as they were still teaching long division and multiplication on paper and those hundred values could get the rest.
Ah ok. Ours taught long division and multiplication on paper as well. Makes me wonder how they are teaching it now. I was taught memorizing the squares in I think it was "Applied Math" or "Algebra
I". I want to say it was "Applied Math". I started out on a nonacademic track and then switched to academic courses my sophomore year.
"In this galaxy, there’s a mathematical probability of three million Earth-type planets. And in all the universe, three million million galaxies like this. And in all of that, and perhaps more...only
one of each of us. Don’t destroy the one named Kirk." - Dr. Leonard McCoy, "Balance of Terror", Star Trek: The Original Series.
Brian0787 wrote:
MatchboxVagabond wrote:
No, we just did the times table to 10x10 as they were still teaching long division and multiplication on paper and those hundred values could get the rest.
Ah ok. Ours taught long division and multiplication on paper as well. Makes me wonder how they are teaching it now. I was taught memorizing the squares in I think it was "Applied Math" or "Algebra
I". I want to say it was "Applied Math". I started out on a nonacademic track and then switched to academic courses my sophomore year.
TBH, it's been a few years since I had any involvement in math professionally, but it was quite honestly terrifying to have students that were at least a year into college math that couldn't do any
of it without a calculator. One student in particular was in calculus 3 and couldn't do any arithmetic without a calculator. It's incredibly time consuming to have to go for the calculator for every
problem and it makes quick estimates about the reasonableness of a given result a lot harder.
Memorization of squares is kind of iffy in terms of any real value, any of the integers being multiplied together up to about 144 is somewhat useful, beyond that, students have a lot of other things
they can memorize that are of more value.
The way it's rationalized is that you'll always have a calculator with you and the concepts are important. The concepts were always important, but you've got calculators that can do basic
approximations of simple calculus problems for under $20, but that shouldn't get students in some programs off the hook from knowing how to do calculus.
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Brian0787 wrote:
Did anyone have to memorize squares to 30?
Not I.
Then again, squares are not impossible for me to work out in my head.
No love for Hamas, Hezbollah, Iranian Leadership, Islamic Jihad, other Islamic terrorist groups, OR their supporters and sympathizers.
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MatchboxVagabond wrote:
Brian0787 wrote:
MatchboxVagabond wrote:
No, we just did the times table to 10x10 as they were still teaching long division and multiplication on paper and those hundred values could get the rest.
Ah ok. Ours taught long division and multiplication on paper as well. Makes me wonder how they are teaching it now. I was taught memorizing the squares in I think it was "Applied Math" or "Algebra
I". I want to say it was "Applied Math". I started out on a nonacademic track and then switched to academic courses my sophomore year.
TBH, it's been a few years since I had any involvement in math professionally, but it was quite honestly terrifying to have students that were at least a year into college math that couldn't do any
of it without a calculator. One student in particular was in calculus 3 and couldn't do any arithmetic without a calculator. It's incredibly time consuming to have to go for the calculator for every
problem and it makes quick estimates about the reasonableness of a given result a lot harder.
Memorization of squares is kind of iffy in terms of any real value, any of the integers being multiplied together up to about 144 is somewhat useful, beyond that, students have a lot of other things
they can memorize that are of more value.
The way it's rationalized is that you'll always have a calculator with you and the concepts are important. The concepts were always important, but you've got calculators that can do basic
approximations of simple calculus problems for under $20, but that shouldn't get students in some programs off the hook from knowing how to do calculus.
I agree with you. I knew the squares up to 12 I think before that and for some reason he wanted us to memorize the squares past that up to 25 with up to 30 being extra credit. It came in help
somewhat as I ran into some squares higher than 20 and didn't need to pull out a calc to know what the root was. I absolutely agree with you about it being kind of scary that students can't do
anything without a calculator. I agree about the use of calculators in Calc too. I had a TI-89 graphing calculator but just mainly used the graph functions sometimes but I knew how to do some graphs
without it.
"In this galaxy, there’s a mathematical probability of three million Earth-type planets. And in all the universe, three million million galaxies like this. And in all of that, and perhaps more...only
one of each of us. Don’t destroy the one named Kirk." - Dr. Leonard McCoy, "Balance of Terror", Star Trek: The Original Series.
Emu Egg
Joined: 25 Aug 2024
Age: 33
Gender: Male
Posts: 5
Location: Colorado
Ah ok. Ours taught long division and multiplication on paper as well. Makes me wonder how they are teaching it now. I was taught memorizing the squares in I think it was "Applied Math" or "Algebra
I". I want to say it was "Applied Math". I started out on a nonacademic track and then switched to academic courses my sophomore year.[/quote]
TBH, it's been a few years since I had any involvement in math professionally, but it was quite honestly terrifying to have students that were at least a year into college math that couldn't do any
of it without a calculator. One student in particular was in calculus 3 and couldn't do any arithmetic without a calculator. It's incredibly time consuming to have to go for the calculator for every
problem and it makes quick estimates about the reasonableness of a given result a lot harder.
Memorization of squares is kind of iffy in terms of any real value, any of the integers being multiplied together up to about 144 is somewhat useful, beyond that, students have a lot of other things
they can memorize that are of more value.
The way it's rationalized is that you'll always have a calculator with you and the concepts are important. The concepts were always important, but you've got calculators that can do basic
approximations of simple calculus problems for under $20, but that shouldn't get students in some programs off the hook from knowing how to do calculus.[/quote]
I agree with you. I knew the squares up to 12 I think before that and for some reason he wanted us to memorize the squares past that up to 25 with up to 30 being extra credit. It came in help
somewhat as I ran into some squares higher than 20 and didn't need to pull out a calc to know what the root was. I absolutely agree with you about it being kind of scary that students can't do
anything without a calculator. I agree about the use of calculators in Calc too. I had a TI-89 graphing calculator but just mainly used the graph functions sometimes but I knew how to do some graphs
without it.[/quote]
Growing up we had to learn to memorize them up to 10x10, but then learned up to 12x12 because I liked math? Later ended up memorizing up to 16x16 because 256 or 2^8 which is the max number of numbers
(so 0-255) a single byte can represent in a computer.
I can't really see the benefit of memorizing past a point, since if you actually learn and memorize concepts and how to calculate you can always just put in a bit of work to get to the correct
Also agree about the use of calculators as well, I think they have their place and are useful to expedite things once you truly understand the concepts.
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rcostheta wrote:
Growing up we had to learn to memorize them up to 10x10, but then learned up to 12x12 because I liked math? Later ended up memorizing up to 16x16 because 256 or 2^8 which is the max number of numbers
(so 0-255) a single byte can represent in a computer.
I can't really see the benefit of memorizing past a point, since if you actually learn and memorize concepts and how to calculate you can always just put in a bit of work to get to the correct
Also agree about the use of calculators as well, I think they have their place and are useful to expedite things once you truly understand the concepts.
Ah ok, very cool. That makes sense with memorizing 16x16 and I heard about 256 being the max number of numbers a single byte can represent.
"In this galaxy, there’s a mathematical probability of three million Earth-type planets. And in all the universe, three million million galaxies like this. And in all of that, and perhaps more...only
one of each of us. Don’t destroy the one named Kirk." - Dr. Leonard McCoy, "Balance of Terror", Star Trek: The Original Series.
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Not me.
The reason why I suck at maths is because I really don't even have the language comprehension to understand what's being pointed out.
However, when I suddenly started playing with numbers, that's when I actually learned multiplication is just addition of an addition.
But there are no encounters with 'playing with numbers' during classes at all.
It's just example problem then solution. Doesn't explain why is that the solution.
Most at the time all I see is steps 1 and last; and nothing in between.
Also no, memorization just won't do with me at all.
I did not have the compensatory basis related to patterns towards numbers like I do with words that I can turn into knowing spellings in elementary until I found that multiplication does has a
Again, no one ever pointed it out for me.
Let alone actually able to communicate an already very abstract concept that isn't easily communicated to someone like me who just doesn't have enough language comprehension to learn well with a
largely verbal medium of learning.
Joined: 19 Aug 2024
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traven wrote:
in highschool??
we used slide rules
Ah, very neat! My math teacher in high school told us about slide rules. I wish we could have had some hands on time with them
"In this galaxy, there’s a mathematical probability of three million Earth-type planets. And in all the universe, three million million galaxies like this. And in all of that, and perhaps more...only
one of each of us. Don’t destroy the one named Kirk." - Dr. Leonard McCoy, "Balance of Terror", Star Trek: The Original Series.
Joined: 19 Aug 2024
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Gender: Male
Posts: 597
Location: Pennsylvania, United States
Edna3362 wrote:
Not me.
The reason why I suck at maths is because I really don't even have the language comprehension to understand what's being pointed out.
However, when I suddenly started playing with numbers, that's when I actually learned multiplication is just addition of an addition.
But there are no encounters with 'playing with numbers' during classes at all.
It's just example problem then solution. Doesn't explain why is that the solution.
Most at the time all I see is steps 1 and last; and nothing in between.
Also no, memorization just won't do with me at all.
I did not have the compensatory basis related to patterns towards numbers like I do with words that I can turn into knowing spellings in elementary until I found that multiplication does has a
Again, no one ever pointed it out for me.
Let alone actually able to communicate an already very abstract concept that isn't easily communicated to someone like me who just doesn't have enough language comprehension to learn well with a
largely verbal medium of learning.
I understand what you are saying! You like to understand the process of how the answer came about and not just seeing the example problem and then the solution itself. I had alot of those questions
with Algebra in high school and different theorems. I could memorize them but knowing how they came about would help understand the concepts better. I completely understand where you are coming from
"In this galaxy, there’s a mathematical probability of three million Earth-type planets. And in all the universe, three million million galaxies like this. And in all of that, and perhaps more...only
one of each of us. Don’t destroy the one named Kirk." - Dr. Leonard McCoy, "Balance of Terror", Star Trek: The Original Series.
Mona Pereth
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Brian0787 wrote:
Ah, very neat! My math teacher in high school told us about slide rules. I wish we could have had some hands on time with them
I'm glad I learned about slide rules in high school too. Good way to get an intuitive feel for logarithms.
- Autistic in NYC - Resources and new ideas for the autistic adult community in the New York City metro area.
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Mona Pereth wrote:
Brian0787 wrote:
Ah, very neat! My math teacher in high school told us about slide rules. I wish we could have had some hands on time with them
I'm glad I learned about slide rules in high school too. Good way to get an intuitive feel for logarithms.
That's cool you got to use them! I think my Math teacher in high school brought one in I seem to remember. I remember him mentioning the logarithmic scale they used too!
"In this galaxy, there’s a mathematical probability of three million Earth-type planets. And in all the universe, three million million galaxies like this. And in all of that, and perhaps more...only
one of each of us. Don’t destroy the one named Kirk." - Dr. Leonard McCoy, "Balance of Terror", Star Trek: The Original Series.
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I didn't do algebra at school only long multiplication, addition, subtraction. I was even put a year behind at school and only left with foundational qualifications. The maths teaching I had was
worse than useless. What I needed to learn the necessary maths wouldn't have been invented for another 40+ years after my school years. I did however manage to memorise the times tables from the 1s
to the 12 times tables. I have later remembered the first 50 squares at 57-58 years old, I can do quadratic equations by factorising, completing the square and quadratic formula. I was taught a
little of that at college but I dropped out of that because of my the seizures were getting frequent and thinking that sort of thing was triggering the seizures which was diagnosed in 1987 as
epilepsy. Much later afterwards I managed to teach myself quadratic equations, sums and differences of cubes, evaluating limits differentiation and integration which is calculus. The teaching myself
higher maths online could be done safely now that I changed my epilepsy meds to one that gives a reliable remission or that stops the seizures before they happen allowing me to learn a lot more years
after school than during it. At school the teachers really didn't think I was the sharpest knife in the drawer. Ableism/neurobigotry. My main interest at college was chemisty though and maths is very
important in physics and chemisty.
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We only memorised the squares up to 13. I think that was because the 12 times table is acually pretty useful, and we were taught the 13 times table up to 13 to make it clear that "this does continue
after 12, you know"!
You're so vain
I bet you think this sig is about you | {"url":"https://wrongplanet.net/forums/viewtopic.php?t=422220&sid=1a442b2909269bc30457dc89b9647495","timestamp":"2024-11-06T17:46:25Z","content_type":"application/xhtml+xml","content_length":"70124","record_id":"<urn:uuid:7f314d3f-047e-4b79-8e6a-ef7f2a806b38>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00024.warc.gz"} |
September 2020
arXiv:physics/0410002, coauthor: A. V. Gavrilin
IEEE Trans. Appl. Supercond., 15, 2, 1439-1443 (2005), coauthors: A. V. Gavrilin, W. S. Marshall
In: Quantum Foundations, Probability and Information, A. Khrennikov, T. Bourama (Eds.), p. 1 (2018)
arXiv:1903.05171, coauthor – A.V. Gavrilin
Patent Application; coauthor – A.V. Gavrilin
Also available at
United States Patent & Trademark Office
(enter the verification code and then application number 11/517915) | {"url":"http://akhmeteli.org/?paged=2&m=202009","timestamp":"2024-11-04T04:24:03Z","content_type":"text/html","content_length":"82067","record_id":"<urn:uuid:896fd3cc-c06b-492b-b1a8-b8e4ef13d389>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00034.warc.gz"} |
Combat - Wiki - Milky Way IdleCombat
Increases max HP by 10 per level. Experience is gained from taking damage and partially from avoiding damage. Avoided damage is any damage that was completely evaded or damage that was reduced
through resistances and armor.
Max HP starts at 100 including the first level in stamina and increases by 10 every level in stamina. HP can also be increased by equipping pouches| which are made through the Tailoring skill.
HP is lost when taking damage in combat and can be recovered through HP regen, consumables from Cooking and abilities such as Heal. HP is also reset to max when out of combat or respawning after
HP Regen
HP recovers at a rate of 1% of max HP every 10 seconds. Any fractional values are discarded so recovery rate only rises every 100 HP (10 stamina) at 1% regen. HP regen % can be raised using Stamina
Increases max MP by 10 per level. Experience from while using abilities. Abilities are special attacks that use up Mana, though the amount used is based on the ability
Max MP starts at 100 including the first level in intelligence and increases by 10 every level in stamina.
Mp Regen
HP recovers at a rate of 1% of max HP every 10 seconds.
Increases melee accuracy and base attack speed. It also increases base attack speed. Experience is gained from dealing stab and slash damage.
Increases max melee damage. Experience is gained from dealing smash and slash damage, and a small amount from piercing.
Increases evasion, armor and elemental resistances. Experience is gained from dodging or blocking damage. A successful evasion completely negates incoming damage while armor and elemental resistances
reduces the damage.
Increases ranged accuracy, ranged damage, and magic evasion. Experience is gained from dealing ranged damage. Ranged attacks can critical strike for max damage and the critical chance increases with
hit chance.
Increases magic accuracy, magic damage and elemental resistances. Experience is gained from dealing magic damage or from the HP amount recovered through magic abilities. | {"url":"https://wiki.milkywayidle.com/wiki/Combat?mobileaction=toggle_view_mobile","timestamp":"2024-11-04T23:12:09Z","content_type":"text/html","content_length":"33350","record_id":"<urn:uuid:064701ca-c5ec-4841-a972-b1a27ebfd0ff>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00505.warc.gz"} |
How to Use Goal Seek Functions in a C# .NET Excel XLSX App
How to Use Goal Seek Functions in a C# .NET Excel XLSX App
What You Will Need
• Document Solutions for Excel
• Visual Studio 2022
• .NET 6+
Controls Referenced
• Document Solutions for Excel, C# .NET Excel Library
• Documentation | Online Goal Seek Demo
Tutorial Concept
Learn step-by-step how to implement Goal Seek using a C# .NET Excel API to achieve your desired results efficiently. Perfect for developers aiming to optimize sales projections.
Goal setting helps identify break-even points and guides decision-making for product scaling.
In today’s competitive business landscape, determining the right sales target provides a clear financial direction and helps align all business activities toward achieving specific goals. The Excel
“What-If” feature is an excellent tool for finding perfect sales targets through various options like Scenario Manager, Goal Seek, and Data Table, which helps in analysis, forecasting, and goal
In this blog, we will use the Document Solutions for Excel (DsExcel) API to create a calculator. This calculator will determine the number of units we need to sell for a product to earn a specified
profit. We will implement this solution in a .NET application, utilizing DsExcel’s Goal Seek feature introduced in the recent v7.2 release. The implementation plan is as follows:
• Add Product Information to Excel Sheet Using C#
• Apply Profit/Loss Calculation Formula Using C#
• Apply Goal Seek Method in DsExcel
Add Product Information to Excel Sheet Using C#
To build this calculator, we need some initial details, such as the total investment amount, the product’s unit price, and variable cost. Let’s assume we invest $10,000,000 in producing the product
with a selling price of $5,499 per unit and a variable cost of $1,100 per unit.
The DsExcel code to set these details in the sheet is as follows:
worksheet.Range["C2:E3"].Value = "Product Unit Calculator for Desired Profit";
worksheet.Range["B5"].Value = "Investment";
worksheet.Range["E5"].Value = "Desired Profit";
worksheet.Range["B8"].Value = "Product Unit Price";
worksheet.Range["B9"].Value = "Variable Cost per Unit";
worksheet.Range["E8"].Value = "Number of Units to Sell";
worksheet.Range["C5"].Value = 10000000; // Investment amount
worksheet.Range["C8"].Value = 5499; // Product unit price
worksheet.Range["C9"].Value = 1100; // Variable cost per unit
After applying formatting to the sheet, these values will appear like below:
Apply Profit/Loss Calculation Formula Using C#
Now, it is time to calculate the number of units that we need to sell to earn a profit of $1,000,000. For this, we need to apply a formula to calculate the profit/loss based on product units and
other parameters discussed above. For example, let’s use a random number for a unit and apply the following formula using DsExcel to show the earned profit/loss amount.
worksheet.Range["F5"].Formula = "=((C8+C9)*F8)-C5";
worksheet.Range["F8"].Value = 100; // Initial number of units to sell
Apply Goal Seek Method in DsExcel
Our goal is to calculate the number of units required to generate a profit of $1,000,000. To do this, we’ll use the latest GoalSeek method in DsExcel, but first, let’s break down how this method
The GoalSeek method helps you find the exact input value needed to achieve a specific result in a formula. This method is called from the cell where our formula is located (which should be a single
cell). When using this method, we provide two key pieces of information: the goal (the result we want) and the changingCell (the input that will be adjusted to meet that goal).
• The formula in the calling range (the cell where GoalSeek is applied) must result in a numeric value.
• The changingCell parameter cannot contain a formula; it must be a cell with an input value that can be adjusted.
With that in mind, it’s time to apply the GoalSeek method to our data and calculate the number of units we need to sell to achieve a profit of $1,000,000. We’ll apply the GoalSeek on cell F5, which
contains the formula for calculating profit. We’ll set 1,000,000 as the goal and use cell F8 as the changing cell, which will update based on the set goal.
The DsExcel code to apply the GoalSeek method is as follows:
worksheet.Range["F5"].GoalSeek(1000000, worksheet.Range["F8"]);
That’s it! We have successfully calculated the exact number of units that need to be sold to achieve the $1,000,000 profit using the Goal Seek method. Below is a screenshot showing the result:
You can also refer to this sample file implementing this exact method. Feel free to try it out with your product details and see how easily you can calculate the required units to reach your desired
profit target.
In this blog, we learned how to use Excel’s Goal Seek feature in DsExcel to easily calculate the number of units needed to reach a specific profit goal. With DsExcel, you can simplify complex Excel
tasks and add powerful automation to your .NET apps.
Exciting news! In our next major release, v8, we are adding support for the What-If Scenario feature, which expands the possibilities for advanced data analysis in Excel through the DsExcel API.
If you have any questions or feedback about this tutorial, feel free to share them in the comments below!
More References: | {"url":"https://medium.com/mesciusinc/how-to-use-goal-seek-functions-in-a-c-net-excel-xlsx-app-3a60221f64e5","timestamp":"2024-11-09T03:31:00Z","content_type":"text/html","content_length":"137201","record_id":"<urn:uuid:a55294f6-384f-4eea-85d2-91ecb788e3f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00058.warc.gz"} |
Square Mil Converter
Square Mil [mil2] Output
1 square mil in ankanam is equal to 9.6450617283951e-11
1 square mil in aana is equal to 2.0290560831101e-11
1 square mil in acre is equal to 1.5942236697094e-13
1 square mil in arpent is equal to 1.8870415714807e-13
1 square mil in are is equal to 6.4516e-12
1 square mil in barn is equal to 6451600000000000000
1 square mil in bigha [assam] is equal to 4.8225308641975e-13
1 square mil in bigha [west bengal] is equal to 4.8225308641975e-13
1 square mil in bigha [uttar pradesh] is equal to 2.5720164609053e-13
1 square mil in bigha [madhya pradesh] is equal to 5.787037037037e-13
1 square mil in bigha [rajasthan] is equal to 2.5507601265177e-13
1 square mil in bigha [bihar] is equal to 2.5512286717283e-13
1 square mil in bigha [gujrat] is equal to 3.9855626976839e-13
1 square mil in bigha [himachal pradesh] is equal to 7.9711253953678e-13
1 square mil in bigha [nepal] is equal to 9.525986892242e-14
1 square mil in biswa [uttar pradesh] is equal to 5.1440329218107e-12
1 square mil in bovate is equal to 1.0752666666667e-14
1 square mil in bunder is equal to 6.4516e-14
1 square mil in caballeria is equal to 1.4336888888889e-15
1 square mil in caballeria [cuba] is equal to 4.8074515648286e-15
1 square mil in caballeria [spain] is equal to 1.6129e-15
1 square mil in carreau is equal to 5.0012403100775e-14
1 square mil in carucate is equal to 1.3274897119342e-15
1 square mil in cawnie is equal to 1.1947407407407e-13
1 square mil in cent is equal to 1.5942236697094e-11
1 square mil in centiare is equal to 6.4516e-10
1 square mil in circular foot is equal to 8.8419332258162e-9
1 square mil in circular inch is equal to 0.0000012732383948401
1 square mil in cong is equal to 6.4516e-13
1 square mil in cover is equal to 2.3912527798369e-13
1 square mil in cuerda is equal to 1.6416284987277e-13
1 square mil in chatak is equal to 1.5432098765432e-10
1 square mil in decimal is equal to 1.5942236697094e-11
1 square mil in dekare is equal to 6.4516042561233e-13
1 square mil in dismil is equal to 1.5942236697094e-11
1 square mil in dhur [tripura] is equal to 1.929012345679e-9
1 square mil in dhur [nepal] is equal to 3.8103947568968e-11
1 square mil in dunam is equal to 6.4516e-13
1 square mil in drone is equal to 2.5117348251029e-14
1 square mil in fanega is equal to 1.0033592534992e-13
1 square mil in farthingdale is equal to 6.3750988142292e-13
1 square mil in feddan is equal to 1.5477899705916e-13
1 square mil in ganda is equal to 8.0375514403292e-12
1 square mil in gaj is equal to 7.7160493827161e-10
1 square mil in gajam is equal to 7.7160493827161e-10
1 square mil in guntha is equal to 6.3769003162943e-12
1 square mil in ghumaon is equal to 1.5942250790736e-13
1 square mil in ground is equal to 2.8935185185185e-12
1 square mil in hacienda is equal to 7.2004464285714e-18
1 square mil in hectare is equal to 6.4516e-14
1 square mil in hide is equal to 1.3274897119342e-15
1 square mil in hout is equal to 4.5393417556031e-13
1 square mil in hundred is equal to 1.3274897119342e-17
1 square mil in jerib is equal to 3.1913807189542e-13
1 square mil in jutro is equal to 1.1210425716768e-13
1 square mil in katha [bangladesh] is equal to 9.6450617283951e-12
1 square mil in kanal is equal to 1.2753800632589e-12
1 square mil in kani is equal to 4.0187757201646e-13
1 square mil in kara is equal to 3.2150205761317e-11
1 square mil in kappland is equal to 4.1822896408661e-12
1 square mil in killa is equal to 1.5942250790736e-13
1 square mil in kranta is equal to 9.6450617283951e-11
1 square mil in kuli is equal to 4.8225308641975e-11
1 square mil in kuncham is equal to 1.5942250790736e-12
1 square mil in lecha is equal to 4.8225308641975e-11
1 square mil in labor is equal to 8.9999981353839e-16
1 square mil in legua is equal to 3.5999992541535e-17
1 square mil in manzana [argentina] is equal to 6.4516e-14
1 square mil in manzana [costa rica] is equal to 9.2311302396923e-14
1 square mil in marla is equal to 2.5507601265177e-11
1 square mil in morgen [germany] is equal to 2.58064e-13
1 square mil in morgen [south africa] is equal to 7.5307575580717e-14
1 square mil in mu is equal to 9.677399951613e-13
1 square mil in murabba is equal to 6.3768946788374e-15
1 square mil in mutthi is equal to 5.1440329218107e-11
1 square mil in ngarn is equal to 1.6129e-12
1 square mil in nali is equal to 3.2150205761317e-12
1 square mil in oxgang is equal to 1.0752666666667e-14
1 square mil in paisa is equal to 8.1164614825204e-11
1 square mil in perche is equal to 1.8870415714807e-11
1 square mil in parappu is equal to 2.550757871535e-12
1 square mil in pyong is equal to 1.95148215366e-10
1 square mil in rai is equal to 4.03225e-13
1 square mil in rood is equal to 6.3769003162943e-13
1 square mil in ropani is equal to 1.2681600519438e-12
1 square mil in satak is equal to 1.5942236697094e-11
1 square mil in section is equal to 2.4909766860524e-16
1 square mil in sitio is equal to 3.5842222222222e-17
1 square mil in square is equal to 6.9444444444444e-11
1 square mil in square angstrom is equal to 64516000000
1 square mil in square astronomical units is equal to 2.882813900904e-32
1 square mil in square attometer is equal to 6.4516e+26
1 square mil in square bicron is equal to 645160000000000
1 square mil in square centimeter is equal to 0.0000064516
1 square mil in square chain is equal to 1.5942185484941e-12
1 square mil in square cubit is equal to 3.0864197530864e-9
1 square mil in square decimeter is equal to 6.4516e-8
1 square mil in square dekameter is equal to 6.4516e-12
1 square mil in square digit is equal to 0.0000017777777777778
1 square mil in square exameter is equal to 6.4516e-46
1 square mil in square fathom is equal to 1.929012345679e-10
1 square mil in square femtometer is equal to 645160000000000000000
1 square mil in square fermi is equal to 645160000000000000000
1 square mil in square feet is equal to 6.9444444444444e-9
1 square mil in square furlong is equal to 1.5942236697094e-14
1 square mil in square gigameter is equal to 6.4516e-28
1 square mil in square hectometer is equal to 6.4516e-14
1 square mil in square inch is equal to 0.000001
1 square mil in square league is equal to 2.767740830046e-17
1 square mil in square light year is equal to 7.2080493955751e-42
1 square mil in square kilometer is equal to 6.4516e-16
1 square mil in square megameter is equal to 6.4516e-22
1 square mil in square meter is equal to 6.4516e-10
1 square mil in square microinch is equal to 999999.12
1 square mil in square micrometer is equal to 645.16
1 square mil in square micromicron is equal to 645160000000000
1 square mil in square micron is equal to 645.16
1 square mil in square mile is equal to 2.4909766860524e-16
1 square mil in square millimeter is equal to 0.00064516
1 square mil in square nanometer is equal to 645160000
1 square mil in square nautical league is equal to 2.0899839891858e-17
1 square mil in square nautical mile is equal to 1.8809839309625e-16
1 square mil in square paris foot is equal to 6.1152606635071e-9
1 square mil in square parsec is equal to 6.775888109918e-43
1 square mil in perch is equal to 2.5507601265177e-11
1 square mil in square perche is equal to 1.263235101155e-11
1 square mil in square petameter is equal to 6.4516e-40
1 square mil in square picometer is equal to 645160000000000
1 square mil in square pole is equal to 2.5507601265177e-11
1 square mil in square rod is equal to 2.5507503078921e-11
1 square mil in square terameter is equal to 6.4516e-34
1 square mil in square thou is equal to 1
1 square mil in square yard is equal to 7.7160493827161e-10
1 square mil in square yoctometer is equal to 6.4516e+38
1 square mil in square yottameter is equal to 6.4516e-58
1 square mil in stang is equal to 2.3815430047988e-13
1 square mil in stremma is equal to 6.4516e-13
1 square mil in sarsai is equal to 2.2956841138659e-10
1 square mil in tarea is equal to 1.0260178117048e-12
1 square mil in tatami is equal to 3.903200435598e-10
1 square mil in tonde land is equal to 1.1696156635243e-13
1 square mil in tsubo is equal to 1.951600217799e-10
1 square mil in township is equal to 6.9193735664469e-18
1 square mil in tunnland is equal to 1.3069443319018e-13
1 square mil in vaar is equal to 7.716049382716e-10
1 square mil in virgate is equal to 5.3763333333333e-15
1 square mil in veli is equal to 8.0375514403292e-14
1 square mil in pari is equal to 6.3769003162943e-14
1 square mil in sangam is equal to 2.5507601265177e-13
1 square mil in kottah [bangladesh] is equal to 9.6450617283951e-12
1 square mil in gunta is equal to 6.3769003162943e-12
1 square mil in point is equal to 1.5942375226171e-11
1 square mil in lourak is equal to 1.2753800632589e-13
1 square mil in loukhai is equal to 5.1015202530354e-13
1 square mil in loushal is equal to 1.0203040506071e-12
1 square mil in tong is equal to 2.0406081012142e-12
1 square mil in kuzhi is equal to 4.8225308641975e-11
1 square mil in chadara is equal to 6.9444444444444e-11
1 square mil in veesam is equal to 7.716049382716e-10
1 square mil in lacham is equal to 2.550757871535e-12
1 square mil in katha [nepal] is equal to 1.9051973784484e-12
1 square mil in katha [assam] is equal to 2.4112654320988e-12
1 square mil in katha [bihar] is equal to 5.1024573434566e-12
1 square mil in dhur [bihar] is equal to 1.0204914686913e-10
1 square mil in dhurki is equal to 2.0409829373826e-9 | {"url":"https://hextobinary.com/unit/area/from/sqmil","timestamp":"2024-11-15T01:41:31Z","content_type":"text/html","content_length":"171586","record_id":"<urn:uuid:296c074f-791f-4ad8-83eb-3a8c2aa3decd>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00693.warc.gz"} |
Connectir is an R package principally for conducting Connectome-Wide Association Studies (CWAS) using Multivariate-Distance Matrix Regression (MDMR). CWAS with MDMR attempts to find regions of the
brain with functional connectivity patterns that are significantly associated with a phenotype. For instance, if you have two groups (ADHD and Controls) and each subject has a resting-state fMRI
scan, then CWAS-MDMR would find brain regions whose connectivity patterns significantly differentiate the two groups. Additional post-hoc analyses such as seed-based correlation analyses would be
needed to discern the specific connections and direction of any group difference between ADHD and Controls with CWAS-MDMR.
Please contact me if you have any questions.
Table of Contents
1. How to run a vanilla CWAS analysis; details can be found on the wiki.
Quick Approach
1. Install R) and optionally Rstudio.
2. Install the relevant packages within R including connectir using my script connectir_install.R.
Details and Troubleshooting
Parallel Matrix Algebra Operations
There are two ways to parallelize the analyses. One approach is to divide your workflow into smaller chunks and run those separately (like separate processes). This comes with the R packages
installed with connectir_install.R. Another approach is to run each matrix algebra operation (e.g., dot product) in parallel, which we go into detail in this section. Below I detail different linear
algebra libraries and linking them to R. Note this section is still under development.
Intel MKL
If you have Windows, Ubuntu, or RedHat/Centos, you can install Revolution R. This is a version of R compiled with Intel MKL by the company Revolution Analytics available free for academic use. You
can get it from here.
Another option is to compile and install R linked with Intel MKL on your own. Here is a good and quick tutorial.
You can also install R via my own script that links R with a parallel matrix algebra library called openblas. This script is in the Rinstall repo and is called install.py.
Another option for linux is to download repositories. A good/quick tutorial can be found here.
Installing Connectir and Other R Packages
After R is setup, there are several packages within R that need to be installed. To do this, please run connectir_install.R. After downloading (or copying and pasting) this script to your machine,
you can run it with Rscript connectir_install.R. On certain linux systems, you need to ensure you have libcurl and libxml installed.
This script is also a work in progress, please contact me if you have trouble.
Here we give a vanilla run of CWAS-MDMR and further details can be found on the wiki. I also go through these steps in our recent resting-state conference poster (2014).
Subject Distances
connectir_subdist.R \
-i functional_path_list.txt \
--automask1 \
--brainmask1 standard_grey_matter.nii.gz \
--bg standard_brain_4mm.nii.gz \
--memlimit 20 -c 3 -t 4 \
• -i: List of your input functional images. Can be nifti (nii or nii.gz) containing voxelwise time-series or text files containing region/parcellation time-series (columns=regions and rows=
• --automask1: Will generate the group mask containing only voxels that have non-zero values (i.e., variance) across all participants.
• --brainmask1: An additional prior mask. We tend to use a 25% probability grey matter mask in MNI152 standard space. You can find these on the CPAC website.
• --bg: This is used to determine writing of output voxelwise files and also in the future will be used to generate images of the results. Since my data here is assumed to be voxelwise in 4mm
space, the standard reference image is also in 4mm space.
• --memlimit: The memory (RAM) limit of the processes in GB. Here it is set to 20GB.
• -c: Number of parallel jobs/forks to run in parallel.
• -t: Number of parallel linear algebra operations.
• Finally the last argument gives the full path to the output directory.
Multivariate Distance Matrix Regression (MDMR)
For the options that are the same as before (--memlimit, -c, -t), I will not repeat the description here.
connectir_mdmr.R \
-i subject_distances_outdir \
--formula FSIQ + Age + Sex + meanFD \
--model model_evs.csv \
--factors2perm FSIQ \
--memlimit 8 -c 3 -t 4 \
--save-perms --ignoreprocerror \
• -i: Input path to your subject distances directory
• --formula:: We use R's formula syntax. Each variable represents a column in your model file (i.e., a factor or explanatory variable). The + indicates to combine the different variables in one
model. If you want to do an interaction you can use *, which will generate the main effects and interaction. To only look at the interaction (for instance between Age and Sex), you would use : as
in Age:Sex.
• --model: This is your model file containing all your explanatory variables and covariates in each column. Each row would correspond to each subject/scan in the same order as the input functional
file list in connectir_subdist.R. So ensure that this model file as the rows (scans/subjects) correspond to the rows in -i file list of connectir_subdist.R.
• factors2perm: This indicates the variables in your formula to permute and calculate significance estimates. The other variables will be treated as covariates. In this case, we were interested in
full-scale IQ (FSIQ).
• --saveperms: This outputs all the permuted F-statistics into a file. This is needed for now to later calculate (in a separate script), permuted cluster correction.
• --ignoreprocerror: Sometimes the code tries to estimate the maximum number of cores on your machine and does this wrong. This option ignores the error that's thrown because of this potential
• The final argument is the output directory for the MDMR script. This should just be the directory name (not the path). The path is set to the subject distances directory given with the -i option. | {"url":"http://czarrar.github.io/connectir/","timestamp":"2024-11-14T00:03:51Z","content_type":"text/html","content_length":"13446","record_id":"<urn:uuid:ff081335-839a-4ffa-a3d8-15edbe6eec64>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00402.warc.gz"} |
37% Rule
37% Rule - Balancing Analysis with Action
Last week at work, I had the opportunity to have an interesting conversation with two of my colleagues. The problem at hand was quite straightforward. We had one opening for a technical position to
fill and were to kick off our recruitment campaign as usual. But just like any other recruiter would expect, we wanted the best candidate for the position.
The challenge, however, was that the number of CVs we would receive and the time it would take to get enough CVs were variables beyond our control. On top of that, my two friends who were in that
discussion had several thought-provoking questions. What if we commit to a candidate too early? What if someone better applies later? What if we miss our best candidate just because we kept on
shopping? How long should we keep looking before making the decision?
By the end of the conversation we made the decision that we must just find the "right balance", hit that sweet spot based on our "judgement" and just make the call. But on my way home the same
evening I couldn’t shake the feeling that there had to be a methodology to address these kinds of problems. And there is: the 37% rule.
The 37% Rule
In its simplest form, the 37% rule recommends that we spend the initial 37% of our time exploring our options without committing to anything. And beyond that point, pick the next best thing that
comes your way which is better than everything you have seen before. Now, as you can understand, this rule can be applied to many similar problems that we encounter in life. Recruitment, apartment
hunting, shopping and even in dating. The math behind this is pretty complicated, and it relates to the mathematical branch of optimal stopping.
Application of the "Rule"
Applying this "rule" to real-life situations can vary depending on the problem at hand. I realised that applying the 37% rule to a time duration can be common for many similar problem categories.
Imagine the scenario below to grasp the concept thoroughly.
• You are looking for an apartment.
• You have given yourself a period of 4 weeks. That is 28 days.
• Within the first 10 days (37%), you find a few attractive apartments. Now, if you know the rule, you don't have to worry about letting go and keep searching during that period.
• However, the experience and knowledge you gained from that exploration will help you set a benchmark in your search for the 'ideal apartment'.
• From the 11th day onwards, be prepared to commit to the next best apartment you see, which is better than everything you have seen before. So you better carry your chequebook when stepping out
for the apartment hunt.
However, an important point to note is that this is purely a probability-based mathematical recommendation. This is not going to guarantee the best outcome but provide guidance to achaive highest
probability in that endeavour.
In real life, there can be obvious scenarios where you might find the best thing you have been looking for in that particular context on your first attempt itself, and you might feel very confident
about it. In such absolutely confident scenarios, the adaption of the rule is totally up to you, and the math should not govern such situations. But in a generalized situation, spending almost 1/3 of
the time exploring and making a decision based on such a framework will help many other situations, preventing us from analysis paralysis.
Further Reading
I found the following articles interesting and to be doing a good job in explaining the concept with better examples:
The 37 Percent Rule: The Mathematical Trick for Making Much Better Decisions
Mathematicians suggest the “37% rule” for your life’s biggest decisions | {"url":"https://aroshjayamanna.com/37-rule-balancing-analysis-with-action","timestamp":"2024-11-13T17:23:50Z","content_type":"text/html","content_length":"131413","record_id":"<urn:uuid:a0d93488-9012-4e0d-9228-c91e6652fb92>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00113.warc.gz"} |
Accessing and Modifying the Model Data
This example shows how to access or edit parameter values and metadata in LTI objects.
Accessing Data
The tf, zpk, ss, and frd commands create LTI objects that store model data in a single MATLAB® variable. This data includes model-specific parameters (e.g., A,B,C,D matrices for state-space models)
as well as generic metadata such as input and output names. The data is arranged into a fixed set of data fields called properties.
You can access model data in the following ways:
• The get command
• Structure-like dot notation
• Data retrieval commands
For illustration purposes, create the SISO transfer function (TF):
G = tf([1 2],[1 3 10],'inputdelay',3)
G =
s + 2
exp(-3*s) * --------------
s^2 + 3 s + 10
Continuous-time transfer function.
To see all properties of the TF object G, type
Numerator: {[0 1 2]}
Denominator: {[1 3 10]}
Variable: 's'
IODelay: 0
InputDelay: 3
OutputDelay: 0
InputName: {''}
InputUnit: {''}
InputGroup: [1x1 struct]
OutputName: {''}
OutputUnit: {''}
OutputGroup: [1x1 struct]
Notes: [0x1 string]
UserData: []
Name: ''
Ts: 0
TimeUnit: 'seconds'
SamplingGrid: [1x1 struct]
The first four properties Numerator, Denominator, IODelay, and Variable are specific to the TF representation. The remaining properties are common to all LTI representations. You can use help
tf.Numerator to get more information on the Numerator property and similarly for the other properties.
To retrieve the value of a particular property, use
G.InputDelay % get input delay value
You can use abbreviations for property names as long as they are unambiguous, for example:
G.iod % get transport delay value
Quick Data Retrieval
You can also retrieve all model parameters at once using tfdata, zpkdata, ssdata, or frdata. For example:
[Numerator,Denominator,Ts] = tfdata(G)
Numerator = 1x1 cell array
{[0 1 2]}
Denominator = 1x1 cell array
{[1 3 10]}
Note that the numerator and denominator are returned as cell arrays. This is consistent with the MIMO case where Numerator and Denominator contain cell arrays of numerator and denominator polynomials
(with one entry per I/O pair). For SISO transfer functions, you can return the numerator and denominator data as vectors by using a flag, for example:
[Numerator,Denominator] = tfdata(G,'v')
Editing Data
You can modify the data stored in LTI objects by editing the corresponding property values with set or dot notation. For example, for the transfer function G created above,
changes the sample time from 0 to 1, which redefines the model as discrete:
G =
z + 2
z^(-3) * --------------
z^2 + 3 z + 10
Sample time: 1 seconds
Discrete-time transfer function.
The set command is equivalent to dot assignment, but also lets you set multiple properties at once:
G.Ts = 0.1;
G.Variable = 'q';
G =
q + 2
q^(-3) * --------------
q^2 + 3 q + 10
Sample time: 0.1 seconds
Discrete-time transfer function.
Sensitivity Analysis Example
Using model editing together with LTI array support, you can easily investigate sensitivity to parameter variations. For example, consider the second-order transfer function
$H\left(s\right)=\frac{s+5}{{s}^{2}+2\zeta s+5}$.
You can investigate the effect of the damping parameter zeta on the frequency response by creating three models with different zeta values and comparing their Bode responses:
s = tf('s');
% Create 3 transfer functions with Numerator = s+5 and Denominator = 1
H = repsys(s+5,[1 1 3]);
% Specify denominators using 3 different zeta values
zeta = [1 .5 .2];
for k = 1:3
H(:,:,k).Denominator = [1 2*zeta(k) 5]; % zeta(k) -> k-th model
% Plot Bode response | {"url":"https://au.mathworks.com/help/control/ug/accessing-and-modifying-the-model-data.html","timestamp":"2024-11-03T22:36:56Z","content_type":"text/html","content_length":"76573","record_id":"<urn:uuid:de005d14-8987-404a-a776-26a61988e304>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00452.warc.gz"} |
Recent Progress in Kinetic and Integro-Differential Equations
Schedule for: 22w5101 - Recent Progress in Kinetic and Integro-Differential Equations
Beginning on Sunday, November 6 and ending Friday November 11, 2022
All times in Banff, Alberta time, MDT (UTC-6).
Sunday, November 6
16:00 - 17:30 Check-in begins at 16:00 on Sunday and is open 24 hours (Front Desk - Professional Development Centre)
Dinner ↓
17:30 - 19:30 A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.
(Vistas Dining Room)
20:00 - 22:00 Informal gathering (TCPL Foyer)
Monday, November 7
Breakfast ↓
- Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
Introduction and Welcome by BIRS Staff ↓
- A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.
(TCPL 201)
Robert Strain: Non-negativity of a local classical solution to the Relativistic Boltzmann Equation without Angular Cut-off ↓
09:00 This talk concerns the relativistic Boltzmann equation without angular cutoff. Global in time unique solutions close to equilibrium were built in Jang-Strain (Ann. PDE, 2022). However the
- non-negativity of those solutions remained an open problem. Now we establish local wellposedness and non-negativity for solutions to the special relativistic Boltzmann equation without angular
09:45 cutoff. The solution lies in an appropriate fractional Sobolev type space. In addition, as a corollary our results provide a rigorous proof for the non-negativity of the classical solutions of
Jang-Strain (Ann. PDE, 2022) in the perturbative setting nearby the relativistic Maxwellian. This is a joint work with Jin Woo Jang.
(TCPL 201)
- Coffee Break (TCPL Foyer)
Matias Delgadino: Boltzmann to Landau from the Gradient Flow Perspective ↓
- In this talk, we revisit the grazing collision limit from the Boltzmann equation to the Landau equations utilizing their recent reinterpretations as gradient flows. We utilize the framework of
11:00 Γ-convergence of gradient flows technique introduced by Sandier and Serfaty.
Raphael Winter: Deceleration of a point charge interacting with the screened Vlasov-Poisson system ↓
We consider an infinitely extended (screened) Vlasov-Poisson plamsa on $\R3$ coupled to a point charge. The well-posedness of this problem has been studied thoroughly in recent years, while
11:10 little is known about Landau damping in this setting. Contrary to other results in nonlinear Landau damping, the dynamics are driven by the non-trivial electric field $E[F]$ of the plasma, even
- for large times $t\gg 1$. We rigorously prove the validity of the `stopping power theory' in physics, which predicts a decrease of the velocity $V(t)$ of the point charge given by $\rh{\dot{V}
11:55 \sim -|V|^{-3} V}$. This formula was first predicted by Bohr (1915), and has since become a standard tool in physics. Our result holds for all initial velocities $V_0>0$ larger than a threshold
value $\ol{V}>0$ and remains valid until (i) the particle slows down to velocity $\ol{V}$, or (ii) the time is exponentially long compared to the velocity of the charge, i.e. $T= \exp(V(T))$.
(TCPL 201)
Lunch ↓
- Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
Hongjie Dong: Sobolev estimates for fractional PDEs ↓
- I will discuss some recent results on Sobolev estimates for fractional elliptic and parabolic equations with or without weights. We considered equations with time fractional derivatives of the
14:45 Caputo type, or with nonlocal derivatives in the space variables, or both. This is based on joint work with Doyoon Kim, Pilgu Jung (Korea University), and Yanze Liu (Brown).
(TCPL 201)
- Coffee Break (TCPL Foyer)
Stanley Snelson: Existence of classical solutions for the non-cutoff Boltzmann equation with irregular initial data ↓
15:30 The non-cutoff Boltzmann equation is known to have a regularizing effect on solutions because of the fractional diffusion induced by the collision operator. This suggests that classical
- solutions should exist even for irregular (e.g. lying in a zeroth-order space) initial data. Such an existence result has previously been shown in the close-to-equilibrium and space-homogeneous
16:15 settings, and in this talk, we discuss the extension to the general case: spatially varying initial data that is far from equilibrium. Our classical solutions have initial data in an L^\infty(R
^6) space with mild polynomial weight in the velocity variable. This talk is based on joint work with Henderson and Tarfulea.
(TCPL 201)
Dinner ↓
- A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.
(Vistas Dining Room)
Tuesday, November 8
Breakfast ↓
- Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
Weiran Sun: Asymptotic Preserving Method for Multiscale Levy-Fokker-Planck ↓
- In this talk, we present a recent result that shows an operator splitting scheme for the multiscale Levy-Fokker-Planck equation is asymptotic preserving (AP). The analysis is carried out by
09:45 separating the parameter domain, which generalizes the traditional AP method when estimates are performed in a uniform way.
(TCPL 201)
- Coffee Break (TCPL Foyer)
Li Wang: Blow up or not? A preliminary study for kinetic granular equation ↓
10:30 The kinetic description of granular media is through a Boltzmann type equation with a nonlocal nonlinear collision operator that is of the same form as in the continuum equation for aggregation
- diffusion dynamics. While the singular behavior of the continuum equation is well studied in the literature, the extension to the kinetic equation is highly nontrivial. The main question is
11:00 whether the singularity formed in velocity direction will be enhanced or mitigated by the shear. Here we present some preliminary study by a careful numerical investigation and a heuristic
Olga Turanova: Approximating degenerate diffusion via nonlocal equations ↓
11:10 In this talk, I'll describe a deterministic particle method for the weighted porous medium equation. The key idea behind the method is to approximate the PDE via certain highly nonlocal
- continuity equations. The formulation of the method and the proof of its convergence rely on the Wasserstein gradient flow formulation of the aforementioned PDEs. This is based on joint work
11:55 with Katy Craig, Karthik Elamvazhuthi, and Matt Haberland.
(TCPL 201)
Lunch ↓
- Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
Alexis Vasseur: Boundary vorticity estimate for the Navier-Stokes equation and control of layer separation in the inviscid limit ↓
- We provide a new boundary estimate on the vorticity for the incompressible Navier-Stokes equation endowed with no-slip boundary condition. The estimate is rescalable through the inviscid limit.
14:45 It provides a control on the layer separation at the inviscid Kato double limit, which is consistent with the Layer separation predictions via convex integration.
- Coffee Break (TCPL Foyer)
Christopher Henderson: Two results on the local well-posedness of collisional kinetic equations ↓
The Landau and Boltzmann equations are nonlocal, nonlinear equations for which (large data) global well-posedness is an extremely difficult problem that is nearly completely open. Recently,
15:30 with Snelson and Tarfulea, we have pursued a program to understand a more tractable, related question: what are the weakest conditions for which local well-posedness holds and what quantities
- prevent blow-up at finite times? I will discuss two pieces of this program that were established in collaboration with Weinan Wang: (1) existence of solutions starting from initial data that
16:15 decays "slowly" in velocity -and- (2) "irregular" Schauder estimates and their application to uniqueness of solutions with almost no initial regularity in velocity. The talk will focus on
interesting pieces of the proofs (instead of slogging through a priori estimates!) such as a simple version of the proof in (1) in certain parameter regimes.
(TCPL 201)
Dinner ↓
- A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.
(Vistas Dining Room)
Wednesday, November 9
Breakfast ↓
- Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
Cyril Imbert: Local regularity for the Landau-Coulomb equation ↓
This talk deals with the space-homogenous Landau equation with very soft potentials, including the Coulomb case. This nonlinear equation is of parabolic type with diffusion matrix given by the
09:00 convolution product of the solution with the matrix $a_{ij}(z)=|z|^\gamma (|z|^2\delta_{ij}−z_iz_j)$ for $\gamma \in [−3,−2)$. We derive local truncated entropy estimates and use them to
- establish two facts. Firstly, we prove that the set of singular points (in time and velocity) for the weak solutions constructed as in [C. Villani, Arch. Rational Mech. Anal. 143 (1998),
09:45 273-307] has zero $P^{m_∗}$ parabolic Hausdorff measure with $m_∗ :=1+\frac{5}{2} |2+\gamma|$. Secondly, we prove that if such a weak solution is axisymmetric, then it is smooth away from the
symmetry axis. In particular, radially symmetric weak solutions are smooth away from the origin.
(TCPL 201)
- Coffee Break (TCPL Foyer)
Natasa Pavlovic: A binary-ternary Boltzmann equation: origins of the equation and moments of solutions ↓
In this talk we will discuss a generalization of the Boltzmann equation that takes into account both binary and ternary interactions of particles. In particular, the first part of the talk will
10:30 focus on the joint work with Ioakeim Ampatzoglou on a derivation of a binary-ternary Boltzmann equation describing the kinetic properties of a dense hard spheres gas, where particles undergo
- either binary or ternary instantaneous interactions, while preserving momentum and energy. This will be followed by a discussions of a recent result obtain in collaboration with Ioakeim
11:00 Ampatzoglou, Irene Gamba and Maja Tasković on generation and propagation in time of polynomial and exponential moments, as well as global well-posedness of the homogeneous binary-ternary
Boltzmann equation. In particular, we show that this equation is “better behaved” compared to the homogeneous version of the (binary) Boltzmann equation or the homogeneous version of the purely
ternary Boltzmann equation.
Andrei Tarfulea: Uniqueness for solutions of the non-cutoff Boltzmann equation in an irregular class ↓
11:10 Building on an improved local existence (and smoothing) result for the inhomogeneous Boltzmann equation, we show that, under mild assumptions on the initial data, the smooth solution obtained
- is unique in a broad class of weak solutions (L^1 in time, L^\infty(R^6) with polynomial velocity weights). We show this by propagating a H\"older modulus of continuity in space and velocity.
11:55 This then yields H\"older continuity in time by controlling time differences. Finally, we obtain weak-strong uniqueness through a suitable energy method. This talk is based on joint work with
Henderson and Snelson.
(TCPL 201)
Lunch ↓
- Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
- Free Afternoon (Banff National Park)
Group tour of The Banff Centre ↓
- Meet in the PDC front desk for a guided tour of The Banff Centre campus.
(PDC Front Desk)
Dinner ↓
- A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.
(Vistas Dining Room)
Thursday, November 10
Breakfast ↓
- Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
Luis Silvestre: Holder continuity up to the boundary for kinetic equations ↓
- We consider a kinetic Fokker-Planck equation with rough coefficients and the spatial variable restricted to a bounded domain. There are recent results concerning interior Holder estimates for
09:45 this class of equations following techniques by De Giorgi, Nash and Moser. In this talk, we discuss the regularity of the solutions on the boundary.
(TCPL 201)
- Coffee Break (TCPL Foyer)
Laurent Desvillettes: Some new results about the Landau-Fermi-Dirac equation ↓
- In a work in collaboration with Ricardo Alonso, Véronique Bagland and Bertrand Lods, we investigate the properties of the spatially homogeneous Landau equation for fermions, in the case of
11:00 soft potentials. We propose regularity and large time behavior results which are uniform with respect to the quantum parameter, so that they also hold in the classical case
Havva Yoldaş: Quantitative hypocoercivity estimates based on Harris-type theorems ↓
11:10 Kinetic equations arising in biology and social sciences often have non-explicit steady states unlike the ones coming from mathematical physics such as Boltzmann-type equations. This makes it
- hard to use classical hypocoercivity techniques to study the long-time behaviour of these equations. Particularly, it is difficult to obtain Poincaré-type inequalities. Harris-type theorems
11:55 present an alternative approach since they are based on controlling the behaviour of moments rather than Poincaré-type inequalities, thus we look at the point-wise bounds rather than integral
controls of operators. I will talk about Harris-type theorems with a couple of examples from mathematical physics and biology.
(TCPL 201)
Group Photo ↓
- BIRS staff will meet you at the TCPL foyer to take a group photo.
(TCPL Foyer)
Lunch ↓
- Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
Moritz Kassmann: The Neumann problem for nonlocal operators ↓
- We study the Neumann problem for nonlocal operators in bounded domains with prescribed data in the complement of the domain. We introduce corresponding function spaces and trace resp.
14:45 extension results. Finally, we present the probabilistic interpretation of the problem.
(TCPL 201)
- Coffee Break (TCPL Foyer)
Jin Woo Jang: On the temperature distribution of a body heated by radiation ↓
15:30 This talk introduces the stationary radiative transfer equation coupled with a non-local temperature equation in the local thermodynamic equilibrium. Via the system, we study the temperature
- distribution of a body when the heat is transmitted only by radiation. The heat transferred by convection and conduction is ignored. We prove that the system with the incoming boundary
16:15 condition admits a solution in a generic case when both the emission/absorption and the scattering of interacting radiation are considered. We also introduce the entropy production formula of
the system for the uniqueness of the solution. This is joint work with Juan J. L. Velázquez.
(TCPL 201)
Dinner ↓
- A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.
(Vistas Dining Room)
Friday, November 11
Breakfast ↓
07:00 -
08:45 Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
10:00 - Coffee Break (TCPL Foyer)
Checkout by 11AM ↓
10:30 -
11:00 5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.
(Front Desk - Professional Development Centre)
12:00 - Lunch from 11:30 to 13:30 (Vistas Dining Room) | {"url":"http://archytas.birs.ca/events/2022/5-day-workshops/22w5101/schedule","timestamp":"2024-11-10T01:25:51Z","content_type":"application/xhtml+xml","content_length":"42098","record_id":"<urn:uuid:3617a317-2a66-44a7-bff4-9ea5f720e616>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00194.warc.gz"} |
Developmental Math Emporium
Learning Outcomes
• Identify polynomials, monomials, binomials, and trinomials
• Determine the degree of polynomials
Polynomials come in many forms. They can vary by how many terms, or monomials, make up the polynomial and they also can vary by the degrees of the monomials in the polynomial. In this section, we
will look at different ways that we classify polynomials. First, we will classify polynomials by the number of terms in the polynomial and then we will classify them by the monomial with the largest
Identify Polynomials, Monomials, Binomials, and Trinomials
A monomial, or a sum and/or difference of monomials, is called a polynomial. A polynomial containing two terms, such as [latex]2x - 9[/latex], is called a binomial. A polynomial containing three
terms, such as [latex]-3{x}^{2}+8x - 7[/latex], is called a trinomial.
polynomial—A monomial, or two or more monomials, combined by addition or subtraction (“poly” means many)
monomial—A polynomial with exactly one term (“mono” means one)
binomial— A polynomial with exactly two terms (“bi” means two)
trinomial—A polynomial with exactly three terms (“tri” means three)
Here are some examples of polynomials:
Polynomial [latex]b+1[/latex] [latex]4{y}^{2}-7y+2[/latex] [latex]5{x}^{5}-4{x}^{4}+{x}^{3}+8{x}^{2}-9x+1[/latex]
Monomial [latex]5[/latex] [latex]4{b}^{2}[/latex] [latex]-9{x}^{3}[/latex]
Binomial [latex]3a - 7[/latex] [latex]{y}^{2}-9[/latex] [latex]17{x}^{3}+14{x}^{2}[/latex]
Trinomial [latex]{x}^{2}-5x+6[/latex] [latex]4{y}^{2}-7y+2[/latex] [latex]5{a}^{4}-3{a}^{3}+a[/latex]
Notice that every monomial, binomial, and trinomial is also a polynomial. They are special members of the family of polynomials and so they have special names. We use the words ‘monomial’,
‘binomial’, and ‘trinomial’ when referring to these special polynomials and just call all the rest ‘polynomials’.
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:
1. [latex]8{x}^{2}-7x - 9[/latex]
2. [latex]-5{a}^{4}[/latex]
3. [latex]{x}^{4}-7{x}^{3}-6{x}^{2}+5x+2[/latex]
4. [latex]11 - 4{y}^{3}[/latex]
5. [latex]n[/latex]
Polynomial Number of terms Type
1. [latex]8{x}^{2}-7x - 9[/latex] [latex]3[/latex] Trinomial
2. [latex]-5{a}^{4}[/latex] [latex]1[/latex] Monomial
3. [latex]{x}^{4}-7{x}^{3}-6{x}^{2}+5x+2[/latex] [latex]5[/latex] Polynomial
4. [latex]11 - 4{y}^{3}[/latex] [latex]2[/latex] Binomial
5. [latex]n[/latex] [latex]1[/latex] Monomial
For the following expressions, determine whether they are a polynomial. If so, categorize them as a monomial, binomial, or trinomial.
1. [latex]\frac{x-3}{1-x}+x^2[/latex]
2. [latex]t^2+2t-3[/latex]
3. [latex]x^3+\frac{x}{8}[/latex]
4. [latex]\frac{\sqrt{y}}{2}-y-1[/latex]
Show Solution
try it
In the following video, you will be shown more examples of how to identify and categorize polynomials.
Determining the Degree of Polynomials
We can find the degree of a polynomial by identifying the highest power of the variable that occurs in the polynomial. Polynomials can be classified by the degree of the polynomial. The degree of a
polynomial is the degree of its highest degree term. So the degree of [latex]2x^{3}+3x^{2}+8x+5[/latex] is 3.
A polynomial is said to be written in standard form when the terms are arranged from the highest degree to the lowest degree. When it is written in standard form it is easy to determine the degree of
the polynomial. The term with the highest degree is called the leading term because it is written first in standard form. The coefficient of the leading term is called the leading coefficient.
How to: Given a polynomial expression, identify the degree and leading coefficient
1. Find the highest power of the variable (usually x) to determine the degree.
2. Identify the term containing the highest power of the variable to find the leading term.
3. Identify the coefficient of the leading term.
Degree of a Polynomial
The degree of a term is the exponent of its variable.
The degree of a constant is [latex]0[/latex].
The degree of a polynomial is the highest degree of all its terms.
When the coefficient of a polynomial term is [latex]0[/latex], you usually do not write the term at all (because [latex]0[/latex] times anything is [latex]0[/latex], and adding [latex]0[/latex] does
not change the value).
A term without a variable is called a constant term, and the degree of that term is [latex]0[/latex]. In the polynomial [latex]3x+13[/latex], we could have written the polynomial as [latex]3x^{1}
+13x^{0}[/latex]. Although this is not how we would normally write this, it allows us to see that [latex]13[/latex] is the constant term because its degree is 0 and the degree of [latex]3x[/latex]
is 1. The degree of this binomial is 1.
If a polynomial does not have a constant term, like in the polynomial [latex]14x^{3}+3x[/latex] we would say that the constant term is [latex]0[/latex].
Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms.
Remember: Any base written without an exponent has an implied exponent of [latex]1[/latex].
Find the degree of the following polynomials:
1. [latex]4x[/latex]
2. [latex]3{x}^{3}-5x+7[/latex]
3. [latex]-11[/latex]
4. [latex]-6{x}^{2}+9x - 3[/latex]
5. [latex]8x+2[/latex]
Show Solution
Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in standard form. Look back at the polynomials in
the previous example. Notice that they are all written in standard form. Get in the habit of writing the term with the highest degree first.
try it
For the following polynomials, identify the degree, the leading term, and the leading coefficient.
1. [latex]3+2{x}^{2}-4{x}^{3}[/latex]
2. [latex]5{t}^{5}-2{t}^{3}+7t[/latex]
3. [latex]6p-{p}^{3}-2[/latex]
Show Solution
In the following video, we will identify the terms, leading coefficient, and degree of a polynomial. | {"url":"https://courses.lumenlearning.com/wm-developmentalemporium/chapter/classifying-polynomials/","timestamp":"2024-11-09T22:41:26Z","content_type":"text/html","content_length":"60637","record_id":"<urn:uuid:8f47b867-26b1-4550-8aea-e933292c5d4c>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00448.warc.gz"} |
Polynomial time approximation schemes for geometric k-clustering
We deal with the problem of clustering data points. Given n points in a larger set (for example, R^d) endowed with a distance function (for example, L^2 distance), we would like to partition the data
set into k disjoint clusters, each with a `cluster center', so as to minimize the sum over all data points of the distance between the point and the center of the cluster containing the point. The
problem is provably NP-hard in some high dimensional geometric settings, even for k = 2. We give polynomial time approximation schemes for this problem in several settings, including the binary cube
{0, 1}^d with Hamming distance, and R^d either with L^1 distance, or with L^2 distance, or with the square of L^2 distance. In all these settings, the best previous results were constant factor
approximation guarantees. We note that our problem is similar in flavor to the k-median problem (and the related facility location problem), which has been considered in graph-theoretic and fixed
dimensional geometric settings, where it becomes hard when k is part of the input. In contrast, we study the problem when k is fixed, but the dimension is part of the input. Our algorithms are based
on a dimension reduction construction for the Hamming cube, which may be of independent interest.
Dive into the research topics of 'Polynomial time approximation schemes for geometric k-clustering'. Together they form a unique fingerprint. | {"url":"https://cris.huji.ac.il/en/publications/polynomial-time-approximation-schemes-for-geometric-k-clustering","timestamp":"2024-11-03T00:45:42Z","content_type":"text/html","content_length":"47068","record_id":"<urn:uuid:62040b0d-0144-4154-873d-672957febd9f>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00513.warc.gz"} |
d I
Lesson 11
Making a Model for Data
Let’s model with functions.
11.1: What Function Could It Be?
What do you notice? What do you wonder?
11.2: Heating Up
Here is a graph of data showing the temperature of a bottle of water after it has been removed from the refrigerator.
For the function types assigned by your teacher:
1. Apply a sequence of transformations to your function so that it matches the data as well as possible.
2. How well does your model fit the data? Make adjustments as needed. | {"url":"https://im.kendallhunt.com/HS/students/3/5/11/index.html","timestamp":"2024-11-06T01:48:26Z","content_type":"text/html","content_length":"69973","record_id":"<urn:uuid:1151d292-480f-4a58-b628-2446a64cea8b>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00635.warc.gz"} |
Model Method......Questions and Answers to Kids Math Problems
Model Method - Questions and Answers
Question posted by Ban from Singapore:
Grade/Level: 3
Question solved by Model Method: In a school, 5 times as many pupils signed up for the frisbee club as for the skating club. after 60 pupils transferred from the frisbee club to the skating club,
both classes have the same number of pupils.
a) how many pupils signed up for the skating club at first?
b) how many pupils signed up for both classes?
Step 1: Draw 5 boxes to represent the number of pupils who sign up for frisbee club and 1 box to represent the number of pupils who sign up for skating club.
Step 2: Since the 2 clubs had the same number of pupils after the transfer of 60 pupils, we "shift" 2 boxes from frisbee club's model to skating club's model thereby making the 2 model equal length.
We can also label the 2 boxes shifted as 60 pupils.
Step 3: To find the number of pupils who sign up for skating club at first, we just need to work out what the value of 1 box/unit is. To find the number of pupils who sign up for both clubs, we just
need to work out what the value of 6 boxes/units is.
2 units ---------- 60
1 unit ---------- 60 / 2 = 30
6 units ---------- 30 x 6 = 180
(a) 30 pupils sign up for skating club at first.
(b) 180 pupils sign up for both clubs.
Go To Top - Model Method - Questions and Answers | {"url":"http://www.teach-kids-math-by-model-method.com/model-method-questions-and-answers-350.html","timestamp":"2024-11-09T04:17:46Z","content_type":"text/html","content_length":"24602","record_id":"<urn:uuid:5374fdd6-b71f-495a-b8a3-e02610c2e43d>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00803.warc.gz"} |
How to use ChatGPT for Learning Math Fast - TestFellow
How to use ChatGPT for Learning Math Fast
In the field of mathematics, ChatGPT has emerged as a powerful tool that offers unique opportunities to accelerate the learning process. With its advanced capabilities in natural language processing
and artificial intelligence, ChatGPT provides personalized assistance and solutions to help students grasp mathematical concepts more efficiently.
In this article, we will explore how ChatGPT can be utilized effectively to learn math fast, offering insights into its features, benefits, and various prompts designed specifically for math
learning. Discover how ChatGPT can become your trusted companion in mastering the fascinating world of mathematics.
What is ChatGPT?
ChatGPT is an advanced technology known as a language model developed by OpenAI. It uses artificial intelligence to understand and generate human-like text responses. In simple terms, ChatGPT is like
a smart computer program that can have conversations with people. It can understand questions, provide information, and even help with specific tasks.
Benefits of ChatGPT for Learning Math
When it comes to learning math, ChatGPT provides several advantages that can greatly enhance your learning experience. By leveraging the power of ChatGPT, you can receive customized support, improve
your math proficiency, and approach mathematical concepts with confidence. Let’s dive into each of these benefits and discover how ChatGPT can help you learn math more effectively.
1. Personalized Assistance:
ChatGPT offers personalized assistance in learning math. It can understand your specific needs and provide tailored explanations and examples to help you grasp mathematical concepts more effectively.
2. Instant Feedback:
When practicing math problems, ChatGPT can provide immediate feedback on your solutions. It can identify mistakes, suggest alternative approaches, and help you understand where you went wrong,
allowing for quick learning and improvement.
3. Step-by-Step Solutions:
When facing a challenging math problem, ChatGPT acts as your personal guide, breaking down complex equations into manageable steps. It simplifies the problem-solving process, allowing you to follow
along and grasp each step with ease.
4. Enhanced Problem Solving:
ChatGPT can assist you in developing strong problem-solving skills. It can provide you with various strategies, tips, and techniques to approach different types of math problems, empowering you to
become a more confident and efficient problem solver.
5. Convenient and Accessible:
ChatGPT is available anytime and anywhere, making it a convenient tool for learning math. Whether you’re studying at home or on the go, you can access ChatGPT on your computer or mobile device.
ChatGPT Prompts for Learning Math Fast
Here we will explain some useful prompts with the prompt format, descriptions, and example responses. Students can use these prompts to learn math efficiently and effectively.
1. General Math Tips and Techniques
Prompt: “Act as a [Math Expert]”
Description: This prompt allows students to seek general advice and strategies from ChatGPT on enhancing their math skills. ChatGPT can provide valuable insights, such as problem-solving techniques,
time management tips, and effective study strategies, to help students excel in their math learning journey.
Example Prompt: “Act as a math expert and provide guidance on how i can learn math fast.”
Example Response:
2. Step-by-Step Problem Solving
Prompt Format: “Provide step-by-step solutions to a [math problem].”
Example Prompt: “Provide a step-by-step solution to the equation: 2x + 5 = 11.”
Description: This prompt enables students to request ChatGPT’s assistance in solving specific math problems. By providing step-by-step solutions, ChatGPT helps students understand the problem-solving
process and gain insights into the logical steps required to arrive at the correct answer.
Example Response:
3. Practice Math Problems
Prompt: “Create practice problems on [math topic].”
Description: With this prompt, students can engage with ChatGPT to generate custom practice problems tailored to a specific math topic, such as algebra. They can solve these problems and receive
immediate feedback from ChatGPT.
Example Prompt: “Create practice problems on algebra.”
Example Response:
4. Math Project Assistance:
Prompt: “Help me with my math project on [topic].”
Description: ChatGPT can be a valuable resource for students working on math projects. Whether it involves analyzing statistical data, graphing functions, or solving complex mathematical problems,
students can seek guidance and assistance from ChatGPT to enhance the quality and accuracy of their math projects.
Example Prompt: “Help me with my math project on creating a math puzzle.”
Example Response:
5. Mathematical Proof
Prompt: “Provide a mathematical proof for [specific theorem or proposition].”
Description: This prompt empowers students to seek detailed explanations and logical reasoning behind mathematical statements by requesting a proof for a specific theorem or proposition. ChatGPT can
guide students through the thought process and steps involved in constructing a proof, helping them develop a deeper understanding of mathematical logic and reasoning. This prompt is beneficial for
students interested in advanced mathematical concepts and formal proofs.
Example Prompt: “Provide a mathematical proof for the Pythagorean Theorem.”
Example Response:
ChatGPT Prompts for Solving Complex Math Problems
ChatGPT’s advanced capabilities enable it to solve complex math problems easily and offer step-by-step solutions. From algebra to calculus, geometry to trigonometry, and statistics to probability,
students can rely on ChatGPT’s expertise to gain insights and master challenging mathematical concepts. Here, we will see some example prompts with solutions for solving complex math problems.
1. Algebra Problem
Prompt Format: Solve for [variable]
Example Prompt: Solve for x, 5x + 3 = 2x^2 – 7
Description: By using this prompt, students can enhance their algebraic problem-solving skills. The provided algebraic equations with variables challenge students to apply various techniques such as
simplification, factoring, and solving for ‘x.’
2. Solving Complex Integration problems
Example Prompt: Problem: Solve ∮(2z^3 + z) dz, where C is the circle |z| = 3 in the counterclockwise direction.
Description: This prompt challenges students to perform complex integration, evaluating the given complex function along a specified closed curve in the complex plane. Solving such problems requires
understanding the properties of complex functions, the Cauchy Integral Formula, and contour integration techniques.
3. Solving Logic Puzzles
Prompt: Write your problem [Find the value/measurement of….]
Example Prompt: Problem: A ladder is placed against a vertical wall. The ladder is 50 feet long and reaches a window 30 feet above the ground. Without moving the ladder’s base, the ladder can also
reach a window on the opposite side of the wall, which is 20 feet above the ground. What is the width of the wall?
Description: By solving this ladder and wall problem, students can learn the practical application of the Pythagorean theorem in a real-life context.
4. Geometry Problem
Example Prompt: Problem: In the figure shown below, ABCD is a cyclic quadrilateral (inscribed in a circle). The measures of angles A, B, C, and D are in arithmetic progression (AP) with a common
difference of 20 degrees. If angle A is an acute angle and its measure is greater than 30 degrees, find the measure of angle D.
Description: By solving this problem, students enhance their geometric reasoning and critical thinking skills, gaining proficiency in applying mathematical concepts to real-world geometric
5. Trigonometry Problem
Example Prompt: Problem: In triangle XYZ, angle X = 30 degrees, side XY measures 8 units, and side YZ measures 12 units. Determine the length of side XZ and the area of triangle XYZ.
Description: By solving this complex trigonometry problem involving the Law of Cosines and the Law of Sines, students can deepen their understanding of trigonometric concepts and their applications
in solving real-world geometry problems.
6. Calculus Problem
Example Prompt: Problem: Find the indefinite integral of the function f(x) = (2x^3 + 5x^2 – 4x + 3) / (x^2), with respect to x.
Description: In this calculus problem, students are tasked with finding the indefinite integral of a rational function. By applying partial fraction decomposition and integration techniques, students
strengthen their skills in solving complex integrals. This exercise provides valuable practice for handling rational functions and prepares students for more advanced calculus concepts.
Closing Remarks
ChatGPT is a valuable tool for fast and effective math learning. Its role as a math expert, provision of step-by-step solutions, and customization for practicing complex problems empower students to
sharpen their math skills efficiently.
Moreover, the convenience and accessibility of ChatGPT make it a personalized and indispensable tool, offering invaluable support to learners of all levels as they navigate the world of mathematics
with confidence and clarity.
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Study of Regularization Techniques of Linear Models and Its Roles
Introduction to Regularization
During the Machine Learning model building, the Regularization Techniques is an unavoidable and important step to improve the model prediction and reduce errors. This is also called the Shrinkage
method. Which we use to add the penalty term to control the complex model to avoid overfitting by reducing the variance.
Let’s discuss the available methods, implementation, and benefits of those in detail here.
The too many parameters/attributes/features along with data and those permutations and combinations of attributes would be sufficient, to capture the possible relationship within dependent and
independent variables.
To understand the relationship between the target and available independent variables with several permutations and combinations. For this exercise certainly, we need adequate data in terms of
records or datasets for our analysis, hope you are aware of that.
If you have fewer data with huge attributes the depth vs breadth analysis there might lead to that not all possible permutations and combinations among the dependent and independent variables. So
those missing values force good or bad to your model. Of course, we can call out this circumstance as Curse of Dimensionality. Here we are looking for these aspects from data along with parameters/
Curse of Dimensionality is not directly mean that too many dimensions, this is the lack of possible permutation and combination.
In another way round the missing data and gap generates empty space, so we couldn’t connect the dots and create the perfect model. It means that the algorithm cannot understand the data and spread
across with given space or empty, with multi-dimensional mode and meets with kind of relationship between dependent and independent variables and predicting the future data. If you try to visualize
this, it would be really complex format and difficult to follow.
During the training, you will get the above-said observation, but during the testing, the new and not exposed data combination to model’s accuracy will jump across and it suffers from error, because
of variance [variance error] and not fit for production move and risk for prediction.
Due to the too many Dimensions with too few data, the algorithm would build the best fit with peaks and deep-down dells in the observation along with the high magnitude of coefficient its leads to
overfitting and is not suitable for production. [drastic fluctuation in surface inclination]
To understand or implement these techniques, we should understand the cost function of your linear models.
Understanding the Regression Graph
The below graph represents the entire parameters existing in the LR model and is very self-explanatory.
Significance Of Cost Function
Cost function/Error function: Takes slope-intercept (m and c) values and returns the error value/cost value. It shows the error between predicted outcomes is compared with the actual outcomes. It
explains how your model is inaccurate in its prediction.
It is used to estimate how badly models are performing for the given dataset and its dimensions.
Why is cost function important in machine learning? Yes, the cost function helps us reach the optimal solution, So how can we do this. will see all possible methods and simple steps using Python
This function helps us to a figure-out best straight line by minimizing the error
The best fit line is that line where the sum of squared errors around the line is minimized
Regularization Techniques
Let’s discuss the available Regularization techniques and followed by the implementation
1. Ridge Regression (L2 Regularization):
Basically here, we’re going to minimize the sum of squared errors and sum of the squared coefficients (β). In the background,
the coefficients (β) with a large magnitude will generate the graph peak and
deep slope, to suppress this we’re using the lambda (λ) use to be called a
Penalty Factor and help us to get a smooth surface instead of an irregular-graph. Ridge Regression is used to push the coefficients(β) value nearing zero in terms of magnitude. This is L2
regularization, since its adding a penalty-equivalent to the Square-of-the Magnitude of coefficients.
Ridge Regression = Loss function + Regularized term
2. Lasso Regression (L1 Regularization):
This is very similar to Ridge Regression, with little difference in Penalty Factor that coefficient is magnitude instead of squared. In which there are possibilities of many coefficients becoming
zero, so that corresponding attribute/features become zero and dropped from the list, this ultimately reduces the dimensions and supports for dimensionality reduction. So which deciding that those
attributes/features are not suitable as predators for predicting target value. This is L1 regularization, because of adding the Absolute-Value as penalty-equivalent to the magnitude of coefficients.
Lasso Regression = Loss function + Regularized term
3. Characteristics of Lambda
λ = 0 λ => Minimal λ =>High
Lambda or Penalty No impact on coefficients(β) and model would be Overfit. Generalised model and acceptable accuracy and eligible Very high impact on coefficients (β) and leading to
Factor (λ) Not suitable for Test and underfit. Ultimately
for Production Train. Fit for Production not fit for Production.
Remember one thing that the Ridge never make coefficients into zero, Lasso will do. So, you can use the second one for feature selection.
Impact of Regularization
The below graphical representation clearly indicates the best fitment.
4. Elastic-Net Regression Regularization:
Even though Python provides excellent libraries, we should understand the mathematics behind this. Here is the detailed derivation for your reference.
5. Pictorial representation of Regularization Techniques
Mathematical approach for L1 and L2
Even though Python provides excellent libraries and straightforward coding, we should understand the mathematics behind this. Here is the detailed derivation for your reference.
Let’s have below multi-linear regression dataset and its equation
As we know Multi-Linear-Regression
y=β[0]+ β[1 ]x[1+] β[2 ]x[2+………………+] β[n ]x[n] —————–1
y[i]= β[0+ ][Σ] β[i] x[i ]—————–2
Σ y[i]– β[0]– [Σ] β[i] x[i]
Cost/Loss function: [Σ{] y[i]– β[0]– [Σ] β[i] x[ij]}^2—————–3
Regularized term: [λΣ] β[i]^2—————-4
Ridge Regression = Loss function + Regularized term—————–5
Put 3 and 4 in 5
Ridge Regression = [Σ {] y[i]– β[0]– [Σ] β[i] x[ij]}^2+ [λ Σ] β[i]^2
Lasso Regression = [Σ {] y[i]– β[0]– [Σ] β[i] x[ij]}^2+ [λ Σ] |β[i]|
• x ==> independent variables
• y ==> target variables
• β ==> coefficients
• λ ==> penalty-factor
How coefficients(β) are calculated internally
Code for Regularization
Let’s take Automobile – Predictive Analysis and apply the L1 and L2 and how it helps model score.
Objective: Predicting the Mileage/Miles Per Gallon (mpg) of a car using given features of the car.
print("Import required libraries")
%matplotlib inline
import numpy as np
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
from sklearn.linear_model import Ridge
from sklearn.linear_model import Lasso
from sklearn.metrics import r2_score
Python Code:
import pandas as pd
print(" Using auto-mpg dataset ")
df_cars = pd.read_csv("auto-mpg.csv")
Using auto-mpg dataset
EDA: Will do little EDA (Exploratory Data Analysis), to understand the dataset
print(" Info Of the Data Set")
1. we could see that the features and its data type, along with Null constraints.
2. Horsepower and name features are objects in the given data set. have to take care of during the modelling.
Data Cleaning/Wrangling:
Is the process of cleaning and consolidating the complex data sets for easy access and analysis.
• Action:
□ replace(‘?’,’NaN’)
□ Converting “horsepower” Object type into int
df_cars.horsepower = df_cars.horsepower.str.replace('?','NaN').astype(float)
df_cars.horsepower = df_cars.horsepower.astype(int)
print(" After Cleaning and type covertion in the Data Set")
Scale all the columns successfully done
Train and Test Split
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X_scaled, y_scaled, test_size=0.25, random_state=1)
1. We could see that the features/column/fields and its data type, along with Null count
2. horsepower is now int type and name still as an object type in the given data set, since this column not going to support either way as preditors.
# Statistics of the data
# Skewness and kurtosis
print("Skewness: %f" %df_cars['mpg'].skew())
print("Kurtosis: %f" %df_cars['mpg'].kurt())
Output: Look at the curve and how it is distributed across and see the same.
Skewness: 0.457066
Kurtosis: -0.510781
sns_plot = sns.distplot(df_cars["mpg"])
Output: Look at the heatmap
There is a strong NEGATIVE correlation between mpg and below features
• Displacement
• Horsepower
• Weight
• Cylinders
So, if those variables increases, the mpg will decrease.
Feature Selection
print("Predictor variables")
X = df_cars.drop('mpg', axis=1)
print("Dependent variable")
y = df_cars[['mpg']]
Output: Here is the Feature Selection
Predictor variables
[‘cylinders’, ‘displacement’, ‘horsepower’, ‘weight’, ‘acceleration’, ‘model_year’, ‘origin_america’, ‘origin_asia’, ‘origin_europe’]
Dependent variable
Scaling the feature to align the data
from sklearn import preprocessing
print("Scale all the columns successfully done")
X_scaled = preprocessing.scale(X)
X_scaled = pd.DataFrame(X_scaled, columns=X.columns)
y_scaled = preprocessing.scale(y)
y_scaled = pd.DataFrame(y_scaled, columns=y.columns)
Scale all the columns successfully done
Train and Test Split
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X_scaled, y_scaled, test_size=0.25, random_state=1)
LinearRegression Fit and finding the coefficient.
regression_model = LinearRegression()
regression_model.fit(X_train, y_train)
for idcoff, columnname in enumerate(X_train.columns):
print("The coefficient for {} is {}".format(columnname, regression_model.coef_[0][idcoff]))
Output: Try to understand the coefficient (β[i])
The coefficient for cylinders is -0.08627732236942003
The coefficient for displacement is 0.385244857729236
The coefficient for horsepower is -0.10297215401481062
The coefficient for weight is -0.7987498466220165
The coefficient for acceleration is 0.023089636890550748
The coefficient for model_year is 0.3962256595226441
The coefficient for origin_america is 0.3761300367522465
The coefficient for origin_asia is 0.43102736614202025
The coefficient for origin_europe is 0.4412719522838424
intercept = regression_model.intercept_[0]
print("The intercept for our model is {}".format(intercept))
The intercept for our model is 0.015545728908811594
Scores (LR)
print(regression_model.score(X_train, y_train))
print(regression_model.score(X_test, y_test))
Now, will apply regularization techniques and review the scores and impact of the techniques on the model.
Create a Regularized RIDGE Model and coefficients.
ridge = Ridge(alpha=.3)
print ("Ridge model:", (ridge.coef_))
Output: Compare with LR model coefficient
Ridge model: [[-0.07274955 0.3508473 -0.10462368 -0.78067332 0.01896661 0.39439233
0.29378926 0.36094062 0.37375046]]
Create a Regularized LASSO Model and coefficients
lasso = Lasso(alpha=0.1)
print ("Lasso model:", (lasso.coef_))
Output: Compare with LR model coefficient and RIDGE, Here you could see that the few coefficients and zeroed (0) and during the fitment, they are excluded from the feature list.
Lasso model: [-0. -0. -0.01262531 -0.6098498 0. 0.29478559
-0.03712132 0. 0. ]
Scores (RIDGE)
print(ridge.score(X_train, y_train))
print(ridge.score(X_test, y_test))
Scores (LASSO)
print(lasso.score(X_train, y_train))
print(lasso.score(X_test, y_test))
LR RIDGE (L2) LASSO (L1)
81% 81.4% 79.0%
84.5% 83.0%
Certainly, there is an impact on the model due to the Regularization L2 and L1.
Compare L2 and L1 Regularization
Hope after seeing the code level implementation, you could able to relate the importance of regularization techniques and their influence on the model improvements. As a final touch let’s compare the
L1 & L2.
Ridge Regression (L2) Lasso Regression(L1)
Quite accurate and keep all features More Accurate than Ridge
λ ==> Sum of the squares of coefficient λ ==> Sum of the absolute of coefficient.
The coefficient can be not to zeroed, but rounded The coefficient can be zeroed
Variable selection and keeping all variables Model selection by dropping coefficient
Differentiable and leading for gradient descent calculation Not differentiable
Model fitment justification during training and testing
• Model is doing strongly at training set and poorly in test set means we’re at OVERFIT
• Model is doing poor at both (Training and Testing) means we’re at UNDERFIT
• Model is doing better and considers ways in both (Training and Test), means we’re at the RIGHT FIT
I hope, what we have discussed so for, would really help you all how and why regularization techniques are important and inescapable while building a model. Thanks for your valuable time in reading
this article. Will get back to you with some interesting topics. Until then Bye! Cheers! Shanthababu.
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ELI5: Product (mathematics)
Product in mathematics is like when you multiply two or more numbers together. For example, if you had 3 and 4, the product would be 12 (3 x 4 = 12). The product is the answer that you get when you
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Acta Mathematica, 1993
Acta Mathematica, 61 - 1993. ISSN 0236-5294
Download (127MB) | Preview
1993 / 1-2. szám Yoneda, K.: Uniqueness theorems for Walsh series under a strong condition Di Concilio, A. - Naimpally, S. A.: Uniform continuity in sequentially uniform spaces Albert, G. E. - Wade,
W. R.: Haar systems for compact geometries Ishii, H.: Transition layer phenomena of the solutions of boundary value problems for differential equations with discontinuity Sato, R.: An ergodic maximal
equality for nonsingular flows and applications Feigelstock, S.: On fissible modules Balázs, Katherine - Kilgore, T.: Interpolation and simultaneous mean convergence of derivatives Ashraf, M. -
Quadri, M. A.: Some conditions for the commutativity of rings Criscuolo, Giuliana - Mastroianni, G.: Estimates of the Shepard interpolatory procedure Przemski, M.: A decomposition of continuity and
-continuity Joó I.: On the control of a net of strings Lengvárszky Zs.: Independent subsets in modular lattices of breadth two Komjáth P. - Shelah, S.: A consistent edge partition theorem for
infinite graphs Karsai J.: On the asymptotic behaviour of the solutions of a second order linear differential equation with small damping Joó M.: On additive functions Gát G.: Investigations of
certain operators with respect to the Vilenkin system Bognár M.: On pseudomanifolds with boundary. II Losonczi L.: Measurable solutions of functional equations of sum form 1993 / 3-4. szám Argyros,
I. K.: Some methods for finding error bounds for Newton-like methods under mild differentiability conditions Dikranjan, D. - Giuli, E.: Epis in categories of convergence spaces Kannappan, P. L. -
Sahoo, P. K.: Sum form equations of multiplicative type Chen, H. L. - Chui, C. K.: On a generalized Euler spline and its applications to the study of convergence in cardinal interpolation and
solutions of certain extremal problems Sánchez Ruiz, L. M.: On some topological vector spaces related to the general open mapping theorem Wolke, D.: Some applications to zero density theorems for
L-functions Sommer, M. - Strauss, H.: Order of strong uniqueness in best L-approximation by spline spaces Kasana, H. S.: On approximation of unbounded functions by linear combinations of modified
Szász-Mirakian operators Colzani, L.: Expansions in Legendre polynomials and Lagrange interpolation Joó I.: On the control of a circular membrane. I Horváth M.: Uniform estimations of the Green
function for the singular Schrödinger operator Indlekofer, K.-H. - Kátai I.: On the distribution of translates of additive functions Szabados J.: On the order of magnitude of fundamental polynomials
of hermite interpolation Buczolich, Z. - Evans, M. J. - Humke, P. D.: Approximate high order smoothness
Item Type: Journal
Publisher: Akadémiai Kiadó
Additional Information: Acta Mathematica Hungarica, Volume 61. (1993)
Subjects: Q Science / természettudomány > QA Mathematics / matematika
Depositing User: Kötegelt Import
Date Deposited: 30 Aug 2016 10:17
Last Modified: 16 Dec 2016 20:48
URI: https://real-j.mtak.hu/id/eprint/7458
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14.8 ounces to grams
Convert 14.8 Ounces to Grams (oz to gm) with our conversion calculator. 14.8 ounces to grams equals 419.572965439661 oz.
Enter ounces to convert to grams.
Formula for Converting Ounces to Grams (Oz to Gm):
grams = ounces * 28.3495
By multiplying the number of grams by 28.3495, you can easily obtain the equivalent weight in grams from ounces.
Converting ounces to grams is a common task that many people encounter, especially when dealing with recipes, scientific measurements, or everyday tasks. Understanding the conversion factor is
essential for accurate measurements. In this case, the conversion factor from ounces to grams is 28.3495. This means that one ounce is equivalent to approximately 28.3495 grams.
To convert ounces to grams, you can use the following formula:
Grams = Ounces × 28.3495
Let’s break down the conversion of 14.8 ounces to grams step-by-step:
1. Start with the number of ounces you want to convert: 14.8 ounces.
2. Use the conversion factor: 28.3495 grams per ounce.
3. Multiply the number of ounces by the conversion factor: 14.8 × 28.3495.
4. Perform the calculation: 14.8 × 28.3495 = 419.98 grams.
5. Round the result to two decimal places for practical use: 419.98 grams.
This means that 14.8 ounces is equal to approximately 419.98 grams. This conversion is particularly important as it bridges the gap between the metric and imperial systems, allowing for seamless
communication and understanding in various fields.
Practical examples of where this conversion might be useful include:
• Cooking: Many recipes, especially those from different countries, may list ingredients in grams. Knowing how to convert ounces to grams ensures you can follow the recipe accurately.
• Scientific Measurements: In laboratories, precise measurements are crucial. Converting ounces to grams can help in preparing solutions or measuring substances accurately.
• Everyday Use: Whether you’re weighing food items, calculating shipping weights, or managing dietary needs, understanding how to convert ounces to grams can simplify your tasks.
In conclusion, converting 14.8 ounces to grams is a straightforward process that can enhance your accuracy in various applications. By using the conversion factor of 28.3495, you can easily switch
between these two measurement systems, making your life a little easier!
Here are 10 items that weigh close to 14.8 ounces to grams –
• Standard Football
Shape: Prolate spheroid
Dimensions: Approximately 11 inches long, 22 inches in circumference at the center
Usage: Used in American football games, training, and recreational play.
Random Fact: The first footballs were made from inflated pig bladders, which were later encased in leather.
• Medium-Sized Watermelon
Shape: Spherical
Dimensions: About 9-10 inches in diameter
Usage: Consumed fresh, in salads, or as juice; popular in summer picnics.
Random Fact: Watermelons are 92% water, making them a refreshing treat on hot days.
• Standard Laptop
Shape: Rectangular
Dimensions: Typically around 15 inches wide, 10 inches deep, and 1 inch thick
Usage: Used for computing tasks, browsing the internet, and entertainment.
Random Fact: The first laptop was released in 1981 and weighed over 5 pounds!
• Large Bag of Dog Food
Shape: Rectangular bag
Dimensions: Approximately 18 inches tall, 12 inches wide, and 6 inches deep
Usage: Used to feed dogs; typically contains dry kibble.
Random Fact: The first commercial dog food was created in 1860 and was made from horse meat.
• Standard Yoga Mat
Shape: Rectangular
Dimensions: Usually 68 inches long and 24 inches wide
Usage: Used for yoga practice, stretching, and other floor exercises.
Random Fact: The first yoga mats were made from natural rubber, providing excellent grip.
• Medium-Sized Pumpkin
Shape: Round
Dimensions: About 10-12 inches in diameter
Usage: Used for cooking, baking, and decoration, especially during Halloween.
Random Fact: Pumpkins are technically a fruit and belong to the gourd family.
• Standard Basketball
Shape: Spherical
Dimensions: Approximately 29.5 inches in circumference
Usage: Used in basketball games and practice.
Random Fact: The first basketball was made from a brown leather material and was used in 1891.
• Large Book
Shape: Rectangular
Dimensions: About 9 inches wide, 12 inches tall, and 2 inches thick
Usage: Used for reading, studying, and reference.
Random Fact: The largest book in the world is over 5 feet tall and weighs more than 1,400 pounds!
• Standard Pillow
Shape: Rectangular
Dimensions: Typically 20 inches wide and 26 inches long
Usage: Used for sleeping and providing neck support.
Random Fact: The oldest known pillows were made of stone and were used in ancient Mesopotamia.
• Medium-Sized Bag of Flour
Shape: Rectangular bag
Dimensions: Approximately 12 inches tall, 8 inches wide, and 4 inches deep
Usage: Used in baking and cooking.
Random Fact: Flour has been a staple in human diets for thousands of years, dating back to ancient civilizations.
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Vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars.
Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements,
called vector axioms. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space.
Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector
spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems
of linear equations.
Vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. This means that, for two vector spaces with the same
dimension, the properties that depend only on the vector-space structure are exactly the same (technically the vector spaces are isomorphic). A vector space is finite-dimensional if its dimension is
a natural number. Otherwise, it is infinite-dimensional, and its dimension is an infinite cardinal. Finite-dimensional vector spaces occur naturally in geometry and related areas.
Infinite-dimensional vector spaces occur in many areas of mathematics. For example, polynomial rings are countably infinite-dimensional vector spaces, and many function spaces have the cardinality of
the continuum as a dimension.
Many vector spaces that are considered in mathematics are also endowed with other structures. This is the case of algebras, which include field extensions, polynomial rings, associative algebras and
Lie algebras. This is also the case of topological vector spaces, which include function spaces, inner product spaces, normed spaces, Hilbert spaces and Banach spaces.
Definition and basic properties
In this article, vectors are represented in boldface to distinguish them from scalars.
A vector space over a field F is a set V together with two binary operations that satisfy the eight axioms listed below. In this context, the elements of V are commonly called vectors, and the
elements of F are called scalars.
The first operation, called vector addition or simply addition assigns to any two vectors v and w in V a third vector in V which is commonly written as v + w, and called the sum of these two vectors.
The second operation, called scalar multiplication，assigns to any scalar a in F and any vector v in V another vector in V, which is denoted av.
For having a vector space, the eight following axioms must be satisfied for every u, v and w in V, and a and b in F.
Axiom Meaning
Associativity of vector addition u + (v + w) = (u + v) + w
Commutativity of vector addition u + v = v + u
Identity element of vector addition There exists an element 0 ∈ V, called the zero vector, such that v + 0 = v for all v ∈ V.
Inverse elements of vector addition For every v ∈ V, there exists an element −v ∈ V, called the additive inverse of v, such that v + (−v) = 0.
Compatibility of scalar multiplication with field multiplication a(bv) = (ab)v [nb 3]
Identity element of scalar multiplication 1v = v, where 1 denotes the multiplicative identity in F.
Distributivity of scalar multiplication with respect to vector addition a(u + v) = au + av
Distributivity of scalar multiplication with respect to field addition (a + b)v = av + bv
When the scalar field is the real numbers the vector space is called a real vector space. When the scalar field is the complex numbers, the vector space is called a complex vector space. These two
cases are the most common ones, but vector spaces with scalars in an arbitrary field F are also commonly considered. Such a vector space is called an F-vector space or a vector space over F.
An equivalent definition of a vector space can be given, which is much more concise but less elementary: the first four axioms (related to vector addition) say that a vector space is an abelian group
under addition, and the four remaining axioms (related to the scalar multiplication), say that this operation defines a ring homomorphism from the field F into the endomorphism ring of this group.
Subtraction of two vectors can be defined as
v - w = v + (- w )
Direct consequences of the axioms include that, for every s ∈ in F and v ∈ F one has
• 0v= 0
• s0 = 0
• (-1)v = -v
• sv = implies that s=0 or v=0 | {"url":"https://garden.keerah.com/facts/science/mathematics/vector-space/","timestamp":"2024-11-13T21:26:18Z","content_type":"text/html","content_length":"88301","record_id":"<urn:uuid:0e6cfe58-36b9-47ab-a88b-c63dee80212b>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00082.warc.gz"} |
7.4 - The Model | STAT 415
What do \(a\) and \(b\) estimate? Section
So far, we've formulated the idea, as well as the theory, behind least squares estimation. But, now we have a little problem. When we derived formulas for the least squares estimates of the intercept
\(a\) and the slope \(b\), we never addressed for what parameters \(a\) and \(b\) serve as estimates. It is a crucial topic that deserves our attention. Let's investigate the answer by considering
the (linear) relationship between high school grade point averages (GPAs) and scores on a college entrance exam, such as the ACT exam. Well, let's actually center the high school GPAs so that if \(x
\) denotes the high school GPA, then \(x-\bar{x}\) is the centered high school GPA. Here's what a plot of \(x-\bar{x}\), the centered high school GPA, and \(y\), the college entrance test score might
look like:
Well, okay, so that plot deserves some explanation:
So far, in summary, we are assuming two things. First, among the entire population of college students, there is some unknown linear relationship between \(\mu_y\), (or alternatively \(E(Y)\)), the
average college entrance score, and \(x-\bar{x}\), centered high school GPA. That is:
Second, individual students deviate from the mean college entrance test score of the population of students having the same centered high school GPA by some unknown amount \(\epsilon_i\). That is, if
\(Y_i\) denotes the college entrance test score for student \(i\), then:
Unfortunately, we don't have the luxury of collecting data on all of the college students in the population. So, we can never know the population intercept \(\alpha\) or the population slope \(\beta
\). The best we can do is estimate \(\alpha\) and \(\beta\) by taking a random sample from the population of college students. Suppose we randomly select fifteen students from the population, in
which three students have a centered high school GPA of −2, three students have a centered high school GPA of −1, and so on. We can use those fifteen data points to determine the best fitting (least
squares) line:
Now, our least squares line isn't going to be perfect, but it should do a pretty good job of estimating the true unknown population line:
That's it in a nutshell. The intercept \(a\) and the slope \(b\) of the least squares regression line estimate, respectively, the intercept \(\alpha\) and the slope \(\beta\) of the unknown
population line. The only assumption we make in doing so is that the relationship between the predictor \(x\) and the response \(y\) is linear.
Now, if we want to derive confidence intervals for \(\alpha\) and \(\beta\), as we are going to want to do on the next page, we are going to have to make a few more assumptions. That's where the
simple linear regression model comes to the rescue.
The Simple Linear Regression Model Section
So that we can have properly drawn normal curves, let's borrow (steal?) an example from the textbook called Applied Linear Regression Models (4th edition, by Kutner, Nachtsheim, and Neter). Consider
the relationship between \(x\), the number of bids contracting companies prepare, and \(y\), the number of hours it takes to prepare the bids:
A couple of things to note about this graph. Note that again, the mean number of hours, \(E(Y)\), is assumed to be linearly related to \(X\), the number of bids prepared. That's the first assumption.
The textbook authors even go as far as to specify the values of typically unknown \(\alpha\) and \(\beta\). In this case, \(\alpha\) is 9.5 and \(\beta\) is 2.1.
Note that if \(X=45\) bids are prepared, then the expected number of hours it took to prepare the bids is:
In one case, it took a contracting company 108 hours to prepare 45 bids. In that case, the error \(\epsilon_i\) is 4. That is:
The normal curves drawn for each value of \(X\) are meant to suggest that the error terms \(\epsilon_i\), and therefore the responses \(Y_i\), are normally distributed. That's a second assumption.
Did you also notice that the two normal curves in the plot are drawn to have the same shape? That suggests that each population (as defined by \(X\)) has a common variance. That's a third assumption.
That is, the errors, \(\epsilon_i\), and therefore the responses \(Y_i\), have equal variances for all \(x\) values.
There's one more assumption that is made that is difficult to depict on a graph. That's the one that concerns the independence of the error terms. Let's summarize!
In short, the simple linear regression model states that the following four conditions must hold:
• The mean of the responses, \(E(Y_i)\), is a Linear function of the \(x_i\).
• The errors, \(\epsilon_i\), and hence the responses \(Y_i\), are Independent.
• The errors, \(\epsilon_i\), and hence the responses \(Y_i\), are Normally distributed.
• The errors, \(\epsilon_i\), and hence the responses \(Y_i\), have Equal variances (\(\sigma^2\)) for all \(x\) values.
Did you happen to notice that each of the four conditions is capitalized and emphasized in red? And, did you happen to notice that the capital letters spell L-I-N-E? Do you get it? We are
investigating least squares regression lines, and the model effectively spells the word line! You might find this mnemonic an easy way to remember the four conditions.
Maximum Likelihood Estimators of \(\alpha\) and \(\beta\) Section
We know that \(a\) and \(b\):
\(\displaystyle{a=\bar{y} \text{ and } b=\dfrac{\sum\limits_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{\sum\limits_{i=1}^n (x_i-\bar{x})^2}}\)
are ("least squares") estimators of \(\alpha\) and \(\beta\) that minimize the sum of the squared prediction errors. It turns out though that \(a\) and \(b\) are also maximum likelihood estimators of
\(\alpha\) and \(\beta\) providing the four conditions of the simple linear regression model hold true.
If the four conditions of the simple linear regression model hold true, then:
\(\displaystyle{a=\bar{y}\text{ and }b=\dfrac{\sum\limits_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{\sum\limits_{i=1}^n (x_i-\bar{x})^2}}\)
are maximum likelihood estimators of \(\alpha\) and \(\beta\).
The simple linear regression model, in short, states that the errors \(\epsilon_i\) are independent and normally distributed with mean 0 and variance \(\sigma^2\). That is:
\(\epsilon_i \sim N(0,\sigma^2)\)
The linearity condition:
therefore implies that:
\(Y_i \sim N(\alpha+\beta(x_i-\bar{x}),\sigma^2)\)
Therefore, the likelihood function is:
\(\displaystyle{L_{Y_i}(\alpha,\beta,\sigma^2)=\prod\limits_{i=1}^n \dfrac{1}{\sqrt{2\pi}\sigma} \text{exp}\left[-\dfrac{(Y_i-\alpha-\beta(x_i-\bar{x}))^2}{2\sigma^2}\right]}\)
which can be rewritten as:
\(\displaystyle{L=(2\pi)^{-n/2}(\sigma^2)^{-n/2}\text{exp}\left[-\dfrac{1}{2\sigma^2} \sum\limits_{i=1}^n (Y_i-\alpha-\beta(x_i-\bar{x}))^2\right]}\)
Taking the log of both sides, we get:
\(\displaystyle{\text{log}L=-\dfrac{n}{2}\text{log}(2\pi)-\dfrac{n}{2}\text{log}(\sigma^2)-\dfrac{1}{2\sigma^2} \sum\limits_{i=1}^n (Y_i-\alpha-\beta(x_i-\bar{x}))^2} \)
Now, that negative sign in front of that summation on the right hand side:
\(\color{black}\text{log}L=-\dfrac{n}{2} \text{log} (2\pi)-\dfrac{n}{2}\text{log}\left(\sigma^{2}\right)\color{blue}\boxed{\color{black}-}\color{black}\dfrac{1}{2\sigma^{2}} \color{blue}\boxed{\color
tells us that the only way we can maximize \(\text{log}L(\alpha,\beta,\sigma^2)\) with respect to \(\alpha\) and \(\beta\) is if we minimize:
\(\displaystyle{\sum\limits_{i=1}^n (Y_i-\alpha-\beta(x_i-\bar{x}))^2}\)
with respect to \(\alpha\) and \(\beta\). Hey, but that's just the least squares criterion! Therefore, the ML estimators of \(\alpha\) and \(\beta\) must be the same as the least squares estimators \
(\alpha\) and \(\beta\). That is:
\(\displaystyle{a=\bar{y}\text{ and }b=\dfrac{\sum\limits_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{\sum\limits_{i=1}^n (x_i-\bar{x})^2}}\)
are maximum likelihood estimators of \(\alpha\) and \(\beta\) under the assumption that the error terms are independent, normally distributed with mean 0 and variance \(\sigma^2\). As was to be
What about the (Unknown) Variance \(\sigma^2\)? Section
In short, the variance \(\sigma^2\) quantifies how much the responses (\(y\)) vary around the (unknown) mean regression line \(E(Y)\). Now, why should we care about the magnitude of the variance \(\
sigma^2\)? The following example might help to illuminate the answer to that question.
We know that there is a perfect relationship between degrees Celsius (C) and degrees Fahrenheit (F), namely:
Suppose we are unfortunate, however, and therefore don't know the relationship. We might attempt to learn about the relationship by collecting some temperature data and calculating a least squares
regression line. When all is said and done, which brand of thermometers do you think would yield more precise future predictions of the temperature in Fahrenheit? The one whose data are plotted on
the left? Or the one whose data are plotted on the right?
As you can see, for the plot on the left, the Fahrenheit temperatures do not vary or "bounce" much around the estimated regression line. For the plot on the right, on the other hand, the Fahrenheit
temperatures do vary or "bounce" quite a bit around the estimated regression line. It seems reasonable to conclude then that the brand of thermometers on the left will yield more precise future
predictions of the temperature in Fahrenheit.
Now, the variance \(\sigma^2\) is, of course, an unknown population parameter. The only way we can attempt to quantify the variance is to estimate it. In the case in which we had one population, say
the (normal) population of IQ scores:
we would estimate the population variance \(\sigma^2\) using the sample variance:
\(s^2=\dfrac{\sum\limits_{i=1}^n (Y_i-\bar{Y})^2}{n-1}\)
We have learned that \(s^2\) is an unbiased estimator of \(\sigma^2\), the variance of the one population. But what if we no longer have just one population, but instead have many populations? In our
bids and hours example, there is a population for every value of \(x\):
In this case, we have to estimate \(\sigma^2\), the (common) variance of the many populations. There are two possibilities − one is a biased estimator, and one is an unbiased estimator.
The maximum likelihood estimator of \(\sigma^2\) is:
\(\hat{\sigma}^2=\dfrac{\sum\limits_{i=1}^n (Y_i-\hat{Y}_i)^2}{n}\)
(It is a biased estimator of \(\sigma^2\), the common variance of the many populations.
We have previously shown that the log of the likelihood function is:
\(\text{log}L=-\dfrac{n}{2}\text{log}(2\pi)-\dfrac{n}{2}\text{log}(\sigma^2)-\dfrac{1}{2\sigma^2} \sum\limits_{i=1}^n (Y_i-\alpha-\beta(x_i-\bar{x}))^2 \)
To maximize the log likelihood, we have to take the partial derivative of the log likelihood with respect to \(\sigma^2\). Doing so, we get:
\(\dfrac{\partial L_{Y_i}(\alpha,\beta,\sigma^2)}{\partial \sigma^2}=-\dfrac{n}{2\sigma^2}-\dfrac{1}{2}\sum (Y_i-\alpha-\beta(x_i-\bar{x}))^2 \cdot \left(- \dfrac{1}{(\sigma^2)^2}\right)\)
Setting the derivative equal to 0, and multiplying through by \(2\sigma^4\):
\(\frac{\partial L_{Y_{i}}\left(\alpha, \beta, \sigma^{2}\right)}{\partial \sigma^{2}}=\left[-\frac{n}{2 \sigma^{2}}-\frac{1}{2} \sum\left(Y_{i}-\alpha-\beta\left(x_{i}-\bar{x}\right)\right)^{2} \
cdot-\frac{1}{\left(\sigma^{2}\right)^{2}} \stackrel{\operatorname{SET}}{\equiv} 0\right] 2\left(\sigma^{2}\right)^{2}\)
we get:
\(-n\sigma^2+\sum (Y_i-\alpha-\beta(x_i-\bar{x}))^2 =0\)
And, solving for and putting a hat on \(\sigma^2\), as well as replacing \(\alpha\) and \(\beta\) with their ML estimators, we get:
\(\hat{\sigma}^2=\dfrac{\sum (Y_i-\alpha-\beta(x_i-\bar{x}))^2 }{n}=\dfrac{\sum(Y_i-\hat{Y}_i)^2}{n}\)
As was to be proved!
Mean Square Error
The mean square error, on the other hand:
is an unbiased estimator of \(\sigma^2\), the common variance of the many populations.
We'll need to use these estimators of \(\sigma^2\) when we derive confidence intervals for \(\alpha\) and \(\beta\) on the next page. | {"url":"https://online.stat.psu.edu/stat415/lesson/7/7.4","timestamp":"2024-11-05T07:38:00Z","content_type":"text/html","content_length":"189089","record_id":"<urn:uuid:1ebde199-31d7-43d2-8941-053c07b4f479>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00705.warc.gz"} |
PID Control
PID Controller (2DOF)
Continuous-time or discrete-time two-degree-of-freedom PID controller
Simulink / Continuous
The PID Controller (2DOF) block implements a two-degree-of-freedom PID controller (PID, PI, or PD). The block is identical to the Discrete PID Controller (2DOF) block with the Time domain parameter
set to Continuous-time.
The block generates an output signal based on the difference between a reference signal and a measured system output. The block computes a weighted difference signal for the proportional and
derivative actions according to the setpoint weights (b and c) that you specify. The block output is the sum of the proportional, integral, and derivative actions on the respective difference
signals, where each action is weighted according to the gain parameters P, I, and D. A first-order pole filters the derivative action.
The block supports several controller types and structures. Configurable options in the block include:
• Controller type (PID, PI, or PD) — See the Controller parameter.
• Controller form (Parallel or Ideal) — See the Form parameter.
• Time domain (continuous or discrete) — See the Time domain parameter.
• Initial conditions and reset trigger — See the Source and External reset parameters.
• Output saturation limits and built-in anti-windup mechanism — See the Limit output parameter.
• Signal tracking for bumpless control transfer and multiloop control — See the Enable tracking mode parameter.
As you change these options, the internal structure of the block changes by activating different variant subsystems. (See Implement Variations in Separate Hierarchy Using Variant Subsystems.) To
examine the internal structure of the block and its variant subsystems, right-click the block and select Mask > Look Under Mask.
Control Configuration
In one common implementation, the PID Controller block operates in the feedforward path of a feedback loop.
For a single-input block that accepts an error signal (a difference between a setpoint and a system output), see PID Controller.
PID Gain Tuning
The PID controller coefficients and the setpoint weights are tunable either manually or automatically. Automatic tuning requires Simulink^® Control Design™ software. For more information about
automatic tuning, see the Select tuning method parameter.
Ref — Reference signal
scalar | vector
Reference signal for plant to follow, as shown.
When the reference signal is a vector, the block acts separately on each signal, vectorizing the PID coefficients and producing a vector output signal of the same dimensions. You can specify the PID
coefficients and some other parameters as vectors of the same dimensions as the input signal. Doing so is equivalent to specifying a separate PID controller for each entry in the input signal.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point
Port_1( y ) — Measured system output
scalar | vector
Feedback signal for the controller, from the plant output.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point
ydot — Externally sourced derivative
scalar | vector
Since R2024a
Supply the derivative of the plant signal y directly as an input to the block. This is helpful when you have the derivative signal available in your model and want to skip the computation of the
derivative inside the block.
To enable this input port, select a controller type that has derivative action and enable Use externally sourced derivative.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point
P — Proportional gain
scalar | vector
Proportional gain, provided from a source external to the block. External gain input is useful, for example, when you want to map a different PID parameterization to the PID gains of the block. You
can also use external gain input to implement gain-scheduled PID control. In gain-scheduled control, you determine the PID coefficients by logic or other calculation in your model and feed them to
the block.
To enable this port, set Controller parameters Source to external.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point
I — Integral gain
scalar | vector
Integral gain, provided from a source external to the block. External gain input is useful, for example, when you want to map a different PID parameterization to the PID gains of the block. You can
also use external gain input to implement gain-scheduled PID control. In gain-scheduled control, you determine the PID coefficients by logic or other calculation in your model and feed them to the
When you supply gains externally, time variations in the integral gain are also integrated. This result occurs because of the way the PID gains are implemented within the block. For details, see the
Controller parameters Source parameter.
To enable this port, set Controller parameters Source to external, and set Controller to a controller type that has integral action.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point
I*T[s] — Integral gain multiplied by sample time
scalar | vector
For discrete-time controllers, integral gain multiplied by the controller sample time, provided from a source external to the block. External gain input is useful, for example, when you want to map a
different PID parameterization to the PID gains of the block. You can also use external gain input to implement gain-scheduled PID control. In gain-scheduled control, you determine the PID
coefficients by logic or other calculations in your model and feed them to the block.
PID tuning tools, such as the PID Tuner app and Closed-Loop PID Autotuner block, tune the gain I but not I*T[s]. Therefore, multiply the integral gain value you obtain from a tuning tool by the
sample time before you supply it to this port.
When you use I*T[s] instead of I, the block requires fewer calculations to perform integration. This improves the execution time of the generated code.
For continuous-time controllers, disable Use I*Ts and use the I port instead.
To enable this port, set Controller parameters Source to external, set Controller to a controller type that has integral action, and enable the Use I*Ts parameter.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point
D — Derivative gain
scalar | vector
Derivative gain, provided from a source external to the block. External gain input is useful, for example, when you want to map a different PID parameterization to the PID gains of the block. You can
also use external gain input to implement gain-scheduled PID control. In gain-scheduled control, you determine the PID coefficients by logic or other calculation in your model and feed them to the
When you supply gains externally, time variations in the derivative gain are also differentiated. This result occurs because of the way the PID gains are implemented within the block. For details,
see the Controller parameters Source parameter.
To enable this port, set Controller parameters Source to external, and set Controller to a controller type that has derivative action.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point
N — Filter coefficient
scalar | vector
Derivative filter coefficient, provided from a source external to the block. External coefficient input is useful, for example, when you want to map a different PID parameterization to the PID gains
of the block. You can also use the external input to implement gain-scheduled PID control. In gain-scheduled control, you determine the PID coefficients by logic or other calculation in your model
and feed them to the block.
To enable this port, set Controller parameters Source to external, and set Controller to a controller type that has a filtered derivative.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point
b — Proportional setpoint weight
scalar | vector
Proportional setpoint weight, provided from a source external to the block. External input is useful, for example, when you want to map a different PID parameterization to the PID gains of the block.
You can also use the external input to implement gain-scheduled PID control. In gain-scheduled control, you determine the PID coefficients by logic or other calculation in your model and feed them to
the block.
To enable this port, set Controller parameters Source to external.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point
c — Derivative setpoint weight
scalar | vector
Derivative setpoint weight, provided from a source external to the block. External input is useful, for example, when you want to map a different PID parameterization to the PID gains of the block.
You can also use the external input to implement gain-scheduled PID control. In gain-scheduled control, you determine the PID coefficients by logic or other calculation in your model and feed them to
the block.
To enable this port, set Controller parameters Source to external, and set Controller to a controller type that has derivative action.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point
Reset — External reset trigger
Trigger to reset the integrator and filter to their initial conditions. Use the External reset parameter to specify what kind of signal triggers a reset. The port icon indicates the trigger type
specified in that parameter. For example, the following illustration shows a continuous-time PID Controller (2DOF) block with External reset set to rising.
When the trigger occurs, the block resets the integrator and filter to the initial conditions specified by the Integrator Initial condition and Filter Initial condition parameters or the I[0] and D
[0] ports.
To be compliant with the Motor Industry Software Reliability Association (MISRA™) software standard, your model must use Boolean signals to drive the external reset ports of the PID controller block.
To enable this port, set External reset to any value other than none.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point | Boolean
I[0] — Integrator initial condition
scalar | vector
Integrator initial condition, provided from a source external to the block.
To enable this port, set Initial conditions Source to external, and set Controller to a controller type that has integral action.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point
D[0] — Filter initial condition
scalar | vector
Initial condition of the derivative filter, provided from a source external to the block.
To enable this port, set Initial conditions Source to external, and set Controller to a controller type that has derivative action.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point
up — Output saturation upper limit
scalar | vector
Upper limit of the block output, provided from a source external to the block. If the weighted sum of the proportional, integral, and derivative actions exceeds the value provided at this port, the
block output is held at that value.
To enable this port, select Limit output and set the output saturation Source to external.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point
lo — Output saturation lower limit
scalar | vector
Lower limit of the block output, provided from a source external to the block. If the weighted sum of the proportional, integral, and derivative actions goes below the value provided at this port,
the block output is held at that value.
To enable this port, select Limit output and set the output saturation Source to external.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point
TR — Tracking signal
scalar | vector
Signal for controller output to track. When signal tracking is active, the difference between the tracking signal and the block output is fed back to the integrator input. Signal tracking is useful
for implementing bumpless control transfer in systems that switch between two controllers. It can also be useful to prevent block windup in multiloop control systems. For more information, see the
Enable tracking mode parameter.
To enable this port, select the Enable tracking mode parameter.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point
T[DTI] — Discrete-integrator time
Discrete-integrator time, provided as a scalar to the block. You can use your own value of discrete-time integrator sample time that defines the rate at which the block is going to be run either in
Simulink or on external hardware. The value of the discrete-time integrator time should match the average sampling rate of the external interrupts, when the block is used inside a
conditionally-executed subsystem.
In other words, you can specify Ts for any of the integrator methods below such that the value matches the average sampling rate of the external interrupts. In discrete time, the derivative term of
the controller transfer function is:
$D\left[\frac{N}{1+N\alpha \left(z\right)}\right],$
where α(z) depends on the integrator method you specify with this parameter.
Forward Euler
$\alpha \left(z\right)=\frac{{T}_{s}}{z-1}.$
Backward Euler
$\alpha \left(z\right)=\frac{{T}_{s}z}{z-1}.$
$\alpha \left(z\right)=\frac{{T}_{s}}{2}\frac{z+1}{z-1}.$
For more information about discrete-time integration, see the Discrete-Time Integrator block reference page. For more information on conditionally executed subsystems, see Conditionally Executed
Subsystems Overview.
To enable this port, set Time Domain to Discrete-time and select the PID Controller is inside a conditionally executed subsystem option.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
extAW — External anti-windup algorithm
scalar | vector
Since R2024b
Specify a custom anti-windup algorithm at this port. The block provides two built-in anti-windup methods, however, to unwind the integrator, these methods rely on the sum of the block components
exceeding the specified block output limits. If your application has saturations or limits downstream of the PID controller blocks, you can use the extAW input port to implement a custom anti-windup
logic. The block also provides the signal before the integrator at the preInt output port that you can use to implement a custom algorithm.
To enable this port, on the Saturation tab, select Limit output and set Anti-windup Method to external.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
Port_1( u ) — Controller output
scalar | vector
Controller output, generally based on a sum of the input signal, the integral of the input signal, and the derivative of the input signal, weighted by the setpoint weights and by the proportional,
integral, and derivative gain parameters. A first-order pole filters the derivative action. Which terms are present in the controller signal depends on what you select for the Controller parameter.
The base controller transfer function for the current settings is displayed in the Compensator formula section of the block parameters and under the mask. Other parameters modify the block output,
such as saturation limits specified by the Upper Limit and Lower Limit saturation parameters.
The controller output is a vector signal when any of the inputs is a vector signal. In that case, the block acts as N independent PID controllers, where N is the number of signals in the input
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point
preInt — Pre-integrator signal
scalar | vector
Since R2024b
The block outputs the signal before the integrator at this port. Use this signal as an input for the custom anti-windup algorithm you provide at the extAW port.
To enable this port, on the Saturation tab, select Limit output and set Anti-windup Method to external.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point
Controller — Controller type
PID (default) | PI | PD
Specify which of the proportional, integral, and derivative terms are in the controller.
Proportional, integral, and derivative action.
Proportional and integral action only.
Proportional and derivative action only.
The controller output for the current setting is displayed in the Compensator formula section of the block parameters and under the mask.
Programmatic Use
Block Parameter: Controller
Type: string, character vector
Values: "PID", "PI", "PD"
Default: "PID"
Form — Controller structure
Parallel (default) | Ideal
Specify whether the controller structure is parallel or ideal.
The proportional, integral, and derivative gains P, I, and D, are applied independently. For example, for a continuous-time 2-DOF PID controller in parallel form, the controller output u is:
where r is the reference signal, y is the measured plant output signal, and b and c are the setpoint weights.
For a discrete-time 2-DOF controller in parallel form, the controller output is:
$u=P\left(br-y\right)+I\alpha \left(z\right)\left(r-y\right)+D\frac{N}{1+N\beta \left(z\right)}\left(cr-y\right),$
where the Integrator method and Filter method parameters determine α(z) and β(z), respectively.
The proportional gain P acts on the sum of all actions. For example, for a continuous-time 2-DOF PID controller in ideal form, the controller output is:
For a discrete-time 2-DOF PID controller in ideal form, the transfer function is:
$u=P\left[\left(br-y\right)+I\alpha \left(z\right)\left(r-y\right)+D\frac{N}{1+N\beta \left(z\right)}\left(cr-y\right)\right],$
where the Integrator method and Filter method parameters determine α(z) and β(z), respectively.
The controller output for the current settings is displayed in the Compensator formula section of the block parameters and under the mask.
Programmatic Use
Block Parameter: Controller
Type: string, character vector
Values: "Parallel", "Ideal"
Default: "Parallel"
Time domain — Specify continuous-time or discrete-time controller
Continuous-time (default) | Discrete-time
When you select Discrete-time, it is recommended that you specify an explicit sample time for the block. See the Sample time (-1 for inherited) parameter. Selecting Discrete-time also enables the
Integrator method, and Filter method parameters.
When the PID Controller block is in a model with synchronous state control (see the State Control (HDL Coder) block), you cannot select Continuous-time.
The PID Controller (2DOF) and Discrete PID Controller (2DOF) blocks are identical except for the default value of this parameter.
Programmatic Use
Block Parameter: TimeDomain
Type: string, character vector
Values: "Continuous-time", "Discrete-time"
Default: "Continuous-time"
PID Controller is inside a conditionally executed subsystem — Enable the discrete-integrator time port
off (default) | on
For discrete-time PID controllers, enable the discrete-time integrator port to use your own value of discrete-time integrator sample time. To ensure proper integration, use the T[DTI] port to provide
a scalar value of Δt for accurate discrete-time integration.
To enable this parameter, set Time Domain to Discrete-time.
Programmatic Use
Block Parameter: UseExternalTs
Type: string, character vector
Values: "on", "off"
Default: "off"
Sample time (-1 for inherited) — Discrete interval between samples
–1 (default) | positive scalar
Specify a sample time by entering a positive scalar value, such as 0.1. The default discrete sample time of –1 means that the block inherits its sample time from upstream blocks. However, it is
recommended that you set the controller sample time explicitly, especially if you expect the sample time of upstream blocks to change. The effect of the controller coefficients P, I, D, and N depend
on the sample time. Thus, for a given set of coefficient values, changing the sample time changes the performance of the controller.
See Specify Sample Time for more information.
To implement a continuous-time controller, set Time domain to Continuous-time.
If you want to run the block with an externally specified or variable sample time, set this parameter to –1 and put the block in a Triggered Subsystem. Then, trigger the subsystem at the desired
sample time.
To enable this parameter, set Time domain to Discrete-time.
Programmatic Use
Block Parameter: SampleTime
Type: scalar
Values: -1, positive scalar
Default: -1
Integrator method — Method for computing integral in discrete-time controller
Forward Euler (default) | Backward Euler | Trapezoidal
In discrete time, the integral term of the controller transfer function is Ia(z), where a(z) depends on the integrator method you specify with this parameter.
Forward Euler
Forward rectangular (left-hand) approximation,
$\alpha \left(z\right)=\frac{{T}_{s}}{z-1}.$
This method is best for small sampling times, where the Nyquist limit is large compared to the bandwidth of the controller. For larger sampling times, the Forward Euler method can result in
instability, even when discretizing a system that is stable in continuous time.
Backward Euler
Backward rectangular (right-hand) approximation,
$\alpha \left(z\right)=\frac{{T}_{s}z}{z-1}.$
An advantage of the Backward Euler method is that discretizing a stable continuous-time system using this method always yields a stable discrete-time result.
Bilinear approximation,
$\alpha \left(z\right)=\frac{{T}_{s}}{2}\frac{z+1}{z-1}.$
An advantage of the Trapezoidal method is that discretizing a stable continuous-time system using this method always yields a stable discrete-time result. Of all available integration methods,
the Trapezoidal method yields the closest match between frequency-domain properties of the discretized system and the corresponding continuous-time system.
The controller formula for the current setting is displayed in the Compensator formula section of the block parameters and under the mask.
For more information about discrete-time integration, see the Discrete-Time Integrator block reference page.
To enable this parameter, set Time Domain to Discrete-time and set Controller to a controller type with integral action.
Programmatic Use
Block Parameter: IntegratorMethod
Type: string, character vector
Values: "Forward Euler", "Backward Euler", "Trapezoidal"
Default: "Forward Euler"
Filter method — Method for computing derivative in discrete-time controller
Forward Euler (default) | Backward Euler | Trapezoidal
In discrete time, the derivative term of the controller transfer function is:
$D\left[\frac{N}{1+N\alpha \left(z\right)}\right],$
where α(z) depends on the filter method you specify with this parameter.
Forward Euler
Forward rectangular (left-hand) approximation,
$\alpha \left(z\right)=\frac{{T}_{s}}{z-1}.$
This method is best for small sampling times, where the Nyquist limit is large compared to the bandwidth of the controller. For larger sampling times, the Forward Euler method can result in
instability, even when discretizing a system that is stable in continuous time.
Backward Euler
Backward rectangular (right-hand) approximation,
$\alpha \left(z\right)=\frac{{T}_{s}z}{z-1}.$
An advantage of the Backward Euler method is that discretizing a stable continuous-time system using this method always yields a stable discrete-time result.
Bilinear approximation,
$\alpha \left(z\right)=\frac{{T}_{s}}{2}\frac{z+1}{z-1}.$
An advantage of the Trapezoidal method is that discretizing a stable continuous-time system using this method always yields a stable discrete-time result. Of all available integration methods,
the Trapezoidal method yields the closest match between frequency-domain properties of the discretized system and the corresponding continuous-time system.
The controller formula for the current setting is displayed in the Compensator formula section of the block parameters and under the mask.
For more information about discrete-time integration, see the Discrete-Time Integrator block reference page.
To enable this parameter, set Time Domain to Discrete-time and enable Use filtered derivative.
Programmatic Use
Block Parameter: FilterMethod
Type: string, character vector
Values: "Forward Euler", "Backward Euler", "Trapezoidal"
Default: "Forward Euler"
Source — Source for controller gains and filter coefficient
internal (default) | external
Specify the controller gains, filter coefficient, and setpoint weights using the block parameters P, I, D, N, b, and c respectively.
Specify the PID gains, filter coefficient, and setpoint weights externally using block inputs. An additional input port appears on the block for each parameter that is required for the current
controller type.
Enabling external inputs for the parameters allows you to compute their values externally to the block and provide them to the block as signal inputs.
External input is useful, for example, when you want to map a different PID parameterization to the PID gains of the block. You can also use external gain input to implement gain-scheduled PID
control. In gain-scheduled control, you determine the PID gains by logic or other calculation in your model and feed them to the block.
When you supply gains externally, time variations in the integral and derivative gain values are integrated and differentiated, respectively. The derivative setpoint weight c is also differentiated.
This result occurs because in both continuous time and discrete time, the gains are applied to the signal before integration or differentiation. For example, for a continuous-time PID controller with
external inputs, the integrator term is implemented as shown in the following illustration.
Within the block, the input signal u is multiplied by the externally supplied integrator gain, I, before integration. This implementation yields:
${u}_{i}=\int \left(r-y\right)I\text{\hspace{0.17em}}dt.$
Thus, the integrator gain is included in the integral. Similarly, in the derivative term of the block, multiplication by the derivative gain precedes the differentiation, which causes the derivative
gain D and the derivative setpoint weight c to be differentiated.
Programmatic Use
Block Parameter: ControllerParametersSource
Type: string, character vector
Values: "internal", "external"
Default: "internal"
Proportional (P) — Proportional gain
1 (default) | scalar | vector
Specify a finite, real gain value for the proportional gain. When Controller form is:
• Parallel — Proportional action is independent of the integral and derivative actions. For example, for a continuous-time 2-DOF PID controller in parallel form, the controller output u is:
where r is the reference signal, y is the measured plant output signal, and b and c are the setpoint weights.
For a discrete-time 2-DOF controller in parallel form, the controller output is:
$u=P\left(br-y\right)+I\alpha \left(z\right)\left(r-y\right)+D\frac{N}{1+N\beta \left(z\right)}\left(cr-y\right),$
where the Integrator method and Filter method parameters determine α(z) and β(z), respectively.
• Ideal — The proportional gain multiples the integral and derivative terms. For example, for a continuous-time 2-DOF PID controller in ideal form, the controller output is:
For a discrete-time 2-DOF PID controller in ideal form, the transfer function is:
$u=P\left[\left(br-y\right)+I\alpha \left(z\right)\left(r-y\right)+D\frac{N}{1+N\beta \left(z\right)}\left(cr-y\right)\right],$
where the Integrator method and Filter method parameters determine α(z) and β(z), respectively.
Tunable: Yes
To enable this parameter, set the Controller parameters Source to internal.
Programmatic Use
Block Parameter: P
Type: scalar, vector
Default: 1
Integral (I) — Integral gain
1 (default) | scalar | vector
Specify a finite, real gain value for the integral gain.
Tunable: Yes
To enable this parameter, in the Main tab, set the controller-parameters Source to internal, and set Controller to a type that has integral action.
Programmatic Use
Block Parameter: I
Type: scalar, vector
Default: 1
Integral (I*Ts) — Integral gain multiplied by sample time
1 (default) | scalar | vector
For discrete-time controllers, specify a finite, real gain value for the integral gain multiplied by the sample time.
PID tuning tools, such as the PID Tuner app and Closed-Loop PID Autotuner block, tune the gain I but not I*T[s]. Therefore, multiply the integral gain value you obtain from a tuning tool by the
sample time before you write it to this parameter.
When you use I*Ts instead of I, the block requires fewer calculations to perform integration. This improves the execution time of the generated code.
For continuous-time controllers, disable Use I*Ts and use the I parameter instead.
Tunable: No
To enable this parameter, in the Main tab, set the controller-parameters Source to internal, set Controller to a type that has integral action, and enable the Use I*Ts parameter.
Programmatic Use
Block Parameter: I
Type: scalar, vector
Default: 1
Use I*Ts — Use integral gain multiplied by sample time
off (default) | on
For discrete-time controllers with integral action, the block takes the integral gain as an input and multiplies it by the sample time internally as a part of performing the integration. If you
enable this parameter, you explicitly specify integral gain multiplied by sample time as input (I*Ts) in place of the integral gain (I). Doing so reduces the number of internal calculations and is
useful when you want to improve the execution time of your generated code.
If you have enabled signal tracking or the anti-windup mode back-calculation and you enable I*Ts, then you must also set the tracking gain parameter Kt to Kt*Ts and the back-calculation coefficient
Kb to Kb*Ts.
For continuous-time controllers, enabling this parameter has no effect on the integral gain.
To enable this parameter, set Controller to a controller type that has integral action.
Programmatic Use
Block Parameter: UseKiTs
Type: string, character vector
Values: "on", "off"
Default: "on"
Derivative (D) — Derivative gain
0 (default) | scalar | vector
Specify a finite, real gain value for the derivative gain.
Tunable: Yes
To enable this parameter, in the Main tab, set the controller-parameters Source to internal, and set Controller to PID or PD.
Programmatic Use
Block Parameter: D
Type: scalar, vector
Default: 0
Use externally sourced derivative — Specify derivative at block input port
off (default) | on
Since R2024a
Select this option to specify the derivative of the plant signal y directly as an input ydot to the block. This is helpful when you have the derivative signal available in your model and want to skip
the computation of the derivative inside the block.
To enable this option, select a controller type that has derivative action.
Use filtered derivative — Apply filter to derivative term
on (default) | off
For discrete-time PID controllers only, clear this option to replace the filtered derivative with an unfiltered discrete-time differentiator. When you do so, the derivative term of the controller
output becomes:
For continuous-time PID controllers, the derivative term is always filtered.
To enable this parameter, set Time domain to Discrete-time, and set Controller to a type that has a derivative term.
Programmatic Use
Block Parameter: UseFilter
Type: string, character vector
Values: "on", "off"
Default: "on"
Filter coefficient (N) — Derivative filter coefficient
100 (default) | scalar | vector
Specify a finite, real gain value for the filter coefficient. The filter coefficient determines the pole location of the filter in the derivative action of the block. The location of the filter pole
depends on the Time domain parameter.
• When Time domain is Continuous-time, the pole location is s = -N.
• When Time domain is Discrete-time, the pole location depends on the Filter method parameter.
Filter Method Location of Filter Pole
Forward Euler ${z}_{pole}=1-N{T}_{s}$
Backward Euler ${z}_{pole}=\frac{1}{1+N{T}_{s}}$
Trapezoidal ${z}_{pole}=\frac{1-N{T}_{s}/2}{1+N{T}_{s}/2}$
The block does not support N = Inf (ideal unfiltered derivative). When the Time domain is Discrete-time, you can clear Use filtered derivative to remove the derivative filter.
Tunable: Yes
To enable this parameter, in the Main tab, set the controller-parameters Source to internal and set Controller to PID or PD.
Programmatic Use
Block Parameter: N
Type: scalar, vector
Default: 100
Setpoint weight (b) — Proportional setpoint weight
1 (default) | scalar | vector
Setpoint weight on the proportional term of the controller. The proportional term of a 2-DOF controller output is P(br–y), where r is the reference signal and y is the measured plant output. Setting
b to 0 eliminates proportional action on the reference signal, which can reduce overshoot in the system response to step changes in the setpoint. Changing the relative values of b and c changes the
balance between disturbance rejection and setpoint tracking.
Tunable: Yes
To enable this parameter, in the Main tab, set the controller-parameters Source to internal.
Programmatic Use
Block Parameter: b
Type: scalar, vector
Default: 1
Setpoint weight (c) — Derivative setpoint weight
1 (default) | scalar | vector
Setpoint weight on the derivative term of the controller. The derivative term of a 2-DOF controller acts on cr–y, where r is the reference signal and y is the measured plant output. Thus, setting c
to 0 eliminates derivative action on the reference signal, which can reduce transient response to step changes in the setpoint. Setting c to 0 can yield a controller that achieves both effective
disturbance rejection and smooth setpoint tracking without excessive transient response. Changing the relative values of b and c changes the balance between disturbance rejection and setpoint
Tunable: Yes
To enable this parameter, in the Main tab, set the controller-parameters Source to internal and set Controller to a type that has derivative action.
Programmatic Use
Block Parameter: c
Type: scalar, vector
Default: 1
Select tuning method — Tool for automatic tuning of controller coefficients
Transfer Function Based (PID Tuner App) (default) | Frequency Response Based
If you have Simulink Control Design software, you can automatically tune the PID coefficients when they are internal to the block. To do so, use this parameter to select a tuning tool, and click Tune
Transfer Function Based (PID Tuner App)
Use PID Tuner, which lets you interactively tune PID coefficients while examining relevant system responses to validate performance. PID Tuner can tune all the coefficients P, I, D, and N, and
the setpoint coefficients b and c. By default, PID Tuner works with a linearization of your plant model. For models that cannot be linearized, you can tune PID coefficients against a plant model
estimated from simulated or measured response data. For more information, see Design Two-Degree-of-Freedom PID Controllers (Simulink Control Design).
Frequency Response Based
Use Frequency Response Based PID Tuner, which tunes PID controller coefficients based on frequency-response estimation data obtained by simulation. This tuning approach is especially useful for
plants that are not linearizable or that linearize to zero. Frequency Response Based PID Tuner tunes the coefficients P, I, D, and N, but does not tune the setpoint coefficients b and c. For more
information, see Design PID Controller from Plant Frequency-Response Data (Simulink Control Design).
Both of these tuning methods assume a single-loop control configuration. Simulink Control Design software includes other tuning approaches that suit more complex configurations. For information about
other ways to tune a PID Controller block, see Choose a Control Design Approach (Simulink Control Design).
To enable this parameter, in the Main tab, set the controller-parameters Source to internal.
Enable zero-crossing detection — Detect zero crossings on reset and on entering or leaving a saturation state
on (default) | off
Zero-crossing detection can accurately locate signal discontinuities without resorting to excessively small time steps that can lead to lengthy simulation times. If you select Limit output or
activate External reset in your PID Controller block, activating zero-crossing detection can reduce computation time in your simulation. Selecting this parameter activates zero-crossing detection:
• At initial-state reset
• When entering an upper or lower saturation state
• When leaving an upper or lower saturation state
For more information about zero-crossing detection, see Zero-Crossing Detection.
Programmatic Use
Block Parameter: ZeroCross
Type: string, character vector
Values: "on", "off"
Default: "on"
Source — Source for integrator and derivative initial conditions
internal (default) | external
Simulink uses initial conditions to initialize the integrator and derivative-filter (or the unfiltered derivative) output at the start of a simulation or at a specified trigger event. (See the
External reset parameter.) These initial conditions determine the initial block output. Use this parameter to select how to supply the initial condition values to the block.
Specify the initial conditions using the Integrator Initial condition and Filter Initial condition parameters. If Use filtered derivative is not selected, use the Differentiator parameter to
specify the initial condition for the unfiltered differentiator instead of a filter initial condition.
Specify the initial conditions externally using block inputs. Additional input ports I[o] and D[o] appear on the block. If Use filtered derivative is not selected, supply the initial condition
for the unfiltered differentiator at D[o] instead of a filter initial condition.
Programmatic Use
Block Parameter: InitialConditionSource
Type: string, character vector
Values: "internal", "external"
Default: "internal"
Integrator — Integrator initial condition
0 (default) | scalar | vector
Simulink uses the integrator initial condition to initialize the integrator at the start of a simulation or at a specified trigger event (see External reset). The integrator initial condition and the
filter initial condition determine the initial output of the PID controller block.
The integrator initial condition cannot be NaN or Inf.
To use this parameter, in the Initialization tab, set Source to internal, and set Controller to a type that has integral action.
Programmatic Use
Block Parameter: InitialConditionForIntegrator
Type: scalar, vector
Default: 0
Filter — Filter initial condition
0 (default) | scalar | vector
Simulink uses the filter initial condition to initialize the derivative filter at the start of a simulation or at a specified trigger event (see External reset). The integrator initial condition and
the filter initial condition determine the initial output of the PID controller block.
The filter initial condition cannot be NaN or Inf.
To use this parameter, in the Initialization tab, set Source to internal, and use a controller that has a derivative filter.
Programmatic Use
Block Parameter: InitialConditionForFilter
Type: scalar, vector
Default: 0
Differentiator — Initial condition for unfiltered derivative
0 (default) | scalar | vector
When you use an unfiltered derivative, Simulink uses this parameter to initialize the differentiator at the start of a simulation or at a specified trigger event (see External reset). The integrator
initial condition and the derivative initial condition determine the initial output of the PID controller block.
The derivative initial condition cannot be NaN or Inf.
To use this parameter, set Time domain to Discrete-time, clear the Use filtered derivative check box, and in the Initialization tab, set Source to internal.
Programmatic Use
Block Parameter: DifferentiatorICPrevScaledInput
Type: scalar, vector
Default: 0
Initial condition setting — Location at which initial condition is applied
Auto (default) | Output
Use this parameter to specify whether to apply the Integrator Initial condition and Filter Initial condition parameter to the corresponding block state or output. You can change this parameter at the
command line only, using set_param to set the InitialConditionSetting parameter of the block.
Use this option in all situations except when the block is in a triggered subsystem or a function-call subsystem and simplified initialization mode is enabled.
Use this option when the block is in a triggered subsystem or a function-call subsystem and simplified initialization mode is enabled.
For more information about the Initial condition setting parameter, see the Discrete-Time Integrator block.
This parameter is only accessible through programmatic use.
Programmatic Use
Block Parameter: InitialConditionSetting
Type: string, character vector
Values: "Auto", "Output"
Default: "Auto"
External reset — Trigger for resetting integrator and filter values
none (default) | rising | falling | either | level
Specify the trigger condition that causes the block to reset the integrator and filter to initial conditions. (If Use filtered derivative is not selected, the trigger resets the integrator and
differentiator to initial conditions.) Selecting any option other than none enables the Reset port on the block for the external reset signal.
The integrator and filter (or differentiator) outputs are set to initial conditions at the beginning of simulation, and are not reset during simulation.
Reset the outputs when the reset signal has a rising edge.
Reset the outputs when the reset signal has a falling edge.
Reset the outputs when the reset signal either rises or falls.
Reset the outputs when the reset signal either:
□ Is nonzero at the current time step
□ Changes from nonzero at the previous time step to zero at the current time step
This option holds the outputs to the initial conditions while the reset signal is nonzero.
To enable this parameter, set Controller to a type that has derivative or integral action.
Programmatic Use
Block Parameter: ExternalReset
Type: string, character vector
Values: "none", "rising", "falling", "either","level"
Default: "none"
Ignore reset when linearizing — Force linearization to ignore reset
off (default) | on
Select to force Simulink and Simulink Control Design linearization commands to ignore any reset mechanism specified in the External reset parameter. Ignoring reset states allows you to linearize a
model around an operating point even if that operating point causes the block to reset.
Programmatic Use
Block Parameter: IgnoreLimit
Type: string, character vector
Values: "off", "on"
Default: "off"
Enable tracking mode — Activate signal tracking
off (default) | on
Signal tracking lets the block output follow a tracking signal that you provide at the TR port. When signal tracking is active, the difference between the tracking signal and the block output is fed
back to the integrator input with a gain Kt, specified by the Tracking gain (Kt) parameter. Signal tracking has several applications, including bumpless control transfer and avoiding windup in
multiloop control structures.
Bumpless control transfer
Use signal tracking to achieve bumpless control transfer in systems that switch between two controllers. Suppose you want to transfer control between a PID controller and another controller. To do
so, connecting the controller output to the TR input as shown in the following illustration.
For more information, see Bumpless Control Transfer with a Two-Degree-of-Freedom PID Controller.
Multiloop control
Use signal tracking to prevent block windup in multiloop control approaches. For an example illustrating this approach with a 1DOF PID controller, see Prevent Block Windup in Multiloop Control.
To enable this parameter, set Controller to a type that has integral action.
Programmatic Use
Block Parameter: TrackingMode
Type: string, character vector
Values: "off", "on"
Default: "off"
Tracking coefficient (Kt) — Gain of signal-tracking feedback loop
1 (default) | scalar
When you select Enable tracking mode, the difference between the signal TR and the block output is fed back to the integrator input with a gain Kt. Use this parameter to specify the gain in that
feedback loop.
For discrete-time controllers, if you select the Use I*Ts parameter of the block, then set this parameter to the value Kt*Ts, where Kt is the desired gain and Ts is the sample time.
To enable this parameter, select Enable tracking mode.
Programmatic Use
Block Parameter: Kt
Type: scalar
Default: 1
Output saturation
Limit Output — Limit block output to specified saturation values
off (default) | on
Activating this option limits the block output, so that you do not need a separate Saturation block after the controller. It also allows you to activate the anti-windup mechanism built into the block
(see the Anti-windup method parameter). Specify the output saturation limits using the Lower limit and Upper limit parameters. You can also specify the saturation limits externally as block input
Programmatic Use
Block Parameter: LimitOutput
Type: string, character vector
Values: "off", "on"
Default: "off"
Source — Source for output saturation limits
internal (default) | external
Use this parameter to specify how to supply the upper and lower saturation limits of the block output.
Specify the output saturation limits using the Upper limit and Lower limit parameters.
Specify the output saturation limits externally using block input ports. The additional input ports up and lo appear on the block. You can use the input ports to implement the upper and lower
output saturation limits determined by logic or other calculations in the Simulink model and passed to the block.
Programmatic Use
Block Parameter: SatLimitsSource
Type: string, character vector
Values: "internal", "external"
Default: "internal"
Upper limit — Upper saturation limit for block output
Inf (default) | scalar
Specify the upper limit for the block output. The block output is held at the Upper saturation limit whenever the weighted sum of the proportional, integral, and derivative actions exceeds that
To enable this parameter, select Limit output.
Programmatic Use
Block Parameter: UpperSaturationLimit
Type: scalar
Default: Inf
Lower limit — Lower saturation limit for block output
-Inf (default) | scalar
Specify the lower limit for the block output. The block output is held at the Lower saturation limit whenever the weighted sum of the proportional, integral, and derivative actions goes below that
To enable this parameter, select Limit output.
Programmatic Use
Block Parameter: LowerSaturationLimit
Type: scalar
Default: -Inf
Ignore saturation when linearizing — Force linearization to ignore output limits
off (default) | on
Force Simulink and Simulink Control Design linearization commands to ignore block output limits specified in the Upper limit and Lower limit parameters. Ignoring output limits allows you to linearize
a model around an operating point even if that operating point causes the block to exceed the output limits.
To enable this parameter, select the Limit output parameter.
Programmatic Use
Block Parameter: LinearizeAsGain
Type: string, character vector
Values: "off", "on"
Default: "off"
Anti-windup method — Integrator anti-windup method
none (default) | back-calculation | clamping | external
When you select Limit output and the weighted sum of the controller components exceeds the specified output limits, the block output holds at the specified limit. However, the integrator output can
continue to grow (integrator windup), increasing the difference between the block output and the sum of the block components. In other words, the internal signals in the block can be unbounded even
if the output appears bounded by saturation limits. Without a mechanism to prevent integrator windup, two results are possible:
• If the sign of the signal entering the integrator never changes, the integrator continues to integrate until it overflows. The overflow value is the maximum or minimum value for the data type of
the integrator output.
• If the sign of the signal entering the integrator changes once the weighted sum has grown beyond the output limits, it can take a long time to unwind the integrator and return the weighted sum
within the block saturation limit.
In either case, controller performance can suffer. To combat the effects of windup without an anti-windup mechanism, it may be necessary to detune the controller (for example, by reducing the
controller gains), resulting in a sluggish controller. To avoid this problem, activate an anti-windup mechanism using this parameter.
Do not use an anti-windup mechanism.
Unwind the integrator when the block output saturates by feeding back to the integrator the difference between the saturated and unsaturated control signal. The following diagram represents the
back-calculation feedback circuit for a continuous-time controller. To see the actual feedback circuit for your controller configuration, right-click the block and select Mask > Look Under Mask.
Use the Back-calculation coefficient (Kb) parameter to specify the gain of the anti-windup feedback circuit. It is usually satisfactory to set Kb = I, or for controllers with derivative action,
Kb = sqrt(I*D). Back-calculation can be effective for plants with relatively large dead time [1].
Integration stops when the sum of the block components exceeds the output limits and the integrator output and block input have the same sign. Integration resumes when the sum of the block
components exceeds the output limits and the integrator output and block input have opposite sign. Clamping is sometimes referred to as conditional integration.
Clamping can be useful for plants with relatively small dead times, but can yield a poor transient response for large dead times [1].
external (since R2024b)
The built-in anti-windup methods rely on the sum of the block components exceeding the specified block output limits. If your application has saturations or limits downstream of the PID
controller blocks, you can use the extAW input port to implement a custom anti-windup logic. The block also provides the signal before the integrator at the preInt output port that you can use as
input to your custom algorithm.
To enable this parameter, select the Limit output parameter.
Programmatic Use
Block Parameter: AntiWindupMode
Type: string, character vector
Values: "none", "back-calculation","clamping", "external"
Default: "none"
Back-calculation coefficient (Kb) — Gain coefficient of anti-windup feedback loop
1 (default) | scalar
The back-calculation anti-windup method unwinds the integrator when the block output saturates. It does so by feeding back to the integrator the difference between the saturated and unsaturated
control signal. Use the Back-calculation coefficient (Kb) parameter to specify the gain of the anti-windup feedback circuit. For more information, see the Anti-windup method parameter.
For discrete-time controllers, if you select the Use I*Ts parameter of the block, then set this parameter to the value Kb*Ts, where Kb is the desired coefficient and Ts is the sample time.
To enable this parameter, select the Limit output parameter, and set the Anti-windup method parameter to back-calculation.
Programmatic Use
Block Parameter: Kb
Type: scalar
Default: 1
Integrator saturation
Limit Output — Limit integrator output to specified saturation limits
off (default) | on
Enable this parameter to limit the integrator output to be within a specified range. When the integrator output reaches the limits, the integral action turns off to prevent integral windup. Specify
the saturation limits using the Lower limit and Upper limit parameters.
To enable this parameter, set Controller to a controller type that has integral action.
Programmatic Use
Block Parameter: LimitIntegratorOutput
Type: string, character vector
Values: "off", "on"
Default: "off"
Upper limit — Upper saturation limit for integrator
Inf (default) | scalar
Specify the upper limit for the integrator output. The integrator output is held at this value whenever it would otherwise exceed this value.
To enable this parameter, under Integrator saturation, select Limit output.
Programmatic Use
Block Parameter: UpperIntegratorSaturationLimit
Type: scalar
Default: Inf
Lower limit — Lower saturation limit for integrator
-Inf (default) | scalar
Specify the lower limit for the integrator output. The integrator output is held at this value whenever it would otherwise go below this value.
To enable this parameter, under Integrator saturation, select Limit output.
Programmatic Use
Block Parameter: LowerIntegratorSaturationLimit
Type: scalar
Default: -Inf
Data Types
The parameters in this tab are primarily of use in fixed-point code generation using Fixed-Point Designer™. They define how numeric quantities associated with the block are stored and processed when
you generate code.
If you need to configure data types for fixed-point code generation, click Open Fixed-Point Tool and use that tool to configure the rest of the parameters in the tab. For information about using
Fixed-Point Tool, see Autoscaling Data Objects Using the Fixed-Point Tool (Fixed-Point Designer).
After you use Fixed-Point Tool, you can use the parameters in this tab to make adjustments to fixed-point data-type settings if necessary. For each quantity associated with the block, you can
• Floating-point or fixed-point data type, including whether the data type is inherited from upstream values in the block.
• The minimum and maximum values for the quantity, which determine how the quantity is scaled for fixed-point representation.
For assistance in selecting appropriate values, click to open the Data Type Assistant for the corresponding quantity. For more information, see Specify Data Types Using Data Type Assistant.
The specific quantities listed in the Data Types tab vary depending on how you configure the PID controller block. In general, you can configure data types for the following types of quantities:
• Product output — Stores the result of a multiplication carried out under the block mask. For example, P product output stores the output of the gain block that multiplies the block input with the
proportional gain P.
• Parameter — Stores the value of a numeric block parameter, such as P, I, or D.
• Block output — Stores the output of a block that resides under the PID controller block mask. For example, use Integrator output to specify the data type of the output of the block called
Integrator. This block resides under the mask in the Integrator subsystem, and computes integrator term of the controller action.
• Accumulator — Stores values associated with a sum block. For example, SumI2 Accumulator sets the data type of the accumulator associated with the sum block SumI2. This block resides under the
mask in the Back Calculation subsystem of the Anti-Windup subsystem.
In general, you can find the block associated with any listed parameter by looking under the PID Controller block mask and examining its subsystems. You can also use the Model Explorer to search
under the mask for the listed parameter name, such as SumI2. (See Model Explorer.)
Matching Input and Internal Data Types
By default, all data types in the block are set to Inherit: Inherit via internal rule. With this setting, Simulink chooses data types to balance numerical accuracy, performance, and generated code
size, while accounting for the properties of the embedded target hardware.
Under some conditions, incompatibility can occur between data types within the block. For instance, in continuous time, the Integrator block under the mask can accept only signals of type double. If
the block input signal is a type that cannot be converted to double, such as uint16, the internal rules for type inheritance generate an error when you generate code.
To avoid such errors, you can use the Data Types settings to force a data type conversion. For instance, you can explicitly set P product output, I product output, and D product output to double,
ensuring that the signals reaching the continuous-time integrators are of type double.
In general, it is not recommended to use the block in continuous time for code generation applications. However, similar data type errors can occur in discrete time, if you explicitly set some values
to data types that are incompatible with downstream signal constraints within the block. In such cases, use the Data Types settings to ensure that all data types are internally compatible.
Fixed-Point Operational Parameters
Integer rounding mode — Rounding mode for fixed-point operations
Floor (default) | Ceiling | Convergent | Nearest | Round | Simplest | Zero
Specify the rounding mode for fixed-point operations. For more information, see Rounding Modes (Fixed-Point Designer).
Block parameters always round to the nearest representable value. To control the rounding of a block parameter, enter an expression using a MATLAB^® rounding function into the mask field.
Programmatic Use
To set the block parameter value programmatically, use the set_param function.
Parameter: RndMeth
Values: 'Floor' (default) | 'Ceiling' | 'Convergent' | 'Nearest' | 'Round' | 'Simplest' | 'Zero'
Saturate on integer overflow — Method of overflow action
off (default) | on
Specify whether overflows saturate or wrap.
• on — Overflows saturate to either the minimum or maximum value that the data type can represent.
• off — Overflows wrap to the appropriate value that the data type can represent.
For example, the maximum value that the signed 8-bit integer int8 can represent is 127. Any block operation result greater than this maximum value causes overflow of the 8-bit integer.
• With this parameter selected, the block output saturates at 127. Similarly, the block output saturates at a minimum output value of -128.
• With this parameter cleared, the software interprets the overflow-causing value as int8, which can produce an unintended result. For example, a block result of 130 (binary 1000 0010) expressed as
int8 is -126.
• Consider selecting this parameter when your model has a possible overflow and you want explicit saturation protection in the generated code.
• Consider clearing this parameter when you want to optimize efficiency of your generated code. Clearing this parameter also helps you to avoid overspecifying how a block handles out-of-range
signals. For more information, see Troubleshoot Signal Range Errors.
• When you select this parameter, saturation applies to every internal operation on the block, not just the output or result.
• In general, the code generation process can detect when overflow is not possible. In this case, the code generator does not produce saturation code.
Programmatic Use
To set the block parameter value programmatically, use the set_param function.
Parameter: SaturateOnIntegerOverflow
Values: 'off' (default) | 'on'
Lock data type settings against changes by the fixed-point tools — Prevent fixed-point tools from overriding data types
off (default) | on
Select this parameter to prevent the fixed-point tools from overriding the data types you specify on this block. For more information, see Lock the Output Data Type Setting (Fixed-Point Designer).
Programmatic Use
Block Parameter: LockScale
Type: character vector
Values: 'off' | 'on'
Default: 'off'
State Attributes
The parameters in this tab are primarily of use in code generation.
State name (e.g., 'position') — Name for continuous-time filter and integrator states
'' (default) | character vector
Assign a unique name to the state associated with the integrator or the filter, for continuous-time PID controllers. (For information about state names in a discrete-time PID controller, see the
State name parameter.) The state name is used, for example:
• For the corresponding variable in generated code
• As part of the storage name when logging states during simulation
• For the corresponding state in a linear model obtain by linearizing the block
A valid state name begins with an alphabetic or underscore character, followed by alphanumeric or underscore characters.
To enable this parameter, set Time domain to Continuous-time.
Programmatic Use
Parameter: IntegratorContinuousStateAttributes, FilterContinuousStateAttributes
Type: character vector
Default: ''
State name — Names for discrete-time filter and integrator states
empty string (default) | string | character vector
Assign a unique name to the state associated with the integrator or the filter, for discrete-time PID controllers. (For information about state names in a continuous-time PID controller, see the
State name (e.g., 'position') parameter.)
A valid state name begins with an alphabetic or underscore character, followed by alphanumeric or underscore characters. The state name is used, for example:
• For the corresponding variable in generated code
• As part of the storage name when logging states during simulation
• For the corresponding state in a linear model obtain by linearizing the block
For more information about the use of state names in code generation, see C Data Code Interface Configuration for Model Interface Elements (Simulink Coder).
To enable this parameter, set Time domain to Discrete-time.
Programmatic Use
Parameter: IntegratorStateIdentifier, FilterStateIdentifier
Type: string, character vector
Default: ""
State name must resolve to Simulink signal object — Require that state name resolve to a signal object
off (default) | on
Select this parameter to require that the discrete-time integrator or filter state name resolves to a Simulink signal object.
To enable this parameter for the discrete-time integrator or filter state:
1. Set Time domain to Discrete-time.
2. Specify a value for the integrator or filter State name.
3. Set the model configuration parameter Signal resolution to a value other than None.
Programmatic Use
Block Parameter: IntegratorStateMustResolveToSignalObject, FilterStateMustResolveToSignalObject
Type: string, character vector
Values: "off", "on"
Default: "off"
Block Characteristics
Data Types double | fixed point | integer | single
Direct Feedthrough no
Multidimensional Signals no
Variable-Size Signals no
Zero-Crossing Detection no
More About
Decomposition of 2-DOF PID Controllers
A 2-DOF PID controller can be interpreted as a PID controller with a prefilter, or a PID controller with a feedforward element.
Prefilter Decomposition
In parallel form, a two-degree-of-freedom PID controller can be equivalently modeled by the following block diagram, where C is a single degree-of-freedom PID controller and F is a prefilter on the
reference signal.
Ref is the reference signal, y is the feedback from the measured system output, and u is the controller output. For a continuous-time 2-DOF PID controller in parallel form, the transfer functions for
F and C are
$\begin{array}{l}{F}_{par}\left(s\right)=\frac{\left(bP+cDN\right){s}^{2}+\left(bPN+I\right)s+IN}{\left(P+DN\right){s}^{2}+\left(PN+I\right)s+IN},\\ {C}_{par}\left(s\right)=\frac{\left(P+DN\right){s}
where b and c are the setpoint weights.
For a 2-DOF PID controller in ideal form, the transfer functions are
$\begin{array}{l}{F}_{id}\left(s\right)=\frac{\left(b+cDN\right){s}^{2}+\left(bN+I\right)s+IN}{\left(1+DN\right){s}^{2}+\left(N+I\right)s+IN},\\ {C}_{id}\left(s\right)=P\frac{\left(1+DN\right){s}^{2}
A similar decomposition applies for a discrete-time 2-DOF controller.
Feedforward Decomposition
Alternatively, the parallel two-degree-of-freedom PID controller can be modeled by the following block diagram.
In this realization, Q acts as feedforward conditioning on the reference signal. For a continuous-time 2-DOF PID controller in parallel form, the transfer function for Q is
For a 2-DOF PID controller in ideal form, the transfer function is
The transfer functions for C are the same as in the filter decomposition.
A similar decomposition applies for a discrete-time 2-DOF controller.
[1] Visioli, A., "Modified Anti-Windup Scheme for PID Controllers," IEE Proceedings - Control Theory and Applications, Vol. 150, Number 1, January 2003
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
For continuous-time PID controllers (Time domain set to Continuous-time):
• Consider using Model Discretizer to map continuous-time blocks to discrete equivalents that support code generation. To access Model Discretizer, in the Apps tab, under Control Systems, click
Model Discretizer.
• Not recommended for production code.
For discrete-time PID controllers (Time domain set to Discrete-time):
• Depends on absolute time when placed inside a triggered subsystem hierarchy.
• Generated code relies on memcpy or memset functions (string.h) under certain conditions.
PLC Code Generation
Generate Structured Text code using Simulink® PLC Coder™.
Fixed-Point Conversion
Design and simulate fixed-point systems using Fixed-Point Designer™.
Fixed-point code generation is supported for discrete-time PID controllers only (Time domain set to Discrete-time).
Version History
Introduced in R2009b
R2024b: Specify anti-windup algorithm externally using new ports
The block now allows you to specify an anti-windup algorithm externally using a new input port extAW. The block also provides the signal before the integrator at the preInt output port that you can
use as input for the custom algorithm. The PID controller blocks provide two built-in anti-windup methods, however, to unwind the integrator, these methods rely on the sum of the block components
exceeding the specified block output limits. If your application has saturations or limits downstream of the PID controller blocks, you can use the new extAW and preInt ports to implement a custom
anti-windup logic. To enable the new ports, on the Saturation tab, select Limit Output and set Anti-windup Method to external.
R2024a: Use derivative signal from external source
The PID controller blocks now allow you to supply the derivative of the plant signal y directly as an input to the block. This is helpful when you have the derivative signal available in your model
and want to skip the computation of the derivative inside the block.
To enable the input port for supplying the derivative, select a controller type that has derivative action and enable the Use externally sourced derivative parameter.
R2022b: Issues error when integrator and filter initial conditions lie outside saturation limits
The block now issues an error when the integrator or filter initial condition value lies outside the output saturation limits. In previous releases, the block did not issue an error when these
initial conditions had such values.
If this change impacts your model, update the PID integrator or filter initial condition values such that they are within the output saturation limits.
R2021b: ReferenceBlock parameter returns different path
Starting in R2021b, the get_param function returns a different value for the ReferenceBlock parameter. The ReferenceBlock parameter is a property common to all Simulink blocks and gives the path of
the library block to which a block links. The PID Controller (2DOF) and Discrete PID Controller (2DOF) blocks now link to 'slpidlib/PID Controller (2DOF)'. Previously, the blocks linked to 'pid_lib/
PID Controller (2DOF)'.
This change does not affect any other functionality or workflows. You can still use the previous path with the set_param function.
R2020b: ReferenceBlock parameter returns different path
Starting in R2020b, the get_param function returns a different value for the ReferenceBlock parameter. The ReferenceBlock parameter is a property common to all Simulink blocks and gives the path of
the library block to which a block links. The PID Controller (2DOF) and Discrete PID Controller (2DOF) blocks now link to 'pid_lib/PID Controller (2DOF)'. Previously, the blocks linked to 'simulink/
Continuous/PID Controller (2DOF)'.
This change does not affect any other functionality or workflows. You can still use the previous path with the set_param function. | {"url":"https://it.mathworks.com/help/simulink/slref/pidcontroller2dof.html","timestamp":"2024-11-10T08:52:22Z","content_type":"text/html","content_length":"335020","record_id":"<urn:uuid:666df4c8-2dc7-467d-a270-fb79ef6376ff>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00304.warc.gz"} |
The Rate of Convergence of the Augmented Lagrangian Method for Nonlinear Semidefinite Programming
We analyze the rate of local convergence of the augmented Lagrangian method for nonlinear semidefinite optimization. The presence of the positive semidefinite cone constraint requires extensive tools
such as the singular value decomposition of matrices, an implicit function theorem for semismooth functions, and certain variational analysis on the projection operator in the symmetric-matrix space.
Without requiring strict complementarity, we prove that, under the constraint nondegeneracy condition and the strong second order sufficient condition, the rate is proportional to $1/c$, where $c$ is
the penalty parameter that exceeds a threshold $\overline{c}>0$.
Technical Report, Department of Mathematics, National University of Singapore, January 2006.
View The Rate of Convergence of the Augmented Lagrangian Method for Nonlinear Semidefinite Programming | {"url":"https://optimization-online.org/2006/01/1294/","timestamp":"2024-11-12T06:09:29Z","content_type":"text/html","content_length":"84439","record_id":"<urn:uuid:1dddef21-8d43-40b0-bae3-79e263d97350>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00538.warc.gz"} |
Sixth Grade
An angle is a geometric figure formed by two rays that extend from the same point, called the vertex. Angles are measured in degrees, and they are used to measure the amount of rotation or turn
between two rays.
Types of Angles
Angles are measured in degrees, and a protractor is used to measure and draw angles. The symbol for degrees is °.
Angle Relationships
There are several important angle relationships to understand, including complementary angles (two angles that add up to 90 degrees), supplementary angles (two angles that add up to 180 degrees), and
vertically opposite angles (pairs of angles formed by two intersecting lines).
Angle Properties
Angles have properties that affect their relationships with other angles and geometric figures, such as parallel lines, triangles, and polygons.
Practice Problems
1. What type of angle is formed when the measure is 45 degrees?
Answer: An acute angle.
2. If one angle measures 60 degrees, what is the measure of its complement?
Answer: The complement of a 60-degree angle is 30 degrees.
Study Tips
• Practice using a protractor to measure and draw angles.
• Understand the properties and relationships of angles in various geometric figures.
• Look for real-life examples of angles in the environment to reinforce understanding.
• Practice identifying and classifying angles in different contexts.
Understanding angles is essential for geometry and various other fields of mathematics and science. Mastering the concept of angles will provide a solid foundation for more advanced topics in the | {"url":"https://newpathworksheets.com/math/grade-6/algebraic-equations-1?dictionary=angle&did=229","timestamp":"2024-11-08T06:15:15Z","content_type":"text/html","content_length":"46471","record_id":"<urn:uuid:52d23737-fefa-4c64-ad3a-d60a028347e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00632.warc.gz"} |
The Asymptotic Convergence-Rate of Q-learningThe Asymptotic Convergence-Rate of Q-learning
The Asymptotic Convergence-Rate of Q-learning
The Asymptotic Convergence-Rate of Q-learning
The asymptotic rate of convergence of Q-learning is Ο( 1/t^R(1-γ) ), if R(1-γ)<0.5, where R=P[min]/P[max], P is state-action occupation frequency.
|Q[t] (x,a) − Q*(x,a)| < B/t^R(1-γ)
Convergence-rate is the difference between True value and Optimum value, i.e., the smaller it is, the faster convergence Q-learning is. We hope the Ο( 1/t^R(1-γ) ) should be as small as possible,
which means the R is bigger, i.e., the on-policy distribution is higher, the state space should be smaller. | {"url":"https://blogs.cuit.columbia.edu/zp2130/the_asymptotic_convergence-rate_of_q-learning/","timestamp":"2024-11-11T04:03:03Z","content_type":"text/html","content_length":"53377","record_id":"<urn:uuid:ba0acc40-b03f-4fe3-be5b-5745841a15b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00604.warc.gz"} |
Simple Trigonometric Equations
Lesson Video: Simple Trigonometric Equations Mathematics • First Year of Secondary School
In this video, we will learn how to find angle measures given interval and function values.
Video Transcript
In this video, we will learn how to find the general solution of a trigonometric equation or how to solve it over a specified interval. A trigonometric equation is an equation which involves at least
one of the following: a trig function such as sine, cosine, and tangent; a reciprocal trig function, cosecant, secant, and cotangent; or an inverse of any of these. Some of the simpler examples of
such equations can be solved without using a calculator. In these cases, we will use our knowledge of the special angles together with the symmetry and periodicity of the sine, cosine, and tangent
We will begin this video by recalling the exact values for the sine, cosine, and tangent of a number of special angles. It is important that we are able to recall the sine, cosine, and tangent of
zero, 30, 45, 60, and 90 degrees. It is also important to know the corresponding values of these angles in radians. Whilst we’ll not consider the proofs of these in any detail in this video, it is
important to recall that they come from our knowledge of right-angle trigonometry and the Pythagorean theorem. The exact values of sine, cosine, and tangent of the angles given are shown. Recalling
the identity tan 𝜃 is equal to sin 𝜃 over cos 𝜃, we can calculate the tangent of any of these angles by dividing the value of the sine of the angle by the value of the cosine of the angle.
In our first example, we will demonstrate how to use the symmetry of the graph of the sine function alongside this table of values to find all solutions to a simple trig equation.
What is the general solution of sin 𝜃 is equal to root two over two?
In order to find the general solution to a trigonometric equation, we begin by finding a particular solution. In this case, the table of exact trig values can help. For any angle 𝜃 given in radian
measure, the exact values of the sine function are as shown. We observe that the sin of 𝜋 by four radians is equal to root two over two. This means that 𝜃 equals 𝜋 over four is a particular solution
to the equation sin 𝜃 equals root two over two. To find further solutions, we sketch the graph of 𝑦 equals sin 𝜃 between zero and two 𝜋. The solutions to sin 𝜃 equals root two over two are found by
adding the line 𝑦 equals root two over two to the diagram.
We notice that this intersects the curve twice between zero and two 𝜋. The first point of intersection corresponds to the solution 𝜋 over four. Since the sine curve has symmetry about 𝜋 over two in
the interval from zero to 𝜋, the second solution is found by subtracting 𝜋 over four from 𝜋. This is equal to three 𝜋 over four. We now have two solutions to the equation sin 𝜃 equals root two over
two: 𝜃 equals 𝜋 over four and 𝜃 equals three 𝜋 over four. Recalling that the sine function is periodic with a period of 360 degrees or two 𝜋 radians, we can find the general solution. Firstly, we
have 𝜃 is equal to 𝜋 over four plus two 𝑛𝜋 — we can write it like this because further solutions are found by adding or subtracting multiples of two 𝜋 or 360 degrees — and secondly, three 𝜋 over four
plus two 𝑛𝜋, where 𝑛 is an integer.
In this question, we demonstrated how to interpret the symmetry of the graph of the sine function to find all solutions to an equation. Another way of extending the domain of the sine function is the
unit circle. Recalling that the unit circle is centered at the origin with a radius of one unit, we can work out the sine of any angle 𝜃 by starting at the point one, zero and traveling along the
circumference of the circle in a counterclockwise direction until the angle that is formed between this point, the origin, and the positive 𝑥-axis is equal to 𝜃. If this point has coordinates 𝑥, 𝑦,
then sin 𝜃 is equal to the value of 𝑦. The value of the 𝑦-coordinate is positive in both the first and second quadrants. Hence, the value of sin 𝜃 will also be positive in these quadrants.
Since the unit circle has reflection symmetry about the 𝑦-axis, we can see that sin 𝜃 is equal to sin of 180 degrees minus 𝜃. By continuing to move along the circumference of the unit circle, we see
that sin 𝜃 is also equal to sin of 360 plus 𝜃 for all values of 𝜃. These results can be generalized as shown where the set of all solutions to sin 𝜃 equals 𝐶 is 𝜃 equals 𝜃 sub one plus 360𝑛 and 𝜃
equals 180 minus 𝜃 sub one plus 360𝑛, where 𝑛 is an integer. Note that if 𝜃 is measured in radians, 360 degrees is replaced by two 𝜋 and 180 degrees by 𝜋. Whilst we might feel inclined to memorize
these formulae, in practice, it could be much more effective to sketch the graph of the function or the unit circle.
In our next example, we will look at how to use the symmetry of the graph of the cosine function to solve a trigonometric equation.
Find the set of values satisfying the cos of 𝜃 minus 105 is equal to negative a half, where 𝜃 is greater than zero degrees and less than 360 degrees.
In order to find the solutions to a trig equation in a given interval, we begin by finding a particular solution. In this case, the table of exact trigonometric values can help. We will first
redefine the argument of the function by letting 𝛼 equal 𝜃 minus 105 such that the cos of 𝛼 is equal to negative one-half and 𝜃 is equal to 𝛼 plus 105. We can then amend the interval over which our
solutions are valid by adding 105 to each part of the inequality; 𝛼 is greater than 105 degrees and less than 465 degrees. Filling in the table for the exact values of cos 𝛼, we can see that cos 𝛼
equals a half when 𝛼 is 60 degrees. However, there are no values of 𝛼 in the table such that cos 𝛼 is equal to negative a half.
By sketching the graph of the cosine function along with the lines 𝑦 equals one-half and 𝑦 equals negative a half, we can find the associated value of 𝛼. It appears on the graph that there might be
three values between 105 and 465 degrees. As the graph has rotational symmetry between zero and 180 degrees about 90 degrees, zero, the first solution is equal to 180 minus 60. This is equal to 120
degrees, which lies in the required interval. Next, using symmetry of the curve, we have 𝛼 is equal to 180 plus 60. This is equal to 240 degrees, which also lies in the given interval. The third
solution corresponds to 120 plus 360 degrees. However, this value of 480 degrees lies outside of our interval for 𝛼. Hence, the solutions to cos 𝛼 equals negative one-half are 𝛼 equals 120 degrees
and 𝛼 equals 240 degrees.
We can now calculate the corresponding values of 𝜃. 120 plus 105 is equal to 225, and 240 plus 105 is 345. The set of values that satisfy cos of 𝜃 minus 105 equals negative one-half are 225 degrees
and 345 degrees. An alternative technique to find the particular solution to cos of 𝛼 equals negative one-half is to use the inverse cosine function such that 𝛼 is equal to the inverse cos of
negative one-half, which is equal to 120 degrees. From this point, we would use the same steps to find the other solutions. This could also have been done using the unit circle, which would lead us
to the general rule: the cos of 𝜃 is equal to the cos of 360 degrees minus 𝜃.
Using the symmetry of the unit circle and periodicity of the cosine function, we can quote formulas for the general solution to equations involving this function. In the same way as we have already
seen for the sine function, then the set of all solutions to cos 𝜃 equals 𝐶 is 𝜃 is equal to 𝜃 sub one plus 360𝑛 and 𝜃 is equal to 360 minus 𝜃 sub one plus 360𝑛 for all integer values of 𝑛. Once
again, if 𝜃 is measured in radians, we replace 360 degrees with two 𝜋 radians.
We saw in the previous question how to solve a trig equation where the argument of the function has been transformed in some way. We will now look at a similar version of this involving the tangent
Find the set of values satisfying the tan of two 𝑥 plus 𝜋 over five is equal to negative one, where 𝑥 is greater than or equal to zero and less than or equal to two 𝜋.
To solve this equation, we’ll begin by redefining the argument as this will allow us to use the symmetry of the tangent function. We will let 𝜃 equal two 𝑥 plus 𝜋 over five. This means that we need
to solve the tan of 𝜃 equals negative one where 𝜃 is greater than or equal to 𝜋 over five and less than or equal to 21𝜋 over five as we multiply each part of the inequality by two and then add 𝜋 over
five. Next, we recall that for 𝜃 measured in radians, the exact values of tan 𝜃 are as shown. We see that tan of 𝜋 over four is equal to one. Next, we will sketch the graph of 𝑦 equals the tan of 𝜃.
We will then add the horizontal lines where 𝑦 equals one and 𝑦 equals negative one.
Due to the rotational symmetry of the tangent function, the first solution occurs when 𝜃 is equal to 𝜋 minus 𝜋 over four. This is equal to three 𝜋 over four. As the function is periodic with a period
of 𝜋 radians, we can find the remaining solutions by adding multiples of 𝜋 to this value. Firstly, three 𝜋 over four plus 𝜋 is equal to seven 𝜋 over four. We also have solutions 11𝜋 over four and 15𝜋
over four. These are the four points of intersection shown on the graph. Clearing some space and rewriting our four solutions for 𝜃, we can now calculate the values of 𝑥. As 𝜃 is equal to two 𝑥 plus
𝜋 over five, two 𝑥 is equal to 𝜃 minus 𝜋 over five. Dividing through by two, we have 𝑥 is equal to 𝜃 over two minus 𝜋 over 10.
We can now substitute each of our values of 𝜃 into this equation. This gives us four values of 𝑥 equal to 11𝜋 over 40, 31𝜋 over 40, 51𝜋 over 40, and 71𝜋 over 40. This is the set of values that
satisfies the equation tan of two 𝑥 plus 𝜋 over five equals negative one where 𝑥 lies between zero and two 𝜋 inclusive.
As we have done for the sine and cosine functions, we can now quote the general solutions to equations involving the tangent function. When 𝜃 is measured in degrees, the solutions are 𝜃 is equal to 𝜃
sub one plus 180𝑛 where 𝑛 is an integer. And if 𝜃 is measured in radians, we have 𝜃 is equal to 𝜃 sub one plus 𝑛𝜋 where, once again, 𝑛 is an integer.
In this video, we’ve only considered the standard trigonometric functions sine, cosine, and tangent. Whilst we’ll not cover them here, it is important to understand that the process holds for the
reciprocal functions cosecant, secant, and cotangent.
We will now recap the key points from this video. We can solve simple trigonometric equations using tables of exact values or the inverse trig functions. To help us calculate all solutions to a given
equation in a specified range, we can draw the graph of the necessary trig function or use the unit circle. The symmetry and periodicity of the sine, cosine, and tangent functions allow us to
calculate further solutions to trigonometric equations or general solutions involving integer multiples of 360 degrees or two 𝜋 radians for sine and cosine and 180 degrees or 𝜋 radians for tangent. | {"url":"https://www.nagwa.com/en/videos/792163631761/","timestamp":"2024-11-12T00:12:54Z","content_type":"text/html","content_length":"273790","record_id":"<urn:uuid:19673ce7-deac-40eb-a096-919ac551c785>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00277.warc.gz"} |
Laziness: Clojure vs Haskell
Last week I punted on randomness, and just made my genetic_search function take a [Double]. While that was convenient, it is unfortunately not as general as I thought at the time. I'm still in the
process of learning Haskell, and I got confused between laziness in Clojure and laziness in Haskell.
So how do they differ? The one-word answer is "purity", but let me try to expand on that a little bit. Clojure has the "seq" abstraction, which is defined by the two functions first and next, where
first gives you an element and next gives you another seq if there are more elements to be had, or nil if there are none. When I think of a lazy list, I think in terms of Clojure seqs, even though
Clojure lists are actually not lazy.
How is that different from Haskell? In Clojure, a lazy seq is one where the elements are explicitly produced on demand, and then cached. This sounds a lot like laziness in Haskell, except for one
crucial difference: Clojure does not mind how these elements are produced, and in particular, whether that involves any kind of side effects.
In Haskell, on the other hand, laziness means that the elements of a list will be computed on demand, but that only applies to pure computation. There is no lazy side effect in Haskell.
Here is a short Clojure program to illustrate this notion further. This program will ask the user for positive integers and then print a running total.
First, we start with a seemingly pure function which, given a possibly lazy seq of numbers, returns its sum so far:
(defn sum-so-far
(reductions + 0 ls))
Testing this out yields the expected behaviour:
t.core=> (sum-so-far [1 2 3 4 5])
(0 1 3 6 10 15)
The range function, with no argument, returns an infinite lazy seq of integers, starting with 0. We can use our sum-so-far function on it:
t.core=> (take 10 (sum-so-far (range)))
(0 0 1 3 6 10 15 21 28 36)
We can also construct a lazy seq by asking the user for input:
(defn read-nums
(print "Please enter a number: ")
(cons (Integer/parseInt (read-line)) (lazy-seq (read-nums))))
Testing it works as expected:
t.core=> (take 5 (read-nums))
Please enter a number: 1
Please enter a number: 2
Please enter a number: 3
Please enter a number: 4
Please enter a number: 5
(1 2 3 4 5)
Similarly, we can easily compute the running sum:
t.core=> (take 5 (sum-so-far (read-nums)))
Please enter a number: 1
Please enter a number: 2
Please enter a number: 3
Please enter a number: 4
(0 1 3 6 10)
So we have this one function, sum-so-far, that can handle any Clojure seq, regardless of how it is processed, and produces a new seq itself. Such a function is better thought of as a filter acting on
a stream than as a function taking an argument and returning a result.
Let's look at the Haskell equivalent. The sum-so-far function seems easy enough:
sum_so_far :: [Int] -> [Int]
sum_so_far is =
loop 0 is
loop :: Int -> [Int] -> [Int]
loop sum [] = [sum]
loop sum (h:t) = sum : loop (sum + h) t
I don't know if Haskell has a direct equivalent to Clojure's reductions, but that's not the point here and it's easy enough to code our own. This obvisouly works as intended on both finite and
infinite lists:
*Main> sum_so_far [1, 2, 3, 4, 5]
*Main> let ints = 1:map (+1) ints
*Main> take 10 $ sum_so_far ints
But what about read-nums? It's not too hard to replicate the beginning of the function:
read_nums :: IO [Int]
read_nums = do
int <- read <$> getLine
return int : ???
How can we replace those ???? Well, the : ("cons") function needs a list as its second argument, so we could try constructing that:
read_nums :: IO [Int]
read_nums = do
int <- read <$> getLine
tl <- read_nums
return $ int : tl
That typechecks. But the whole point of monads is to sequence computation: there is no way we can reach the return line without having first produced the entire tl, and thus we're not lazy anymore
and can't pile on more processing on top of this.
What other option do we have? Perhaps Hoogle knows of a functiont that would help here? The type we'd need would look something like:
Int -> IO [Int] -> IO [Int]
so we could replace : and use that instead. Hoogle does find a result for that, but it's not quite what we need here. We could of course write a function with that signature easily enough:
ignore_tl :: Int -> IO [Int] -> IO [Int]
ignore_tl i _ = pure [i]
but that's obviously not what we want.
Are we stuck? Let's take a step back. The solution here is to realize that Clojure seqs are not the same as Haskell lists. Instead of thinking of sum-so-far as a function, let's go back to the idea
of thinking of it as a filter between two streams. What would it take to construct such a filter in Haskell? We'd need a type with the following operations:
• Produce an element in the "output" stream.
• Request an element from the "input" stream.
• Let my consumer know that I will not be producing further elements.
The decoupling Clojure gives us is to be completely independent of how the input elements are produced and to produce the output elements on-demand. Let's model this a bit more precisely. We need a
data definition OnDemand that represents a filter between two streams of elements. The filter could change the type, so we'll make it take two type parameters: an input one and an output one. We
start with:
data OnDemand input output
Next, we need to be able to express "here is an element" and "there are no more elements". We can take a page from the List book and express those exactly like Nil and Cons:
= Halt
| Out output (OnDemand input output)
Finally, we need to be able to ask for a new element, wait for it, and then keep going. This phrasing suggests we need to suspend the current computation to give our supplier the opportunity to
manufacture an input, and keep going after that. A generally good way to model suspended computations is with a continuation:
| In (Maybe input -> OnDemand input output)
where the input parameter is wrapped in a Maybe because the input stream may have ran out.
Using this definition, we can rewrite our sum_so_far as a filter:
{-# LANGUAGE LambdaCase #-}
{- ... -}
sum_so_far :: OnDemand Int Int
sum_so_far =
loop 0
loop :: Int -> OnDemand Int Int
loop sum = Out sum (In $ \case
Nothing -> Halt
Just n -> loop (sum + n))
We can make this work on lists again with a simple adapter:
convert_list :: OnDemand input out -> [input] -> [out]
convert_list = \case
Halt -> \_ -> []
Out out cont -> \ls -> out : convert_list cont ls
In f -> \case
[] -> convert_list (f Nothing) []
hd:tl -> convert_list (f $ Just hd) tl
and we can use (convert_list sum_so_far) as we did before, with both finite and infinite lists:
*Main> (convert_list sum_so_far) [1, 2, 3, 4, 5]
*Main> let ints = 1:map (+1) ints
*Main> take 10 $ (convert_list sum_so_far) ints
But let's stay in the realm of streams for a bit. First, let's define a simple function to produce a stream from a list:
out_list :: [a] -> OnDemand () a
out_list [] = Halt
out_list (h:t) = Out h (out_list t)
Then, let's define a function to drain a stream into a list:
drain :: OnDemand () b -> [b]
drain = \case
Halt -> []
Out b kont -> b : drain kont
In f -> drain $ f Nothing
Now, we can do fun stuff like
*Main> drain $ out_list [1, 2, 3, 4]
Ok, so maybe that's not so much fun yet. We'd like to be able to express the equivalent of take 10 $ sum_so_far $ ints. Let's first work on each of these pieces. We can get a stream of naturals with
ints :: OnDemand () Int
ints = loop 1
loop n = Out n (loop (n + 1))
and we can limit a stream to a given number of elements with:
take_od :: Int -> OnDemand a a
take_od 0 = Halt
take_od n = In (\case
Nothing -> Halt
Just a -> Out a (take_od $ n - 1))
We now have all the pieces. What's the equivalent of $? We need to take two filters and return a filter than combines them. Here is the code:
join :: OnDemand a b -> OnDemand b c -> OnDemand a c
join od1 od2 = case od2 of
Halt -> Halt
Out c kont -> Out c (join od1 kont)
In f -> case od1 of
Halt -> join od1 (f Nothing)
Out b kont -> join kont (f $ Just b)
In g -> In (\ma -> join (g ma) od2)
We start from the outer filter. If that one says to stop, we don't need to look into any more input from the inner filter. If we have an output ready, we can just produce that. So far, so good. What
happens if the outer filter needs an input? Well, in that case, we need to look at the inner one. Does it have an output ready? If so, we can just feed that into the outer filter. Is it halted? We
can feed that information into the outer filter by calling its continuation with Nothing. Finally, if the inner filter itself is also waiting for an input, we have no other choice but to ask for more
input from the context.
We can now have a bit more fun:
*Main> drain $ ints `join` sum_so_far `join` take_od 20
This may look like the beginnings of a useful abstraction. But can we do IO with it?
Let's try to write read_nums. We still cannot write a OnDemand () (IO Int) that would be useful, just like we could not write a useful IO [Int]. But the whole point of this OnDemand stuff is to do
operations one at a time. So let's define a function that gets a single integer:
import qualified Text.Read
{- ... -}
read_num :: IO (Maybe Int)
read_num = Text.Read.readMaybe <$> getLine
We cannot create an infinite stream of lazy IO actions. We've already gone through that rabbit hole. But what we can do is define a function that will run a filter within the IO context and generate
all the required elements on demand:
process :: IO (Maybe a) -> OnDemand a b -> IO [b]
process io = \case
Halt -> return []
Out hd k -> do
tl <- process io k
return $ hd : tl
In f -> do
input <- io
process io (f input)
Now we can use the exat same sum_so_far filter with values coming from pure and impure contexts:
*Main> drain $ ints `join` sum_so_far `join` take_od 10
*Main> process read_num $ sum_so_far `join` take_od 5
Here is the full code for reference:
{-# LANGUAGE LambdaCase #-}
module Main where
import qualified Text.Read
data OnDemand a b
= Halt
| Out b (OnDemand a b)
| In (Maybe a -> OnDemand a b)
sum_so_far :: OnDemand Int Int
sum_so_far =
loop 0
loop :: Int -> OnDemand Int Int
loop sum = Out sum (In $ \case
Nothing -> Halt
Just n -> loop (sum + n))
convert_list :: OnDemand input out -> [input] -> [out]
convert_list = \case
Halt -> \_ -> []
Out out cont -> \ls -> out : convert_list cont ls
In f -> \case
[] -> convert_list (f Nothing) []
hd:tl -> convert_list (f $ Just hd) tl
out_list :: [a] -> OnDemand () a
out_list [] = Halt
out_list (h:t) = Out h (out_list t)
drain :: OnDemand () b -> [b]
drain = \case
Halt -> []
Out b kont -> b : drain kont
In f -> drain $ f Nothing
ints :: OnDemand () Int
ints = loop 1
loop n = Out n (loop (n + 1))
take_od :: Int -> OnDemand a a
take_od 0 = Halt
take_od n = In (\case
Nothing -> Halt
Just a -> Out a (take_od $ n - 1))
join :: OnDemand a b -> OnDemand b c -> OnDemand a c
join od1 od2 = case od2 of
Halt -> Halt
Out c kont -> Out c (join od1 kont)
In f -> case od1 of
Halt -> join od1 (f Nothing)
Out b kont -> join kont (f $ Just b)
In g -> In (\ma -> join (g ma) od2)
print_od :: Show b => OnDemand a b -> IO ()
print_od = \case
Halt -> return ()
In _ -> print "Error: missing input"
Out b k -> do
print b
print_od k
read_num :: IO (Maybe Int)
read_num = Text.Read.readMaybe <$> getLine
process :: IO (Maybe a) -> OnDemand a b -> IO [b]
process io = \case
Halt -> return []
Out hd k -> do
tl <- process io k
return $ hd : tl
In f -> do
input <- io
process io (f input)
main :: IO ()
main = do
let same_filter = sum_so_far `join` take_od 10
let pure_call = drain $ ints `join` same_filter
sidef_call <- process read_num $ same_filter
print pure_call
print sidef_call
And here is a sample invocation:
$ stack run <<< $(seq 1 9)
So, how hard is it to map all of these learnings to our little genetic algorithm from last week? A lot easier than it may seem, actually. First, we need to add the OnDemand data definition:
+data OnDemand a b
+ = Halt
+ | Out b (OnDemand a b)
+ | In (a -> OnDemand a b)
Next, we need to change the exec_random function: since we're going to ask for random values from our caller explicitly, we don't need to carry around a list anymore. In fact, we don't need to carry
any state around anymore, which makes this monad look almost unnecessary. Still, it offers a slightly nicer syntax for client functions (GetRand instead of explicit continuations). It's also quite
nice that almost none of the functions that use the monad need to change here.
-exec_random :: WithRandom a -> [Double] -> ([Double] -> a -> b) -> b
-exec_random m s cont = case m of
- Bind ma f -> exec_random ma s (\s a -> exec_random (f a) s cont)
- Return a -> cont s a
- GetRand -> cont (tail s) (head s)
+exec_random :: WithRandom a -> (a -> OnDemand Double b) -> OnDemand Double b
+exec_random m cont = case m of
+ Bind ma f -> exec_random ma (\a -> exec_random (f a) cont)
+ Return a -> cont a
+ GetRand -> In (\r -> cont r)
The biggest change is the signature of the main genetic_search function: instead of getting a [Double] as the last input and returning a [(solution, Double)], we now just return a OnDemand Double
(solution, Double).
- -> [Double]
- -> [(solution, Double)]
-genetic_search fitness mutate crossover make_solution rnd =
- map head $ exec_random init
- rnd
- (\rnd prev -> loop prev rnd)
+ -> OnDemand Double (solution, Double)
+genetic_search fitness mutate crossover make_solution =
+ exec_random init (\prev -> Out (head prev) (loop prev))
- loop :: [(solution, Double)] -> [Double] -> [[(solution, Double)]]
- loop prev rnd = prev : exec_random (step prev)
- rnd
- (\rnd next -> loop next rnd)
+ loop :: [(solution, Double)] -> OnDemand Double (solution, Double)
+ loop prev = exec_random (step prev) (\next -> Out (head next) (loop next))
The changes here are mostly trivial: we just remove the manual threading of the random list, and add one explicit Out to the core loop.
Finally, we of course need to change the call in the main function to actually drive the new version and provide random numbers on demand. This is a fairly trivial loop:
- print $ map snd
- $ take 40
- $ genetic_search fitness mutate crossover mk_sol rands
+ loop rands 40 $ genetic_search fitness mutate crossover mk_sol
+ where
+ loop :: [Double] -> Int -> OnDemand Double ((Double, Double), Double) -> IO ()
+ loop rs n od =
+ if n == 0
+ then return ()
+ else case od of
+ Halt -> return ()
+ In f -> do
+ next_rand <- pure (head rs)
+ loop (tail rs) n (f next_rand)
+ Out v k -> do
+ print v
+ loop rs (n - 1) k
Obviously in an ideal scenario the next_rand <- pure (head rs) could be more complex; the point here is just to illustrate that we can do any IO we want to produce the next random element. The full,
updated code can be found here (diff). | {"url":"https://cuddly-octo-palm-tree.com/posts/2021-03-28-lazy-io/","timestamp":"2024-11-14T17:32:02Z","content_type":"application/xhtml+xml","content_length":"28157","record_id":"<urn:uuid:b8902e67-f353-4413-aa10-7ee502582376>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00686.warc.gz"} |
If a game of chance consists of spinning an arrow which is equa-Turito
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If a game of chance consists of spinning an arrow which is equally likely to come to rest pointing to one of the number 1,2,3……12 , then the probability that it will point to an odd number is:
We have to find the probability of getting odd number if the spinning arrow's resting points are from 1 to 12. Firstly we will find the total number of outcomes which is 12 as outcomes can be any
number from 1 to 12, then we will find the number of possible outcomes i.e. the number of odd numbers. Further we will find the probability by dividing number of possible outcomes by total number of
Total number of outcome = 12
No. possible outcome(no. odd nos. from 1 to 12 ) = 6
Probability =
So, the probability of getting the odd number is
Odd number is the numbers which are not divisible by 2.
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Addition Of Negative Numbers Worksheet
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Addition Of Negative Numbers Worksheet
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Adding Negative Numbers Worksheet
Adding Negative Numbers Worksheet
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14 Best Images Of Adding Positive And Negative Numbers Worksheet Adding And Subtracting
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Year 6 Maths Worksheets Negative Numbers Kidsworksheetfun
Adding Negative Numbers Worksheet Preschool Printable Sheet
11 Best Images Of Working With Negative Numbers Worksheet Adding Negative Numbers Worksheet
Addition With Negative Number Worksheet FREE Download
Adding And Subtracting Negative Numbers Worksheets
Here are the rules for adding or subtracting negative numbers Adding a positive number is addition e g 4 2 4 2 6 Subtracting a negative number is addition e g 4 2 4 2 6 Adding a negative number is
subtraction e g 4 2 4 2 2
Negative Numbers Worksheet Math Salamanders
Hub Page Welcome to our Negative Numbers Worksheets hub page On this page you will find links to all of our worksheets and resources about negative numbers Need help practicing adding subtracting
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Here are the rules for adding or subtracting negative numbers Adding a positive number is addition e g 4 2 4 2 6 Subtracting a negative number is addition e g 4 2 4 2 6 Adding a negative number is
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Polynomials Involving Cost, Revenue, and Profit
Learning Outcomes
• Write polynomials involving cost, revenue, and profit
In this section, we will see that polynomials are sometimes used to describe cost and revenue.
Profit is typically defined in business as the difference between the amount of money earned (revenue) by producing a certain number of items and the amount of money it takes to produce that number
of items. When you are in business, you definitely want to see profit, so it is important to know what your cost and revenue is.
For example, let’s say that the cost to a manufacturer to produce a certain number of things is C and the revenue generated by selling those things is R. The profit, P, can then be defined as
P = R-C
The example we will work with is a hypothetical cell phone manufacturer whose cost to manufacture x number of phones is [latex]C=2000x+750,000[/latex], and the Revenue generated from manufacturing x
number of cell phones is [latex]R=-0.09x^2+7000x[/latex].
Define a Profit polynomial for the hypothetical cell phone manufacturer.
Show Solution
Mathematical models are great when you use them to learn important information. The cell phone manufacturing company can use the profit equation to find out how much profit they will make given x
number of phones are manufactured. In the next example, we will explore some profit values for this company.
Given the following numbers of cell phones manufactured, find the profit for the cell phone manufacturer:
1. x = [latex]100[/latex] phones
2. x = [latex]25,000[/latex] phones
3. x= [latex]60,000[/latex] phones
Interpret your results.
Show Solution
Try It
In the video that follows, we present another example of finding a polynomial profit equation.
We have shown that profit can be modeled with a polynomial, and that the profit a company can make based on a business model like this has its bounds. | {"url":"https://courses.lumenlearning.com/wm-developmentalemporium/chapter/read-applications-of-multiplying-and-dividing-polynomials/","timestamp":"2024-11-03T19:29:27Z","content_type":"text/html","content_length":"54115","record_id":"<urn:uuid:b7d2d107-5b8c-49fb-8faf-634317be5e85>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00188.warc.gz"} |
Calling Sequence
Algebraic.disposeallows the expression represented by an Algebraic to be collected by the Maple garbage collector
Calling SequenceDescriptionExamples
void dispose() throws MapleException The dispose functions allows a Maple expression that is represented by an Algebraic object to be collected by the Maple garbage collector. Once dispose has been
called on an Algebraic, it is an error to call any member function other than isDisposed. Every Algebraic object represents a Maple expression. The Algebraic objects keep the corresponding Maple
expression from getting collected by the Maple garbage collector. When the Java garbage collector collects the Algebraic object, the link between it and the Maple expression is broken, and Maple may
collect the expression. In some situations the Java garbage collector may be too slow in collecting the Algebraic objects. When this happens, Maple is unable to reuse the memory and so it must
allocate more. Calling dispose on objects that are no longer needed may fix this problem.
import com.maplesoft.openmaple.*;import com.maplesoft.externalcall.MapleException;class Example{ public static void main( String notused[] ) throws MapleException { String[] mapleArgs = { "java" };
Engine engine = new Engine( mapleArgs, new EngineCallBacksDefault(), null, null ); Algebraic a1; int i; for ( i = 0; i < 100000; i++ ) { a1 = engine.evaluate( "Array( 1..10^4 ):" ); a1.dispose(); }
Numeric n = (Numeric)engine.evaluate( "kernelopts( bytesalloc ):" ); System.out.println( "Total Bytes Alloc: "+n ); }}
Executing this code should produce output similar to the following, with the bytes used messages removed.
Total Bytes Alloc: 6945544
See AlsoExternalCalling/Java/MapleExceptionOpenMapleOpenMaple/Java/AlgebraicOpenMaple/Java/Algebraic/isDisposedOpenMaple/Java/APIOpenMaple/Java/Engine/evaluateOpenMaple/Java/memory | {"url":"https://maplesoft.com/support/help/content/5976/OpenMaple-Java-Algebraic-dispose.mw","timestamp":"2024-11-14T07:57:32Z","content_type":"application/xml","content_length":"20530","record_id":"<urn:uuid:9350a328-98af-474e-8368-22188884f255>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00188.warc.gz"} |
Lively Logic - Filtering and highlighting
Filtering and highlighting
You can add a filter to control which entries from your dataset will be displayed in a chart or series. You can use filters to segment your dataset or highlight important values.
To add a filter, first select your chart. If it’s a table or a category graph, you’ll see the Filter Dataset pop-up menu in the inspector bar. If it’s an X-Y graph, each series can use a different
dataset, so you’ll find the button in the panels for configuring series. For more about charts and the inspector bar, see the “Customizing charts” guide.
Click the Filter Only show entries where… and pick a dataset field and a condition. The condition will be tested for every value in that field, and only dataset entries that meet the condition will
be displayed in the chart or series.
For tables and category graphs, the filter applies to the whole chart. For X-Y graph series, each series can have different filters.
You can add multiple conditions by clicking the + button in the filter panel. With multiple conditions, only dataset entries that meet all conditions will be displayed. To remove a condition, click
the − button. To disable a filter altogether, remove every condition or choose Show all dataset entries.
Highlighting values
Filters can be especially useful in X-Y graphs, since each series can have different filters. You can add multiple series with different filters to label certain values or to highlight a specific
subset of the data.
For example, you could display all the dataset’s values in a line graph series, and use a filter on a text series to add labels to the outliers.
Or you could color-code a scatterplot by adding a series for each color, each with a different filter to show a different subset of the data. In the image below, the scatterplot has two series, both
using the same dataset and the same fields for the X and Y values. However, the red series has a filter applied to draw attention to a specific segment of the data.
Filtering with formulas
When setting a filter condition, you can choose to test the condition against the result of a formula, instead of against the values of a dataset field. This lets you achieve more complex effects
than you could by testing against a dataset field directly. To filter with a formula, choose Formula… instead of a dataset field, and enter a formula. For example, the following formula returns a
number for each entry representing the difference, measured in standard deviations, between the entry’s Weight value and the mean:
ABS($"Weight" - AVERAGE($"Weight")) / STD($"Weight")
By using this formula with a filter condition, you could highlight values where the difference was greater than a particular number of standard deviations.
For more about formulas, see the “Writing formulas” guide. | {"url":"https://livelylogic.com/guides/filtering-and-highlighting","timestamp":"2024-11-10T01:22:05Z","content_type":"text/html","content_length":"7423","record_id":"<urn:uuid:61665131-9b24-4218-8e20-8f9513d6c38c>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00464.warc.gz"} |
Integrate $\int_0^1\int_{\sqrt{x}}^{1}e^{y^3}dydx$
Change the order of integration and compute this integral
Answers can only be viewed under the following conditions:
1. The questioner was satisfied with and accepted the answer, or
2. The answer was evaluated as being 100% correct by the judge.
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to earn up to a 50% commission on every question that your affiliated users ask or answer. | {"url":"https://matchmaticians.com/questions/y5ioaz/integrate-int-0-1-int-sqrt-x-1-e-y-3-dydx-multivariable","timestamp":"2024-11-08T15:45:46Z","content_type":"text/html","content_length":"74558","record_id":"<urn:uuid:d1fd8721-9517-445b-96f5-d8cd66fc4135>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00844.warc.gz"} |
Basic Steps for Forecasting
You have gone through the Air Passenger Traffic problem statement and understood that it is the time series forecasting problem. To solve this problem, we have a set of basic steps of forecasting
that are needed to be followed. Let’s quickly understand them from Chiranjoy.
You just learnt about the basic steps involved in any forecasting problem. These were –
• Define the problem
• Collect the data
• Analyze the data
• Build and evaluate the forecast model
The one thing to keep in mind before moving forward is that there are some caveats associated with a time series forecasting. These caveats revolve around the steps you learnt about while defining
the problem.
• The Granularity Rule: The more aggregate your forecasts, the more accurate you are in your predictions simply because aggregated data has lesser variance and hence, lesser noise. As a thought
experiment, suppose you work at ABC, an online entertainment streaming service, and you want to predict the number of views for a few newly launched TV show in Mumbai for the next one year. Now,
would you be more accurate in your predictions if you predicted at the city-level or if you go at an area-level? Obviously, accurately predicting the views from each area might be difficult but
when you sum up the number of views for each area and present your final predictions at a city-level, your predictions might be surprisingly accurate. This is because, for some areas, you might
have predicted lower views than the actual whereas, for some, the number of predicted views might be higher. And when you sum all of these up, the noise and variance cancel each other out,
leaving you with a good prediction. Hence, you should not make predictions at very granular levels.
• The Frequency Rule: This rule tells you to keep updating your forecasts regularly to capture any new information that comes in. Let’s continue with the ABC, an online entertainment streaming
service, an example where the problem is to predict the number of views for a newly launched TV show in Mumbai for the next year. Now, if you keep the frequency too low, you might not be able to
capture accurately the new information coming in. For example, say, your frequency for updating the forecasts is 3 months. Now, due to the COVID-19 pandemic, the residents may be locked in their
homes for around 2-3 months during which the number of views will significantly increase. Now, if the frequency of your forecast is only 3 months, you will not be able to capture the increase in
views which may incur significant losses and lead to mismanagement.
• The Horizon Rule: When you have the horizon planned for a large number of months into the future, you are more likely to be accurate in the earlier months as compared to the later ones. Let’s
again go back to ABC, an online entertainment streaming service, example. Suppose that the online entertainment streaming service made a prediction for the number of views for the next 6 months
in December 2019. Now, it may have been quite accurate for the first two months, but due to the unforeseen COVID-19 situation, the actual number of view in the next couple of months would have
been significantly higher than predicted because of everyone staying at home. The farther ahead we go into the future, the more uncertain we are about the forecasts.
Now that you have understood the steps in defining the problem, let’s apply them to the air passenger traffic problem.
• Quantity: Number of passengers
• Granularity: Flights from city A to city B; i.e., flights for a particular route
• Frequency: Monthly
• Horizon: 1 year (12 months) | {"url":"https://www.internetknowledgehub.com/basic-steps-for-forecasting/","timestamp":"2024-11-09T22:27:54Z","content_type":"text/html","content_length":"81844","record_id":"<urn:uuid:2a1b4477-f9b4-4da5-b9e5-10b040771477>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00850.warc.gz"} |
Querying Large Collections of Semistructured Data
University of Waterloo
An increasing amount of data is published as semistructured documents formatted with presentational markup. Examples include data objects such as mathematical expressions encoded with MathML or web
pages encoded with XHTML. Our intention is to improve the state of the art in retrieving, manipulating, or mining such data. We focus first on mathematics retrieval, which is appealing in various
domains, such as education, digital libraries, engineering, patent documents, and medical sciences. Capturing the similarity of mathematical expressions also greatly enhances document classification
in such domains. Unlike text retrieval, where keywords carry enough semantics to distinguish text documents and rank them, math symbols do not contain much semantic information on their own.
Unfortunately, considering the structure of mathematical expressions to calculate relevance scores of documents results in ranking algorithms that are computationally more expensive than the typical
ranking algorithms employed for text documents. As a result, current math retrieval systems either limit themselves to exact matches, or they ignore the structure completely; they sacrifice either
recall or precision for efficiency. We propose instead an efficient end-to-end math retrieval system based on a structural similarity ranking algorithm. We describe novel optimization techniques to
reduce the index size and the query processing time. Thus, with the proposed optimizations, mathematical contents can be fully exploited to rank documents in response to mathematical queries. We
demonstrate the effectiveness and the efficiency of our solution experimentally, using a special-purpose testbed that we developed for evaluating math retrieval systems. We finally extend our
retrieval system to accommodate rich queries that consist of combinations of math expressions and textual keywords. As a second focal point, we address the problem of recognizing structural
repetitions in typical web documents. Most web pages use presentational markup standards, in which the tags control the formatting of documents rather than semantically describing their contents.
Hence, their structures typically contain more irregularities than descriptive (data-oriented) markup languages. Even though applications would greatly benefit from a grammar inference algorithm that
captures structure to make it explicit, the existing algorithms for XML schema inference, which target data-oriented markup, are ineffective in inferring grammars for web documents with
presentational markup. There is currently no general-purpose grammar inference framework that can handle irregularities commonly found in web documents and that can operate with only a few examples.
Although inferring grammars for individual web pages has been partially addressed by data extraction tools, the existing solutions rely on simplifying assumptions that limit their application. Hence,
we describe a principled approach to the problem by defining a class of grammars that can be inferred from very small sample sets and can capture the structure of most web documents. The
effectiveness of this approach, together with a comparison against various classes of grammars including DTDs and XSDs, is demonstrated through extensive experiments on web documents. We finally use
the proposed grammar inference framework to extend our math retrieval system and to optimize it further.
Mathematics retrieval, Search, Semistructured data, Query Language, XML, Grammar inference | {"url":"https://uwspace.uwaterloo.ca/items/8cc71393-114e-4778-8a56-729f54aa6f8a","timestamp":"2024-11-12T03:25:10Z","content_type":"text/html","content_length":"432124","record_id":"<urn:uuid:5d30c4e9-cfc5-4973-88ca-591b637905e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00438.warc.gz"} |
What is Newton second law of motion with example?
What is Newton second law of motion with example?
Newton’s Second Law of Motion says that acceleration (gaining speed) happens when a force acts on a mass (object). Riding your bicycle is a good example of this law of motion at work. When you push
on the pedals, your bicycle accelerates. You are increasing the speed of the bicycle by applying force to the pedals.
What are 10 examples of Newton’s second law?
10 Examples of Newton’s Second Law of Motion in Everyday Life
• Pushing a Car and a Truck.
• Pushing a Shopping Cart.
• Two People Walking Together.
• Hitting a Ball.
• Rocket Launch.
• Car Crash.
• Object thrown from a Height.
• Karate Player Breaking Slab of Bricks.
What is Newton’s 2nd law provide an example and formula?
Newton’s second law: F = ma The momentum of a body is equal to the product of its mass and its velocity. Momentum, like velocity, is a vector quantity, having both magnitude and direction. A force
applied to a body can change the magnitude of the momentum or its direction or both.
How do you explain Newton’s second law of motion?
Newton’s second law says that when a constant force acts on a massive body, it causes it to accelerate, i.e., to change its velocity, at a constant rate. In the simplest case, a force applied to an
object at rest causes it to accelerate in the direction of the force.
What are 2 examples of Newton’s third law?
Other examples of Newton’s third law are easy to find:
• As a professor paces in front of a whiteboard, he exerts a force backward on the floor.
• A car accelerates forward because the ground pushes forward on the drive wheels, in reaction to the drive wheels pushing backward on the ground.
What are 3 examples of Newton’s third law?
While Rowing a boat, when you want to move forward on a boat, you paddle by pushing the water backwards, causing you to move forward. While Walking, You push the floor or the surface you are walking
on with your toes, And the surface pushes your legs up, helping you to lift your legs up.
What is Newton’s second law class 9?
Newton’s Second Law of motion states that the rate of change of momentum of an object is proportional to the applied unbalanced force in the direction of the force. ie., F=ma.
What is Newton’s second law called?
Newton’s second law of motion is F = ma, or force is equal to mass times acceleration. Learn how to use the formula to calculate acceleration.
Which is the best example of Newton’s third law?
Examples of Newton’s third law of motion are ubiquitous in everyday life. For example, when you jump, your legs apply a force to the ground, and the ground applies and equal and opposite reaction
force that propels you into the air. Engineers apply Newton’s third law when designing rockets and other projectile devices.
What are 5 examples of Newton’s third law?
Newton’s third law of motion examples
• Pulling an elastic band.
• Swimming or rowing a boat.
• Static friction while pushing an object.
• Walking.
• Standing on the ground or sitting on a chair.
• The upward thrust of a rocket.
• Resting against a wall or tree.
• Slingshot.
Which is an example of the second law of Newton?
In the second law of Newton, Known as the Fundamental Principle of Dynamics, the scientist states that the larger the mass of an object, the more force will be required to accelerate it. That is, the
acceleration of the object is directly proportional to the net force acting on it and inversely proportional to that of the object.
Why is Ma not a force in newton’s second law?
One advantage of writing Newton’s second law in this form is that it makes people less likely to think that ma —mass times acceleration—is a specific force on an object. The expression ma is not a
force, ma is what the net force equals. Looking at the form of Newton’s second law shown above,…
Why do we plug vertical forces into Newton’s second law?
When using these equations we must be careful to only plug horizontal forces into the horizontal form of Newton’s second law and to plug vertical forces into the vertical form of Newton’s second law.
We do this because horizontal forces only affect the horizontal acceleration and vertical forces only affect the vertical acceleration.
How is force related to the second law of motion?
Force is equal to the rate of change of momentum. For a constant mass, force equals mass times acceleration. Newton’s second law of motion, unlike the first law of motion pertains to the behavior of
objects for which all existing forces are unbalanced. | {"url":"https://draftlessig.org/what-is-newton-second-law-of-motion-with-example/","timestamp":"2024-11-08T22:10:46Z","content_type":"text/html","content_length":"71416","record_id":"<urn:uuid:ce4e71aa-352c-445c-ab4e-549a0dae95fa>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00286.warc.gz"} |
Potential impact of missing outcome data on treatment effects in systematic reviews: imputation study
Potential impact of missing outcome data on treatment effects in systematic reviews: imputation study
BMJ 2020
370 doi: https://doi.org/10.1136/bmj.m2898 (Published 26 August 2020) Cite this as: BMJ 2020;370:m2898
This article has a correction. Please see:
1. Correspondence to: E A Akl ea32{at}aub.edu.lb (or @elie__akl on Twitter)
Objective To assess the risk of bias associated with missing outcome data in systematic reviews.
Setting Systematic reviews.
Population 100 systematic reviews that included a group level meta-analysis with a statistically significant effect on a patient important dichotomous efficacy outcome.
Main outcome measures Median percentage change in the relative effect estimate when applying each of the following assumption (four commonly discussed but implausible assumptions (best case scenario,
none had the event, all had the event, and worst case scenario) and four plausible assumptions for missing data based on the informative missingness odds ratio (IMOR) approach (IMOR 1.5 (least
stringent), IMOR 2, IMOR 3, IMOR 5 (most stringent)); percentage of meta-analyses that crossed the threshold of the null effect for each method; and percentage of meta-analyses that qualitatively
changed direction of effect for each method. Sensitivity analyses based on the eight different methods of handling missing data were conducted.
Results 100 systematic reviews with 653 randomised controlled trials were included. When applying the implausible but commonly discussed assumptions, the median change in the relative effect estimate
varied from 0% to 30.4%. The percentage of meta-analyses crossing the threshold of the null effect varied from 1% (best case scenario) to 60% (worst case scenario), and 26% changed direction with the
worst case scenario. When applying the plausible assumptions, the median percentage change in relative effect estimate varied from 1.4% to 7.0%. The percentage of meta-analyses crossing the threshold
of the null effect varied from 6% (IMOR 1.5) to 22% (IMOR 5) of meta-analyses, and 2% changed direction with the most stringent (IMOR 5).
Conclusion Even when applying plausible assumptions to the outcomes of participants with definite missing data, the average change in pooled relative effect estimate is substantive, and almost a
quarter (22%) of meta-analyses crossed the threshold of the null effect. Systematic review authors should present the potential impact of missing outcome data on their effect estimates and use this
to inform their overall GRADE (grading of recommendations assessment, development, and evaluation) ratings of risk of bias and their interpretation of the results.
Despite efforts to reduce the occurrence of missing outcome data in clinical trials,1 this is still common. Across six methodological surveys the percentage of randomised controlled trials with
missing outcome data ranged from 63% to 100%,234567 and the average proportion of participants with missing outcome data among trials reporting missing data ranged from 6% to 24%.2345678 Among 235
randomised controlled trials with statistically significant results published in leading medical journals, one in three lost statistical significance when making plausible assumptions about the
outcomes of participants with missing data.2 Another study comparing different approaches to modelling binary outcomes with missing data in an alcohol clinical trial yielded different results with
various amount of bias depending on the approach and missing data scenario.9
The extent of missing outcome data in randomised controlled trials contributes to the risk of bias of meta-analyses involving those trials. Additional factors that might bias the results of
meta-analysis include the methods used by contributing randomised controlled trials to handle missing data and the transparency of reporting missing data (for each arm and follow-up time point). To
explore the impact on risk of bias, the GRADE (grading of recommendations assessment, development, and evaluation) working group recommends conducting sensitivity analyses using assumptions regarding
the outcomes of patients with missing outcome data.10 A recent empirical comparison of bayesian modelling strategies for missing binary outcome data in network meta-analysis found that using
implausible assumptions could negatively affect network meta-analysis estimates.11 No methodological study has yet assessed the impact of different assumptions about missing data on the robustness of
the pooled relative effect in a large representative sample of published systematic reviews of pairwise comparisons.
One challenge when handling missing data is the lack of clarity in trial reports on whether participants have missing outcome data.12 We recently published guidance for authors of systematic reviews
on how to identify and classify participants with missing outcome data in the trial reports.13 The Cochrane Handbook acknowledges that attempts to deal with missing data in systematic reviews are
often hampered by incomplete reporting of missing outcome data by trialists.14 A recent methodological survey among 638 randomised controlled trials reported that the median percentage of
participants with unclear follow-up status was 9.7% (interquartile range 4.1-19.9%),8 and that when authors explicitly reported not following-up participants, almost half did not specify how they
handled missing data in their analysis.
We assessed risk of bias associated with missing outcome data in systematic reviews with two interventions by quantifying the change in effect estimate when applying different methods of handling
missing outcome data; examining how these methods alter crossing the threshold of the null effect of pooled effect estimates; qualitatively changing the direction of effect; and exploring the
potential effect on heterogeneity of each of these approaches.4
This study is part of a larger project examining methodological problems related to missing data in systematic reviews and randomised controlled trials.15 Our published protocol includes detailed
information on the definitions, eligibility criteria, search strategy, selection process, data abstraction, and data analysis.15 In the appendix (supplementary table 1), we present deviations from
the protocol and the rationale for these deviations.
We defined missing data as outcome data for trial participants that are not available to authors of systematic reviews (from the published randomised controlled trial reports or personal contact with
the trial authors).
In the current study, we collected a random sample of 50 Cochrane and 50 non-Cochrane systematic reviews published in 2012 that reported a group level meta-analysis of a patient important dichotomous
efficacy outcome, with a statistically significant effect estimate (the meta-analysis of interest).16 We used the term original pooled relative effect to refer to the result of the meta-analysis as
reported by the systematic review authors. For the individual trials included in the meta-analyses of interest,8 we abstracted detailed information relevant to the statistical analysis and missing
data and conducted sensitivity meta-analyses based on nine different methods of handling missing data. Our outcomes were the median change of effect estimates across meta-analyses; the percentage of
meta-analyses that crossed the threshold of the null effect with each of these methods; the percentage of meta-analyses that changed direction of effect; and the change in heterogeneity associated
with each of the methods.
Identifying participants with missing data
Since publication of our protocol, we published a guidance for authors of systematic reviews on how to identify and classify participants with missing outcome data in the trial reports depending on
how trial authors report on those categories and handle them in their analyses (table 1).13 The guidance includes a taxonomy of categories of trial participants who might have missing data, along
with a description of those categories (supplementary table 2). The categorisation reflects the wording used in trial reports—that is, the presentation faced by systematic review authors. We used
this guidance to judge the outcome data missingness of categories of participants who might have missing data (ie, whether they have definite, potential, or no missing data).
Data abstraction
Eleven reviewers trained in health research methodology abstracted the data independently and in duplicate. The reviewers met regularly with a core team (EAA, LAK, BD, and AD) to discuss progress and
challenges and develop solutions. We used a pilot tested standardised data abstraction form hosted in an electronic data capture tool, REDCap.17 All reviewers underwent calibration exercises before
data abstraction to improve reliability, and a senior investigator served as a third independent reviewer for resolving disagreements.
From each eligible meta-analysis we abstracted the original (published) pooled relative effect—that is, pooled relative effect measure (relative risk or odds ratio) and the associated 95% confidence
interval, the analysis model used (random effects or fixed effect), and the statistical method used for pooling data (eg, Mantel-Haenszel, Peto).
For each study arm in the randomised controlled trials, we abstracted the numbers of participants randomised, events, participants with definite missing data (according to the suggested guidance on
identifying participants with missing data13), and participants with potentially missing data.
Data analysis
SPSS statistical software, version 21.0 was used to conduct a descriptive analysis of study characteristics of eligible systematic reviews and the associated randomised controlled trials.18 For
categorical variables, we report frequencies and percentages, and for continuous variables that were not normally distributed, we use medians and interquartile ranges.
To explore the robustness of pooled effect estimates reported by systematic reviews, we conducted several sensitivity analyses based on nine different methods of handling missing data, using Stata
software release 1219 (see table 2):
• Complete case analysis1020
• Four implausible but commonly discussed assumptions: best case scenario, none of the participants with missing data had the outcome, all participants with missing data had the outcome, and worst
case scenario.
• Four plausible assumptions using increasingly stringent values for informative missing odds ratio (IMOR) in the intervention arm.2122 IMOR describes the ratio of odds of the outcome among
participants with missing data to the odds of the outcome among observed participants. In other words, to obtain the odds among participants with missing data, the odds is multiplied among the
observed participants with a stringent value (ie, 1.5, 2, 3, and 5). We chose an upper limit of 5 as it represents the highest ratio reported in the literature. One study used a community tracker to
evaluate the incidence of death among participants in scale-up programmes of antiretroviral treatment in Africa who were lost to follow-up.2324 The study found the mortality rate to be five times
higher in patients lost to follow-up compared with patients who were followed up. We did not use an IMOR of 1 because it provides the same effect estimate as the complete case analysis.
Naïve single imputation methods falsely increase precision by treating imputed outcome data as if they were observed, whereas IMORs (or pattern mixture models in general), by imputing risks, increase
uncertainty within a trial to deal with the fact that data have been imputed. Imputing events consists of including participants with missing data in the denominator and making assumptions about
their outcomes in the numerator. This approach might lead to imputing several events as if they were fully observed, leading to possibly problematic narrower confidence intervals than would be the
case. To correct for this, methodologists have developed methods that account for uncertainty associated with imputing missing observations using statistical approaches.212227 The command “metamiss”
in Stata integrates uncertainty within its calculation2526 (see statistical notes in appendix for further details).
We did not consider the nature of outcomes (positive versus negative) during the conduct of the sensitivity analyses best case and worst case scenarios. Instead, we focused in the sensitivity
analyses on challenging the effect estimates against the null value when the best case scenario shifts the effect estimate away from the null value of 1 and the worst case scenario shifts the effect
estimate towards the null value of 1 (irrespective of whether the outcome is positive or negative).
Our analytical approach was executed in one command (metamiss2526) of Stata release 1219 for each sensitivity analysis. Firstly, we recalculated each meta-analysis of interest, for all 100 systematic
reviews, using each method to deal with missing outcome data to generate different sensitivity analysis pooled effects along with the corresponding 95% confidence intervals. We used the same relative
effect measure (relative risk or odds ratio), the same analysis model (random effects or fixed effect), and the same statistical method (eg, Mantel-Haenszel, Peto) as the original meta-analysis of
interest. Secondly, across all included meta-analyses and for each method, we explored the impact of the revised meta-analysis for several outcomes:
Change in relative effect estimate
To quantify the percentage change in relative effect estimate between the sensitivity analysis pooled effect estimate (assumption) and the sensitivity analysis pooled effect estimate (complete case
analysis), we applied the formula (fig 1; see statistical notes in appendix for further details).
We calculated specifically the percentage of meta-analyses with change of relative effect estimate (by direction) between the sensitivity analysis pooled effect estimate (assumption) and the
sensitivity analysis pooled effect estimate (complete case analysis) and the median and interquartile range for the change in relative effect estimate (stratified by direction of change).
This relative change could be an increase or a reduction in effect. For example, a relative increase in relative risk of 25% for the sensitivity analysis pooled effect estimate (worst case scenario)
over the sensitivity analysis pooled effect estimate (complete case analysis) implies that for a relative risk of 0.8 with complete case analysis, the relative risk for the worst case scenario would
be 1, and for a relative risk of 1.6 with complete case analysis, the relative risk for the worst case scenario would be 1.2.
Crossing the threshold of the null effect
For the analysis of the percentage of meta-analyses for which the sensitivity analysis pooled relative effect (assumption) crossed the threshold of the null effect compared with the sensitivity
analysis pooled relative effect (complete case analysis), we restricted the sample to the meta-analyses that did not cross the threshold of the null effect under the complete case analysis method.
Changing direction of pooled relative effect
In the analysis of the percentage of meta-analyses for which the sensitivity analysis pooled relative effect (assumption) changed direction compared with the sensitivity analysis pooled relative
effect (complete case analysis), the direction could change from favouring the intervention to favouring the control or vice versa. For this analysis, we restricted the sample to the meta-analyses
that crossed the threshold of the null effect under the complete case analysis method.
We reproduced the analyses for outcomes when crossing the threshold of the null effect and when changing direction, comparing the sensitivity analysis pooled relative effect to the original pooled
relative effect. In addition, we conducted all the analyses twice: first considering participants with definite missing outcome data, then considering participants with total possible missing outcome
data (see table 1). In addition, we explored how heterogeneity varies across the different methods.
Patient and public involvement
As this project concerns methodology, no patients or public were involved.
Study characteristics of included meta-analyses
We previously reported on the details of the 100 eligible systematic reviews16 and the 653 randomised controlled trials they considered.8Table 3 summarises the characteristics of the 100 included
systematic reviews and the corresponding meta-analyses. Most reported on a drug related outcome (61%) and a non-active control (55%), assessed a morbidity outcome (56%), reported an unfavourable
outcome (73%), used the risk ratio (61%), and applied a fixed effect analysis model (57%) and Mantel-Haenszel statistical methods (77%). The median number of randomised controlled trials per
meta-analysis was 6 (interquartile range 3-8). Eight meta-analyses included randomised controlled trials that reported no missing data.
Missing data in randomised controlled trials
The results of 400 of the 653 randomised controlled trials (63%) mentioned at least one of the predefined categories of participants who might have missing outcome data. Among those 400 trials, the
total median percentage of participants with definite missing outcome data was 5.8% (interquartile range 2.2-14.8%); 3.8% (0-12%) in the intervention arm and 3.4% (0-12%) in the control arm. The
median percentage of participants with potential missing outcome data was 9.7% (4.1-19.9%) and with total possible missing data was 11.7% (5.6-23.7%). Only three trials provided a reason for
missingness (eg, missing at random).
Change in relative effect estimate
Figure 2 shows the change in the relative effect estimate between the sensitivity analysis pooled effect estimate (assumption) and the sensitivity analysis pooled effect estimate (complete case
analysis), when considering participants with definite missing data.
For the four implausible but commonly discussed assumptions, the percentage of meta-analyses with an increased relative effect estimate (shifted away from the null value of 1) was 91% for the best
case scenario assumption, 25% for the assumption that none of the participants with missing data had the event, and 17% for the assumption that all participants with missing data had the event. The
median increase in the relative effect estimate ranged from 0% for the worst case scenario assumption to 18.9% (interquartile range 6.8-38.9%) for the best case scenario assumption. The percentage of
meta-analyses with reduced relative effect estimates (shifted closer towards the null value of 1) was 90% for the worst case scenario assumption, 38% for the assumption that none of the participants
with missing data had the event, and 75% for the assumption that all participants with missing data had the event. The median reduction in the relative effect estimate ranged from 0% for the best
case scenario assumption to 30.4% (interquartile range10.5-77.5%) for the worst case scenario assumption.
For the plausible assumptions based on the IMOR, the percentage of meta-analyses with an increased relative effect estimate was 85% for the least stringent assumption (IMOR 1.5) and 88% for the most
stringent assumption (IMOR 5). The median reduction in relative effect estimate ranged from 1.4% (0.6-3.9%) for the IMOR 1.5 assumption to 7.0% (2.7-18.2%) for the IMOR 5 assumption. The appendix
presents the details of the percentage change in the relative effect estimate when considering participants with total possible missing data (supplementary results C), and stratified by whether the
estimate is less than or greater than 1 under the complete case analysis, using either definite missing data or total possible missing data (supplementary results D table 3).
Crossing the threshold of the null effect and change of direction
Of the 100 meta-analyses, 87 did not cross the threshold of the null effect under the complete case analysis method. Figure 3 shows the number of meta-analyses that crossed the threshold of the null
effect when comparing the sensitivity analysis pooled relative effect (assumption) with the sensitivity analysis pooled relative effect (complete case analysis) for each assumption and considering
participants with definite missing data. For the four implausible but commonly discussed assumptions, the percentage of meta-analyses that crossed the threshold of the null effect ranged from 1%
(best case scenario and none of the participants with missing data had the event) to 18% (all participants with missing data had the event) to 60% (worst case scenario). For the plausible assumptions
based on IMOR, the percentage of meta-analyses that crossed the threshold of the null effect ranged from 6% (least stringent assumption IMOR 1.5) to 22% (most stringent assumption IMOR 5).
The percentage of meta-analyses that changed direction with the two extreme assumptions was 26% for the worst case scenario and 2% for the most stringent assumption IMOR 5.
The appendix presents the results of meta-analyses that crossed the threshold of the null effect1 and changed direction when participants with total possible missing data were considered2
(supplementary results A), and when the sensitivity analysis pooled relative effect (assumption) was compared with the original pooled relative effect (supplementary results B).
Change in heterogeneity
Figure 4 shows the change in heterogeneity across the different methods of handling missing data when considering participants with definite missing data. The median I^2 for the original pooled
relative effect was 0% (interquartile range 0-42%) and for the complete case analysis was 3.6% (0-47%). For the four implausible but commonly discussed assumptions, the median I^2 ranged from 0.4%
(none of the participants with missing data had the event) to 20.3% (all participants with missing data had the event) to 29% (best case scenario) to 58.6% (worst case scenario). For the plausible
assumptions based on IMOR, the percentage of meta-analyses that crossed the threshold of the null effect ranged from 2.4% (least stringent assumption IMOR 1.5) to 4.4% (most stringent assumption IMOR
In the current study, we quantified the change in the effect estimate when applying different methods of handling missing data to the outcomes of participants with definite missing data. When
applying plausible assumptions to the outcomes of participants with definite missing data, the median change in relative effect estimate was as high as 7.0% (interquartile range 2.7-18.2%). When
applying implausible but commonly discussed assumptions, the median change in the relative effect estimate was as large as 30.4% (10.5-77.5%).
We also examined how different methods of handling missing data alter crossing the threshold of the null effect of pooled effect estimates of dichotomous outcomes. Even when applying plausible
assumptions to the outcomes of participants with definite missing data, almost a quarter (22%) of meta-analyses crossed the threshold of the null effect. When applying implausible but commonly
discussed assumptions, the percentage of systematic reviews that crossed the threshold of the null effect was as high as 60% with the worst case scenario.
Strengths and limitations of this study
In this study we assessed the effect of using different assumptions (both commonly discussed and more plausible) on a large number of published meta-analyses of pairwise comparisons targeting a wide
range of clinical areas. Strengths of our study include a detailed approach to assessing participants with missing data, and accounting for participants with potential missing data in our analyses.
We used two statistical approaches to assess the risk of bias associated with missing outcomes: change in effect estimate and crossing the threshold of the null effect. Although the approach of
crossing the threshold of the null effect has been criticised as the basis for decision making,28 we used it to assess the robustness of the meta-analysis effect estimate (ie, when conducting
sensitivity meta-analyses using different methods of handling missing data). We are confident that if results cross the threshold of the null effect, the certainty in evidence should be rated down
owing to risk of bias. The change in effect estimate approach has its own limitation in interpretation (cut-off for topic specific minimally important difference that would vary across a wide range
of topics and outcomes).
A limitation of our study is that we considered only dichotomous outcome data; methods for handling missing continuous data are different and our findings might not be generalisable to systematic
reviews of continuous outcomes.2930 Our sample consisted of systematic reviews that were published in 2012 and these might not reflect more current reviews; however, recent surveys have found that
the reporting, handling, and assessment of risk of bias in relation to missing data has not improved over the date of our search.7313233 For further confirmation that current practice is unlikely to
have changed, we assessed the reporting and handling of missing data in a sample of recently published systematic reviews. To make our selection of a sample of systematic reviews reproducible, we
ordered systematic reviews published from January 2020 in the chronological order of their publication and selected the first 15 that met our eligibility criteria. Then we applied the same
methodology stated in the methods section to assess the reporting and handling of missing data in the 15 eligible meta-analyses. As with the 2012 sample of systematic reviews reported on here, most
of the systematic reviews published in 2020 did not explicitly plan (in the Methods section) to consider any category of missing data (60%), did not report (in the Results section) data for any
category of missing data (67%), did not report the proportion of missing data for each trial and for each arm (67%), did not explicitly state a specific analytical method for handling missing data
(80%), and did not provide a justification for the analytical method used to handle missing data (100%).
Another limitation was our focus on systematic reviews with statistically significant results. Although this prevented us from assessing the change in effect estimate for meta-analyses with
non-statistically significant results, it allowed us to focus on reviews that are more likely to influence clinical practice.
We acknowledge the possible superiority of using the random IMOR over fixed IMOR. However, the variance or uncertainty increases as a function of the proportion of missingness and the variance of the
IMOR parameter.125 Hence, uncertainty increases even with fixed IMORs although to a smaller extent. Fully introducing uncertainty to IMORs by using random IMOR (although it was not feasible for this
study) would exaggerate our results. Specifically, we would expect a further increase in the proportion of meta-analyses in which the confidence interval would cross the line of no effect and thus
lose statistical significance, larger within study variance, and further downgrading of the certainty of the evidence. Hence, the switch to random IMOR would only reinforce the inferences from our
work that we already make.
Interpretation of findings
Experience and acceptance for using plausible assumptions is growing as a result of face validity.1134 The advantage of the IMOR approach is that it provides a tool for review authors to challenge
the robustness of effect estimates by applying increasingly stringent assumptions.25 This approach allows the review authors to choose the IMOR value or range of values based on their clinical
judgment or expert opinion. Only if the effect estimate is robust to the most conservative and plausible scenario can it be concluded that the evidence is at low risk of bias from missing data
without additional sensitivity analyses. The selection of the assumptions about missing data should be determined a priori with researchers blinded to the extent of missing data in the included
trials to avoid data dredging. Thus we should rate down for missing data when it might change inferences. Inferences could be that treatment is of benefit (versus not of benefit) or that treatment
achieves an important benefit. That would require a threshold of minimal clinically important difference and a check to see if the threshold was crossed initially and then after accounting for
minimal clinically important difference. We could not do that unless we knew the minimal clinically important difference.
Almost a quarter (22%) of meta-analyses crossed the threshold of the null effect when we used a conservative approach to test robustness (ie, applying plausible assumptions to the outcomes of
participants with definite missing data). When using the same conservative approach, up to a quarter of meta-analyses had a change of at least 18% in their relative effect estimates (based on the 75^
th centile for IMOR 5, see median and interquartile range of IMOR 5 in fig 2). These findings mean that a substantive percentage of meta-analyses is at serious risk of bias associated with missing
outcome data. Findings such as these should lead systematic review authors to rate down the certainty of evidence for risk of bias. Our results highlight the importance of minimising missing data for
clinical trials3536 by better reporting and handling of missing data.81637
With the two assumptions that all participants with missing data had the event and none of the participants with missing data had the event, the size and direction of the confidence intervals are
unpredictable; sometimes the effect estimate is shifted closer to 1 and sometimes away from 1. Thus, these assumptions are not helpful for challenging the robustness of the effect estimate. By
design, the effect estimate using the best case scenario assumption is shifted in the opposite direction of challenging the robustness and should not be used for that purpose. The worst case scenario
consistently challenges the robustness of the effect estimate as it is shifted towards the null effect, but the implausibility of its underlying assumptions makes it a poor choice for sensitivity
analyses. If the effect estimate is robust to the worst case scenario it can be concluded that the evidence is at low risk of bias from missing data, without proceeding with further sensitivity
As for the change in heterogeneity, since with the implausible but common assumptions, the size and direction of the confidence intervals are unpredictable, consequently the change in I^2 was
observed to be unpredictable. As for the plausible assumptions, since uncertainty is taken into account, confidence intervals tend to be wider and more overlapping, leading to low I^2 values. Future
studies might need to study whether a certain change in I^2 in a sensitivity analysis is worth rating down the certainty of evidence as a result of inconsistency.
Conclusion and implications
The findings of this study show the potential impact of missing data on the results of systematic reviews. This has implications when both the risk of bias associated with missing outcome data is
assessed and the extent of missing outcome data in clinical trials needs to be reduced.
Systematic review authors should present the potential impact of missing outcome data on their effect estimates and, when these suggest lack of robustness of the results, rate down the certainty of
evidence for risk of bias. For practical purposes, authors of systematic reviews might wish to use statistical software that allows running assumptions about missing data (eg, Stata). As for users of
the medical literature, a rule of thumb on how to judge risk of bias associated with missing outcome data at the trial level is needed. This would account for factors such as percentage of missing
data in each study arm, the ratio of missing data to event rate for each study arm (ie, the higher the ratio, the larger the change), fragility of statistical significance (ie, borderline
significance), the magnitude of the effect estimate (ie, the larger the effect estimate, the smaller the change), and the duration of follow-up (ie, the longer the duration of follow-up, the higher
the percentage of missing data).
We acknowledge that assessing the impact of missing data with crossing the threshold of the null effect might be insensitive. Thus, when using this approach and the threshold of null effect is
crossed, then rate down the certainty for risk of bias associated with missing data. If threshold of null effect is not crossed, it might be valuable to then evaluate the change in effect estimate to
assess whether the relative effect goes from an important to an unimportant effect. If the latter happens, then rate down the certainty for risk of bias associated with missing data. However, the
judgment of whether the change in effect estimate is clinically significant requires using minimal clinically important difference, which varies by clinical question. Thus, it would be ideal to
reproduce this study in specific subjects of medical science with clearly defined minimal clinically important differences and using random IMOR instead of fixed IMOR.
Future research could also validate some of the findings of this study. For example, this study could be reproduced using individual participant data meta-analyses and findings compared with the
current study. In addition, individual participant data meta-analyses would allow testing other imputation methods, such as multiple imputations.
What is already known on this topic
• Missing data on the outcomes of participants in randomised controlled trials might introduce bias in systematic reviews
• To assess that risk of bias, the GRADE (grading of recommendations assessment, development, and evaluation) working group recommends challenging the robustness of the meta-analysis effect
estimate by conducting sensitivity analyses with different methods of handling missing data
What this study adds
• Even when applying plausible assumptions to the outcomes of participants with definite missing data, the average change in pooled relative effect estimate is substantive, whereas almost a quarter
(22%) of meta-analyses crossed the threshold of the null effect
• Systematic review authors should present the potential impact of missing outcome data on their effect estimates and use this to inform their overall GRADE ratings of risk of bias and their
interpretation of the results
• Contributors: LAK, AMK, GG, HJS, and EAA conceived and designed the paper. LAK and BD developed the full text screening form. LAK developed the data abstraction form. LAK, BD, YC, LCL, AA, LL,
RM, SK, RW, JWB, and AD abstracted the data. LAK and AMK analysed the data. LAK, LH, RJPMS, and EAA interpreted the results. LAK, AMK, LH, RJPMS, and EAA drafted the manuscript. All authors
revised and approved the final manuscript. EAA is the lead author and guarantor. LAK attests that all listed authors meet authorship criteria and that no others meeting the criteria have been
• Funding: This study was funded by the Cochrane Methods Innovation Fund. The funder had no role in considering the study design or in the collection, analysis, interpretation of data, writing of
the report, or decision to submit the article for publication.
• Competing interest: All authors have completed the ICMJE uniform disclosure form at www.icmje.org/coi_disclosure.pdf and declare: support from the Cochrane Methods Innovation Fund for the
submitted work, no financial relationships with any organisations that might have an interest in the submitted work in the previous three years; no other relationships or activities that could
appear to have influenced the submitted work.
• Ethical approval: Not required.
• Data sharing: Data are available on reasonable request from the corresponding author at ea32@aub.edu.lb. Proposals requesting data access will need to specify how it is planned to use the data.
• The lead author EAA (the manuscript’s guarantor) affirms that the manuscript is an honest, accurate, and transparent account of the study being reported; that no important aspects of the study
have been omitted; and that any discrepancies from the study as planned (and, if relevant, registered) have been explained.
• Dissemination to participants and related patient and public communities: Lay information on the key results of the study will be made available on request from the corresponding author.
• This manuscript has been deposited as a preprint.
This is an Open Access article distributed in accordance with the Creative Commons Attribution Non Commercial (CC BY-NC 4.0) license, which permits others to distribute, remix, adapt, build upon this
work non-commercially, and license their derivative works on different terms, provided the original work is properly cited and the use is non-commercial. See: http://creativecommons.org/licenses/ | {"url":"https://www.bmj.com/content/370/bmj.m2898?tab=article-alert","timestamp":"2024-11-11T17:55:25Z","content_type":"text/html","content_length":"263461","record_id":"<urn:uuid:8b456361-d3c7-4df8-8f06-d2491aeff554>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00836.warc.gz"} |
Provides a MATLAB-like plotting framework.
pylab combines pyplot with numpy into a single namespace. This is convenient for interactive work, but for programming it is recommended that the namespaces be kept separate, e.g.:
import numpy as np
import matplotlib.pyplot as plt
x = np.arange(0, 5, 0.1);
y = np.sin(x)
plt.plot(x, y)
matplotlib.pyplot.acorr(x, hold=None, data=None, **kwargs)
Plot the autocorrelation of x.
x : sequence of scalar
hold : boolean, optional, default: True
detrend : callable, optional, default: mlab.detrend_none
x is detrended by the detrend callable. Default is no normalization.
normed : boolean, optional, default: True
if True, normalize the data by the autocorrelation at the 0-th lag.
usevlines : boolean, optional, default: True
if True, Axes.vlines is used to plot the vertical lines from the origin to the acorr. Otherwise, Axes.plot is used.
maxlags : integer, optional, default: 10
number of lags to show. If None, will return all 2 * len(x) - 1 lags.
(lags, c, line, b) : where:
• lags are a length 2`maxlags+1 lag vector.
Returns: • c is the 2`maxlags+1 auto correlation vectorI
• line is a Line2D instance returned by plot.
• b is the x-axis.
Other Parameters:
linestyle : Line2D prop, optional, default: None
Only used if usevlines is False.
marker : string, optional, default: ?o?
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?x?.
Additional kwargs: hold = [True|False] overrides default hold state
xcorr is top graph, and acorr is bottom graph.
(Source code, png, hires.png, pdf)
matplotlib.pyplot.angle_spectrum(x, Fs=None, Fc=None, window=None, pad_to=None, sides=None, hold=None, data=None, **kwargs)
Plot the angle spectrum.
Call signature:
angle_spectrum(x, Fs=2, Fc=0, window=mlab.window_hanning,
pad_to=None, sides='default', **kwargs)
Compute the angle spectrum (wrapped phase spectrum) of x. Data is padded to a length of pad_to and the windowing function window is applied to the signal.
x: 1-D array or sequence
Array or sequence containing the data
Keyword arguments:
Fs: scalar
The sampling frequency (samples per time unit). It is used to calculate the Fourier frequencies, freqs, in cycles per time unit. The default value is 2.
window: callable or ndarray
A function or a vector of length NFFT. To create window vectors see window_hanning(), window_none(), numpy.blackman(), numpy.hamming(), numpy.bartlett(), scipy.signal(),
scipy.signal.get_window(), etc. The default is window_hanning(). If a function is passed as the argument, it must take a data segment as an argument and return the windowed version of the
sides: [ ?default? | ?onesided? | ?twosided? ]
Specifies which sides of the spectrum to return. Default gives the default behavior, which returns one-sided for real data and both for complex data. ?onesided? forces the return of a
one-sided spectrum, while ?twosided? forces two-sided.
pad_to: integer
The number of points to which the data segment is padded when performing the FFT. While not increasing the actual resolution of the spectrum (the minimum distance between resolvable peaks),
this can give more points in the plot, allowing for more detail. This corresponds to the n parameter in the call to fft(). The default is None, which sets pad_to equal to the length of the
input signal (i.e. no padding).
Fc: integer
The center frequency of x (defaults to 0), which offsets the x extents of the plot to reflect the frequency range used when a signal is acquired and then filtered and downsampled to
Returns the tuple (spectrum, freqs, line):
spectrum: 1-D array
The values for the angle spectrum in radians (real valued)
freqs: 1-D array
The frequencies corresponding to the elements in spectrum
line: a Line2D instance
The line created by this function
kwargs control the Line2D properties:
Property Description
agg_filter unknown
alpha float (0.0 transparent through 1.0 opaque)
animated [True | False]
antialiased or aa [True | False]
axes an Axes instance
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
color or c any matplotlib color
contains a callable function
dash_capstyle [?butt? | ?round? | ?projecting?]
dash_joinstyle [?miter? | ?round? | ?bevel?]
dashes sequence of on/off ink in points
drawstyle [?default? | ?steps? | ?steps-pre? | ?steps-mid? | ?steps-post?]
figure a matplotlib.figure.Figure instance
fillstyle [?full? | ?left? | ?right? | ?bottom? | ?top? | ?none?]
gid an id string
label string or anything printable with ?%s? conversion.
linestyle or ls [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or lw float value in points
marker A valid marker style
markeredgecolor or mec any matplotlib color
markeredgewidth or mew float value in points
markerfacecolor or mfc any matplotlib color
markerfacecoloralt or mfcalt any matplotlib color
markersize or ms float
markevery [None | int | length-2 tuple of int | slice | list/array of int | float | length-2 tuple of float]
path_effects unknown
picker float distance in points or callable pick function fn(artist, event)
pickradius float distance in points
rasterized [True | False | None]
sketch_params unknown
snap unknown
solid_capstyle [?butt? | ?round? | ?projecting?]
solid_joinstyle [?miter? | ?round? | ?bevel?]
transform a matplotlib.transforms.Transform instance
url a url string
visible [True | False]
xdata 1D array
ydata 1D array
zorder any number
(Source code, png, hires.png, pdf)
See also
angle_spectrum() plots the magnitudes of the corresponding frequencies.
phase_spectrum() plots the unwrapped version of this function.
specgram() can plot the angle spectrum of segments within the signal in a colormap.
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?x?.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.annotate(*args, **kwargs)
Annotate the point xy with text s.
Additional kwargs are passed to Text.
s : str
The text of the annotation
xy : iterable
Length 2 sequence specifying the (x,y) point to annotate
xytext : iterable, optional
Length 2 sequence specifying the (x,y) to place the text at. If None, defaults to xy.
xycoords : str, Artist, Transform, callable or tuple, optional
The coordinate system that xy is given in.
For a str the allowed values are:
Property Description
?figure points? points from the lower left of the figure
?figure pixels? pixels from the lower left of the figure
?figure fraction? fraction of figure from lower left
?axes points? points from lower left corner of axes
?axes pixels? pixels from lower left corner of axes
?axes fraction? fraction of axes from lower left
?data? use the coordinate system of the object being annotated (default)
?polar? (theta,r) if not native ?data? coordinates
If a Artist object is passed in the units are fraction if it?s bounding box.
If a Transform object is passed in use that to transform xy to screen coordinates
If a callable it must take a RendererBase object as input and return a Transform or Bbox object
If a tuple must be length 2 tuple of str, Artist, Transform or callable objects. The first transform is used for the x coordinate and the second for y.
See Annotating Axes for more details.
Defaults to 'data'
textcoords : str, Artist, Transform, callable or tuple, optional
The coordinate system that xytext is given, which may be different than the coordinate system used for xy.
All xycoords values are valid as well as the following strings:
Property Description
?offset points? offset (in points) from the xy value
?offset pixels? offset (in pixels) from the xy value
defaults to the input of xycoords
Parameters: arrowprops : dict, optional
If not None, properties used to draw a FancyArrowPatch arrow between xy and xytext.
If arrowprops does not contain the key 'arrowstyle' the allowed keys are:
Key Description
width the width of the arrow in points
headwidth the width of the base of the arrow head in points
headlength the length of the arrow head in points
shrink fraction of total length to ?shrink? from both ends
? any key to matplotlib.patches.FancyArrowPatch
If the arrowprops contains the key 'arrowstyle' the above keys are forbidden. The allowed values of 'arrowstyle' are:
Name Attrs
'-' None
'->' head_length=0.4,head_width=0.2
'-[' widthB=1.0,lengthB=0.2,angleB=None
'|-|' widthA=1.0,widthB=1.0
'-|>' head_length=0.4,head_width=0.2
'<-' head_length=0.4,head_width=0.2
'<->' head_length=0.4,head_width=0.2
'<|-' head_length=0.4,head_width=0.2
'<|-|>' head_length=0.4,head_width=0.2
'fancy' head_length=0.4,head_width=0.4,tail_width=0.4
'simple' head_length=0.5,head_width=0.5,tail_width=0.2
'wedge' tail_width=0.3,shrink_factor=0.5
Valid keys for FancyArrowPatch are:
Key Description
arrowstyle the arrow style
connectionstyle the connection style
relpos default is (0.5, 0.5)
patchA default is bounding box of the text
patchB default is None
shrinkA default is 2 points
shrinkB default is 2 points
mutation_scale default is text size (in points)
mutation_aspect default is 1.
? any key for matplotlib.patches.PathPatch
Defaults to None
annotation_clip : bool, optional
Controls the visibility of the annotation when it goes outside the axes area.
If True, the annotation will only be drawn when the xy is inside the axes. If False, the annotation will always be drawn regardless of its position.
The default is None, which behave as True only if xycoords is ?data?.
Returns: Annotation
matplotlib.pyplot.arrow(x, y, dx, dy, hold=None, **kwargs)
Add an arrow to the axes.
Call signature:
arrow(x, y, dx, dy, **kwargs)
Draws arrow on specified axis from (x, y) to (x + dx, y + dy). Uses FancyArrow patch to construct the arrow.
The resulting arrow is affected by the axes aspect ratio and limits. This may produce an arrow whose head is not square with its stem. To create an arrow whose head is square with its stem, use
annotate() for example:
ax.annotate("", xy=(0.5, 0.5), xytext=(0, 0),
Optional kwargs control the arrow construction and properties:
Constructor arguments
width: float (default: 0.001)
width of full arrow tail
length_includes_head: [True | False] (default: False)
True if head is to be counted in calculating the length.
head_width: float or None (default: 3*width)
total width of the full arrow head
head_length: float or None (default: 1.5 * head_width)
length of arrow head
shape: [?full?, ?left?, ?right?] (default: ?full?)
draw the left-half, right-half, or full arrow
overhang: float (default: 0)
fraction that the arrow is swept back (0 overhang means triangular shape). Can be negative or greater than one.
head_starts_at_zero: [True | False] (default: False)
if True, the head starts being drawn at coordinate 0 instead of ending at coordinate 0.
Other valid kwargs (inherited from Patch) are:
Property Description
agg_filter unknown
alpha float or None
animated [True | False]
antialiased or aa [True | False] or None for default
axes an Axes instance
capstyle [?butt? | ?round? | ?projecting?]
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
color matplotlib color spec
contains a callable function
edgecolor or ec mpl color spec, or None for default, or ?none? for no color
facecolor or fc mpl color spec, or None for default, or ?none? for no color
figure a matplotlib.figure.Figure instance
fill [True | False]
gid an id string
hatch [?/? | ?\? | ?|? | ?-? | ?+? | ?x? | ?o? | ?O? | ?.? | ?*?]
joinstyle [?miter? | ?round? | ?bevel?]
label string or anything printable with ?%s? conversion.
linestyle or ls [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or lw float or None for default
path_effects unknown
picker [None|float|boolean|callable]
rasterized [True | False | None]
sketch_params unknown
snap unknown
transform Transform instance
url a url string
visible [True | False]
zorder any number
(Source code, png, hires.png, pdf)
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.autoscale(enable=True, axis='both', tight=None)
Autoscale the axis view to the data (toggle).
Convenience method for simple axis view autoscaling. It turns autoscaling on or off, and then, if autoscaling for either axis is on, it performs the autoscaling on the specified axis or axes.
enable: [True | False | None]
True (default) turns autoscaling on, False turns it off. None leaves the autoscaling state unchanged.
axis: [?x? | ?y? | ?both?]
which axis to operate on; default is ?both?
tight: [True | False | None]
If True, set view limits to data limits; if False, let the locator and margins expand the view limits; if None, use tight scaling if the only artist is an image, otherwise treat tight as
False. The tight setting is retained for future autoscaling until it is explicitly changed.
Returns None.
set the default colormap to autumn and apply to current image if any. See help(colormaps) for more information
matplotlib.pyplot.axes(*args, **kwargs)
Add an axes to the figure.
The axes is added at position rect specified by:
□ axes() by itself creates a default full subplot(111) window axis.
□ axes(rect, axisbg='w') where rect = [left, bottom, width, height] in normalized (0, 1) units. axisbg is the background color for the axis, default white.
□ axes(h) where h is an axes instance makes h the current axis. An Axes instance is returned.
kwarg Accepts Description
axisbg color the axes background color
frameon [True|False] display the frame?
sharex otherax current axes shares xaxis attribute with otherax
sharey otherax current axes shares yaxis attribute with otherax
polar [True|False] use a polar axes?
aspect [str | num] [?equal?, ?auto?] or a number. If a number the ratio of x-unit/y-unit in screen-space. Also see set_aspect().
□ examples/pylab_examples/axes_demo.py places custom axes.
□ examples/pylab_examples/shared_axis_demo.py uses sharex and sharey.
matplotlib.pyplot.axhline(y=0, xmin=0, xmax=1, hold=None, **kwargs)
Add a horizontal line across the axis.
y : scalar, optional, default: 0
y position in data coordinates of the horizontal line.
xmin : scalar, optional, default: 0
Should be between 0 and 1, 0 being the far left of the plot, 1 the far right of the plot.
xmax : scalar, optional, default: 1
Should be between 0 and 1, 0 being the far left of the plot, 1 the far right of the plot.
Returns: Line2D
See also
for example plot and source code
kwargs are passed to Line2D and can be used to control the line properties.
□ draw a thick red hline at ?y? = 0 that spans the xrange:
>>> axhline(linewidth=4, color='r')
□ draw a default hline at ?y? = 1 that spans the xrange:
>>> axhline(y=1)
□ draw a default hline at ?y? = .5 that spans the middle half of the xrange:
>>> axhline(y=.5, xmin=0.25, xmax=0.75)
Valid kwargs are Line2D properties, with the exception of ?transform?:
Property Description
agg_filter unknown
alpha float (0.0 transparent through 1.0 opaque)
animated [True | False]
antialiased or aa [True | False]
axes an Axes instance
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
color or c any matplotlib color
contains a callable function
dash_capstyle [?butt? | ?round? | ?projecting?]
dash_joinstyle [?miter? | ?round? | ?bevel?]
dashes sequence of on/off ink in points
drawstyle [?default? | ?steps? | ?steps-pre? | ?steps-mid? | ?steps-post?]
figure a matplotlib.figure.Figure instance
fillstyle [?full? | ?left? | ?right? | ?bottom? | ?top? | ?none?]
gid an id string
label string or anything printable with ?%s? conversion.
linestyle or ls [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or lw float value in points
marker A valid marker style
markeredgecolor or mec any matplotlib color
markeredgewidth or mew float value in points
markerfacecolor or mfc any matplotlib color
markerfacecoloralt or mfcalt any matplotlib color
markersize or ms float
markevery [None | int | length-2 tuple of int | slice | list/array of int | float | length-2 tuple of float]
path_effects unknown
picker float distance in points or callable pick function fn(artist, event)
pickradius float distance in points
rasterized [True | False | None]
sketch_params unknown
snap unknown
solid_capstyle [?butt? | ?round? | ?projecting?]
solid_joinstyle [?miter? | ?round? | ?bevel?]
transform a matplotlib.transforms.Transform instance
url a url string
visible [True | False]
xdata 1D array
ydata 1D array
zorder any number
matplotlib.pyplot.axhspan(ymin, ymax, xmin=0, xmax=1, hold=None, **kwargs)
Add a horizontal span (rectangle) across the axis.
Call signature:
axhspan(ymin, ymax, xmin=0, xmax=1, **kwargs)
y coords are in data units and x coords are in axes (relative 0-1) units.
Draw a horizontal span (rectangle) from ymin to ymax. With the default values of xmin = 0 and xmax = 1, this always spans the xrange, regardless of the xlim settings, even if you change them,
e.g., with the set_xlim() command. That is, the horizontal extent is in axes coords: 0=left, 0.5=middle, 1.0=right but the y location is in data coordinates.
Return value is a matplotlib.patches.Polygon instance.
□ draw a gray rectangle from y = 0.25-0.75 that spans the horizontal extent of the axes:
>>> axhspan(0.25, 0.75, facecolor='0.5', alpha=0.5)
Valid kwargs are Polygon properties:
Property Description
agg_filter unknown
alpha float or None
animated [True | False]
antialiased or aa [True | False] or None for default
axes an Axes instance
capstyle [?butt? | ?round? | ?projecting?]
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
color matplotlib color spec
contains a callable function
edgecolor or ec mpl color spec, or None for default, or ?none? for no color
facecolor or fc mpl color spec, or None for default, or ?none? for no color
figure a matplotlib.figure.Figure instance
fill [True | False]
gid an id string
hatch [?/? | ?\? | ?|? | ?-? | ?+? | ?x? | ?o? | ?O? | ?.? | ?*?]
joinstyle [?miter? | ?round? | ?bevel?]
label string or anything printable with ?%s? conversion.
linestyle or ls [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or lw float or None for default
path_effects unknown
picker [None|float|boolean|callable]
rasterized [True | False | None]
sketch_params unknown
snap unknown
transform Transform instance
url a url string
visible [True | False]
zorder any number
(Source code, png, hires.png, pdf)
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.axis(*v, **kwargs)
Convenience method to get or set axis properties.
Calling with no arguments:
>>> axis()
returns the current axes limits [xmin, xmax, ymin, ymax].:
>>> axis(v)
sets the min and max of the x and y axes, with v = [xmin, xmax, ymin, ymax].:
>>> axis('off')
turns off the axis lines and labels.:
>>> axis('equal')
changes limits of x or y axis so that equal increments of x and y have the same length; a circle is circular.:
>>> axis('scaled')
achieves the same result by changing the dimensions of the plot box instead of the axis data limits.:
>>> axis('tight')
changes x and y axis limits such that all data is shown. If all data is already shown, it will move it to the center of the figure without modifying (xmax - xmin) or (ymax - ymin). Note this is
slightly different than in MATLAB.:
>>> axis('image')
is ?scaled? with the axis limits equal to the data limits.:
>>> axis('auto')
>>> axis('normal')
are deprecated. They restore default behavior; axis limits are automatically scaled to make the data fit comfortably within the plot box.
if len(*v)==0, you can pass in xmin, xmax, ymin, ymax as kwargs selectively to alter just those limits without changing the others.
>>> axis('square')
changes the limit ranges (xmax-xmin) and (ymax-ymin) of the x and y axes to be the same, and have the same scaling, resulting in a square plot.
The xmin, xmax, ymin, ymax tuple is returned
See also
xlim(), ylim()
For setting the x- and y-limits individually.
matplotlib.pyplot.axvline(x=0, ymin=0, ymax=1, hold=None, **kwargs)
Add a vertical line across the axes.
x : scalar, optional, default: 0
x position in data coordinates of the vertical line.
ymin : scalar, optional, default: 0
Should be between 0 and 1, 0 being the bottom of the plot, 1 the top of the plot.
ymax : scalar, optional, default: 1
Should be between 0 and 1, 0 being the bottom of the plot, 1 the top of the plot.
Returns: Line2D
See also
for example plot and source code
□ draw a thick red vline at x = 0 that spans the yrange:
>>> axvline(linewidth=4, color='r')
□ draw a default vline at x = 1 that spans the yrange:
>>> axvline(x=1)
□ draw a default vline at x = .5 that spans the middle half of the yrange:
>>> axvline(x=.5, ymin=0.25, ymax=0.75)
Valid kwargs are Line2D properties, with the exception of ?transform?:
Property Description
agg_filter unknown
alpha float (0.0 transparent through 1.0 opaque)
animated [True | False]
antialiased or aa [True | False]
axes an Axes instance
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
color or c any matplotlib color
contains a callable function
dash_capstyle [?butt? | ?round? | ?projecting?]
dash_joinstyle [?miter? | ?round? | ?bevel?]
dashes sequence of on/off ink in points
drawstyle [?default? | ?steps? | ?steps-pre? | ?steps-mid? | ?steps-post?]
figure a matplotlib.figure.Figure instance
fillstyle [?full? | ?left? | ?right? | ?bottom? | ?top? | ?none?]
gid an id string
label string or anything printable with ?%s? conversion.
linestyle or ls [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or lw float value in points
marker A valid marker style
markeredgecolor or mec any matplotlib color
markeredgewidth or mew float value in points
markerfacecolor or mfc any matplotlib color
markerfacecoloralt or mfcalt any matplotlib color
markersize or ms float
markevery [None | int | length-2 tuple of int | slice | list/array of int | float | length-2 tuple of float]
path_effects unknown
picker float distance in points or callable pick function fn(artist, event)
pickradius float distance in points
rasterized [True | False | None]
sketch_params unknown
snap unknown
solid_capstyle [?butt? | ?round? | ?projecting?]
solid_joinstyle [?miter? | ?round? | ?bevel?]
transform a matplotlib.transforms.Transform instance
url a url string
visible [True | False]
xdata 1D array
ydata 1D array
zorder any number
matplotlib.pyplot.axvspan(xmin, xmax, ymin=0, ymax=1, hold=None, **kwargs)
Add a vertical span (rectangle) across the axes.
Call signature:
axvspan(xmin, xmax, ymin=0, ymax=1, **kwargs)
x coords are in data units and y coords are in axes (relative 0-1) units.
Draw a vertical span (rectangle) from xmin to xmax. With the default values of ymin = 0 and ymax = 1, this always spans the yrange, regardless of the ylim settings, even if you change them, e.g.,
with the set_ylim() command. That is, the vertical extent is in axes coords: 0=bottom, 0.5=middle, 1.0=top but the y location is in data coordinates.
Return value is the matplotlib.patches.Polygon instance.
□ draw a vertical green translucent rectangle from x=1.25 to 1.55 that spans the yrange of the axes:
>>> axvspan(1.25, 1.55, facecolor='g', alpha=0.5)
Valid kwargs are Polygon properties:
Property Description
agg_filter unknown
alpha float or None
animated [True | False]
antialiased or aa [True | False] or None for default
axes an Axes instance
capstyle [?butt? | ?round? | ?projecting?]
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
color matplotlib color spec
contains a callable function
edgecolor or ec mpl color spec, or None for default, or ?none? for no color
facecolor or fc mpl color spec, or None for default, or ?none? for no color
figure a matplotlib.figure.Figure instance
fill [True | False]
gid an id string
hatch [?/? | ?\? | ?|? | ?-? | ?+? | ?x? | ?o? | ?O? | ?.? | ?*?]
joinstyle [?miter? | ?round? | ?bevel?]
label string or anything printable with ?%s? conversion.
linestyle or ls [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or lw float or None for default
path_effects unknown
picker [None|float|boolean|callable]
rasterized [True | False | None]
sketch_params unknown
snap unknown
transform Transform instance
url a url string
visible [True | False]
zorder any number
See also
for example plot and source code
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.bar(left, height, width=0.8, bottom=None, hold=None, data=None, **kwargs)
Make a bar plot.
Make a bar plot with rectangles bounded by:
left, left + width, bottom, bottom + height
(left, right, bottom and top edges)
left : sequence of scalars
the x coordinates of the left sides of the bars
height : sequence of scalars
the heights of the bars
width : scalar or array-like, optional
the width(s) of the bars default: 0.8
bottom : scalar or array-like, optional
the y coordinate(s) of the bars default: None
color : scalar or array-like, optional
the colors of the bar faces
edgecolor : scalar or array-like, optional
the colors of the bar edges
linewidth : scalar or array-like, optional
width of bar edge(s). If None, use default linewidth; If 0, don?t draw edges. default: None
tick_label : string or array-like, optional
the tick labels of the bars default: None
Parameters: xerr : scalar or array-like, optional
if not None, will be used to generate errorbar(s) on the bar chart default: None
yerr : scalar or array-like, optional
if not None, will be used to generate errorbar(s) on the bar chart default: None
ecolor : scalar or array-like, optional
specifies the color of errorbar(s) default: None
capsize : scalar, optional
determines the length in points of the error bar caps default: None, which will take the value from the errorbar.capsize rcParam.
error_kw : dict, optional
dictionary of kwargs to be passed to errorbar method. ecolor and capsize may be specified here rather than as independent kwargs.
align : {?edge?, ?center?}, optional
If ?edge?, aligns bars by their left edges (for vertical bars) and by their bottom edges (for horizontal bars). If ?center?, interpret the left argument as the coordinates of the
centers of the bars. To align on the align bars on the right edge pass a negative width.
orientation : {?vertical?, ?horizontal?}, optional
The orientation of the bars.
log : boolean, optional
If true, sets the axis to be log scale. default: False
bars : matplotlib.container.BarContainer
Container with all of the bars + errorbars
See also
Plot a horizontal bar plot.
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?linewidth?, ?bottom?, ?yerr?, ?height?, ?xerr?, ?edgecolor?, ?tick_label?, ?color?, ?left?, ?width?, ?ecolor?.
Additional kwargs: hold = [True|False] overrides default hold state
Example: A stacked bar chart.
(Source code, png, hires.png, pdf)
matplotlib.pyplot.barbs(*args, **kw)
Plot a 2-D field of barbs.
Call signatures:
barb(U, V, **kw)
barb(U, V, C, **kw)
barb(X, Y, U, V, **kw)
barb(X, Y, U, V, C, **kw)
X, Y:
The x and y coordinates of the barb locations (default is head of barb; see pivot kwarg)
U, V:
Give the x and y components of the barb shaft
An optional array used to map colors to the barbs
All arguments may be 1-D or 2-D arrays or sequences. If X and Y are absent, they will be generated as a uniform grid. If U and V are 2-D arrays but X and Y are 1-D, and if len(X) and len(Y) match
the column and row dimensions of U, then X and Y will be expanded with numpy.meshgrid().
U, V, C may be masked arrays, but masked X, Y are not supported at present.
Keyword arguments:
Length of the barb in points; the other parts of the barb are scaled against this. Default is 9
pivot: [ ?tip? | ?middle? ]
The part of the arrow that is at the grid point; the arrow rotates about this point, hence the name pivot. Default is ?tip?
barbcolor: [ color | color sequence ]
Specifies the color all parts of the barb except any flags. This parameter is analagous to the edgecolor parameter for polygons, which can be used instead. However this parameter will
override facecolor.
flagcolor: [ color | color sequence ]
Specifies the color of any flags on the barb. This parameter is analagous to the facecolor parameter for polygons, which can be used instead. However this parameter will override facecolor.
If this is not set (and C has not either) then flagcolor will be set to match barbcolor so that the barb has a uniform color. If C has been set, flagcolor has no effect.
A dictionary of coefficients specifying the ratio of a given feature to the length of the barb. Only those values one wishes to override need to be included. These features include:
☆ ?spacing? - space between features (flags, full/half barbs)
☆ ?height? - height (distance from shaft to top) of a flag or full barb
☆ ?width? - width of a flag, twice the width of a full barb
☆ ?emptybarb? - radius of the circle used for low magnitudes
A flag on whether the empty barbs (circles) that are drawn should be filled with the flag color. If they are not filled, they will be drawn such that no color is applied to the center.
Default is False
A flag to indicate whether the vector magnitude should be rounded when allocating barb components. If True, the magnitude is rounded to the nearest multiple of the half-barb increment. If
False, the magnitude is simply truncated to the next lowest multiple. Default is True
A dictionary of increments specifying values to associate with different parts of the barb. Only those values one wishes to override need to be included.
☆ ?half? - half barbs (Default is 5)
☆ ?full? - full barbs (Default is 10)
☆ ?flag? - flags (default is 50)
Either a single boolean flag or an array of booleans. Single boolean indicates whether the lines and flags should point opposite to normal for all barbs. An array (which should be the same
size as the other data arrays) indicates whether to flip for each individual barb. Normal behavior is for the barbs and lines to point right (comes from wind barbs having these features point
towards low pressure in the Northern Hemisphere.) Default is False
Barbs are traditionally used in meteorology as a way to plot the speed and direction of wind observations, but can technically be used to plot any two dimensional vector quantity. As opposed to
arrows, which give vector magnitude by the length of the arrow, the barbs give more quantitative information about the vector magnitude by putting slanted lines or a triangle for various
increments in magnitude, as show schematically below:
: /\ \
: / \ \
: / \ \ \
: / \ \ \
: ------------------------------
The largest increment is given by a triangle (or ?flag?). After those come full lines (barbs). The smallest increment is a half line. There is only, of course, ever at most 1 half line. If the
magnitude is small and only needs a single half-line and no full lines or triangles, the half-line is offset from the end of the barb so that it can be easily distinguished from barbs with a
single full line. The magnitude for the barb shown above would nominally be 65, using the standard increments of 50, 10, and 5.
linewidths and edgecolors can be used to customize the barb. Additional PolyCollection keyword arguments:
Property Description
agg_filter unknown
alpha float or None
animated [True | False]
antialiased or antialiaseds Boolean or sequence of booleans
array unknown
axes an Axes instance
clim a length 2 sequence of floats
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
cmap a colormap or registered colormap name
color matplotlib color arg or sequence of rgba tuples
contains a callable function
edgecolor or edgecolors matplotlib color spec or sequence of specs
facecolor or facecolors matplotlib color spec or sequence of specs
figure a matplotlib.figure.Figure instance
gid an id string
hatch [ ?/? | ?\? | ?|? | ?-? | ?+? | ?x? | ?o? | ?O? | ?.? | ?*? ]
label string or anything printable with ?%s? conversion.
linestyle or linestyles or dashes [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or linewidths or lw float or sequence of floats
norm unknown
offset_position unknown
offsets float or sequence of floats
path_effects unknown
picker [None|float|boolean|callable]
pickradius unknown
rasterized [True | False | None]
sketch_params unknown
snap unknown
transform Transform instance
url a url string
urls unknown
visible [True | False]
zorder any number
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All positional and all keyword arguments.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.barh(bottom, width, height=0.8, left=None, hold=None, **kwargs)
Make a horizontal bar plot.
Make a horizontal bar plot with rectangles bounded by:
left, left + width, bottom, bottom + height
(left, right, bottom and top edges)
bottom, width, height, and left can be either scalars or sequences
bottom : scalar or array-like
the y coordinate(s) of the bars
width : scalar or array-like
the width(s) of the bars
height : sequence of scalars, optional, default: 0.8
the heights of the bars
left : sequence of scalars
the x coordinates of the left sides of the bars
Returns: matplotlib.patches.Rectangle instances.
Other Parameters:
color : scalar or array-like, optional
the colors of the bars
edgecolor : scalar or array-like, optional
the colors of the bar edges
linewidth : scalar or array-like, optional, default: None
width of bar edge(s). If None, use default linewidth; If 0, don?t draw edges.
tick_label : string or array-like, optional, default: None
the tick labels of the bars
xerr : scalar or array-like, optional, default: None
if not None, will be used to generate errorbar(s) on the bar chart
yerr : scalar or array-like, optional, default: None
if not None, will be used to generate errorbar(s) on the bar chart
ecolor : scalar or array-like, optional, default: None
specifies the color of errorbar(s)
capsize : scalar, optional
determines the length in points of the error bar caps default: None, which will take the value from the errorbar.capsize rcParam.
error_kw :
dictionary of kwargs to be passed to errorbar method. ecolor and capsize may be specified here rather than as independent kwargs.
align : [?edge? | ?center?], optional, default: ?edge?
If edge, aligns bars by their left edges (for vertical bars) and by their bottom edges (for horizontal bars). If center, interpret the left argument as the coordinates of the centers
of the bars.
log : boolean, optional, default: False
If true, sets the axis to be log scale
See also
Plot a vertical bar plot.
The optional arguments color, edgecolor, linewidth, xerr, and yerr can be either scalars or sequences of length equal to the number of bars. This enables you to use bar as the basis for stacked
bar charts, or candlestick plots. Detail: xerr and yerr are passed directly to errorbar(), so they can also have shape 2xN for independent specification of lower and upper errors.
Other optional kwargs:
Property Description
agg_filter unknown
alpha float or None
animated [True | False]
antialiased or aa [True | False] or None for default
axes an Axes instance
capstyle [?butt? | ?round? | ?projecting?]
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
color matplotlib color spec
contains a callable function
edgecolor or ec mpl color spec, or None for default, or ?none? for no color
facecolor or fc mpl color spec, or None for default, or ?none? for no color
figure a matplotlib.figure.Figure instance
fill [True | False]
gid an id string
hatch [?/? | ?\? | ?|? | ?-? | ?+? | ?x? | ?o? | ?O? | ?.? | ?*?]
joinstyle [?miter? | ?round? | ?bevel?]
label string or anything printable with ?%s? conversion.
linestyle or ls [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or lw float or None for default
path_effects unknown
picker [None|float|boolean|callable]
rasterized [True | False | None]
sketch_params unknown
snap unknown
transform Transform instance
url a url string
visible [True | False]
zorder any number
set the default colormap to bone and apply to current image if any. See help(colormaps) for more information
Turn the axes box on or off. on may be a boolean or a string, ?on? or ?off?.
If on is None, toggle state.
matplotlib.pyplot.boxplot(x, notch=None, sym=None, vert=None, whis=None, positions=None, widths=None, patch_artist=None, bootstrap=None, usermedians=None, conf_intervals=None, meanline=None,
showmeans=None, showcaps=None, showbox=None, showfliers=None, boxprops=None, labels=None, flierprops=None, medianprops=None, meanprops=None, capprops=None, whiskerprops=None, manage_xticks=True,
hold=None, data=None)
Make a box and whisker plot.
Call signature:
boxplot(self, x, notch=None, sym=None, vert=None, whis=None,
positions=None, widths=None, patch_artist=False,
bootstrap=None, usermedians=None, conf_intervals=None,
meanline=False, showmeans=False, showcaps=True,
showbox=True, showfliers=True, boxprops=None,
labels=None, flierprops=None, medianprops=None,
meanprops=None, capprops=None, whiskerprops=None,
manage_xticks=True, autorange=False):
Make a box and whisker plot for each column of x or each vector in sequence x. The box extends from the lower to upper quartile values of the data, with a line at the median. The whiskers extend
from the box to show the range of the data. Flier points are those past the end of the whiskers.
x : Array or a sequence of vectors.
The input data.
notch : bool, optional (False)
If True, will produce a notched box plot. Otherwise, a rectangular boxplot is produced.
sym : str, optional
The default symbol for flier points. Enter an empty string (??) if you don?t want to show fliers. If None, then the fliers default to ?b+? If you want more control use the flierprops
vert : bool, optional (True)
If True (default), makes the boxes vertical. If False, everything is drawn horizontally.
whis : float, sequence, or string (default = 1.5)
As a float, determines the reach of the whiskers past the first and third quartiles (e.g., Q3 + whis*IQR, IQR = interquartile range, Q3-Q1). Beyond the whiskers, data are considered
outliers and are plotted as individual points. Set this to an unreasonably high value to force the whiskers to show the min and max values. Alternatively, set this to an ascending
sequence of percentile (e.g., [5, 95]) to set the whiskers at specific percentiles of the data. Finally, whis can be the string 'range' to force the whiskers to the min and max of the
bootstrap : int, optional
Specifies whether to bootstrap the confidence intervals around the median for notched boxplots. If bootstrap is None, no bootstrapping is performed, and notches are calculated using a
Gaussian-based asymptotic approximation (see McGill, R., Tukey, J.W., and Larsen, W.A., 1978, and Kendall and Stuart, 1967). Otherwise, bootstrap specifies the number of times to
bootstrap the median to determine its 95% confidence intervals. Values between 1000 and 10000 are recommended.
usermedians : array-like, optional
An array or sequence whose first dimension (or length) is compatible with x. This overrides the medians computed by matplotlib for each element of usermedians that is not None. When
an element of usermedians is None, the median will be computed by matplotlib as normal.
conf_intervals : array-like, optional
Array or sequence whose first dimension (or length) is compatible with x and whose second dimension is 2. When the an element of conf_intervals is not None, the notch locations
computed by matplotlib are overridden (provided notch is True). When an element of conf_intervals is None, the notches are computed by the method specified by the other kwargs (e.g.,
positions : array-like, optional
Sets the positions of the boxes. The ticks and limits are automatically set to match the positions. Defaults to range(1, N+1) where N is the number of boxes to be drawn.
widths : scalar or array-like
Sets the width of each box either with a scalar or a sequence. The default is 0.5, or 0.15*(distance between extreme positions), if that is smaller.
patch_artist : bool, optional (False)
If False produces boxes with the Line2D artist. Otherwise, boxes and drawn with Patch artists.
labels : sequence, optional
Labels for each dataset. Length must be compatible with dimensions of x.
manage_xticks : bool, optional (True)
If the function should adjust the xlim and xtick locations.
autorange : bool, optional (False)
When True and the data are distributed such that the 25th and 75th percentiles are equal, whis is set to 'range' such that the whisker ends are at the minimum and maximum of the data.
meanline : bool, optional (False)
If True (and showmeans is True), will try to render the mean as a line spanning the full width of the box according to meanprops (see below). Not recommended if shownotches is also
True. Otherwise, means will be shown as points.
result : dict
A dictionary mapping each component of the boxplot to a list of the matplotlib.lines.Line2D instances created. That dictionary has the following keys (assuming vertical boxplots):
• boxes: the main body of the boxplot showing the quartiles and the median?s confidence intervals if enabled.
Returns: • medians: horizontal lines at the median of each box.
• whiskers: the vertical lines extending to the most extreme, non-outlier data points.
• caps: the horizontal lines at the ends of the whiskers.
• fliers: points representing data that extend beyond the whiskers (fliers).
• means: points or lines representing the means.
Other Parameters:
The following boolean options toggle the drawing of individual
components of the boxplots:
• showcaps: the caps on the ends of whiskers (default is True)
• showbox: the central box (default is True)
• showfliers: the outliers beyond the caps (default is True)
• showmeans: the arithmetic means (default is False)
The remaining options can accept dictionaries that specify the
style of the individual artists:
• capprops
• boxprops
• whiskerprops
• flierprops
• medianprops
• meanprops
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All positional and all keyword arguments.
Additional kwargs: hold = [True|False] overrides default hold state
(Source code, png, hires.png, pdf)
(png, hires.png, pdf)
matplotlib.pyplot.broken_barh(xranges, yrange, hold=None, data=None, **kwargs)
Plot horizontal bars.
Call signature:
broken_barh(self, xranges, yrange, **kwargs)
A collection of horizontal bars spanning yrange with a sequence of xranges.
Required arguments:
Argument Description
xranges sequence of (xmin, xwidth)
yrange sequence of (ymin, ywidth)
kwargs are matplotlib.collections.BrokenBarHCollection properties:
Property Description
agg_filter unknown
alpha float or None
animated [True | False]
antialiased or antialiaseds Boolean or sequence of booleans
array unknown
axes an Axes instance
clim a length 2 sequence of floats
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
cmap a colormap or registered colormap name
color matplotlib color arg or sequence of rgba tuples
contains a callable function
edgecolor or edgecolors matplotlib color spec or sequence of specs
facecolor or facecolors matplotlib color spec or sequence of specs
figure a matplotlib.figure.Figure instance
gid an id string
hatch [ ?/? | ?\? | ?|? | ?-? | ?+? | ?x? | ?o? | ?O? | ?.? | ?*? ]
label string or anything printable with ?%s? conversion.
linestyle or linestyles or dashes [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or linewidths or lw float or sequence of floats
norm unknown
offset_position unknown
offsets float or sequence of floats
path_effects unknown
picker [None|float|boolean|callable]
pickradius unknown
rasterized [True | False | None]
sketch_params unknown
snap unknown
transform Transform instance
url a url string
urls unknown
visible [True | False]
zorder any number
these can either be a single argument, i.e.,:
facecolors = 'black'
or a sequence of arguments for the various bars, i.e.,:
facecolors = ('black', 'red', 'green')
(Source code, png, hires.png, pdf)
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All positional and all keyword arguments.
Additional kwargs: hold = [True|False] overrides default hold state
Clear the current axes.
matplotlib.pyplot.clabel(CS, *args, **kwargs)
Label a contour plot.
Call signature:
clabel(cs, **kwargs)
Adds labels to line contours in cs, where cs is a ContourSet object returned by contour.
clabel(cs, v, **kwargs)
only labels contours listed in v.
Optional keyword arguments:
size in points or relative size e.g., ?smaller?, ?x-large?
☆ if None, the color of each label matches the color of the corresponding contour
☆ if one string color, e.g., colors = ?r? or colors = ?red?, all labels will be plotted in this color
☆ if a tuple of matplotlib color args (string, float, rgb, etc), different labels will be plotted in different colors in the order specified
controls whether the underlying contour is removed or not. Default is True.
space in pixels to leave on each side of label when placing inline. Defaults to 5. This spacing will be exact for labels at locations where the contour is straight, less so for labels on
curved contours.
a format string for the label. Default is ?%1.3f? Alternatively, this can be a dictionary matching contour levels with arbitrary strings to use for each contour level (i.e., fmt[level]=
string), or it can be any callable, such as a Formatter instance, that returns a string when called with a numeric contour level.
if True, contour labels will be placed manually using mouse clicks. Click the first button near a contour to add a label, click the second button (or potentially both mouse buttons at once)
to finish adding labels. The third button can be used to remove the last label added, but only if labels are not inline. Alternatively, the keyboard can be used to select label locations
(enter to end label placement, delete or backspace act like the third mouse button, and any other key will select a label location).
manual can be an iterable object of x,y tuples. Contour labels will be created as if mouse is clicked at each x,y positions.
if True (default), label rotations will always be plus or minus 90 degrees from level.
if True (default is False), ClabelText class (instead of matplotlib.Text) is used to create labels. ClabelText recalculates rotation angles of texts during the drawing time, therefore this
can be used if aspect of the axes changes.
Additional kwargs: hold = [True|False] overrides default hold state
Clear the current figure.
matplotlib.pyplot.clim(vmin=None, vmax=None)
Set the color limits of the current image.
To apply clim to all axes images do:
clim(0, 0.5)
If either vmin or vmax is None, the image min/max respectively will be used for color scaling.
If you want to set the clim of multiple images, use, for example:
for im in gca().get_images():
im.set_clim(0, 0.05)
Close a figure window.
close() by itself closes the current figure
close(h) where h is a Figure instance, closes that figure
close(num) closes figure number num
close(name) where name is a string, closes figure with that label
close('all') closes all the figure windows
matplotlib.pyplot.cohere(x, y, NFFT=256, Fs=2, Fc=0, detrend=, window=, noverlap=0, pad_to=None, sides='default', scale_by_freq=None, hold=None, data=None, **kwargs)
Plot the coherence between x and y.
Call signature:
cohere(x, y, NFFT=256, Fs=2, Fc=0, detrend = mlab.detrend_none,
window = mlab.window_hanning, noverlap=0, pad_to=None,
sides='default', scale_by_freq=None, **kwargs)
Plot the coherence between x and y. Coherence is the normalized cross spectral density:
Keyword arguments:
Fs: scalar
The sampling frequency (samples per time unit). It is used to calculate the Fourier frequencies, freqs, in cycles per time unit. The default value is 2.
window: callable or ndarray
A function or a vector of length NFFT. To create window vectors see window_hanning(), window_none(), numpy.blackman(), numpy.hamming(), numpy.bartlett(), scipy.signal(),
scipy.signal.get_window(), etc. The default is window_hanning(). If a function is passed as the argument, it must take a data segment as an argument and return the windowed version of the
sides: [ ?default? | ?onesided? | ?twosided? ]
Specifies which sides of the spectrum to return. Default gives the default behavior, which returns one-sided for real data and both for complex data. ?onesided? forces the return of a
one-sided spectrum, while ?twosided? forces two-sided.
pad_to: integer
The number of points to which the data segment is padded when performing the FFT. This can be different from NFFT, which specifies the number of data points used. While not increasing the
actual resolution of the spectrum (the minimum distance between resolvable peaks), this can give more points in the plot, allowing for more detail. This corresponds to the n parameter in the
call to fft(). The default is None, which sets pad_to equal to NFFT
NFFT: integer
The number of data points used in each block for the FFT. A power 2 is most efficient. The default value is 256. This should NOT be used to get zero padding, or the scaling of the result will
be incorrect. Use pad_to for this instead.
detrend: [ ?default? | ?constant? | ?mean? | ?linear? | ?none?] or
The function applied to each segment before fft-ing, designed to remove the mean or linear trend. Unlike in MATLAB, where the detrend parameter is a vector, in matplotlib is it a function.
The pylab module defines detrend_none(), detrend_mean(), and detrend_linear(), but you can use a custom function as well. You can also use a string to choose one of the functions. ?default?,
?constant?, and ?mean? call detrend_mean(). ?linear? calls detrend_linear(). ?none? calls detrend_none().
scale_by_freq: boolean
Specifies whether the resulting density values should be scaled by the scaling frequency, which gives density in units of Hz^-1. This allows for integration over the returned frequency
values. The default is True for MATLAB compatibility.
noverlap: integer
The number of points of overlap between blocks. The default value is 0 (no overlap).
Fc: integer
The center frequency of x (defaults to 0), which offsets the x extents of the plot to reflect the frequency range used when a signal is acquired and then filtered and downsampled to
The return value is a tuple (Cxy, f), where f are the frequencies of the coherence vector.
kwargs are applied to the lines.
□ Bendat & Piersol ? Random Data: Analysis and Measurement Procedures, John Wiley & Sons (1986)
kwargs control the Line2D properties of the coherence plot:
Property Description
agg_filter unknown
alpha float (0.0 transparent through 1.0 opaque)
animated [True | False]
antialiased or aa [True | False]
axes an Axes instance
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
color or c any matplotlib color
contains a callable function
dash_capstyle [?butt? | ?round? | ?projecting?]
dash_joinstyle [?miter? | ?round? | ?bevel?]
dashes sequence of on/off ink in points
drawstyle [?default? | ?steps? | ?steps-pre? | ?steps-mid? | ?steps-post?]
figure a matplotlib.figure.Figure instance
fillstyle [?full? | ?left? | ?right? | ?bottom? | ?top? | ?none?]
gid an id string
label string or anything printable with ?%s? conversion.
linestyle or ls [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or lw float value in points
marker A valid marker style
markeredgecolor or mec any matplotlib color
markeredgewidth or mew float value in points
markerfacecolor or mfc any matplotlib color
markerfacecoloralt or mfcalt any matplotlib color
markersize or ms float
markevery [None | int | length-2 tuple of int | slice | list/array of int | float | length-2 tuple of float]
path_effects unknown
picker float distance in points or callable pick function fn(artist, event)
pickradius float distance in points
rasterized [True | False | None]
sketch_params unknown
snap unknown
solid_capstyle [?butt? | ?round? | ?projecting?]
solid_joinstyle [?miter? | ?round? | ?bevel?]
transform a matplotlib.transforms.Transform instance
url a url string
visible [True | False]
xdata 1D array
ydata 1D array
zorder any number
(Source code, png, hires.png, pdf)
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?y?, ?x?.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.colorbar(mappable=None, cax=None, ax=None, **kw)
Add a colorbar to a plot.
Function signatures for the pyplot interface; all but the first are also method signatures for the colorbar() method:
colorbar(mappable, **kwargs)
colorbar(mappable, cax=cax, **kwargs)
colorbar(mappable, ax=ax, **kwargs)
the Image, ContourSet, etc. to which the colorbar applies; this argument is mandatory for the colorbar() method but optional for the colorbar() function, which sets the default to the current
keyword arguments:
None | axes object into which the colorbar will be drawn
None | parent axes object(s) from which space for a new colorbar axes will be stolen. If a list of axes is given they will all be resized to make room for the colorbar axes.
False | If cax is None, a new cax is created as an instance of Axes. If ax is an instance of Subplot and use_gridspec is True, cax is created as an instance of Subplot using the grid_spec
Additional keyword arguments are of two kinds:
axes properties:
Property Description
orientation vertical or horizontal
fraction 0.15; fraction of original axes to use for colorbar
pad 0.05 if vertical, 0.15 if horizontal; fraction of original axes between colorbar and new image axes
shrink 1.0; fraction by which to shrink the colorbar
aspect 20; ratio of long to short dimensions
anchor (0.0, 0.5) if vertical; (0.5, 1.0) if horizontal; the anchor point of the colorbar axes
panchor (1.0, 0.5) if vertical; (0.5, 0.0) if horizontal; the anchor point of the colorbar parent axes. If False, the parent axes? anchor will be unchanged
colorbar properties:
Property Description
extend [ ?neither? | ?both? | ?min? | ?max? ] If not ?neither?, make pointed end(s) for out-of- range values. These are set for a given colormap using the colormap set_under and set_over
[ None | ?auto? | length | lengths ] If set to None, both the minimum and maximum triangular colorbar extensions with have a length of 5% of the interior colorbar length (this is the
default setting). If set to ?auto?, makes the triangular colorbar extensions the same lengths as the interior boxes (when spacing is set to ?uniform?) or the same lengths as the
extendfrac respective adjacent interior boxes (when spacing is set to ?proportional?). If a scalar, indicates the length of both the minimum and maximum triangular colorbar extensions as a
fraction of the interior colorbar length. A two-element sequence of fractions may also be given, indicating the lengths of the minimum and maximum colorbar extensions respectively as a
fraction of the interior colorbar length.
extendrect [ False | True ] If False the minimum and maximum colorbar extensions will be triangular (the default). If True the extensions will be rectangular.
spacing [ ?uniform? | ?proportional? ] Uniform spacing gives each discrete color the same space; proportional makes the space proportional to the data interval.
ticks [ None | list of ticks | Locator object ] If None, ticks are determined automatically from the input.
format [ None | format string | Formatter object ] If None, the ScalarFormatter is used. If a format string is given, e.g., ?%.3f?, that is used. An alternative Formatter object may be given
drawedges [ False | True ] If true, draw lines at color boundaries.
The following will probably be useful only in the context of indexed colors (that is, when the mappable has norm=NoNorm()), or other unusual circumstances.
Property Description
boundaries None or a sequence
values None or a sequence which must be of length 1 less than the sequence of boundaries. For each region delimited by adjacent entries in boundaries, the color mapped to the corresponding
value in values will be used.
If mappable is a ContourSet, its extend kwarg is included automatically.
Note that the shrink kwarg provides a simple way to keep a vertical colorbar, for example, from being taller than the axes of the mappable to which the colorbar is attached; but it is a manual
method requiring some trial and error. If the colorbar is too tall (or a horizontal colorbar is too wide) use a smaller value of shrink.
For more precise control, you can manually specify the positions of the axes objects in which the mappable and the colorbar are drawn. In this case, do not use any of the axes properties kwargs.
It is known that some vector graphics viewer (svg and pdf) renders white gaps between segments of the colorbar. This is due to bugs in the viewers not matplotlib. As a workaround the colorbar can
be rendered with overlapping segments:
cbar = colorbar()
However this has negative consequences in other circumstances. Particularly with semi transparent images (alpha < 1) and colorbar extensions and is not enabled by default see (issue #1188).
Colorbar instance; see also its base class, ColorbarBase. Call the set_label() method to label the colorbar.
This is a do-nothing function to provide you with help on how matplotlib handles colors.
Commands which take color arguments can use several formats to specify the colors. For the basic built-in colors, you can use a single letter
Alias Color
?b? blue
?g? green
?r? red
?c? cyan
?m? magenta
?y? yellow
?k? black
?w? white
For a greater range of colors, you have two options. You can specify the color using an html hex string, as in:
color = '#eeefff'
or you can pass an R,G,B tuple, where each of R,G,B are in the range [0,1].
You can also use any legal html name for a color, for example:
color = 'red'
color = 'burlywood'
color = 'chartreuse'
The example below creates a subplot with a dark slate gray background:
subplot(111, axisbg=(0.1843, 0.3098, 0.3098))
Here is an example that creates a pale turquoise title:
title('Is this the best color?', color='#afeeee')
matplotlib.pyplot.connect(s, func)
Connect event with string s to func. The signature of func is:
def func(event)
where event is a matplotlib.backend_bases.Event. The following events are recognized
□ ?button_press_event?
□ ?button_release_event?
□ ?draw_event?
□ ?key_press_event?
□ ?key_release_event?
□ ?motion_notify_event?
□ ?pick_event?
□ ?resize_event?
□ ?scroll_event?
□ ?figure_enter_event?,
□ ?figure_leave_event?,
□ ?axes_enter_event?,
□ ?axes_leave_event?
□ ?close_event?
For the location events (button and key press/release), if the mouse is over the axes, the variable event.inaxes will be set to the Axes the event occurs is over, and additionally, the variables
event.xdata and event.ydata will be defined. This is the mouse location in data coords. See KeyEvent and MouseEvent for more info.
Return value is a connection id that can be used with mpl_disconnect().
Example usage:
def on_press(event):
print('you pressed', event.button, event.xdata, event.ydata)
cid = canvas.mpl_connect('button_press_event', on_press)
matplotlib.pyplot.contour(*args, **kwargs)
Plot contours.
contour() and contourf() draw contour lines and filled contours, respectively. Except as noted, function signatures and return values are the same for both versions.
contourf() differs from the MATLAB version in that it does not draw the polygon edges. To draw edges, add line contours with calls to contour().
Call signatures:
make a contour plot of an array Z. The level values are chosen automatically.
X, Y specify the (x, y) coordinates of the surface
contour up to N automatically-chosen levels.
draw contour lines at the values specified in sequence V, which must be in increasing order.
contourf(..., V)
fill the len(V)-1 regions between the values in V, which must be in increasing order.
contour(Z, **kwargs)
Use keyword args to control colors, linewidth, origin, cmap ... see below for more details.
X and Y must both be 2-D with the same shape as Z, or they must both be 1-D such that len(X) is the number of columns in Z and len(Y) is the number of rows in Z.
C = contour(...) returns a QuadContourSet object.
Optional keyword arguments:
corner_mask: [ True | False | ?legacy? ]
Enable/disable corner masking, which only has an effect if Z is a masked array. If False, any quad touching a masked point is masked out. If True, only the triangular corners of quads nearest
those points are always masked out, other triangular corners comprising three unmasked points are contoured as usual. If ?legacy?, the old contouring algorithm is used, which is equivalent to
False and is deprecated, only remaining whilst the new algorithm is tested fully.
If not specified, the default is taken from rcParams[?contour.corner_mask?], which is True unless it has been modified.
colors: [ None | string | (mpl_colors) ]
If None, the colormap specified by cmap will be used.
If a string, like ?r? or ?red?, all levels will be plotted in this color.
If a tuple of matplotlib color args (string, float, rgb, etc), different levels will be plotted in different colors in the order specified.
alpha: float
The alpha blending value
cmap: [ None | Colormap ]
A cm Colormap instance or None. If cmap is None and colors is None, a default Colormap is used.
norm: [ None | Normalize ]
A matplotlib.colors.Normalize instance for scaling data values to colors. If norm is None and colors is None, the default linear scaling is used.
vmin, vmax: [ None | scalar ]
If not None, either or both of these values will be supplied to the matplotlib.colors.Normalize instance, overriding the default color scaling based on levels.
levels: [level0, level1, ..., leveln]
A list of floating point numbers indicating the level curves to draw, in increasing order; e.g., to draw just the zero contour pass levels=[0]
origin: [ None | ?upper? | ?lower? | ?image? ]
If None, the first value of Z will correspond to the lower left corner, location (0,0). If ?image?, the rc value for image.origin will be used.
This keyword is not active if X and Y are specified in the call to contour.
extent: [ None | (x0,x1,y0,y1) ]
If origin is not None, then extent is interpreted as in matplotlib.pyplot.imshow(): it gives the outer pixel boundaries. In this case, the position of Z[0,0] is the center of the pixel, not a
corner. If origin is None, then (x0, y0) is the position of Z[0,0], and (x1, y1) is the position of Z[-1,-1].
This keyword is not active if X and Y are specified in the call to contour.
locator: [ None | ticker.Locator subclass ]
If locator is None, the default MaxNLocator is used. The locator is used to determine the contour levels if they are not given explicitly via the V argument.
extend: [ ?neither? | ?both? | ?min? | ?max? ]
Unless this is ?neither?, contour levels are automatically added to one or both ends of the range so that all data are included. These added ranges are then mapped to the special colormap
values which default to the ends of the colormap range, but can be set via matplotlib.colors.Colormap.set_under() and matplotlib.colors.Colormap.set_over() methods.
xunits, yunits: [ None | registered units ]
Override axis units by specifying an instance of a matplotlib.units.ConversionInterface.
antialiased: [ True | False ]
enable antialiasing, overriding the defaults. For filled contours, the default is True. For line contours, it is taken from rcParams[?lines.antialiased?].
nchunk: [ 0 | integer ]
If 0, no subdivision of the domain. Specify a positive integer to divide the domain into subdomains of nchunk by nchunk quads. Chunking reduces the maximum length of polygons generated by the
contouring algorithm which reduces the rendering workload passed on to the backend and also requires slightly less RAM. It can however introduce rendering artifacts at chunk boundaries
depending on the backend, the antialiased flag and value of alpha.
contour-only keyword arguments:
linewidths: [ None | number | tuple of numbers ]
If linewidths is None, the default width in lines.linewidth in matplotlibrc is used.
If a number, all levels will be plotted with this linewidth.
If a tuple, different levels will be plotted with different linewidths in the order specified.
linestyles: [ None | ?solid? | ?dashed? | ?dashdot? | ?dotted? ]
If linestyles is None, the default is ?solid? unless the lines are monochrome. In that case, negative contours will take their linestyle from the matplotlibrc contour.negative_linestyle
linestyles can also be an iterable of the above strings specifying a set of linestyles to be used. If this iterable is shorter than the number of contour levels it will be repeated as
contourf-only keyword arguments:
A list of cross hatch patterns to use on the filled areas. If None, no hatching will be added to the contour. Hatching is supported in the PostScript, PDF, SVG and Agg backends only.
Note: contourf fills intervals that are closed at the top; that is, for boundaries z1 and z2, the filled region is:
z1 < z <= z2
There is one exception: if the lowest boundary coincides with the minimum value of the z array, then that minimum value will be included in the lowest interval.
(Source code, png, hires.png, pdf)
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.contourf(*args, **kwargs)
Plot contours.
contour() and contourf() draw contour lines and filled contours, respectively. Except as noted, function signatures and return values are the same for both versions.
contourf() differs from the MATLAB version in that it does not draw the polygon edges. To draw edges, add line contours with calls to contour().
Call signatures:
make a contour plot of an array Z. The level values are chosen automatically.
X, Y specify the (x, y) coordinates of the surface
contour up to N automatically-chosen levels.
draw contour lines at the values specified in sequence V, which must be in increasing order.
contourf(..., V)
fill the len(V)-1 regions between the values in V, which must be in increasing order.
contour(Z, **kwargs)
Use keyword args to control colors, linewidth, origin, cmap ... see below for more details.
X and Y must both be 2-D with the same shape as Z, or they must both be 1-D such that len(X) is the number of columns in Z and len(Y) is the number of rows in Z.
C = contour(...) returns a QuadContourSet object.
Optional keyword arguments:
corner_mask: [ True | False | ?legacy? ]
Enable/disable corner masking, which only has an effect if Z is a masked array. If False, any quad touching a masked point is masked out. If True, only the triangular corners of quads nearest
those points are always masked out, other triangular corners comprising three unmasked points are contoured as usual. If ?legacy?, the old contouring algorithm is used, which is equivalent to
False and is deprecated, only remaining whilst the new algorithm is tested fully.
If not specified, the default is taken from rcParams[?contour.corner_mask?], which is True unless it has been modified.
colors: [ None | string | (mpl_colors) ]
If None, the colormap specified by cmap will be used.
If a string, like ?r? or ?red?, all levels will be plotted in this color.
If a tuple of matplotlib color args (string, float, rgb, etc), different levels will be plotted in different colors in the order specified.
alpha: float
The alpha blending value
cmap: [ None | Colormap ]
A cm Colormap instance or None. If cmap is None and colors is None, a default Colormap is used.
norm: [ None | Normalize ]
A matplotlib.colors.Normalize instance for scaling data values to colors. If norm is None and colors is None, the default linear scaling is used.
vmin, vmax: [ None | scalar ]
If not None, either or both of these values will be supplied to the matplotlib.colors.Normalize instance, overriding the default color scaling based on levels.
levels: [level0, level1, ..., leveln]
A list of floating point numbers indicating the level curves to draw, in increasing order; e.g., to draw just the zero contour pass levels=[0]
origin: [ None | ?upper? | ?lower? | ?image? ]
If None, the first value of Z will correspond to the lower left corner, location (0,0). If ?image?, the rc value for image.origin will be used.
This keyword is not active if X and Y are specified in the call to contour.
extent: [ None | (x0,x1,y0,y1) ]
If origin is not None, then extent is interpreted as in matplotlib.pyplot.imshow(): it gives the outer pixel boundaries. In this case, the position of Z[0,0] is the center of the pixel, not a
corner. If origin is None, then (x0, y0) is the position of Z[0,0], and (x1, y1) is the position of Z[-1,-1].
This keyword is not active if X and Y are specified in the call to contour.
locator: [ None | ticker.Locator subclass ]
If locator is None, the default MaxNLocator is used. The locator is used to determine the contour levels if they are not given explicitly via the V argument.
extend: [ ?neither? | ?both? | ?min? | ?max? ]
Unless this is ?neither?, contour levels are automatically added to one or both ends of the range so that all data are included. These added ranges are then mapped to the special colormap
values which default to the ends of the colormap range, but can be set via matplotlib.colors.Colormap.set_under() and matplotlib.colors.Colormap.set_over() methods.
xunits, yunits: [ None | registered units ]
Override axis units by specifying an instance of a matplotlib.units.ConversionInterface.
antialiased: [ True | False ]
enable antialiasing, overriding the defaults. For filled contours, the default is True. For line contours, it is taken from rcParams[?lines.antialiased?].
nchunk: [ 0 | integer ]
If 0, no subdivision of the domain. Specify a positive integer to divide the domain into subdomains of nchunk by nchunk quads. Chunking reduces the maximum length of polygons generated by the
contouring algorithm which reduces the rendering workload passed on to the backend and also requires slightly less RAM. It can however introduce rendering artifacts at chunk boundaries
depending on the backend, the antialiased flag and value of alpha.
contour-only keyword arguments:
linewidths: [ None | number | tuple of numbers ]
If linewidths is None, the default width in lines.linewidth in matplotlibrc is used.
If a number, all levels will be plotted with this linewidth.
If a tuple, different levels will be plotted with different linewidths in the order specified.
linestyles: [ None | ?solid? | ?dashed? | ?dashdot? | ?dotted? ]
If linestyles is None, the default is ?solid? unless the lines are monochrome. In that case, negative contours will take their linestyle from the matplotlibrc contour.negative_linestyle
linestyles can also be an iterable of the above strings specifying a set of linestyles to be used. If this iterable is shorter than the number of contour levels it will be repeated as
contourf-only keyword arguments:
A list of cross hatch patterns to use on the filled areas. If None, no hatching will be added to the contour. Hatching is supported in the PostScript, PDF, SVG and Agg backends only.
Note: contourf fills intervals that are closed at the top; that is, for boundaries z1 and z2, the filled region is:
z1 < z <= z2
There is one exception: if the lowest boundary coincides with the minimum value of the z array, then that minimum value will be included in the lowest interval.
(Source code, png, hires.png, pdf)
Additional kwargs: hold = [True|False] overrides default hold state
set the default colormap to cool and apply to current image if any. See help(colormaps) for more information
set the default colormap to copper and apply to current image if any. See help(colormaps) for more information
matplotlib.pyplot.csd(x, y, NFFT=None, Fs=None, Fc=None, detrend=None, window=None, noverlap=None, pad_to=None, sides=None, scale_by_freq=None, return_line=None, hold=None, data=None, **kwargs)
Plot the cross-spectral density.
Call signature:
csd(x, y, NFFT=256, Fs=2, Fc=0, detrend=mlab.detrend_none,
window=mlab.window_hanning, noverlap=0, pad_to=None,
sides='default', scale_by_freq=None, return_line=None, **kwargs)
The cross spectral density x and y are divided into NFFT length segments. Each segment is detrended by function detrend and windowed by function window. noverlap gives the length of the overlap
between segments. The product of the direct FFTs of x and y are averaged over each segment to compute
If len(x) < NFFT or len(y) < NFFT, they will be zero padded to NFFT.
x, y: 1-D arrays or sequences
Arrays or sequences containing the data
Keyword arguments:
Fs: scalar
The sampling frequency (samples per time unit). It is used to calculate the Fourier frequencies, freqs, in cycles per time unit. The default value is 2.
window: callable or ndarray
A function or a vector of length NFFT. To create window vectors see window_hanning(), window_none(), numpy.blackman(), numpy.hamming(), numpy.bartlett(), scipy.signal(),
scipy.signal.get_window(), etc. The default is window_hanning(). If a function is passed as the argument, it must take a data segment as an argument and return the windowed version of the
sides: [ ?default? | ?onesided? | ?twosided? ]
Specifies which sides of the spectrum to return. Default gives the default behavior, which returns one-sided for real data and both for complex data. ?onesided? forces the return of a
one-sided spectrum, while ?twosided? forces two-sided.
pad_to: integer
The number of points to which the data segment is padded when performing the FFT. This can be different from NFFT, which specifies the number of data points used. While not increasing the
actual resolution of the spectrum (the minimum distance between resolvable peaks), this can give more points in the plot, allowing for more detail. This corresponds to the n parameter in the
call to fft(). The default is None, which sets pad_to equal to NFFT
NFFT: integer
The number of data points used in each block for the FFT. A power 2 is most efficient. The default value is 256. This should NOT be used to get zero padding, or the scaling of the result will
be incorrect. Use pad_to for this instead.
detrend: [ ?default? | ?constant? | ?mean? | ?linear? | ?none?] or
The function applied to each segment before fft-ing, designed to remove the mean or linear trend. Unlike in MATLAB, where the detrend parameter is a vector, in matplotlib is it a function.
The pylab module defines detrend_none(), detrend_mean(), and detrend_linear(), but you can use a custom function as well. You can also use a string to choose one of the functions. ?default?,
?constant?, and ?mean? call detrend_mean(). ?linear? calls detrend_linear(). ?none? calls detrend_none().
scale_by_freq: boolean
Specifies whether the resulting density values should be scaled by the scaling frequency, which gives density in units of Hz^-1. This allows for integration over the returned frequency
values. The default is True for MATLAB compatibility.
noverlap: integer
The number of points of overlap between segments. The default value is 0 (no overlap).
Fc: integer
The center frequency of x (defaults to 0), which offsets the x extents of the plot to reflect the frequency range used when a signal is acquired and then filtered and downsampled to
return_line: bool
Whether to include the line object plotted in the returned values. Default is False.
If return_line is False, returns the tuple (Pxy, freqs). If return_line is True, returns the tuple (Pxy, freqs. line):
Pxy: 1-D array
The values for the cross spectrum P_{xy} before scaling (complex valued)
freqs: 1-D array
The frequencies corresponding to the elements in Pxy
line: a Line2D instance
The line created by this function. Only returend if return_line is True.
For plotting, the power is plotted as P_{xy} itself is returned.
Bendat & Piersol ? Random Data: Analysis and Measurement Procedures, John Wiley & Sons (1986)
kwargs control the Line2D properties:
Property Description
agg_filter unknown
alpha float (0.0 transparent through 1.0 opaque)
animated [True | False]
antialiased or aa [True | False]
axes an Axes instance
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
color or c any matplotlib color
contains a callable function
dash_capstyle [?butt? | ?round? | ?projecting?]
dash_joinstyle [?miter? | ?round? | ?bevel?]
dashes sequence of on/off ink in points
drawstyle [?default? | ?steps? | ?steps-pre? | ?steps-mid? | ?steps-post?]
figure a matplotlib.figure.Figure instance
fillstyle [?full? | ?left? | ?right? | ?bottom? | ?top? | ?none?]
gid an id string
label string or anything printable with ?%s? conversion.
linestyle or ls [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or lw float value in points
marker A valid marker style
markeredgecolor or mec any matplotlib color
markeredgewidth or mew float value in points
markerfacecolor or mfc any matplotlib color
markerfacecoloralt or mfcalt any matplotlib color
markersize or ms float
markevery [None | int | length-2 tuple of int | slice | list/array of int | float | length-2 tuple of float]
path_effects unknown
picker float distance in points or callable pick function fn(artist, event)
pickradius float distance in points
rasterized [True | False | None]
sketch_params unknown
snap unknown
solid_capstyle [?butt? | ?round? | ?projecting?]
solid_joinstyle [?miter? | ?round? | ?bevel?]
transform a matplotlib.transforms.Transform instance
url a url string
visible [True | False]
xdata 1D array
ydata 1D array
zorder any number
(Source code, png, hires.png, pdf)
See also
psd() is the equivalent to setting y=x.
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?y?, ?x?.
Additional kwargs: hold = [True|False] overrides default hold state
Remove an axes from the current figure. If ax doesn?t exist, an error will be raised.
delaxes(): delete the current axes
Disconnect callback id cid
Example usage:
cid = canvas.mpl_connect('button_press_event', on_press)
Redraw the current figure.
This is used in interactive mode to update a figure that has been altered, but not automatically re-drawn. This should be only rarely needed, but there may be ways to modify the state of a figure
with out marking it as stale. Please report these cases as bugs.
A more object-oriented alternative, given any Figure instance, fig, that was created using a pyplot function, is:
matplotlib.pyplot.errorbar(x, y, yerr=None, xerr=None, fmt='', ecolor=None, elinewidth=None, capsize=None, barsabove=False, lolims=False, uplims=False, xlolims=False, xuplims=False, errorevery=1,
capthick=None, hold=None, data=None, **kwargs)
Plot an errorbar graph.
Call signature:
errorbar(x, y, yerr=None, xerr=None,
fmt='', ecolor=None, elinewidth=None, capsize=None,
barsabove=False, lolims=False, uplims=False,
xlolims=False, xuplims=False, errorevery=1,
Plot x versus y with error deltas in yerr and xerr. Vertical errorbars are plotted if yerr is not None. Horizontal errorbars are plotted if xerr is not None.
x, y, xerr, and yerr can all be scalars, which plots a single error bar at x, y.
Optional keyword arguments:
xerr/yerr: [ scalar | N, Nx1, or 2xN array-like ]
If a scalar number, len(N) array-like object, or an Nx1 array-like object, errorbars are drawn at +/-value relative to the data.
If a sequence of shape 2xN, errorbars are drawn at -row1 and +row2 relative to the data.
fmt: [ ?? | ?none? | plot format string ]
The plot format symbol. If fmt is ?none? (case-insensitive), only the errorbars are plotted. This is used for adding errorbars to a bar plot, for example. Default is ??, an empty plot format
string; properties are then identical to the defaults for plot().
ecolor: [ None | mpl color ]
A matplotlib color arg which gives the color the errorbar lines; if None, use the color of the line connecting the markers.
elinewidth: scalar
The linewidth of the errorbar lines. If None, use the linewidth.
capsize: scalar
The length of the error bar caps in points; if None, it will take the value from errorbar.capsize rcParam.
capthick: scalar
An alias kwarg to markeredgewidth (a.k.a. - mew). This setting is a more sensible name for the property that controls the thickness of the error bar cap in points. For backwards
compatibility, if mew or markeredgewidth are given, then they will over-ride capthick. This may change in future releases.
barsabove: [ True | False ]
if True, will plot the errorbars above the plot symbols. Default is below.
lolims / uplims / xlolims / xuplims: [ False | True ]
These arguments can be used to indicate that a value gives only upper/lower limits. In that case a caret symbol is used to indicate this. lims-arguments may be of the same type as xerr and
yerr. To use limits with inverted axes, set_xlim() or set_ylim() must be called before errorbar().
errorevery: positive integer
subsamples the errorbars. e.g., if errorevery=5, errorbars for every 5-th datapoint will be plotted. The data plot itself still shows all data points.
All other keyword arguments are passed on to the plot command for the markers. For example, this code makes big red squares with thick green edges:
x,y,yerr = rand(3,10)
errorbar(x, y, yerr, marker='s',
mfc='red', mec='green', ms=20, mew=4)
where mfc, mec, ms and mew are aliases for the longer property names, markerfacecolor, markeredgecolor, markersize and markeredgewith.
valid kwargs for the marker properties are
Property Description
agg_filter unknown
alpha float (0.0 transparent through 1.0 opaque)
animated [True | False]
antialiased or aa [True | False]
axes an Axes instance
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
color or c any matplotlib color
contains a callable function
dash_capstyle [?butt? | ?round? | ?projecting?]
dash_joinstyle [?miter? | ?round? | ?bevel?]
dashes sequence of on/off ink in points
drawstyle [?default? | ?steps? | ?steps-pre? | ?steps-mid? | ?steps-post?]
figure a matplotlib.figure.Figure instance
fillstyle [?full? | ?left? | ?right? | ?bottom? | ?top? | ?none?]
gid an id string
label string or anything printable with ?%s? conversion.
linestyle or ls [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or lw float value in points
marker A valid marker style
markeredgecolor or mec any matplotlib color
markeredgewidth or mew float value in points
markerfacecolor or mfc any matplotlib color
markerfacecoloralt or mfcalt any matplotlib color
markersize or ms float
markevery [None | int | length-2 tuple of int | slice | list/array of int | float | length-2 tuple of float]
path_effects unknown
picker float distance in points or callable pick function fn(artist, event)
pickradius float distance in points
rasterized [True | False | None]
sketch_params unknown
snap unknown
solid_capstyle [?butt? | ?round? | ?projecting?]
solid_joinstyle [?miter? | ?round? | ?bevel?]
transform a matplotlib.transforms.Transform instance
url a url string
visible [True | False]
xdata 1D array
ydata 1D array
zorder any number
Returns (plotline, caplines, barlinecols):
plotline: Line2D instance
x, y plot markers and/or line
caplines: list of error bar cap
Line2D instances
barlinecols: list of
LineCollection instances for the horizontal and vertical error ranges.
(Source code, png, hires.png, pdf)
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?xerr?, ?y?, ?yerr?, ?x?.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.eventplot(positions, orientation='horizontal', lineoffsets=1, linelengths=1, linewidths=None, colors=None, linestyles='solid', hold=None, data=None, **kwargs)
Plot identical parallel lines at specific positions.
Call signature:
eventplot(positions, orientation='horizontal', lineoffsets=0,
linelengths=1, linewidths=None, color =None,
Plot parallel lines at the given positions. positions should be a 1D or 2D array-like object, with each row corresponding to a row or column of lines.
This type of plot is commonly used in neuroscience for representing neural events, where it is commonly called a spike raster, dot raster, or raster plot.
However, it is useful in any situation where you wish to show the timing or position of multiple sets of discrete events, such as the arrival times of people to a business on each day of the
month or the date of hurricanes each year of the last century.
: [ ?horizonal? | ?vertical? ]
?horizonal? : the lines will be vertical and arranged in rows ?vertical? : lines will be horizontal and arranged in columns
lineoffsets :
A float or array-like containing floats.
linelengths :
A float or array-like containing floats.
linewidths :
A float or array-like containing floats.
must be a sequence of RGBA tuples (e.g., arbitrary color strings, etc, not allowed) or a list of such sequences
linestyles :
[ ?solid? | ?dashed? | ?dashdot? | ?dotted? ] or an array of these values
For linelengths, linewidths, colors, and linestyles, if only a single value is given, that value is applied to all lines. If an array-like is given, it must have the same length as positions, and
each value will be applied to the corresponding row or column in positions.
Returns a list of matplotlib.collections.EventCollection objects that were added.
kwargs are LineCollection properties:
Property Description
agg_filter unknown
alpha float or None
animated [True | False]
antialiased or antialiaseds Boolean or sequence of booleans
array unknown
axes an Axes instance
clim a length 2 sequence of floats
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
cmap a colormap or registered colormap name
color matplotlib color arg or sequence of rgba tuples
contains a callable function
edgecolor or edgecolors matplotlib color spec or sequence of specs
facecolor or facecolors matplotlib color spec or sequence of specs
figure a matplotlib.figure.Figure instance
gid an id string
hatch [ ?/? | ?\? | ?|? | ?-? | ?+? | ?x? | ?o? | ?O? | ?.? | ?*? ]
label string or anything printable with ?%s? conversion.
linestyle or linestyles or dashes [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or linewidths or lw float or sequence of floats
norm unknown
offset_position unknown
offsets float or sequence of floats
path_effects unknown
paths unknown
picker [None|float|boolean|callable]
pickradius unknown
rasterized [True | False | None]
segments unknown
sketch_params unknown
snap unknown
transform Transform instance
url a url string
urls unknown
verts unknown
visible [True | False]
zorder any number
(Source code, png, hires.png, pdf)
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
• All arguments with the following names: ?linewidths?, ?linelengths?, ?lineoffsets?, ?colors?, ?positions?, ?linestyles?.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.figimage(*args, **kwargs)
Adds a non-resampled image to the figure.
call signatures:
figimage(X, **kwargs)
adds a non-resampled array X to the figure.
figimage(X, xo, yo)
with pixel offsets xo, yo,
X must be a float array:
□ If X is MxN, assume luminance (grayscale)
□ If X is MxNx3, assume RGB
□ If X is MxNx4, assume RGBA
Optional keyword arguments:
Keyword Description
resize a boolean, True or False. If ?True?, then re-size the Figure to match the given image size.
xo or yo An integer, the x and y image offset in pixels
cmap a matplotlib.colors.Colormap instance, e.g., cm.jet. If None, default to the rc image.cmap value
norm a matplotlib.colors.Normalize instance. The default is normalization(). This scales luminance -> 0-1
vmin| are used to scale a luminance image to 0-1. If either is None, the min and max of the luminance values will be used. Note if you pass a norm instance, the settings for vmin and vmax will
vmax be ignored.
alpha the alpha blending value, default is None
origin [ ?upper? | ?lower? ] Indicates where the [0,0] index of the array is in the upper left or lower left corner of the axes. Defaults to the rc image.origin value
figimage complements the axes image (imshow()) which will be resampled to fit the current axes. If you want a resampled image to fill the entire figure, you can define an Axes with size
An matplotlib.image.FigureImage instance is returned.
(Source code, png, hires.png, pdf)
Additional kwargs are Artist kwargs passed on to FigureImage Addition kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.figlegend(handles, labels, loc, **kwargs)
Place a legend in the figure.
a sequence of strings
a sequence of Line2D or Patch instances
can be a string or an integer specifying the legend location
A matplotlib.legend.Legend instance is returned.
figlegend( (line1, line2, line3),
('label1', 'label2', 'label3'),
'upper right' )
matplotlib.pyplot.figtext(*args, **kwargs)
Add text to figure.
Call signature:
text(x, y, s, fontdict=None, **kwargs)
Add text to figure at location x, y (relative 0-1 coords). See text() for the meaning of the other arguments.
kwargs control the Text properties:
Property Description
agg_filter unknown
alpha float (0.0 transparent through 1.0 opaque)
animated [True | False]
axes an Axes instance
backgroundcolor any matplotlib color
bbox FancyBboxPatch prop dict
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
color any matplotlib color
contains a callable function
family or fontname or fontfamily [FONTNAME | ?serif? | ?sans-serif? | ?cursive? | ?fantasy? | ?monospace? ]
or name
figure a matplotlib.figure.Figure instance
fontproperties or font_properties a matplotlib.font_manager.FontProperties instance
gid an id string
horizontalalignment or ha [ ?center? | ?right? | ?left? ]
label string or anything printable with ?%s? conversion.
linespacing float (multiple of font size)
multialignment [?left? | ?right? | ?center? ]
path_effects unknown
picker [None|float|boolean|callable]
position (x,y)
rasterized [True | False | None]
rotation [ angle in degrees | ?vertical? | ?horizontal? ]
rotation_mode unknown
size or fontsize [size in points | ?xx-small? | ?x-small? | ?small? | ?medium? | ?large? | ?x-large? | ?xx-large? ]
sketch_params unknown
snap unknown
stretch or fontstretch [a numeric value in range 0-1000 | ?ultra-condensed? | ?extra-condensed? | ?condensed? | ?semi-condensed? | ?normal? | ?semi-expanded? | ?expanded? | ?
extra-expanded? | ?ultra-expanded? ]
style or fontstyle [ ?normal? | ?italic? | ?oblique?]
text string or anything printable with ?%s? conversion.
transform Transform instance
url a url string
usetex unknown
variant or fontvariant [ ?normal? | ?small-caps? ]
verticalalignment or ma or va [ ?center? | ?top? | ?bottom? | ?baseline? ]
visible [True | False]
weight or fontweight [a numeric value in range 0-1000 | ?ultralight? | ?light? | ?normal? | ?regular? | ?book? | ?medium? | ?roman? | ?semibold? | ?demibold? | ?demi? | ?bold? | ?
heavy? | ?extra bold? | ?black? ]
wrap unknown
x float
y float
zorder any number
matplotlib.pyplot.figure(num=None, figsize=None, dpi=None, facecolor=None, edgecolor=None, frameon=True, FigureClass=, **kwargs)
Creates a new figure.
num : integer or string, optional, default: none
If not provided, a new figure will be created, and the figure number will be incremented. The figure objects holds this number in a number attribute. If num is provided, and a figure
with this id already exists, make it active, and returns a reference to it. If this figure does not exists, create it and returns it. If num is a string, the window title will be set
to this figure?s num.
figsize : tuple of integers, optional, default: None
width, height in inches. If not provided, defaults to rc figure.figsize.
Parameters: dpi : integer, optional, default: None
resolution of the figure. If not provided, defaults to rc figure.dpi.
facecolor :
the background color. If not provided, defaults to rc figure.facecolor
edgecolor :
the border color. If not provided, defaults to rc figure.edgecolor
figure : Figure
The Figure instance returned will also be passed to new_figure_manager in the backends, which allows to hook custom Figure classes into the pylab interface. Additional kwargs will be
passed to the figure init function.
If you are creating many figures, make sure you explicitly call ?close? on the figures you are not using, because this will enable pylab to properly clean up the memory.
rcParams defines the default values, which can be modified in the matplotlibrc file
matplotlib.pyplot.fill(*args, **kwargs)
Plot filled polygons.
Call signature:
fill(*args, **kwargs)
args is a variable length argument, allowing for multiple x, y pairs with an optional color format string; see plot() for details on the argument parsing. For example, to plot a polygon with
vertices at x, y in blue.:
ax.fill(x,y, 'b' )
An arbitrary number of x, y, color groups can be specified:
ax.fill(x1, y1, 'g', x2, y2, 'r')
Return value is a list of Patch instances that were added.
The same color strings that plot() supports are supported by the fill format string.
If you would like to fill below a curve, e.g., shade a region between 0 and y along x, use fill_between()
The closed kwarg will close the polygon when True (default).
kwargs control the Polygon properties:
Property Description
agg_filter unknown
alpha float or None
animated [True | False]
antialiased or aa [True | False] or None for default
axes an Axes instance
capstyle [?butt? | ?round? | ?projecting?]
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
color matplotlib color spec
contains a callable function
edgecolor or ec mpl color spec, or None for default, or ?none? for no color
facecolor or fc mpl color spec, or None for default, or ?none? for no color
figure a matplotlib.figure.Figure instance
fill [True | False]
gid an id string
hatch [?/? | ?\? | ?|? | ?-? | ?+? | ?x? | ?o? | ?O? | ?.? | ?*?]
joinstyle [?miter? | ?round? | ?bevel?]
label string or anything printable with ?%s? conversion.
linestyle or ls [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or lw float or None for default
path_effects unknown
picker [None|float|boolean|callable]
rasterized [True | False | None]
sketch_params unknown
snap unknown
transform Transform instance
url a url string
visible [True | False]
zorder any number
(Source code, png, hires.png, pdf)
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?y?, ?x?.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.fill_between(x, y1, y2=0, where=None, interpolate=False, step=None, hold=None, data=None, **kwargs)
Make filled polygons between two curves.
Create a PolyCollection filling the regions between y1 and y2 where where==True
x : array
An N-length array of the x data
y1 : array
An N-length array (or scalar) of the y data
y2 : array
An N-length array (or scalar) of the y data
where : array, optional
If None, default to fill between everywhere. If not None, it is an N-length numpy boolean array and the fill will only happen over the regions where where==True.
interpolate : bool, optional
If True, interpolate between the two lines to find the precise point of intersection. Otherwise, the start and end points of the filled region will only occur on explicit values in
the x array.
step : {?pre?, ?post?, ?mid?}, optional
If not None, fill with step logic.
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?y1?, ?x?, ?where?, ?y2?.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.fill_betweenx(y, x1, x2=0, where=None, step=None, hold=None, data=None, **kwargs)
Make filled polygons between two horizontal curves.
Call signature:
fill_betweenx(y, x1, x2=0, where=None, **kwargs)
Create a PolyCollection filling the regions between x1 and x2 where where==True
y : array
An N-length array of the y data
x1 : array
An N-length array (or scalar) of the x data
x2 : array, optional
An N-length array (or scalar) of the x data
where : array, optional
If None, default to fill between everywhere. If not None, it is a N length numpy boolean array and the fill will only happen over the regions where where==True
step : {?pre?, ?post?, ?mid?}, optional
If not None, fill with step logic.
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?x2?, ?y?, ?where?, ?x1?.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.findobj(o=None, match=None, include_self=True)
Find artist objects.
Recursively find all Artist instances contained in self.
match can be
□ None: return all objects contained in artist.
□ function with signature boolean = match(artist) used to filter matches
□ class instance: e.g., Line2D. Only return artists of class type.
If include_self is True (default), include self in the list to be checked for a match.
set the default colormap to flag and apply to current image if any. See help(colormaps) for more information
Get the current Axes instance on the current figure matching the given keyword args, or create one.
See also
The figure?s gca method.
To get the current polar axes on the current figure:
If the current axes doesn?t exist, or isn?t a polar one, the appropriate axes will be created and then returned.
Get a reference to the current figure.
Get the current colorable artist. Specifically, returns the current ScalarMappable instance (image or patch collection), or None if no images or patch collections have been defined. The commands
imshow() and figimage() create Image instances, and the commands pcolor() and scatter() create Collection instances. The current image is an attribute of the current axes, or the nearest earlier
axes in the current figure that contains an image.
Return a list of existing figure labels.
Return a list of existing figure numbers.
Get a sorted list of all of the plotting commands.
matplotlib.pyplot.ginput(*args, **kwargs)
Call signature:
ginput(self, n=1, timeout=30, show_clicks=True,
mouse_add=1, mouse_pop=3, mouse_stop=2)
Blocking call to interact with the figure.
This will wait for n clicks from the user and return a list of the coordinates of each click.
If timeout is zero or negative, does not timeout.
If n is zero or negative, accumulate clicks until a middle click (or potentially both mouse buttons at once) terminates the input.
Right clicking cancels last input.
The buttons used for the various actions (adding points, removing points, terminating the inputs) can be overriden via the arguments mouse_add, mouse_pop and mouse_stop, that give the associated
mouse button: 1 for left, 2 for middle, 3 for right.
The keyboard can also be used to select points in case your mouse does not have one or more of the buttons. The delete and backspace keys act like right clicking (i.e., remove last point), the
enter key terminates input and any other key (not already used by the window manager) selects a point.
set the default colormap to gray and apply to current image if any. See help(colormaps) for more information
matplotlib.pyplot.grid(b=None, which='major', axis='both', **kwargs)
Turn the axes grids on or off.
Call signature:
grid(self, b=None, which='major', axis='both', **kwargs)
Set the axes grids on or off; b is a boolean. (For MATLAB compatibility, b may also be a string, ?on? or ?off?.)
If b is None and len(kwargs)==0, toggle the grid state. If kwargs are supplied, it is assumed that you want a grid and b is thus set to True.
which can be ?major? (default), ?minor?, or ?both? to control whether major tick grids, minor tick grids, or both are affected.
axis can be ?both? (default), ?x?, or ?y? to control which set of gridlines are drawn.
kwargs are used to set the grid line properties, e.g.,:
ax.grid(color='r', linestyle='-', linewidth=2)
Valid Line2D kwargs are
Property Description
agg_filter unknown
alpha float (0.0 transparent through 1.0 opaque)
animated [True | False]
antialiased or aa [True | False]
axes an Axes instance
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
color or c any matplotlib color
contains a callable function
dash_capstyle [?butt? | ?round? | ?projecting?]
dash_joinstyle [?miter? | ?round? | ?bevel?]
dashes sequence of on/off ink in points
drawstyle [?default? | ?steps? | ?steps-pre? | ?steps-mid? | ?steps-post?]
figure a matplotlib.figure.Figure instance
fillstyle [?full? | ?left? | ?right? | ?bottom? | ?top? | ?none?]
gid an id string
label string or anything printable with ?%s? conversion.
linestyle or ls [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or lw float value in points
marker A valid marker style
markeredgecolor or mec any matplotlib color
markeredgewidth or mew float value in points
markerfacecolor or mfc any matplotlib color
markerfacecoloralt or mfcalt any matplotlib color
markersize or ms float
markevery [None | int | length-2 tuple of int | slice | list/array of int | float | length-2 tuple of float]
path_effects unknown
picker float distance in points or callable pick function fn(artist, event)
pickradius float distance in points
rasterized [True | False | None]
sketch_params unknown
snap unknown
solid_capstyle [?butt? | ?round? | ?projecting?]
solid_joinstyle [?miter? | ?round? | ?bevel?]
transform a matplotlib.transforms.Transform instance
url a url string
visible [True | False]
xdata 1D array
ydata 1D array
zorder any number
matplotlib.pyplot.hexbin(x, y, C=None, gridsize=100, bins=None, xscale='linear', yscale='linear', extent=None, cmap=None, norm=None, vmin=None, vmax=None, alpha=None, linewidths=None, edgecolors=
'none', reduce_C_function=, mincnt=None, marginals=False, hold=None, data=None, **kwargs)
Make a hexagonal binning plot.
Call signature:
hexbin(x, y, C = None, gridsize = 100, bins = None,
xscale = 'linear', yscale = 'linear',
cmap=None, norm=None, vmin=None, vmax=None,
alpha=None, linewidths=None, edgecolors='none'
reduce_C_function = np.mean, mincnt=None, marginals=True
Make a hexagonal binning plot of x versus y, where x, y are 1-D sequences of the same length, N. If C is None (the default), this is a histogram of the number of occurences of the observations at
If C is specified, it specifies values at the coordinate (x[i],y[i]). These values are accumulated for each hexagonal bin and then reduced according to reduce_C_function, which defaults to numpy?
s mean function (np.mean). (If C is specified, it must also be a 1-D sequence of the same length as x and y.)
x, y and/or C may be masked arrays, in which case only unmasked points will be plotted.
Optional keyword arguments:
gridsize: [ 100 | integer ]
The number of hexagons in the x-direction, default is 100. The corresponding number of hexagons in the y-direction is chosen such that the hexagons are approximately regular. Alternatively,
gridsize can be a tuple with two elements specifying the number of hexagons in the x-direction and the y-direction.
bins: [ None | ?log? | integer | sequence ]
If None, no binning is applied; the color of each hexagon directly corresponds to its count value.
If ?log?, use a logarithmic scale for the color map. Internally,
If an integer, divide the counts in the specified number of bins, and color the hexagons accordingly.
If a sequence of values, the values of the lower bound of the bins to be used.
xscale: [ ?linear? | ?log? ]
Use a linear or log10 scale on the horizontal axis.
scale: [ ?linear? | ?log? ]
Use a linear or log10 scale on the vertical axis.
mincnt: [ None | a positive integer ]
If not None, only display cells with more than mincnt number of points in the cell
marginals: [ True | False ]
if marginals is True, plot the marginal density as colormapped rectagles along the bottom of the x-axis and left of the y-axis
extent: [ None | scalars (left, right, bottom, top) ]
The limits of the bins. The default assigns the limits based on gridsize, x, y, xscale and yscale.
Other keyword arguments controlling color mapping and normalization arguments:
cmap: [ None | Colormap ]
a matplotlib.colors.Colormap instance. If None, defaults to rc image.cmap.
norm: [ None | Normalize ]
matplotlib.colors.Normalize instance is used to scale luminance data to 0,1.
vmin / vmax: scalar
vmin and vmax are used in conjunction with norm to normalize luminance data. If either are None, the min and max of the color array C is used. Note if you pass a norm instance, your settings
for vmin and vmax will be ignored.
alpha: scalar between 0 and 1, or None
the alpha value for the patches
linewidths: [ None | scalar ]
If None, defaults to rc lines.linewidth. Note that this is a tuple, and if you set the linewidths argument you must set it as a sequence of floats, as required by RegularPolyCollection.
Other keyword arguments controlling the Collection properties:
edgecolors: [ None | 'none' | mpl color | color sequence ]
If 'none', draws the edges in the same color as the fill color. This is the default, as it avoids unsightly unpainted pixels between the hexagons.
If None, draws the outlines in the default color.
If a matplotlib color arg or sequence of rgba tuples, draws the outlines in the specified color.
Here are the standard descriptions of all the Collection kwargs:
Property Description
agg_filter unknown
alpha float or None
animated [True | False]
antialiased or antialiaseds Boolean or sequence of booleans
array unknown
axes an Axes instance
clim a length 2 sequence of floats
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
cmap a colormap or registered colormap name
color matplotlib color arg or sequence of rgba tuples
contains a callable function
edgecolor or edgecolors matplotlib color spec or sequence of specs
facecolor or facecolors matplotlib color spec or sequence of specs
figure a matplotlib.figure.Figure instance
gid an id string
hatch [ ?/? | ?\? | ?|? | ?-? | ?+? | ?x? | ?o? | ?O? | ?.? | ?*? ]
label string or anything printable with ?%s? conversion.
linestyle or linestyles or dashes [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or linewidths or lw float or sequence of floats
norm unknown
offset_position unknown
offsets float or sequence of floats
path_effects unknown
picker [None|float|boolean|callable]
pickradius unknown
rasterized [True | False | None]
sketch_params unknown
snap unknown
transform Transform instance
url a url string
urls unknown
visible [True | False]
zorder any number
The return value is a PolyCollection instance; use get_array() on this PolyCollection to get the counts in each hexagon. If marginals is True, horizontal bar and vertical bar (both
PolyCollections) will be attached to the return collection as attributes hbar and vbar.
(Source code, png, hires.png, pdf)
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?y?, ?x?.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.hist(x, bins=10, range=None, normed=False, weights=None, cumulative=False, bottom=None, histtype='bar', align='mid', orientation='vertical', rwidth=None, log=False, color=None,
label=None, stacked=False, hold=None, data=None, **kwargs)
Plot a histogram.
Compute and draw the histogram of x. The return value is a tuple (n, bins, patches) or ([n0, n1, ...], bins, [patches0, patches1,...]) if the input contains multiple data.
Multiple data can be provided via x as a list of datasets of potentially different length ([x0, x1, ...]), or as a 2-D ndarray in which each column is a dataset. Note that the ndarray form is
transposed relative to the list form.
Masked arrays are not supported at present.
x : (n,) array or sequence of (n,) arrays
Input values, this takes either a single array or a sequency of arrays which are not required to be of the same length
bins : integer or array_like, optional
If an integer is given, bins + 1 bin edges are returned, consistently with numpy.histogram() for numpy version >= 1.3.
Unequally spaced bins are supported if bins is a sequence.
default is 10
range : tuple or None, optional
The lower and upper range of the bins. Lower and upper outliers are ignored. If not provided, range is (x.min(), x.max()). Range has no effect if bins is a sequence.
If bins is a sequence or range is specified, autoscaling is based on the specified bin range instead of the range of x.
Default is None
normed : boolean, optional
If True, the first element of the return tuple will be the counts normalized to form a probability density, i.e., n/(len(x)`dbin), i.e., the integral of the histogram will sum to 1.
If stacked is also True, the sum of the histograms is normalized to 1.
Default is False
weights : (n, ) array_like or None, optional
An array of weights, of the same shape as x. Each value in x only contributes its associated weight towards the bin count (instead of 1). If normed is True, the weights are
normalized, so that the integral of the density over the range remains 1.
Default is None
cumulative : boolean, optional
If True, then a histogram is computed where each bin gives the counts in that bin plus all bins for smaller values. The last bin gives the total number of datapoints. If normed is
also True then the histogram is normalized such that the last bin equals 1. If cumulative evaluates to less than 0 (e.g., -1), the direction of accumulation is reversed. In this case,
if normed is also True, then the histogram is normalized such that the first bin equals 1.
Default is False
bottom : array_like, scalar, or None
Location of the bottom baseline of each bin. If a scalar, the base line for each bin is shifted by the same amount. If an array, each bin is shifted independently and the length of
bottom must match the number of bins. If None, defaults to 0.
Default is None
histtype : {?bar?, ?barstacked?, ?step?, ?stepfilled?}, optional
The type of histogram to draw.
Parameters: • ?bar? is a traditional bar-type histogram. If multiple data are given the bars are aranged side by side.
• ?barstacked? is a bar-type histogram where multiple data are stacked on top of each other.
• ?step? generates a lineplot that is by default unfilled.
• ?stepfilled? generates a lineplot that is by default filled.
Default is ?bar?
align : {?left?, ?mid?, ?right?}, optional
Controls how the histogram is plotted.
• ?left?: bars are centered on the left bin edges.
• ?mid?: bars are centered between the bin edges.
• ?right?: bars are centered on the right bin edges.
Default is ?mid?
orientation : {?horizontal?, ?vertical?}, optional
If ?horizontal?, barh will be used for bar-type histograms and the bottom kwarg will be the left edges.
rwidth : scalar or None, optional
The relative width of the bars as a fraction of the bin width. If None, automatically compute the width.
Ignored if histtype is ?step? or ?stepfilled?.
Default is None
log : boolean, optional
If True, the histogram axis will be set to a log scale. If log is True and x is a 1D array, empty bins will be filtered out and only the non-empty (n, bins, patches) will be returned.
Default is False
color : color or array_like of colors or None, optional
Color spec or sequence of color specs, one per dataset. Default (None) uses the standard line color sequence.
Default is None
label : string or None, optional
String, or sequence of strings to match multiple datasets. Bar charts yield multiple patches per dataset, but only the first gets the label, so that the legend command will work as
default is None
stacked : boolean, optional
If True, multiple data are stacked on top of each other If False multiple data are aranged side by side if histtype is ?bar? or on top of each other if histtype is ?step?
Default is False
n : array or list of arrays
The values of the histogram bins. See normed and weights for a description of the possible semantics. If input x is an array, then this is an array of length nbins. If input is a
sequence arrays [data1, data2,..], then this is a list of arrays with the values of the histograms for each of the arrays in the same order.
bins : array
The edges of the bins. Length nbins + 1 (nbins left edges and right edge of last bin). Always a single array even when multiple data sets are passed in.
patches : list or list of lists
Silent list of individual patches used to create the histogram or list of such list if multiple input datasets.
Other Parameters:
kwargs : Patch properties
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?weights?, ?x?.
Additional kwargs: hold = [True|False] overrides default hold state
(Source code, png, hires.png, pdf)
matplotlib.pyplot.hist2d(x, y, bins=10, range=None, normed=False, weights=None, cmin=None, cmax=None, hold=None, data=None, **kwargs)
Make a 2D histogram plot.
x, y: array_like, shape (n, )
Input values
bins: [None | int | [int, int] | array_like | [array, array]]
The bin specification:
• If int, the number of bins for the two dimensions (nx=ny=bins).
• If [int, int], the number of bins in each dimension (nx, ny = bins).
• If array_like, the bin edges for the two dimensions (x_edges=y_edges=bins).
• If [array, array], the bin edges in each dimension (x_edges, y_edges = bins).
The default value is 10.
range : array_like shape(2, 2), optional, default: None
The leftmost and rightmost edges of the bins along each dimension (if not specified explicitly in the bins parameters): [[xmin, xmax], [ymin, ymax]]. All values outside of this range
Parameters: will be considered outliers and not tallied in the histogram.
normed : boolean, optional, default: False
Normalize histogram.
weights : array_like, shape (n, ), optional, default: None
An array of values w_i weighing each sample (x_i, y_i).
cmin : scalar, optional, default: None
All bins that has count less than cmin will not be displayed and these count values in the return value count histogram will also be set to nan upon return
cmax : scalar, optional, default: None
All bins that has count more than cmax will not be displayed (set to none before passing to imshow) and these count values in the return value count histogram will also be set to nan
upon return
Returns: The return value is (counts, xedges, yedges, Image).
Other Parameters:
kwargs : pcolorfast() properties.
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?y?, ?weights?, ?x?.
Additional kwargs: hold = [True|False] overrides default hold state
(Source code, png, hires.png, pdf)
matplotlib.pyplot.hlines(y, xmin, xmax, colors='k', linestyles='solid', label='', hold=None, data=None, **kwargs)
Plot horizontal lines at each y from xmin to xmax.
y : scalar or sequence of scalar
y-indexes where to plot the lines.
xmin, xmax : scalar or 1D array_like
Parameters: Respective beginning and end of each line. If scalars are provided, all lines will have same length.
colors : array_like of colors, optional, default: ?k?
linestyles : [?solid? | ?dashed? | ?dashdot? | ?dotted?], optional
label : string, optional, default: ??
Returns: lines : LineCollection
Other Parameters:
kwargs : LineCollection properties.
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?xmax?, ?y?, ?xmin?.
Additional kwargs: hold = [True|False] overrides default hold state
(Source code, png, hires.png, pdf)
Set the hold state. If b is None (default), toggle the hold state, else set the hold state to boolean value b:
hold() # toggle hold
hold(True) # hold is on
hold(False) # hold is off
When hold is True, subsequent plot commands will be added to the current axes. When hold is False, the current axes and figure will be cleared on the next plot command.
set the default colormap to hot and apply to current image if any. See help(colormaps) for more information
set the default colormap to hsv and apply to current image if any. See help(colormaps) for more information
matplotlib.pyplot.imread(*args, **kwargs)
Read an image from a file into an array.
fname may be a string path, a valid URL, or a Python file-like object. If using a file object, it must be opened in binary mode.
If format is provided, will try to read file of that type, otherwise the format is deduced from the filename. If nothing can be deduced, PNG is tried.
Return value is a numpy.array. For grayscale images, the return array is MxN. For RGB images, the return value is MxNx3. For RGBA images the return value is MxNx4.
matplotlib can only read PNGs natively, but if PIL is installed, it will use it to load the image and return an array (if possible) which can be used with imshow(). Note, URL strings may not be
compatible with PIL. Check the PIL documentation for more information.
matplotlib.pyplot.imsave(*args, **kwargs)
Save an array as in image file.
The output formats available depend on the backend being used.
A string containing a path to a filename, or a Python file-like object. If format is None and fname is a string, the output format is deduced from the extension of the filename.
An MxN (luminance), MxNx3 (RGB) or MxNx4 (RGBA) array.
Keyword arguments:
vmin/vmax: [ None | scalar ]
vmin and vmax set the color scaling for the image by fixing the values that map to the colormap color limits. If either vmin or vmax is None, that limit is determined from the arr min/max
cmap is a colors.Colormap instance, e.g., cm.jet. If None, default to the rc image.cmap value.
One of the file extensions supported by the active backend. Most backends support png, pdf, ps, eps and svg.
[ ?upper? | ?lower? ] Indicates where the [0,0] index of the array is in the upper left or lower left corner of the axes. Defaults to the rc image.origin value.
The DPI to store in the metadata of the file. This does not affect the resolution of the output image.
matplotlib.pyplot.imshow(X, cmap=None, norm=None, aspect=None, interpolation=None, alpha=None, vmin=None, vmax=None, origin=None, extent=None, shape=None, filternorm=1, filterrad=4.0, imlim=None,
resample=None, url=None, hold=None, data=None, **kwargs)
Display an image on the axes.
X : array_like, shape (n, m) or (n, m, 3) or (n, m, 4)
Display the image in X to current axes. X may be a float array, a uint8 array or a PIL image. If X is an array, it can have the following shapes:
• MxN ? luminance (grayscale, float array only)
• MxNx3 ? RGB (float or uint8 array)
• MxNx4 ? RGBA (float or uint8 array)
The value for each component of MxNx3 and MxNx4 float arrays should be in the range 0.0 to 1.0; MxN float arrays may be normalised.
cmap : Colormap, optional, default: None
If None, default to rc image.cmap value. cmap is ignored when X has RGB(A) information
aspect : [?auto? | ?equal? | scalar], optional, default: None
If ?auto?, changes the image aspect ratio to match that of the axes.
If ?equal?, and extent is None, changes the axes aspect ratio to match that of the image. If extent is not None, the axes aspect ratio is changed to match that of the extent.
If None, default to rc image.aspect value.
interpolation : string, optional, default: None
Acceptable values are ?none?, ?nearest?, ?bilinear?, ?bicubic?, ?spline16?, ?spline36?, ?hanning?, ?hamming?, ?hermite?, ?kaiser?, ?quadric?, ?catrom?, ?gaussian?, ?bessel?, ?
mitchell?, ?sinc?, ?lanczos?
If interpolation is None, default to rc image.interpolation. See also the filternorm and filterrad parameters. If interpolation is ?none?, then no interpolation is performed on the
Agg, ps and pdf backends. Other backends will fall back to ?nearest?.
norm : Normalize, optional, default: None
A Normalize instance is used to scale luminance data to 0, 1. If None, use the default func:normalize. norm is only used if X is an array of floats.
vmin, vmax : scalar, optional, default: None
vmin and vmax are used in conjunction with norm to normalize luminance data. Note if you pass a norm instance, your settings for vmin and vmax will be ignored.
alpha : scalar, optional, default: None
The alpha blending value, between 0 (transparent) and 1 (opaque)
origin : [?upper? | ?lower?], optional, default: None
Place the [0,0] index of the array in the upper left or lower left corner of the axes. If None, default to rc image.origin.
extent : scalars (left, right, bottom, top), optional, default: None
The location, in data-coordinates, of the lower-left and upper-right corners. If None, the image is positioned such that the pixel centers fall on zero-based (row, column) indices.
shape : scalars (columns, rows), optional, default: None
For raw buffer images
filternorm : scalar, optional, default: 1
A parameter for the antigrain image resize filter. From the antigrain documentation, if filternorm = 1, the filter normalizes integer values and corrects the rounding errors. It
doesn?t do anything with the source floating point values, it corrects only integers according to the rule of 1.0 which means that any sum of pixel weights must be equal to 1.0. So,
the filter function must produce a graph of the proper shape.
filterrad : scalar, optional, default: 4.0
The filter radius for filters that have a radius parameter, i.e. when interpolation is one of: ?sinc?, ?lanczos? or ?blackman?
Returns: image : AxesImage
Other Parameters:
kwargs : Artist properties.
See also
Plot a matrix or an array as an image.
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All positional and all keyword arguments.
Additional kwargs: hold = [True|False] overrides default hold state
(Source code, png, hires.png, pdf)
set the default colormap to inferno and apply to current image if any. See help(colormaps) for more information
Install a repl display hook so that any stale figure are automatically redrawn when control is returned to the repl.
This works with IPython terminals and kernels, as well as vanilla python shells.
Turn interactive mode off.
Turn interactive mode on.
Return the hold status of the current axes.
Return status of interactive mode.
set the default colormap to jet and apply to current image if any. See help(colormaps) for more information
matplotlib.pyplot.legend(*args, **kwargs)
Places a legend on the axes.
To make a legend for lines which already exist on the axes (via plot for instance), simply call this function with an iterable of strings, one for each legend item. For example:
ax.plot([1, 2, 3])
ax.legend(['A simple line'])
However, in order to keep the ?label? and the legend element instance together, it is preferable to specify the label either at artist creation, or by calling the set_label() method on the
line, = ax.plot([1, 2, 3], label='Inline label')
# Overwrite the label by calling the method.
line.set_label('Label via method')
Specific lines can be excluded from the automatic legend element selection by defining a label starting with an underscore. This is default for all artists, so calling legend() without any
arguments and without setting the labels manually will result in no legend being drawn.
For full control of which artists have a legend entry, it is possible to pass an iterable of legend artists followed by an iterable of legend labels respectively:
legend((line1, line2, line3), ('label1', 'label2', 'label3'))
loc : int or string or pair of floats, default: ?upper right?
The location of the legend. Possible codes are:
Location String Location Code
?best? 0
?upper right? 1
?upper left? 2
?lower left? 3
?lower right? 4
?right? 5
?center left? 6
?center right? 7
?lower center? 8
?upper center? 9
?center? 10
Alternatively can be a 2-tuple giving x, y of the lower-left corner of the legend in axes coordinates (in which case bbox_to_anchor will be ignored).
bbox_to_anchor : matplotlib.transforms.BboxBase instance or tuple of floats
Specify any arbitrary location for the legend in bbox_transform coordinates (default Axes coordinates).
For example, to put the legend?s upper right hand corner in the center of the axes the following keywords can be used:
loc='upper right', bbox_to_anchor=(0.5, 0.5)
ncol : integer
The number of columns that the legend has. Default is 1.
prop : None or matplotlib.font_manager.FontProperties or dict
The font properties of the legend. If None (default), the current matplotlib.rcParams will be used.
fontsize : int or float or {?xx-small?, ?x-small?, ?small?, ?medium?, ?large?, ?x-large?, ?xx-large?}
Controls the font size of the legend. If the value is numeric the size will be the absolute font size in points. String values are relative to the current default font size. This
argument is only used if prop is not specified.
numpoints : None or int
The number of marker points in the legend when creating a legend entry for a line/matplotlib.lines.Line2D. Default is None which will take the value from the legend.numpoints rcParam.
scatterpoints : None or int
The number of marker points in the legend when creating a legend entry for a scatter plot/ matplotlib.collections.PathCollection. Default is None which will take the value from the
legend.scatterpoints rcParam.
scatteryoffsets : iterable of floats
The vertical offset (relative to the font size) for the markers created for a scatter plot legend entry. 0.0 is at the base the legend text, and 1.0 is at the top. To draw all markers
at the same height, set to [0.5]. Default [0.375, 0.5, 0.3125].
markerscale : None or int or float
The relative size of legend markers compared with the originally drawn ones. Default is None which will take the value from the legend.markerscale rcParam.
Parameters: *markerfirst*: [ *True* | *False* ]
if True, legend marker is placed to the left of the legend label if False, legend marker is placed to the right of the legend label
frameon : None or bool
Control whether a frame should be drawn around the legend. Default is None which will take the value from the legend.frameon rcParam.
fancybox : None or bool
Control whether round edges should be enabled around the FancyBboxPatch which makes up the legend?s background. Default is None which will take the value from the legend.fancybox
shadow : None or bool
Control whether to draw a shadow behind the legend. Default is None which will take the value from the legend.shadow rcParam.
framealpha : None or float
Control the alpha transparency of the legend?s frame. Default is None which will take the value from the legend.framealpha rcParam.
mode : {?expand?, None}
If mode is set to "expand" the legend will be horizontally expanded to fill the axes area (or bbox_to_anchor if defines the legend?s size).
bbox_transform : None or matplotlib.transforms.Transform
The transform for the bounding box (bbox_to_anchor). For a value of None (default) the Axes? transAxes transform will be used.
title : str or None
The legend?s title. Default is no title (None).
borderpad : float or None
The fractional whitespace inside the legend border. Measured in font-size units. Default is None which will take the value from the legend.borderpad rcParam.
labelspacing : float or None
The vertical space between the legend entries. Measured in font-size units. Default is None which will take the value from the legend.labelspacing rcParam.
handlelength : float or None
The length of the legend handles. Measured in font-size units. Default is None which will take the value from the legend.handlelength rcParam.
handletextpad : float or None
The pad between the legend handle and text. Measured in font-size units. Default is None which will take the value from the legend.handletextpad rcParam.
borderaxespad : float or None
The pad between the axes and legend border. Measured in font-size units. Default is None which will take the value from the legend.borderaxespad rcParam.
columnspacing : float or None
The spacing between columns. Measured in font-size units. Default is None which will take the value from the legend.columnspacing rcParam.
handler_map : dict or None
The custom dictionary mapping instances or types to a legend handler. This handler_map updates the default handler map found at matplotlib.legend.Legend.get_legend_handler_map().
Not all kinds of artist are supported by the legend command. See Legend guide for details.
(Source code, png, hires.png, pdf)
matplotlib.pyplot.locator_params(axis='both', tight=None, **kwargs)
Control behavior of tick locators.
Keyword arguments:
[?x? | ?y? | ?both?] Axis on which to operate; default is ?both?.
[True | False | None] Parameter passed to autoscale_view(). Default is None, for no change.
Remaining keyword arguments are passed to directly to the set_params() method.
Typically one might want to reduce the maximum number of ticks and use tight bounds when plotting small subplots, for example:
ax.locator_params(tight=True, nbins=4)
Because the locator is involved in autoscaling, autoscale_view() is called automatically after the parameters are changed.
This presently works only for the MaxNLocator used by default on linear axes, but it may be generalized.
matplotlib.pyplot.loglog(*args, **kwargs)
Make a plot with log scaling on both the x and y axis.
Call signature:
loglog(*args, **kwargs)
loglog() supports all the keyword arguments of plot() and matplotlib.axes.Axes.set_xscale() / matplotlib.axes.Axes.set_yscale().
Notable keyword arguments:
basex/basey: scalar > 1
Base of the x/y logarithm
subsx/subsy: [ None | sequence ]
The location of the minor x/y ticks; None defaults to autosubs, which depend on the number of decades in the plot; see matplotlib.axes.Axes.set_xscale() / matplotlib.axes.Axes.set_yscale()
for details
nonposx/nonposy: [?mask? | ?clip? ]
Non-positive values in x or y can be masked as invalid, or clipped to a very small positive number
The remaining valid kwargs are Line2D properties:
Property Description
agg_filter unknown
alpha float (0.0 transparent through 1.0 opaque)
animated [True | False]
antialiased or aa [True | False]
axes an Axes instance
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
color or c any matplotlib color
contains a callable function
dash_capstyle [?butt? | ?round? | ?projecting?]
dash_joinstyle [?miter? | ?round? | ?bevel?]
dashes sequence of on/off ink in points
drawstyle [?default? | ?steps? | ?steps-pre? | ?steps-mid? | ?steps-post?]
figure a matplotlib.figure.Figure instance
fillstyle [?full? | ?left? | ?right? | ?bottom? | ?top? | ?none?]
gid an id string
label string or anything printable with ?%s? conversion.
linestyle or ls [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or lw float value in points
marker A valid marker style
markeredgecolor or mec any matplotlib color
markeredgewidth or mew float value in points
markerfacecolor or mfc any matplotlib color
markerfacecoloralt or mfcalt any matplotlib color
markersize or ms float
markevery [None | int | length-2 tuple of int | slice | list/array of int | float | length-2 tuple of float]
path_effects unknown
picker float distance in points or callable pick function fn(artist, event)
pickradius float distance in points
rasterized [True | False | None]
sketch_params unknown
snap unknown
solid_capstyle [?butt? | ?round? | ?projecting?]
solid_joinstyle [?miter? | ?round? | ?bevel?]
transform a matplotlib.transforms.Transform instance
url a url string
visible [True | False]
xdata 1D array
ydata 1D array
zorder any number
(Source code, png, hires.png, pdf)
Additional kwargs: hold = [True|False] overrides default hold state
set the default colormap to magma and apply to current image if any. See help(colormaps) for more information
matplotlib.pyplot.magnitude_spectrum(x, Fs=None, Fc=None, window=None, pad_to=None, sides=None, scale=None, hold=None, data=None, **kwargs)
Plot the magnitude spectrum.
Call signature:
magnitude_spectrum(x, Fs=2, Fc=0, window=mlab.window_hanning,
pad_to=None, sides='default', **kwargs)
Compute the magnitude spectrum of x. Data is padded to a length of pad_to and the windowing function window is applied to the signal.
x: 1-D array or sequence
Array or sequence containing the data
Keyword arguments:
Fs: scalar
The sampling frequency (samples per time unit). It is used to calculate the Fourier frequencies, freqs, in cycles per time unit. The default value is 2.
window: callable or ndarray
A function or a vector of length NFFT. To create window vectors see window_hanning(), window_none(), numpy.blackman(), numpy.hamming(), numpy.bartlett(), scipy.signal(),
scipy.signal.get_window(), etc. The default is window_hanning(). If a function is passed as the argument, it must take a data segment as an argument and return the windowed version of the
sides: [ ?default? | ?onesided? | ?twosided? ]
Specifies which sides of the spectrum to return. Default gives the default behavior, which returns one-sided for real data and both for complex data. ?onesided? forces the return of a
one-sided spectrum, while ?twosided? forces two-sided.
pad_to: integer
The number of points to which the data segment is padded when performing the FFT. While not increasing the actual resolution of the spectrum (the minimum distance between resolvable peaks),
this can give more points in the plot, allowing for more detail. This corresponds to the n parameter in the call to fft(). The default is None, which sets pad_to equal to the length of the
input signal (i.e. no padding).
scale: [ ?default? | ?linear? | ?dB? ]
The scaling of the values in the spec. ?linear? is no scaling. ?dB? returns the values in dB scale. When mode is ?density?, this is dB power (10 * log10). Otherwise this is dB amplitude
(20 * log10). ?default? is ?linear?.
Fc: integer
The center frequency of x (defaults to 0), which offsets the x extents of the plot to reflect the frequency range used when a signal is acquired and then filtered and downsampled to
Returns the tuple (spectrum, freqs, line):
spectrum: 1-D array
The values for the magnitude spectrum before scaling (real valued)
freqs: 1-D array
The frequencies corresponding to the elements in spectrum
line: a Line2D instance
The line created by this function
kwargs control the Line2D properties:
Property Description
agg_filter unknown
alpha float (0.0 transparent through 1.0 opaque)
animated [True | False]
antialiased or aa [True | False]
axes an Axes instance
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
color or c any matplotlib color
contains a callable function
dash_capstyle [?butt? | ?round? | ?projecting?]
dash_joinstyle [?miter? | ?round? | ?bevel?]
dashes sequence of on/off ink in points
drawstyle [?default? | ?steps? | ?steps-pre? | ?steps-mid? | ?steps-post?]
figure a matplotlib.figure.Figure instance
fillstyle [?full? | ?left? | ?right? | ?bottom? | ?top? | ?none?]
gid an id string
label string or anything printable with ?%s? conversion.
linestyle or ls [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or lw float value in points
marker A valid marker style
markeredgecolor or mec any matplotlib color
markeredgewidth or mew float value in points
markerfacecolor or mfc any matplotlib color
markerfacecoloralt or mfcalt any matplotlib color
markersize or ms float
markevery [None | int | length-2 tuple of int | slice | list/array of int | float | length-2 tuple of float]
path_effects unknown
picker float distance in points or callable pick function fn(artist, event)
pickradius float distance in points
rasterized [True | False | None]
sketch_params unknown
snap unknown
solid_capstyle [?butt? | ?round? | ?projecting?]
solid_joinstyle [?miter? | ?round? | ?bevel?]
transform a matplotlib.transforms.Transform instance
url a url string
visible [True | False]
xdata 1D array
ydata 1D array
zorder any number
(Source code, png, hires.png, pdf)
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?x?.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.margins(*args, **kw)
Set or retrieve autoscaling margins.
returns xmargin, ymargin
margins(xmargin, ymargin)
margins(x=xmargin, y=ymargin)
margins(..., tight=False)
All three forms above set the xmargin and ymargin parameters. All keyword parameters are optional. A single argument specifies both xmargin and ymargin. The tight parameter is passed to
autoscale_view(), which is executed after a margin is changed; the default here is True, on the assumption that when margins are specified, no additional padding to match tick marks is usually
desired. Setting tight to None will preserve the previous setting.
Specifying any margin changes only the autoscaling; for example, if xmargin is not None, then xmargin times the X data interval will be added to each end of that interval before it is used in
matplotlib.pyplot.matshow(A, fignum=None, **kw)
Display an array as a matrix in a new figure window.
The origin is set at the upper left hand corner and rows (first dimension of the array) are displayed horizontally. The aspect ratio of the figure window is that of the array, unless this would
make an excessively short or narrow figure.
Tick labels for the xaxis are placed on top.
With the exception of fignum, keyword arguments are passed to imshow(). You may set the origin kwarg to ?lower? if you want the first row in the array to be at the bottom instead of the top.
fignum: [ None | integer | False ]
By default, matshow() creates a new figure window with automatic numbering. If fignum is given as an integer, the created figure will use this figure number. Because of how matshow() tries to
set the figure aspect ratio to be the one of the array, if you provide the number of an already existing figure, strange things may happen.
If fignum is False or 0, a new figure window will NOT be created.
Remove minor ticks from the current plot.
Display minor ticks on the current plot.
Displaying minor ticks reduces performance; turn them off using minorticks_off() if drawing speed is a problem.
matplotlib.pyplot.over(func, *args, **kwargs)
Call a function with hold(True).
func(*args, **kwargs)
with hold(True) and then restores the hold state.
Pause for interval seconds.
If there is an active figure it will be updated and displayed, and the GUI event loop will run during the pause.
If there is no active figure, or if a non-interactive backend is in use, this executes time.sleep(interval).
This can be used for crude animation. For more complex animation, see matplotlib.animation.
This function is experimental; its behavior may be changed or extended in a future release.
matplotlib.pyplot.pcolor(*args, **kwargs)
Create a pseudocolor plot of a 2-D array.
pcolor can be very slow for large arrays; consider using the similar but much faster pcolormesh() instead.
Call signatures:
pcolor(C, **kwargs)
pcolor(X, Y, C, **kwargs)
C is the array of color values.
X and Y, if given, specify the (x, y) coordinates of the colored quadrilaterals; the quadrilateral for C[i,j] has corners at:
(X[i, j], Y[i, j]),
(X[i, j+1], Y[i, j+1]),
(X[i+1, j], Y[i+1, j]),
(X[i+1, j+1], Y[i+1, j+1]).
Ideally the dimensions of X and Y should be one greater than those of C; if the dimensions are the same, then the last row and column of C will be ignored.
Note that the column index corresponds to the x-coordinate, and the row index corresponds to y; for details, see the Grid Orientation section below.
If either or both of X and Y are 1-D arrays or column vectors, they will be expanded as needed into the appropriate 2-D arrays, making a rectangular grid.
X, Y and C may be masked arrays. If either C[i, j], or one of the vertices surrounding C[i,j] (X or Y at [i, j], [i+1, j], [i, j+1],[i+1, j+1]) is masked, nothing is plotted.
Keyword arguments:
cmap: [ None | Colormap ]
A matplotlib.colors.Colormap instance. If None, use rc settings.
norm: [ None | Normalize ]
An matplotlib.colors.Normalize instance is used to scale luminance data to 0,1. If None, defaults to normalize().
vmin/vmax: [ None | scalar ]
vmin and vmax are used in conjunction with norm to normalize luminance data. If either is None, it is autoscaled to the respective min or max of the color array C. If not None, vmin or vmax
passed in here override any pre-existing values supplied in the norm instance.
shading: [ ?flat? | ?faceted? ]
If ?faceted?, a black grid is drawn around each rectangle; if ?flat?, edges are not drawn. Default is ?flat?, contrary to MATLAB.
This kwarg is deprecated; please use ?edgecolors? instead:
○ shading=?flat? ? edgecolors=?none?
○ shading=?faceted ? edgecolors=?k?
edgecolors: [ None | 'none' | color | color sequence]
If None, the rc setting is used by default.
If 'none', edges will not be visible.
An mpl color or sequence of colors will set the edge color
alpha: 0 <= scalar <= 1 or None
the alpha blending value
snap: bool
Whether to snap the mesh to pixel boundaries.
Return value is a matplotlib.collections.Collection instance.
The grid orientation follows the MATLAB convention: an array C with shape (nrows, ncolumns) is plotted with the column number as X and the row number as Y, increasing up; hence it is plotted the
way the array would be printed, except that the Y axis is reversed. That is, C is taken as C*(*y, x).
Similarly for meshgrid():
x = np.arange(5)
y = np.arange(3)
X, Y = np.meshgrid(x, y)
is equivalent to:
X = array([[0, 1, 2, 3, 4],
[0, 1, 2, 3, 4],
[0, 1, 2, 3, 4]])
Y = array([[0, 0, 0, 0, 0],
[1, 1, 1, 1, 1],
[2, 2, 2, 2, 2]])
so if you have:
C = rand(len(x), len(y))
then you need to transpose C:
pcolor(X, Y, C.T)
MATLAB pcolor() always discards the last row and column of C, but matplotlib displays the last row and column if X and Y are not specified, or if X and Y have one more row and column than C.
kwargs can be used to control the PolyCollection properties:
Property Description
agg_filter unknown
alpha float or None
animated [True | False]
antialiased or antialiaseds Boolean or sequence of booleans
array unknown
axes an Axes instance
clim a length 2 sequence of floats
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
cmap a colormap or registered colormap name
color matplotlib color arg or sequence of rgba tuples
contains a callable function
edgecolor or edgecolors matplotlib color spec or sequence of specs
facecolor or facecolors matplotlib color spec or sequence of specs
figure a matplotlib.figure.Figure instance
gid an id string
hatch [ ?/? | ?\? | ?|? | ?-? | ?+? | ?x? | ?o? | ?O? | ?.? | ?*? ]
label string or anything printable with ?%s? conversion.
linestyle or linestyles or dashes [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or linewidths or lw float or sequence of floats
norm unknown
offset_position unknown
offsets float or sequence of floats
path_effects unknown
picker [None|float|boolean|callable]
pickradius unknown
rasterized [True | False | None]
sketch_params unknown
snap unknown
transform Transform instance
url a url string
urls unknown
visible [True | False]
zorder any number
The default antialiaseds is False if the default edgecolors*=?none? is used. This eliminates artificial lines at patch boundaries, and works regardless of the value of alpha. If *edgecolors is
not ?none?, then the default antialiaseds is taken from rcParams[?patch.antialiased?], which defaults to True. Stroking the edges may be preferred if alpha is 1, but will cause artifacts
See also
For an explanation of the differences between pcolor and pcolormesh.
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All positional and all keyword arguments.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.pcolormesh(*args, **kwargs)
Plot a quadrilateral mesh.
Call signatures:
pcolormesh(X, Y, C)
pcolormesh(C, **kwargs)
Create a pseudocolor plot of a 2-D array.
pcolormesh is similar to pcolor(), but uses a different mechanism and returns a different object; pcolor returns a PolyCollection but pcolormesh returns a QuadMesh. It is much faster, so it is
almost always preferred for large arrays.
C may be a masked array, but X and Y may not. Masked array support is implemented via cmap and norm; in contrast, pcolor() simply does not draw quadrilaterals with masked colors or vertices.
Keyword arguments:
cmap: [ None | Colormap ]
A matplotlib.colors.Colormap instance. If None, use rc settings.
norm: [ None | Normalize ]
A matplotlib.colors.Normalize instance is used to scale luminance data to 0,1. If None, defaults to normalize().
vmin/vmax: [ None | scalar ]
vmin and vmax are used in conjunction with norm to normalize luminance data. If either is None, it is autoscaled to the respective min or max of the color array C. If not None, vmin or vmax
passed in here override any pre-existing values supplied in the norm instance.
shading: [ ?flat? | ?gouraud? ]
?flat? indicates a solid color for each quad. When ?gouraud?, each quad will be Gouraud shaded. When gouraud shading, edgecolors is ignored.
edgecolors: [None | 'None' | 'face' | color |
color sequence]
If None, the rc setting is used by default.
If 'None', edges will not be visible.
If 'face', edges will have the same color as the faces.
An mpl color or sequence of colors will set the edge color
alpha: 0 <= scalar <= 1 or None
the alpha blending value
Return value is a matplotlib.collections.QuadMesh object.
kwargs can be used to control the matplotlib.collections.QuadMesh properties:
Property Description
agg_filter unknown
alpha float or None
animated [True | False]
antialiased or antialiaseds Boolean or sequence of booleans
array unknown
axes an Axes instance
clim a length 2 sequence of floats
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
cmap a colormap or registered colormap name
color matplotlib color arg or sequence of rgba tuples
contains a callable function
edgecolor or edgecolors matplotlib color spec or sequence of specs
facecolor or facecolors matplotlib color spec or sequence of specs
figure a matplotlib.figure.Figure instance
gid an id string
hatch [ ?/? | ?\? | ?|? | ?-? | ?+? | ?x? | ?o? | ?O? | ?.? | ?*? ]
label string or anything printable with ?%s? conversion.
linestyle or linestyles or dashes [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or linewidths or lw float or sequence of floats
norm unknown
offset_position unknown
offsets float or sequence of floats
path_effects unknown
picker [None|float|boolean|callable]
pickradius unknown
rasterized [True | False | None]
sketch_params unknown
snap unknown
transform Transform instance
url a url string
urls unknown
visible [True | False]
zorder any number
See also
For an explanation of the grid orientation and the expansion of 1-D X and/or Y to 2-D arrays.
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All positional and all keyword arguments.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.phase_spectrum(x, Fs=None, Fc=None, window=None, pad_to=None, sides=None, hold=None, data=None, **kwargs)
Plot the phase spectrum.
Call signature:
phase_spectrum(x, Fs=2, Fc=0, window=mlab.window_hanning,
pad_to=None, sides='default', **kwargs)
Compute the phase spectrum (unwrapped angle spectrum) of x. Data is padded to a length of pad_to and the windowing function window is applied to the signal.
x: 1-D array or sequence
Array or sequence containing the data
Keyword arguments:
Fs: scalar
The sampling frequency (samples per time unit). It is used to calculate the Fourier frequencies, freqs, in cycles per time unit. The default value is 2.
window: callable or ndarray
A function or a vector of length NFFT. To create window vectors see window_hanning(), window_none(), numpy.blackman(), numpy.hamming(), numpy.bartlett(), scipy.signal(),
scipy.signal.get_window(), etc. The default is window_hanning(). If a function is passed as the argument, it must take a data segment as an argument and return the windowed version of the
sides: [ ?default? | ?onesided? | ?twosided? ]
Specifies which sides of the spectrum to return. Default gives the default behavior, which returns one-sided for real data and both for complex data. ?onesided? forces the return of a
one-sided spectrum, while ?twosided? forces two-sided.
pad_to: integer
The number of points to which the data segment is padded when performing the FFT. While not increasing the actual resolution of the spectrum (the minimum distance between resolvable peaks),
this can give more points in the plot, allowing for more detail. This corresponds to the n parameter in the call to fft(). The default is None, which sets pad_to equal to the length of the
input signal (i.e. no padding).
Fc: integer
The center frequency of x (defaults to 0), which offsets the x extents of the plot to reflect the frequency range used when a signal is acquired and then filtered and downsampled to
Returns the tuple (spectrum, freqs, line):
spectrum: 1-D array
The values for the phase spectrum in radians (real valued)
freqs: 1-D array
The frequencies corresponding to the elements in spectrum
line: a Line2D instance
The line created by this function
kwargs control the Line2D properties:
Property Description
agg_filter unknown
alpha float (0.0 transparent through 1.0 opaque)
animated [True | False]
antialiased or aa [True | False]
axes an Axes instance
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
color or c any matplotlib color
contains a callable function
dash_capstyle [?butt? | ?round? | ?projecting?]
dash_joinstyle [?miter? | ?round? | ?bevel?]
dashes sequence of on/off ink in points
drawstyle [?default? | ?steps? | ?steps-pre? | ?steps-mid? | ?steps-post?]
figure a matplotlib.figure.Figure instance
fillstyle [?full? | ?left? | ?right? | ?bottom? | ?top? | ?none?]
gid an id string
label string or anything printable with ?%s? conversion.
linestyle or ls [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or lw float value in points
marker A valid marker style
markeredgecolor or mec any matplotlib color
markeredgewidth or mew float value in points
markerfacecolor or mfc any matplotlib color
markerfacecoloralt or mfcalt any matplotlib color
markersize or ms float
markevery [None | int | length-2 tuple of int | slice | list/array of int | float | length-2 tuple of float]
path_effects unknown
picker float distance in points or callable pick function fn(artist, event)
pickradius float distance in points
rasterized [True | False | None]
sketch_params unknown
snap unknown
solid_capstyle [?butt? | ?round? | ?projecting?]
solid_joinstyle [?miter? | ?round? | ?bevel?]
transform a matplotlib.transforms.Transform instance
url a url string
visible [True | False]
xdata 1D array
ydata 1D array
zorder any number
(Source code, png, hires.png, pdf)
See also
magnitude_spectrum() plots the magnitudes of the corresponding frequencies.
angle_spectrum() plots the wrapped version of this function.
specgram() can plot the phase spectrum of segments within the signal in a colormap.
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?x?.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.pie(x, explode=None, labels=None, colors=None, autopct=None, pctdistance=0.6, shadow=False, labeldistance=1.1, startangle=None, radius=None, counterclock=True, wedgeprops=None,
textprops=None, center=(0, 0), frame=False, hold=None, data=None)
Plot a pie chart.
Call signature:
pie(x, explode=None, labels=None,
colors=('b', 'g', 'r', 'c', 'm', 'y', 'k', 'w'),
autopct=None, pctdistance=0.6, shadow=False,
labeldistance=1.1, startangle=None, radius=None,
counterclock=True, wedgeprops=None, textprops=None,
center = (0, 0), frame = False )
Make a pie chart of array x. The fractional area of each wedge is given by x/sum(x). If sum(x) <= 1, then the values of x give the fractional area directly and the array will not be normalized.
The wedges are plotted counterclockwise, by default starting from the x-axis.
Keyword arguments:
explode: [ None | len(x) sequence ]
If not None, is a len(x) array which specifies the fraction of the radius with which to offset each wedge.
colors: [ None | color sequence ]
A sequence of matplotlib color args through which the pie chart will cycle.
labels: [ None | len(x) sequence of strings ]
A sequence of strings providing the labels for each wedge
autopct: [ None | format string | format function ]
If not None, is a string or function used to label the wedges with their numeric value. The label will be placed inside the wedge. If it is a format string, the label will be fmt%pct. If it
is a function, it will be called.
pctdistance: scalar
The ratio between the center of each pie slice and the start of the text generated by autopct. Ignored if autopct is None; default is 0.6.
labeldistance: scalar
The radial distance at which the pie labels are drawn
shadow: [ False | True ]
Draw a shadow beneath the pie.
startangle: [ None | Offset angle ]
If not None, rotates the start of the pie chart by angle degrees counterclockwise from the x-axis.
radius: [ None | scalar ] The radius of the pie, if radius is None it will be set to 1.
counterclock: [ False | True ]
Specify fractions direction, clockwise or counterclockwise.
wedgeprops: [ None | dict of key value pairs ]
Dict of arguments passed to the wedge objects making the pie. For example, you can pass in wedgeprops = { ?linewidth? : 3 } to set the width of the wedge border lines equal to 3. For more
details, look at the doc/arguments of the wedge object. By default clip_on=False.
textprops: [ None | dict of key value pairs ]
Dict of arguments to pass to the text objects.
center: [ (0,0) | sequence of 2 scalars ] Center position of the chart.
frame: [ False | True ]
Plot axes frame with the chart.
The pie chart will probably look best if the figure and axes are square, or the Axes aspect is equal. e.g.:
ax = axes([0.1, 0.1, 0.8, 0.8])
Return value:
If autopct is None, return the tuple (patches, texts):
If autopct is not None, return the tuple (patches, texts, autotexts), where patches and texts are as above, and autotexts is a list of Text instances for the numeric labels.
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?explode?, ?colors?, ?x?, ?labels?.
Additional kwargs: hold = [True|False] overrides default hold state
set the default colormap to pink and apply to current image if any. See help(colormaps) for more information
set the default colormap to plasma and apply to current image if any. See help(colormaps) for more information
matplotlib.pyplot.plot(*args, **kwargs)
Plot lines and/or markers to the Axes. args is a variable length argument, allowing for multiple x, y pairs with an optional format string. For example, each of the following is legal:
plot(x, y) # plot x and y using default line style and color
plot(x, y, 'bo') # plot x and y using blue circle markers
plot(y) # plot y using x as index array 0..N-1
plot(y, 'r+') # ditto, but with red plusses
If x and/or y is 2-dimensional, then the corresponding columns will be plotted.
If used with labeled data, make sure that the color spec is not included as an element in data, as otherwise the last case plot("v","r", data={"v":..., "r":...) can be interpreted as the first
case which would do plot(v, r) using the default line style and color.
If not used with labeled data (i.e., without a data argument), an arbitrary number of x, y, fmt groups can be specified, as in:
a.plot(x1, y1, 'g^', x2, y2, 'g-')
Return value is a list of lines that were added.
By default, each line is assigned a different style specified by a ?style cycle?. To change this behavior, you can edit the axes.prop_cycle rcParam.
The following format string characters are accepted to control the line style or marker:
character description
'-' solid line style
'--' dashed line style
'-.' dash-dot line style
':' dotted line style
'.' point marker
',' pixel marker
'o' circle marker
'v' triangle_down marker
'^' triangle_up marker
'<' triangle_left marker
'>' triangle_right marker
'1' tri_down marker
'2' tri_up marker
'3' tri_left marker
'4' tri_right marker
's' square marker
'p' pentagon marker
'*' star marker
'h' hexagon1 marker
'H' hexagon2 marker
'+' plus marker
'x' x marker
'D' diamond marker
'd' thin_diamond marker
'|' vline marker
'_' hline marker
The following color abbreviations are supported:
character color
?b? blue
?g? green
?r? red
?c? cyan
?m? magenta
?y? yellow
?k? black
?w? white
In addition, you can specify colors in many weird and wonderful ways, including full names ('green'), hex strings ('#008000'), RGB or RGBA tuples ((0,1,0,1)) or grayscale intensities as a string
('0.8'). Of these, the string specifications can be used in place of a fmt group, but the tuple forms can be used only as kwargs.
Line styles and colors are combined in a single format string, as in 'bo' for blue circles.
The kwargs can be used to set line properties (any property that has a set_* method). You can use this to set a line label (for auto legends), linewidth, anitialising, marker face color, etc.
Here is an example:
plot([1,2,3], [1,2,3], 'go-', label='line 1', linewidth=2)
plot([1,2,3], [1,4,9], 'rs', label='line 2')
axis([0, 4, 0, 10])
If you make multiple lines with one plot command, the kwargs apply to all those lines, e.g.:
plot(x1, y1, x2, y2, antialiased=False)
Neither line will be antialiased.
You do not need to use format strings, which are just abbreviations. All of the line properties can be controlled by keyword arguments. For example, you can set the color, marker, linestyle, and
markercolor with:
plot(x, y, color='green', linestyle='dashed', marker='o',
markerfacecolor='blue', markersize=12).
See Line2D for details.
The kwargs are Line2D properties:
Property Description
agg_filter unknown
alpha float (0.0 transparent through 1.0 opaque)
animated [True | False]
antialiased or aa [True | False]
axes an Axes instance
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
color or c any matplotlib color
contains a callable function
dash_capstyle [?butt? | ?round? | ?projecting?]
dash_joinstyle [?miter? | ?round? | ?bevel?]
dashes sequence of on/off ink in points
drawstyle [?default? | ?steps? | ?steps-pre? | ?steps-mid? | ?steps-post?]
figure a matplotlib.figure.Figure instance
fillstyle [?full? | ?left? | ?right? | ?bottom? | ?top? | ?none?]
gid an id string
label string or anything printable with ?%s? conversion.
linestyle or ls [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or lw float value in points
marker A valid marker style
markeredgecolor or mec any matplotlib color
markeredgewidth or mew float value in points
markerfacecolor or mfc any matplotlib color
markerfacecoloralt or mfcalt any matplotlib color
markersize or ms float
markevery [None | int | length-2 tuple of int | slice | list/array of int | float | length-2 tuple of float]
path_effects unknown
picker float distance in points or callable pick function fn(artist, event)
pickradius float distance in points
rasterized [True | False | None]
sketch_params unknown
snap unknown
solid_capstyle [?butt? | ?round? | ?projecting?]
solid_joinstyle [?miter? | ?round? | ?bevel?]
transform a matplotlib.transforms.Transform instance
url a url string
visible [True | False]
xdata 1D array
ydata 1D array
zorder any number
kwargs scalex and scaley, if defined, are passed on to autoscale_view() to determine whether the x and y axes are autoscaled; the default is True.
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?y?, ?x?.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.plot_date(x, y, fmt='o', tz=None, xdate=True, ydate=False, hold=None, data=None, **kwargs)
Plot with data with dates.
Call signature:
plot_date(x, y, fmt='bo', tz=None, xdate=True,
ydate=False, **kwargs)
Similar to the plot() command, except the x or y (or both) data is considered to be dates, and the axis is labeled accordingly.
x and/or y can be a sequence of dates represented as float days since 0001-01-01 UTC.
Keyword arguments:
fmt: string
The plot format string.
tz: [ None | timezone string | tzinfo instance]
The time zone to use in labeling dates. If None, defaults to rc value.
xdate: [ True | False ]
If True, the x-axis will be labeled with dates.
ydate: [ False | True ]
If True, the y-axis will be labeled with dates.
Note if you are using custom date tickers and formatters, it may be necessary to set the formatters/locators after the call to plot_date() since plot_date() will set the default tick locator to
matplotlib.dates.AutoDateLocator (if the tick locator is not already set to a matplotlib.dates.DateLocator instance) and the default tick formatter to matplotlib.dates.AutoDateFormatter (if the
tick formatter is not already set to a matplotlib.dates.DateFormatter instance).
Valid kwargs are Line2D properties:
Property Description
agg_filter unknown
alpha float (0.0 transparent through 1.0 opaque)
animated [True | False]
antialiased or aa [True | False]
axes an Axes instance
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
color or c any matplotlib color
contains a callable function
dash_capstyle [?butt? | ?round? | ?projecting?]
dash_joinstyle [?miter? | ?round? | ?bevel?]
dashes sequence of on/off ink in points
drawstyle [?default? | ?steps? | ?steps-pre? | ?steps-mid? | ?steps-post?]
figure a matplotlib.figure.Figure instance
fillstyle [?full? | ?left? | ?right? | ?bottom? | ?top? | ?none?]
gid an id string
label string or anything printable with ?%s? conversion.
linestyle or ls [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or lw float value in points
marker A valid marker style
markeredgecolor or mec any matplotlib color
markeredgewidth or mew float value in points
markerfacecolor or mfc any matplotlib color
markerfacecoloralt or mfcalt any matplotlib color
markersize or ms float
markevery [None | int | length-2 tuple of int | slice | list/array of int | float | length-2 tuple of float]
path_effects unknown
picker float distance in points or callable pick function fn(artist, event)
pickradius float distance in points
rasterized [True | False | None]
sketch_params unknown
snap unknown
solid_capstyle [?butt? | ?round? | ?projecting?]
solid_joinstyle [?miter? | ?round? | ?bevel?]
transform a matplotlib.transforms.Transform instance
url a url string
visible [True | False]
xdata 1D array
ydata 1D array
zorder any number
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?y?, ?x?.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.plotfile(fname, cols=(0, ), plotfuncs=None, comments='#', skiprows=0, checkrows=5, delimiter=', ', names=None, subplots=True, newfig=True, **kwargs)
Plot the data in in a file.
cols is a sequence of column identifiers to plot. An identifier is either an int or a string. If it is an int, it indicates the column number. If it is a string, it indicates the column header.
matplotlib will make column headers lower case, replace spaces with underscores, and remove all illegal characters; so 'Adj Close*' will have name 'adj_close'.
□ If len(cols) == 1, only that column will be plotted on the y axis.
□ If len(cols) > 1, the first element will be an identifier for data for the x axis and the remaining elements will be the column indexes for multiple subplots if subplots is True (the
default), or for lines in a single subplot if subplots is False.
plotfuncs, if not None, is a dictionary mapping identifier to an Axes plotting function as a string. Default is ?plot?, other choices are ?semilogy?, ?fill?, ?bar?, etc. You must use the same
type of identifier in the cols vector as you use in the plotfuncs dictionary, e.g., integer column numbers in both or column names in both. If subplots is False, then including any function such
as ?semilogy? that changes the axis scaling will set the scaling for all columns.
comments, skiprows, checkrows, delimiter, and names are all passed on to matplotlib.pylab.csv2rec() to load the data into a record array.
If newfig is True, the plot always will be made in a new figure; if False, it will be made in the current figure if one exists, else in a new figure.
kwargs are passed on to plotting functions.
Example usage:
# plot the 2nd and 4th column against the 1st in two subplots
plotfile(fname, (0,1,3))
# plot using column names; specify an alternate plot type for volume
plotfile(fname, ('date', 'volume', 'adj_close'),
plotfuncs={'volume': 'semilogy'})
Note: plotfile is intended as a convenience for quickly plotting data from flat files; it is not intended as an alternative interface to general plotting with pyplot or matplotlib.
matplotlib.pyplot.polar(*args, **kwargs)
Make a polar plot.
call signature:
polar(theta, r, **kwargs)
Multiple theta, r arguments are supported, with format strings, as in plot().
set the default colormap to prism and apply to current image if any. See help(colormaps) for more information
matplotlib.pyplot.psd(x, NFFT=None, Fs=None, Fc=None, detrend=None, window=None, noverlap=None, pad_to=None, sides=None, scale_by_freq=None, return_line=None, hold=None, data=None, **kwargs)
Plot the power spectral density.
Call signature:
psd(x, NFFT=256, Fs=2, Fc=0, detrend=mlab.detrend_none,
window=mlab.window_hanning, noverlap=0, pad_to=None,
sides='default', scale_by_freq=None, return_line=None, **kwargs)
The power spectral density x is divided into NFFT length segments. Each segment is detrended by function detrend and windowed by function window. noverlap gives the length of the overlap between
segments. The
If len(x) < NFFT, it will be zero padded to NFFT.
x: 1-D array or sequence
Array or sequence containing the data
Keyword arguments:
Fs: scalar
The sampling frequency (samples per time unit). It is used to calculate the Fourier frequencies, freqs, in cycles per time unit. The default value is 2.
window: callable or ndarray
A function or a vector of length NFFT. To create window vectors see window_hanning(), window_none(), numpy.blackman(), numpy.hamming(), numpy.bartlett(), scipy.signal(),
scipy.signal.get_window(), etc. The default is window_hanning(). If a function is passed as the argument, it must take a data segment as an argument and return the windowed version of the
sides: [ ?default? | ?onesided? | ?twosided? ]
Specifies which sides of the spectrum to return. Default gives the default behavior, which returns one-sided for real data and both for complex data. ?onesided? forces the return of a
one-sided spectrum, while ?twosided? forces two-sided.
pad_to: integer
The number of points to which the data segment is padded when performing the FFT. This can be different from NFFT, which specifies the number of data points used. While not increasing the
actual resolution of the spectrum (the minimum distance between resolvable peaks), this can give more points in the plot, allowing for more detail. This corresponds to the n parameter in the
call to fft(). The default is None, which sets pad_to equal to NFFT
NFFT: integer
The number of data points used in each block for the FFT. A power 2 is most efficient. The default value is 256. This should NOT be used to get zero padding, or the scaling of the result will
be incorrect. Use pad_to for this instead.
detrend: [ ?default? | ?constant? | ?mean? | ?linear? | ?none?] or
The function applied to each segment before fft-ing, designed to remove the mean or linear trend. Unlike in MATLAB, where the detrend parameter is a vector, in matplotlib is it a function.
The pylab module defines detrend_none(), detrend_mean(), and detrend_linear(), but you can use a custom function as well. You can also use a string to choose one of the functions. ?default?,
?constant?, and ?mean? call detrend_mean(). ?linear? calls detrend_linear(). ?none? calls detrend_none().
scale_by_freq: boolean
Specifies whether the resulting density values should be scaled by the scaling frequency, which gives density in units of Hz^-1. This allows for integration over the returned frequency
values. The default is True for MATLAB compatibility.
noverlap: integer
The number of points of overlap between segments. The default value is 0 (no overlap).
Fc: integer
The center frequency of x (defaults to 0), which offsets the x extents of the plot to reflect the frequency range used when a signal is acquired and then filtered and downsampled to
return_line: bool
Whether to include the line object plotted in the returned values. Default is False.
If return_line is False, returns the tuple (Pxx, freqs). If return_line is True, returns the tuple (Pxx, freqs. line):
Pxx: 1-D array
The values for the power spectrum P_{xx} before scaling (real valued)
freqs: 1-D array
The frequencies corresponding to the elements in Pxx
line: a Line2D instance
The line created by this function. Only returend if return_line is True.
For plotting, the power is plotted as Pxx itself is returned.
Bendat & Piersol ? Random Data: Analysis and Measurement Procedures, John Wiley & Sons (1986)
kwargs control the Line2D properties:
Property Description
agg_filter unknown
alpha float (0.0 transparent through 1.0 opaque)
animated [True | False]
antialiased or aa [True | False]
axes an Axes instance
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
color or c any matplotlib color
contains a callable function
dash_capstyle [?butt? | ?round? | ?projecting?]
dash_joinstyle [?miter? | ?round? | ?bevel?]
dashes sequence of on/off ink in points
drawstyle [?default? | ?steps? | ?steps-pre? | ?steps-mid? | ?steps-post?]
figure a matplotlib.figure.Figure instance
fillstyle [?full? | ?left? | ?right? | ?bottom? | ?top? | ?none?]
gid an id string
label string or anything printable with ?%s? conversion.
linestyle or ls [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or lw float value in points
marker A valid marker style
markeredgecolor or mec any matplotlib color
markeredgewidth or mew float value in points
markerfacecolor or mfc any matplotlib color
markerfacecoloralt or mfcalt any matplotlib color
markersize or ms float
markevery [None | int | length-2 tuple of int | slice | list/array of int | float | length-2 tuple of float]
path_effects unknown
picker float distance in points or callable pick function fn(artist, event)
pickradius float distance in points
rasterized [True | False | None]
sketch_params unknown
snap unknown
solid_capstyle [?butt? | ?round? | ?projecting?]
solid_joinstyle [?miter? | ?round? | ?bevel?]
transform a matplotlib.transforms.Transform instance
url a url string
visible [True | False]
xdata 1D array
ydata 1D array
zorder any number
(Source code, png, hires.png, pdf)
See also
specgram() differs in the default overlap; in not returning the mean of the segment periodograms; in returning the times of the segments; and in plotting a colormap instead of a line.
magnitude_spectrum() plots the magnitude spectrum.
csd() plots the spectral density between two signals.
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?x?.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.quiver(*args, **kw)
Plot a 2-D field of arrows.
call signatures:
quiver(U, V, **kw)
quiver(U, V, C, **kw)
quiver(X, Y, U, V, **kw)
quiver(X, Y, U, V, C, **kw)
X, Y:
The x and y coordinates of the arrow locations (default is tail of arrow; see pivot kwarg)
U, V:
Give the x and y components of the arrow vectors
An optional array used to map colors to the arrows
All arguments may be 1-D or 2-D arrays or sequences. If X and Y are absent, they will be generated as a uniform grid. If U and V are 2-D arrays but X and Y are 1-D, and if len(X) and len(Y) match
the column and row dimensions of U, then X and Y will be expanded with numpy.meshgrid().
U, V, C may be masked arrays, but masked X, Y are not supported at present.
Keyword arguments:
units: [ ?width? | ?height? | ?dots? | ?inches? | ?x? | ?y? | ?xy? ]
Arrow units; the arrow dimensions except for length are in multiples of this unit.
☆ ?width? or ?height?: the width or height of the axes
☆ ?dots? or ?inches?: pixels or inches, based on the figure dpi
☆ ?x?, ?y?, or ?xy?: X, Y, or sqrt(X^2+Y^2) data units
The arrows scale differently depending on the units. For ?x? or ?y?, the arrows get larger as one zooms in; for other units, the arrow size is independent of the zoom state. For ?width or ?
height?, the arrow size increases with the width and height of the axes, respectively, when the window is resized; for ?dots? or ?inches?, resizing does not change the arrows.
angles: [ ?uv? | ?xy? | array ]
With the default ?uv?, the arrow axis aspect ratio is 1, so that if U*==*V the orientation of the arrow on the plot is 45 degrees CCW from the horizontal axis (positive to the right). With ?
xy?, the arrow points from (x,y) to (x+u, y+v). Use this for plotting a gradient field, for example. Alternatively, arbitrary angles may be specified as an array of values in degrees, CCW
from the horizontal axis. Note: inverting a data axis will correspondingly invert the arrows only with angles='xy'.
scale: [ None | float ]
Data units per arrow length unit, e.g., m/s per plot width; a smaller scale parameter makes the arrow longer. If None, a simple autoscaling algorithm is used, based on the average vector
length and the number of vectors. The arrow length unit is given by the scale_units parameter
scale_units: None, or any of the units options.
For example, if scale_units is ?inches?, scale is 2.0, and (u,v) = (1,0), then the vector will be 0.5 inches long. If scale_units is ?width?, then the vector will be half the width of the
If scale_units is ?x? then the vector will be 0.5 x-axis units. To plot vectors in the x-y plane, with u and v having the same units as x and y, use ?angles=?xy?, scale_units=?xy?, scale=1?.
Shaft width in arrow units; default depends on choice of units, above, and number of vectors; a typical starting value is about 0.005 times the width of the plot.
headwidth: scalar
Head width as multiple of shaft width, default is 3
headlength: scalar
Head length as multiple of shaft width, default is 5
headaxislength: scalar
Head length at shaft intersection, default is 4.5
minshaft: scalar
Length below which arrow scales, in units of head length. Do not set this to less than 1, or small arrows will look terrible! Default is 1
minlength: scalar
Minimum length as a multiple of shaft width; if an arrow length is less than this, plot a dot (hexagon) of this diameter instead. Default is 1.
pivot: [ ?tail? | ?mid? | ?middle? | ?tip? ]
The part of the arrow that is at the grid point; the arrow rotates about this point, hence the name pivot.
color: [ color | color sequence ]
This is a synonym for the PolyCollection facecolor kwarg. If C has been set, color has no effect.
The defaults give a slightly swept-back arrow; to make the head a triangle, make headaxislength the same as headlength. To make the arrow more pointed, reduce headwidth or increase headlength and
headaxislength. To make the head smaller relative to the shaft, scale down all the head parameters. You will probably do best to leave minshaft alone.
linewidths and edgecolors can be used to customize the arrow outlines. Additional PolyCollection keyword arguments:
Property Description
agg_filter unknown
alpha float or None
animated [True | False]
antialiased or antialiaseds Boolean or sequence of booleans
array unknown
axes an Axes instance
clim a length 2 sequence of floats
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
cmap a colormap or registered colormap name
color matplotlib color arg or sequence of rgba tuples
contains a callable function
edgecolor or edgecolors matplotlib color spec or sequence of specs
facecolor or facecolors matplotlib color spec or sequence of specs
figure a matplotlib.figure.Figure instance
gid an id string
hatch [ ?/? | ?\? | ?|? | ?-? | ?+? | ?x? | ?o? | ?O? | ?.? | ?*? ]
label string or anything printable with ?%s? conversion.
linestyle or linestyles or dashes [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or linewidths or lw float or sequence of floats
norm unknown
offset_position unknown
offsets float or sequence of floats
path_effects unknown
picker [None|float|boolean|callable]
pickradius unknown
rasterized [True | False | None]
sketch_params unknown
snap unknown
transform Transform instance
url a url string
urls unknown
visible [True | False]
zorder any number
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.quiverkey(*args, **kw)
Add a key to a quiver plot.
Call signature:
quiverkey(Q, X, Y, U, label, **kw)
The Quiver instance returned by a call to quiver.
X, Y:
The location of the key; additional explanation follows.
The length of the key
A string with the length and units of the key
Keyword arguments:
coordinates = [ ?axes? | ?figure? | ?data? | ?inches? ]
Coordinate system and units for X, Y: ?axes? and ?figure? are normalized coordinate systems with 0,0 in the lower left and 1,1 in the upper right; ?data? are the axes data coordinates (used
for the locations of the vectors in the quiver plot itself); ?inches? is position in the figure in inches, with 0,0 at the lower left corner.
overrides face and edge colors from Q.
labelpos = [ ?N? | ?S? | ?E? | ?W? ]
Position the label above, below, to the right, to the left of the arrow, respectively.
Distance in inches between the arrow and the label. Default is 0.1
defaults to default Text color.
A dictionary with keyword arguments accepted by the FontProperties initializer: family, style, variant, size, weight
Any additional keyword arguments are used to override vector properties taken from Q.
The positioning of the key depends on X, Y, coordinates, and labelpos. If labelpos is ?N? or ?S?, X, Y give the position of the middle of the key arrow. If labelpos is ?E?, X, Y positions the
head, and if labelpos is ?W?, X, Y positions the tail; in either of these two cases, X, Y is somewhere in the middle of the arrow+label key object.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.rc(*args, **kwargs)
Set the current rc params. Group is the grouping for the rc, e.g., for lines.linewidth the group is lines, for axes.facecolor, the group is axes, and so on. Group may also be a list or tuple of
group names, e.g., (xtick, ytick). kwargs is a dictionary attribute name/value pairs, e.g.,:
rc('lines', linewidth=2, color='r')
sets the current rc params and is equivalent to:
rcParams['lines.linewidth'] = 2
rcParams['lines.color'] = 'r'
The following aliases are available to save typing for interactive users:
Alias Property
?lw? ?linewidth?
?ls? ?linestyle?
?c? ?color?
?fc? ?facecolor?
?ec? ?edgecolor?
?mew? ?markeredgewidth?
?aa? ?antialiased?
Thus you could abbreviate the above rc command as:
rc('lines', lw=2, c='r')
Note you can use python?s kwargs dictionary facility to store dictionaries of default parameters. e.g., you can customize the font rc as follows:
font = {'family' : 'monospace',
'weight' : 'bold',
'size' : 'larger'}
rc('font', **font) # pass in the font dict as kwargs
This enables you to easily switch between several configurations. Use rcdefaults() to restore the default rc params after changes.
matplotlib.pyplot.rc_context(rc=None, fname=None)
Return a context manager for managing rc settings.
This allows one to do:
with mpl.rc_context(fname='screen.rc'):
plt.plot(x, a)
with mpl.rc_context(fname='print.rc'):
plt.plot(x, b)
plt.plot(x, c)
The ?a? vs ?x? and ?c? vs ?x? plots would have settings from ?screen.rc?, while the ?b? vs ?x? plot would have settings from ?print.rc?.
A dictionary can also be passed to the context manager:
with mpl.rc_context(rc={'text.usetex': True}, fname='screen.rc'):
plt.plot(x, a)
The ?rc? dictionary takes precedence over the settings loaded from ?fname?. Passing a dictionary only is also valid.
Restore the default rc params. These are not the params loaded by the rc file, but mpl?s internal params. See rc_file_defaults for reloading the default params from the rc file
matplotlib.pyplot.rgrids(*args, **kwargs)
Get or set the radial gridlines on a polar plot.
call signatures:
lines, labels = rgrids()
lines, labels = rgrids(radii, labels=None, angle=22.5, **kwargs)
When called with no arguments, rgrid() simply returns the tuple (lines, labels), where lines is an array of radial gridlines (Line2D instances) and labels is an array of tick labels (Text
instances). When called with arguments, the labels will appear at the specified radial distances and angles.
labels, if not None, is a len(radii) list of strings of the labels to use at each angle.
If labels is None, the rformatter will be used
# set the locations of the radial gridlines and labels
lines, labels = rgrids( (0.25, 0.5, 1.0) )
# set the locations and labels of the radial gridlines and labels
lines, labels = rgrids( (0.25, 0.5, 1.0), ('Tom', 'Dick', 'Harry' )
matplotlib.pyplot.savefig(*args, **kwargs)
Save the current figure.
Call signature:
savefig(fname, dpi=None, facecolor='w', edgecolor='w',
orientation='portrait', papertype=None, format=None,
transparent=False, bbox_inches=None, pad_inches=0.1,
The output formats available depend on the backend being used.
A string containing a path to a filename, or a Python file-like object, or possibly some backend-dependent object such as PdfPages.
If format is None and fname is a string, the output format is deduced from the extension of the filename. If the filename has no extension, the value of the rc parameter savefig.format is
If fname is not a string, remember to specify format to ensure that the correct backend is used.
Keyword arguments:
dpi: [ None | scalar > 0 | ?figure?]
The resolution in dots per inch. If None it will default to the value savefig.dpi in the matplotlibrc file. If ?figure? it will set the dpi to be the value of the figure.
facecolor, edgecolor:
the colors of the figure rectangle
orientation: [ ?landscape? | ?portrait? ]
not supported on all backends; currently only on postscript output
One of ?letter?, ?legal?, ?executive?, ?ledger?, ?a0? through ?a10?, ?b0? through ?b10?. Only supported for postscript output.
One of the file extensions supported by the active backend. Most backends support png, pdf, ps, eps and svg.
If True, the axes patches will all be transparent; the figure patch will also be transparent unless facecolor and/or edgecolor are specified via kwargs. This is useful, for example, for
displaying a plot on top of a colored background on a web page. The transparency of these patches will be restored to their original values upon exit of this function.
If True, the figure patch will be colored, if False, the figure background will be transparent. If not provided, the rcParam ?savefig.frameon? will be used.
Bbox in inches. Only the given portion of the figure is saved. If ?tight?, try to figure out the tight bbox of the figure.
Amount of padding around the figure when bbox_inches is ?tight?.
A list of extra artists that will be considered when the tight bbox is calculated.
Set the current Axes instance to ax.
The current Figure is updated to the parent of ax.
matplotlib.pyplot.scatter(x, y, s=20, c=None, marker='o', cmap=None, norm=None, vmin=None, vmax=None, alpha=None, linewidths=None, verts=None, edgecolors=None, hold=None, data=None, **kwargs)
Make a scatter plot of x vs y, where x and y are sequence like objects of the same length.
x, y : array_like, shape (n, )
Input data
s : scalar or array_like, shape (n, ), optional, default: 20
size in points^2.
c : color, sequence, or sequence of color, optional, default: ?b?
c can be a single color format string, or a sequence of color specifications of length N, or a sequence of N numbers to be mapped to colors using the cmap and norm specified via
kwargs (see below). Note that c should not be a single numeric RGB or RGBA sequence because that is indistinguishable from an array of values to be colormapped. c can be a 2-D array
in which the rows are RGB or RGBA, however, including the case of a single row to specify the same color for all points.
marker : MarkerStyle, optional, default: ?o?
See markers for more information on the different styles of markers scatter supports. marker can be either an instance of the class or the text shorthand for a particular marker.
cmap : Colormap, optional, default: None
A Colormap instance or registered name. cmap is only used if c is an array of floats. If None, defaults to rc image.cmap.
norm : Normalize, optional, default: None
A Normalize instance is used to scale luminance data to 0, 1. norm is only used if c is an array of floats. If None, use the default normalize().
vmin, vmax : scalar, optional, default: None
vmin and vmax are used in conjunction with norm to normalize luminance data. If either are None, the min and max of the color array is used. Note if you pass a norm instance, your
settings for vmin and vmax will be ignored.
alpha : scalar, optional, default: None
The alpha blending value, between 0 (transparent) and 1 (opaque)
linewidths : scalar or array_like, optional, default: None
If None, defaults to (lines.linewidth,).
edgecolors : color or sequence of color, optional, default: None
If None, defaults to (patch.edgecolor). If ?face?, the edge color will always be the same as the face color. If it is ?none?, the patch boundary will not be drawn. For non-filled
markers, the edgecolors kwarg is ignored; color is determined by c.
Returns: paths : PathCollection
Other Parameters:
kwargs : Collection properties
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?y?, ?linewidths?, ?color?, ?facecolor?, ?facecolors?, ?edgecolors?, ?c?, ?s?, ?x?.
Additional kwargs: hold = [True|False] overrides default hold state
(Source code, png, hires.png, pdf)
Set the current image. This image will be the target of colormap commands like jet(), hot() or clim()). The current image is an attribute of the current axes.
matplotlib.pyplot.semilogx(*args, **kwargs)
Make a plot with log scaling on the x axis.
Call signature:
semilogx(*args, **kwargs)
semilogx() supports all the keyword arguments of plot() and matplotlib.axes.Axes.set_xscale().
Notable keyword arguments:
basex: scalar > 1
Base of the x logarithm
subsx: [ None | sequence ]
The location of the minor xticks; None defaults to autosubs, which depend on the number of decades in the plot; see set_xscale() for details.
nonposx: [ ?mask? | ?clip? ]
Non-positive values in x can be masked as invalid, or clipped to a very small positive number
The remaining valid kwargs are Line2D properties:
Property Description
agg_filter unknown
alpha float (0.0 transparent through 1.0 opaque)
animated [True | False]
antialiased or aa [True | False]
axes an Axes instance
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
color or c any matplotlib color
contains a callable function
dash_capstyle [?butt? | ?round? | ?projecting?]
dash_joinstyle [?miter? | ?round? | ?bevel?]
dashes sequence of on/off ink in points
drawstyle [?default? | ?steps? | ?steps-pre? | ?steps-mid? | ?steps-post?]
figure a matplotlib.figure.Figure instance
fillstyle [?full? | ?left? | ?right? | ?bottom? | ?top? | ?none?]
gid an id string
label string or anything printable with ?%s? conversion.
linestyle or ls [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or lw float value in points
marker A valid marker style
markeredgecolor or mec any matplotlib color
markeredgewidth or mew float value in points
markerfacecolor or mfc any matplotlib color
markerfacecoloralt or mfcalt any matplotlib color
markersize or ms float
markevery [None | int | length-2 tuple of int | slice | list/array of int | float | length-2 tuple of float]
path_effects unknown
picker float distance in points or callable pick function fn(artist, event)
pickradius float distance in points
rasterized [True | False | None]
sketch_params unknown
snap unknown
solid_capstyle [?butt? | ?round? | ?projecting?]
solid_joinstyle [?miter? | ?round? | ?bevel?]
transform a matplotlib.transforms.Transform instance
url a url string
visible [True | False]
xdata 1D array
ydata 1D array
zorder any number
See also
For example code and figure
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.semilogy(*args, **kwargs)
Make a plot with log scaling on the y axis.
call signature:
semilogy(*args, **kwargs)
semilogy() supports all the keyword arguments of plot() and matplotlib.axes.Axes.set_yscale().
Notable keyword arguments:
basey: scalar > 1
Base of the y logarithm
subsy: [ None | sequence ]
The location of the minor yticks; None defaults to autosubs, which depend on the number of decades in the plot; see set_yscale() for details.
nonposy: [ ?mask? | ?clip? ]
Non-positive values in y can be masked as invalid, or clipped to a very small positive number
The remaining valid kwargs are Line2D properties:
Property Description
agg_filter unknown
alpha float (0.0 transparent through 1.0 opaque)
animated [True | False]
antialiased or aa [True | False]
axes an Axes instance
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
color or c any matplotlib color
contains a callable function
dash_capstyle [?butt? | ?round? | ?projecting?]
dash_joinstyle [?miter? | ?round? | ?bevel?]
dashes sequence of on/off ink in points
drawstyle [?default? | ?steps? | ?steps-pre? | ?steps-mid? | ?steps-post?]
figure a matplotlib.figure.Figure instance
fillstyle [?full? | ?left? | ?right? | ?bottom? | ?top? | ?none?]
gid an id string
label string or anything printable with ?%s? conversion.
linestyle or ls [?solid? | ?dashed?, ?dashdot?, ?dotted? | (offset, on-off-dash-seq) | '-' | '--' | '-.' | ':' | 'None' | ' ' | '']
linewidth or lw float value in points
marker A valid marker style
markeredgecolor or mec any matplotlib color
markeredgewidth or mew float value in points
markerfacecolor or mfc any matplotlib color
markerfacecoloralt or mfcalt any matplotlib color
markersize or ms float
markevery [None | int | length-2 tuple of int | slice | list/array of int | float | length-2 tuple of float]
path_effects unknown
picker float distance in points or callable pick function fn(artist, event)
pickradius float distance in points
rasterized [True | False | None]
sketch_params unknown
snap unknown
solid_capstyle [?butt? | ?round? | ?projecting?]
solid_joinstyle [?miter? | ?round? | ?bevel?]
transform a matplotlib.transforms.Transform instance
url a url string
visible [True | False]
xdata 1D array
ydata 1D array
zorder any number
See also
For example code and figure
Additional kwargs: hold = [True|False] overrides default hold state
Set the default colormap. Applies to the current image if any. See help(colormaps) for more information.
cmap must be a Colormap instance, or the name of a registered colormap.
See matplotlib.cm.register_cmap() and matplotlib.cm.get_cmap().
matplotlib.pyplot.setp(*args, **kwargs)
Set a property on an artist object.
matplotlib supports the use of setp() (?set property?) and getp() to set and get object properties, as well as to do introspection on the object. For example, to set the linestyle of a line to be
dashed, you can do:
>>> line, = plot([1,2,3])
>>> setp(line, linestyle='--')
If you want to know the valid types of arguments, you can provide the name of the property you want to set without a value:
>>> setp(line, 'linestyle')
linestyle: [ '-' | '--' | '-.' | ':' | 'steps' | 'None' ]
If you want to see all the properties that can be set, and their possible values, you can do:
>>> setp(line)
... long output listing omitted
setp() operates on a single instance or a list of instances. If you are in query mode introspecting the possible values, only the first instance in the sequence is used. When actually setting
values, all the instances will be set. e.g., suppose you have a list of two lines, the following will make both lines thicker and red:
>>> x = arange(0,1.0,0.01)
>>> y1 = sin(2*pi*x)
>>> y2 = sin(4*pi*x)
>>> lines = plot(x, y1, x, y2)
>>> setp(lines, linewidth=2, color='r')
setp() works with the MATLAB style string/value pairs or with python kwargs. For example, the following are equivalent:
>>> setp(lines, 'linewidth', 2, 'color', 'r') # MATLAB style
>>> setp(lines, linewidth=2, color='r') # python style
matplotlib.pyplot.show(*args, **kw)
Display a figure. When running in ipython with its pylab mode, display all figures and return to the ipython prompt.
In non-interactive mode, display all figures and block until the figures have been closed; in interactive mode it has no effect unless figures were created prior to a change from non-interactive
to interactive mode (not recommended). In that case it displays the figures but does not block.
A single experimental keyword argument, block, may be set to True or False to override the blocking behavior described above.
matplotlib.pyplot.specgram(x, NFFT=None, Fs=None, Fc=None, detrend=None, window=None, noverlap=None, cmap=None, xextent=None, pad_to=None, sides=None, scale_by_freq=None, mode=None, scale=None, vmin=
None, vmax=None, hold=None, data=None, **kwargs)
Plot a spectrogram.
Call signature:
specgram(x, NFFT=256, Fs=2, Fc=0, detrend=mlab.detrend_none,
window=mlab.window_hanning, noverlap=128,
cmap=None, xextent=None, pad_to=None, sides='default',
scale_by_freq=None, mode='default', scale='default',
Compute and plot a spectrogram of data in x. Data are split into NFFT length segments and the spectrum of each section is computed. The windowing function window is applied to each segment, and
the amount of overlap of each segment is specified with noverlap. The spectrogram is plotted as a colormap (using imshow).
x: 1-D array or sequence
Array or sequence containing the data
Keyword arguments:
Fs: scalar
The sampling frequency (samples per time unit). It is used to calculate the Fourier frequencies, freqs, in cycles per time unit. The default value is 2.
window: callable or ndarray
A function or a vector of length NFFT. To create window vectors see window_hanning(), window_none(), numpy.blackman(), numpy.hamming(), numpy.bartlett(), scipy.signal(),
scipy.signal.get_window(), etc. The default is window_hanning(). If a function is passed as the argument, it must take a data segment as an argument and return the windowed version of the
sides: [ ?default? | ?onesided? | ?twosided? ]
Specifies which sides of the spectrum to return. Default gives the default behavior, which returns one-sided for real data and both for complex data. ?onesided? forces the return of a
one-sided spectrum, while ?twosided? forces two-sided.
pad_to: integer
The number of points to which the data segment is padded when performing the FFT. This can be different from NFFT, which specifies the number of data points used. While not increasing the
actual resolution of the spectrum (the minimum distance between resolvable peaks), this can give more points in the plot, allowing for more detail. This corresponds to the n parameter in the
call to fft(). The default is None, which sets pad_to equal to NFFT
NFFT: integer
The number of data points used in each block for the FFT. A power 2 is most efficient. The default value is 256. This should NOT be used to get zero padding, or the scaling of the result will
be incorrect. Use pad_to for this instead.
detrend: [ ?default? | ?constant? | ?mean? | ?linear? | ?none?] or
The function applied to each segment before fft-ing, designed to remove the mean or linear trend. Unlike in MATLAB, where the detrend parameter is a vector, in matplotlib is it a function.
The pylab module defines detrend_none(), detrend_mean(), and detrend_linear(), but you can use a custom function as well. You can also use a string to choose one of the functions. ?default?,
?constant?, and ?mean? call detrend_mean(). ?linear? calls detrend_linear(). ?none? calls detrend_none().
scale_by_freq: boolean
Specifies whether the resulting density values should be scaled by the scaling frequency, which gives density in units of Hz^-1. This allows for integration over the returned frequency
values. The default is True for MATLAB compatibility.
mode: [ ?default? | ?psd? | ?magnitude? | ?angle? | ?phase? ]
What sort of spectrum to use. Default is ?psd?. which takes the power spectral density. ?complex? returns the complex-valued frequency spectrum. ?magnitude? returns the magnitude
spectrum. ?angle? returns the phase spectrum without unwrapping. ?phase? returns the phase spectrum with unwrapping.
noverlap: integer
The number of points of overlap between blocks. The default value is 128.
scale: [ ?default? | ?linear? | ?dB? ]
The scaling of the values in the spec. ?linear? is no scaling. ?dB? returns the values in dB scale. When mode is ?psd?, this is dB power (10 * log10). Otherwise this is dB amplitude (20 *
log10). ?default? is ?dB? if mode is ?psd? or ?magnitude? and ?linear? otherwise. This must be ?linear? if mode is ?angle? or ?phase?.
Fc: integer
The center frequency of x (defaults to 0), which offsets the x extents of the plot to reflect the frequency range used when a signal is acquired and then filtered and downsampled to
A matplotlib.colors.Colormap instance; if None, use default determined by rc
The image extent along the x-axis. xextent = (xmin,xmax) The default is (0,max(bins)), where bins is the return value from specgram()
Additional kwargs are passed on to imshow which makes the specgram image
detrend and scale_by_freq only apply when mode is set to ?psd?
Returns the tuple (spectrum, freqs, t, im):
spectrum: 2-D array
columns are the periodograms of successive segments
freqs: 1-D array
The frequencies corresponding to the rows in spectrum
t: 1-D array
The times corresponding to midpoints of segments (i.e the columns in spectrum)
im: instance of class AxesImage
The image created by imshow containing the spectrogram
(Source code, png, hires.png, pdf)
See also
psd() differs in the default overlap; in returning the mean of the segment periodograms; in not returning times; and in generating a line plot instead of colormap.
A single spectrum, similar to having a single segment when mode is ?magnitude?. Plots a line instead of a colormap.
A single spectrum, similar to having a single segment when mode is ?angle?. Plots a line instead of a colormap.
A single spectrum, similar to having a single segment when mode is ?phase?. Plots a line instead of a colormap.
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?x?.
Additional kwargs: hold = [True|False] overrides default hold state
set the default colormap to spectral and apply to current image if any. See help(colormaps) for more information
set the default colormap to spring and apply to current image if any. See help(colormaps) for more information
matplotlib.pyplot.spy(Z, precision=0, marker=None, markersize=None, aspect='equal', hold=None, **kwargs)
Plot the sparsity pattern on a 2-D array.
spy(Z) plots the sparsity pattern of the 2-D array Z.
Z : sparse array (n, m)
The array to be plotted.
precision : float, optional, default: 0
If precision is 0, any non-zero value will be plotted; else, values of
For scipy.sparse.spmatrix instances, there is a special case: if precision is ?present?, any value present in the array will be plotted, even if it is identically zero.
origin : [?upper?, ?lower?], optional, default: ?upper?
Place the [0,0] index of the array in the upper left or lower left corner of the axes.
aspect : [?auto? | ?equal? | scalar], optional, default: ?equal?
If ?equal?, and extent is None, changes the axes aspect ratio to match that of the image. If extent is not None, the axes aspect ratio is changed to match that of the extent.
If ?auto?, changes the image aspect ratio to match that of the axes.
If None, default to rc image.aspect value.
Two plotting styles are available: image or marker. Both
are available for full arrays, but only the marker style
works for :class:`scipy.sparse.spmatrix` instances.
If *marker* and *markersize* are *None*, an image will be
returned and any remaining kwargs are passed to
:func:`~matplotlib.pyplot.imshow`; else, a
:class:`~matplotlib.lines.Line2D` object will be returned with
the value of marker determining the marker type, and any
remaining kwargs passed to the
:meth:`~matplotlib.axes.Axes.plot` method.
If *marker* and *markersize* are *None*, useful kwargs include:
* *cmap*
* *alpha*
See also
for image options.
for plotting options
matplotlib.pyplot.stackplot(x, *args, **kwargs)
Draws a stacked area plot.
x : 1d array of dimension N
: 2d array of dimension MxN, OR any number 1d arrays each of dimension
1xN. The data is assumed to be unstacked. Each of the following calls is legal:
stackplot(x, y) # where y is MxN
stackplot(x, y1, y2, y3, y4) # where y1, y2, y3, y4, are all 1xNm
Keyword arguments:
: [?zero?, ?sym?, ?wiggle?, ?weighted_wiggle?]Method used to calculate the baseline. ?zero? is just a simple stacked plot. ?sym? is symmetric around zero and is sometimes called ThemeRiver. ?wiggle?
minimizes the sum of the squared slopes. ?weighted_wiggle? does the same but weights to account for size of each layer. It is also called Streamgraph-layout. More details can be found at http://
labels : A list or tuple of labels to assign to each data series.
: A list or tuple of colors. These will be cycled through andused to colour the stacked areas. All other keyword arguments are passed to fill_between()
Returns r : A list of PolyCollection, one for each element in the stacked area plot.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.stem(*args, **kwargs)
Create a stem plot.
Call signatures:
stem(y, linefmt='b-', markerfmt='bo', basefmt='r-')
stem(x, y, linefmt='b-', markerfmt='bo', basefmt='r-')
A stem plot plots vertical lines (using linefmt) at each x location from the baseline to y, and places a marker there using markerfmt. A horizontal line at 0 is is plotted using basefmt.
If no x values are provided, the default is (0, 1, ..., len(y) - 1)
Return value is a tuple (markerline, stemlines, baseline).
(Source code, png, hires.png, pdf)
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All positional and all keyword arguments.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.step(x, y, *args, **kwargs)
Make a step plot.
Call signature:
step(x, y, *args, **kwargs)
Additional keyword args to step() are the same as those for plot().
x and y must be 1-D sequences, and it is assumed, but not checked, that x is uniformly increasing.
Keyword arguments:
where: [ ?pre? | ?post? | ?mid? ]
If ?pre? (the default), the interval from x[i] to x[i+1] has level y[i+1].
If ?post?, that interval has level y[i].
If ?mid?, the jumps in y occur half-way between the x-values.
Return value is a list of lines that were added.
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?y?, ?x?.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.streamplot(x, y, u, v, density=1, linewidth=None, color=None, cmap=None, norm=None, arrowsize=1, arrowstyle='-|>', minlength=0.1, transform=None, zorder=1, start_points=None, hold=
None, data=None)
Draws streamlines of a vector flow.
x, y
: 1d arrays
an evenly spaced grid.
u, v
: 2d arraysx and y-velocities. Number of rows should match length of y, and the number of columns should match x. density : float or 2-tupleControls the closeness of streamlines. When density = 1,
the domain is divided into a 30x30 grid?density linearly scales this grid. Each cell in the grid can have, at most, one traversing streamline. For different densities in each direction, use
[density_x, density_y]. linewidth : numeric or 2d arrayvary linewidth when given a 2d array with the same shape as velocities. color : matplotlib color code, or 2d arrayStreamline color. When given
an array with the same shape as velocities, color values are converted to colors using cmap. cmap : ColormapColormap used to plot streamlines and arrows. Only necessary when using an array input for
color. norm : NormalizeNormalize object used to scale luminance data to 0, 1. If None, stretch (min, max) to (0, 1). Only necessary when color is an array. arrowsize : floatFactor scale arrow size.
arrowstyle : strArrow style specification. See FancyArrowPatch. minlength : floatMinimum length of streamline in axes coordinates. start_points: Nx2 array Coordinates of starting points for the
streamlines. In data coordinates, the same as the x and y arrays. zorder : intany number
: StreamplotSet
Container object with attributes
This container will probably change in the future to allow changes to the colormap, alpha, etc. for both lines and arrows, but these changes should be backward compatible.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.subplot(*args, **kwargs)
Return a subplot axes positioned by the given grid definition.
Typical call signature:
subplot(nrows, ncols, plot_number)
Where nrows and ncols are used to notionally split the figure into nrows * ncols sub-axes, and plot_number is used to identify the particular subplot that this function is to create within the
notional grid. plot_number starts at 1, increments across rows first and has a maximum of nrows * ncols.
In the case when nrows, ncols and plot_number are all less than 10, a convenience exists, such that the a 3 digit number can be given instead, where the hundreds represent nrows, the tens
represent ncols and the units represent plot_number. For instance:
produces a subaxes in a figure which represents the top plot (i.e. the first) in a 2 row by 1 column notional grid (no grid actually exists, but conceptually this is how the returned subplot has
been positioned).
Creating a new subplot with a position which is entirely inside a pre-existing axes will trigger the larger axes to be deleted:
import matplotlib.pyplot as plt
# plot a line, implicitly creating a subplot(111)
# now create a subplot which represents the top plot of a grid
# with 2 rows and 1 column. Since this subplot will overlap the
# first, the plot (and its axes) previously created, will be removed
plt.subplot(212, axisbg='y') # creates 2nd subplot with yellow background
If you do not want this behavior, use the add_subplot() method or the axes() function instead.
Keyword arguments:
The background color of the subplot, which can be any valid color specifier. See matplotlib.colors for more information.
A boolean flag indicating whether the subplot plot should be a polar projection. Defaults to False.
A string giving the name of a custom projection to be used for the subplot. This projection must have been previously registered. See matplotlib.projections.
See also
For additional information on axes() and subplot() keyword arguments.
For an example
(Source code, png, hires.png, pdf)
matplotlib.pyplot.subplot2grid(shape, loc, rowspan=1, colspan=1, **kwargs)
Create a subplot in a grid. The grid is specified by shape, at location of loc, spanning rowspan, colspan cells in each direction. The index for loc is 0-based.
subplot2grid(shape, loc, rowspan=1, colspan=1)
is identical to
gridspec=GridSpec(shape[0], shape[1])
subplotspec=gridspec.new_subplotspec(loc, rowspan, colspan)
Launch a subplot tool window for a figure.
A matplotlib.widgets.SubplotTool instance is returned.
matplotlib.pyplot.subplots(nrows=1, ncols=1, sharex=False, sharey=False, squeeze=True, subplot_kw=None, gridspec_kw=None, **fig_kw)
Create a figure with a set of subplots already made.
This utility wrapper makes it convenient to create common layouts of subplots, including the enclosing figure object, in a single call.
Keyword arguments:
: int
Number of rows of the subplot grid. Defaults to 1.
: intNumber of columns of the subplot grid. Defaults to 1. sharex : string or boolIf True, the X axis will be shared amongst all subplots. If True and you have multiple rows, the x tick labels on all
but the last row of plots will have visible set to False If a string must be one of ?row?, ?col?, ?all?, or ?none?. ?all? has the same effect as True, ?none? has the same effect as False. If ?row?,
each subplot row will share a X axis. If ?col?, each subplot column will share a X axis and the x tick labels on all but the last row will have visible set to False. sharey : string or boolIf True,
the Y axis will be shared amongst all subplots. If True and you have multiple columns, the y tick labels on all but the first column of plots will have visible set to False If a string must be one of
?row?, ?col?, ?all?, or ?none?. ?all? has the same effect as True, ?none? has the same effect as False. If ?row?, each subplot row will share a Y axis and the y tick labels on all but the first
column will have visible set to False. If ?col?, each subplot column will share a Y axis. squeeze : bool
If True, extra dimensions are squeezed out from the returned axis object:
• if only one subplot is constructed (nrows=ncols=1), the resulting single Axis object is returned as a scalar.
• for Nx1 or 1xN subplots, the returned object is a 1-d numpy object array of Axis objects are returned as numpy 1-d arrays.
• for NxM subplots with N>1 and M>1 are returned as a 2d array.
If False, no squeezing at all is done: the returned axis object is always a 2-d array containing Axis instances, even if it ends up being 1x1.
: dictDict with keywords passed to the add_subplot() call used to create each subplots. gridspec_kw : dictDict with keywords passed to the GridSpec constructor used to create the grid the subplots
are placed on. fig_kw : dictDict with keywords passed to the figure() call. Note that all keywords not recognized above will be automatically included here.
fig, ax : tuple
• fig is the matplotlib.figure.Figure object
• ax can be either a single axis object or an array of axis objects if more than one subplot was created. The dimensions of the resulting array can be controlled with the squeeze keyword, see
x = np.linspace(0, 2*np.pi, 400)
y = np.sin(x**2)
# Just a figure and one subplot
f, ax = plt.subplots()
ax.plot(x, y)
ax.set_title('Simple plot')
# Two subplots, unpack the output array immediately
f, (ax1, ax2) = plt.subplots(1, 2, sharey=True)
ax1.plot(x, y)
ax1.set_title('Sharing Y axis')
ax2.scatter(x, y)
# Four polar axes
plt.subplots(2, 2, subplot_kw=dict(polar=True))
# Share a X axis with each column of subplots
plt.subplots(2, 2, sharex='col')
# Share a Y axis with each row of subplots
plt.subplots(2, 2, sharey='row')
# Share a X and Y axis with all subplots
plt.subplots(2, 2, sharex='all', sharey='all')
# same as
plt.subplots(2, 2, sharex=True, sharey=True)
matplotlib.pyplot.subplots_adjust(*args, **kwargs)
Tune the subplot layout.
call signature:
subplots_adjust(left=None, bottom=None, right=None, top=None,
wspace=None, hspace=None)
The parameter meanings (and suggested defaults) are:
left = 0.125 # the left side of the subplots of the figure
right = 0.9 # the right side of the subplots of the figure
bottom = 0.1 # the bottom of the subplots of the figure
top = 0.9 # the top of the subplots of the figure
wspace = 0.2 # the amount of width reserved for blank space between subplots
hspace = 0.2 # the amount of height reserved for white space between subplots
The actual defaults are controlled by the rc file
set the default colormap to summer and apply to current image if any. See help(colormaps) for more information
matplotlib.pyplot.suptitle(*args, **kwargs)
Add a centered title to the figure.
kwargs are matplotlib.text.Text properties. Using figure coordinates, the defaults are:
: 0.5
The x location of the text in figure coords
: 0.98The y location of the text in figure coords horizontalalignment : ?center?The horizontal alignment of the text verticalalignment : ?top?The vertical alignment of the text
A matplotlib.text.Text instance is returned.
fig.suptitle('this is the figure title', fontsize=12)
Switch the default backend. This feature is experimental, and is only expected to work switching to an image backend. e.g., if you have a bunch of PostScript scripts that you want to run from an
interactive ipython session, you may want to switch to the PS backend before running them to avoid having a bunch of GUI windows popup. If you try to interactively switch from one GUI backend to
another, you will explode.
Calling this command will close all open windows.
Add a table to the current axes.
Call signature:
table(cellText=None, cellColours=None,
cellLoc='right', colWidths=None,
rowLabels=None, rowColours=None, rowLoc='left',
colLabels=None, colColours=None, colLoc='center',
loc='bottom', bbox=None):
Returns a matplotlib.table.Table instance. For finer grained control over tables, use the Table class and add it to the axes with add_table().
Thanks to John Gill for providing the class and table.
kwargs control the Table properties:
Property Description
agg_filter unknown
alpha float (0.0 transparent through 1.0 opaque)
animated [True | False]
axes an Axes instance
clip_box a matplotlib.transforms.Bbox instance
clip_on [True | False]
clip_path [ (Path, Transform) | Patch | None ]
contains a callable function
figure a matplotlib.figure.Figure instance
fontsize a float in points
gid an id string
label string or anything printable with ?%s? conversion.
path_effects unknown
picker [None|float|boolean|callable]
rasterized [True | False | None]
sketch_params unknown
snap unknown
transform Transform instance
url a url string
visible [True | False]
zorder any number
matplotlib.pyplot.text(x, y, s, fontdict=None, withdash=False, **kwargs)
Add text to the axes.
Add text in string s to axis at location x, y, data coordinates.
x, y : scalars
data coordinates
s : string
fontdict : dictionary, optional, default: None
A dictionary to override the default text properties. If fontdict is None, the defaults are determined by your rc parameters.
withdash : boolean, optional, default: False
Creates a TextWithDash instance instead of a Text instance.
Other Parameters:
kwargs : Text properties.
Other miscellaneous text parameters.
Individual keyword arguments can be used to override any given parameter:
>>> text(x, y, s, fontsize=12)
The default transform specifies that text is in data coords, alternatively, you can specify text in axis coords (0,0 is lower-left and 1,1 is upper-right). The example below places text in the
center of the axes:
>>> text(0.5, 0.5,'matplotlib', horizontalalignment='center',
... verticalalignment='center',
... transform=ax.transAxes)
You can put a rectangular box around the text instance (e.g., to set a background color) by using the keyword bbox. bbox is a dictionary of Rectangle properties. For example:
>>> text(x, y, s, bbox=dict(facecolor='red', alpha=0.5))
matplotlib.pyplot.thetagrids(*args, **kwargs)
Get or set the theta locations of the gridlines in a polar plot.
If no arguments are passed, return a tuple (lines, labels) where lines is an array of radial gridlines (Line2D instances) and labels is an array of tick labels (Text instances):
lines, labels = thetagrids()
Otherwise the syntax is:
lines, labels = thetagrids(angles, labels=None, fmt='%d', frac = 1.1)
set the angles at which to place the theta grids (these gridlines are equal along the theta dimension).
angles is in degrees.
labels, if not None, is a len(angles) list of strings of the labels to use at each angle.
If labels is None, the labels will be fmt%angle.
frac is the fraction of the polar axes radius at which to place the label (1 is the edge). e.g., 1.05 is outside the axes and 0.95 is inside the axes.
Return value is a list of tuples (lines, labels):
□ lines are Line2D instances
□ labels are Text instances.
Note that on input, the labels argument is a list of strings, and on output it is a list of Text instances.
# set the locations of the radial gridlines and labels
lines, labels = thetagrids( range(45,360,90) )
# set the locations and labels of the radial gridlines and labels
lines, labels = thetagrids( range(45,360,90), ('NE', 'NW', 'SW','SE') )
matplotlib.pyplot.tick_params(axis='both', **kwargs)
Change the appearance of ticks and tick labels.
Keyword arguments:
: [?x? | ?y? | ?both?]
Axis on which to operate; default is ?both?.
: [True | False]If True, set all parameters to defaults before processing other keyword arguments. Default is False. which : [?major? | ?minor? | ?both?]Default is ?major?; apply arguments to which
ticks. direction : [?in? | ?out? | ?inout?]Puts ticks inside the axes, outside the axes, or both. length Tick length in points. width Tick width in points. color Tick color; accepts any mpl color
spec. pad Distance in points between tick and label. labelsize Tick label font size in points or as a string (e.g., ?large?). labelcolor Tick label color; mpl color spec. colors Changes the tick
color and the label color to the same value: mpl color spec. zorder Tick and label zorder. bottom, top, left, right : [bool | ?on? | ?off?]controls whether to draw the respective ticks. labelbottom,
labeltop, labelleft, labelright Boolean or [?on? | ?off?], controls whether to draw the respective tick labels.
ax.tick_params(direction='out', length=6, width=2, colors='r')
This will make all major ticks be red, pointing out of the box, and with dimensions 6 points by 2 points. Tick labels will also be red.
Change the ScalarFormatter used by default for linear axes.
Optional keyword arguments:
Keyword Description
style [ ?sci? (or ?scientific?) | ?plain? ] plain turns off scientific notation
scilimits (m, n), pair of integers; if style is ?sci?, scientific notation will be used for numbers outside the range 10`m`:sup: to 10`n`:sup:. Use (0,0) to include all numbers.
useOffset [True | False | offset]; if True, the offset will be calculated as needed; if False, no offset will be used; if a numeric offset is specified, it will be used.
axis [ ?x? | ?y? | ?both? ]
useLocale If True, format the number according to the current locale. This affects things such as the character used for the decimal separator. If False, use C-style (English) formatting. The
default setting is controlled by the axes.formatter.use_locale rcparam.
Only the major ticks are affected. If the method is called when the ScalarFormatter is not the Formatter being used, an AttributeError will be raised.
matplotlib.pyplot.tight_layout(pad=1.08, h_pad=None, w_pad=None, rect=None)
Automatically adjust subplot parameters to give specified padding.
: float
padding between the figure edge and the edges of subplots, as a fraction of the font-size.
h_pad, w_pad
: floatpadding (height/width) between edges of adjacent subplots. Defaults to pad_inches. rect : if rect is given, it is interpreted as a rectangle(left, bottom, right, top) in the normalized figure
coordinate that the whole subplots area (including labels) will fit into. Default is (0, 0, 1, 1).
matplotlib.pyplot.title(s, *args, **kwargs)
Set a title of the current axes.
Set one of the three available axes titles. The available titles are positioned above the axes in the center, flush with the left edge, and flush with the right edge.
See also
See text() for adding text to the current axes
label : str
Text to use for the title
fontdict : dict
Parameters: A dictionary controlling the appearance of the title text, the default fontdict is:
{?fontsize?: rcParams[?axes.titlesize?], ?fontweight? : rcParams[?axes.titleweight?], ?verticalalignment?: ?baseline?, ?horizontalalignment?: loc}
loc : {?center?, ?left?, ?right?}, str, optional
Which title to set, defaults to ?center?
text : Text
The matplotlib text instance representing the title
Other Parameters:
kwargs : text properties
Other keyword arguments are text properties, see Text for a list of valid text properties.
matplotlib.pyplot.tricontour(*args, **kwargs)
Draw contours on an unstructured triangular grid. tricontour() and tricontourf() draw contour lines and filled contours, respectively. Except as noted, function signatures and return values are
the same for both versions.
The triangulation can be specified in one of two ways; either:
tricontour(triangulation, ...)
where triangulation is a matplotlib.tri.Triangulation object, or
tricontour(x, y, ...)
tricontour(x, y, triangles, ...)
tricontour(x, y, triangles=triangles, ...)
tricontour(x, y, mask=mask, ...)
tricontour(x, y, triangles, mask=mask, ...)
in which case a Triangulation object will be created. See Triangulation for a explanation of these possibilities.
The remaining arguments may be:
tricontour(..., Z)
where Z is the array of values to contour, one per point in the triangulation. The level values are chosen automatically.
tricontour(..., Z, N)
contour N automatically-chosen levels.
tricontour(..., Z, V)
draw contour lines at the values specified in sequence V, which must be in increasing order.
tricontourf(..., Z, V)
fill the (len(V)-1) regions between the values in V, which must be in increasing order.
tricontour(Z, **kwargs)
Use keyword args to control colors, linewidth, origin, cmap ... see below for more details.
C = tricontour(...) returns a TriContourSet object.
Optional keyword arguments:
colors: [ None | string | (mpl_colors) ]
If None, the colormap specified by cmap will be used.
If a string, like ?r? or ?red?, all levels will be plotted in this color.
If a tuple of matplotlib color args (string, float, rgb, etc), different levels will be plotted in different colors in the order specified.
alpha: float
The alpha blending value
cmap: [ None | Colormap ]
A cm Colormap instance or None. If cmap is None and colors is None, a default Colormap is used.
norm: [ None | Normalize ]
A matplotlib.colors.Normalize instance for scaling data values to colors. If norm is None and colors is None, the default linear scaling is used.
levels [level0, level1, ..., leveln]
A list of floating point numbers indicating the level curves to draw, in increasing order; e.g., to draw just the zero contour pass levels=[0]
origin: [ None | ?upper? | ?lower? | ?image? ]
If None, the first value of Z will correspond to the lower left corner, location (0,0). If ?image?, the rc value for image.origin will be used.
This keyword is not active if X and Y are specified in the call to contour.
extent: [ None | (x0,x1,y0,y1) ]
If origin is not None, then extent is interpreted as in matplotlib.pyplot.imshow(): it gives the outer pixel boundaries. In this case, the position of Z[0,0] is the center of the pixel, not a
corner. If origin is None, then (x0, y0) is the position of Z[0,0], and (x1, y1) is the position of Z[-1,-1].
This keyword is not active if X and Y are specified in the call to contour.
locator: [ None | ticker.Locator subclass ]
If locator is None, the default MaxNLocator is used. The locator is used to determine the contour levels if they are not given explicitly via the V argument.
extend: [ ?neither? | ?both? | ?min? | ?max? ]
Unless this is ?neither?, contour levels are automatically added to one or both ends of the range so that all data are included. These added ranges are then mapped to the special colormap
values which default to the ends of the colormap range, but can be set via matplotlib.colors.Colormap.set_under() and matplotlib.colors.Colormap.set_over() methods.
xunits, yunits: [ None | registered units ]
Override axis units by specifying an instance of a matplotlib.units.ConversionInterface.
tricontour-only keyword arguments:
linewidths: [ None | number | tuple of numbers ]
If linewidths is None, the default width in lines.linewidth in matplotlibrc is used.
If a number, all levels will be plotted with this linewidth.
If a tuple, different levels will be plotted with different linewidths in the order specified
linestyles: [ None | ?solid? | ?dashed? | ?dashdot? | ?dotted? ]
If linestyles is None, the ?solid? is used.
linestyles can also be an iterable of the above strings specifying a set of linestyles to be used. If this iterable is shorter than the number of contour levels it will be repeated as
If contour is using a monochrome colormap and the contour level is less than 0, then the linestyle specified in contour.negative_linestyle in matplotlibrc will be used.
tricontourf-only keyword arguments:
antialiased: [ True | False ]
enable antialiasing
Note: tricontourf fills intervals that are closed at the top; that is, for boundaries z1 and z2, the filled region is:
z1 < z <= z2
There is one exception: if the lowest boundary coincides with the minimum value of the z array, then that minimum value will be included in the lowest interval.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.tricontourf(*args, **kwargs)
Draw contours on an unstructured triangular grid. tricontour() and tricontourf() draw contour lines and filled contours, respectively. Except as noted, function signatures and return values are
the same for both versions.
The triangulation can be specified in one of two ways; either:
tricontour(triangulation, ...)
where triangulation is a matplotlib.tri.Triangulation object, or
tricontour(x, y, ...)
tricontour(x, y, triangles, ...)
tricontour(x, y, triangles=triangles, ...)
tricontour(x, y, mask=mask, ...)
tricontour(x, y, triangles, mask=mask, ...)
in which case a Triangulation object will be created. See Triangulation for a explanation of these possibilities.
The remaining arguments may be:
tricontour(..., Z)
where Z is the array of values to contour, one per point in the triangulation. The level values are chosen automatically.
tricontour(..., Z, N)
contour N automatically-chosen levels.
tricontour(..., Z, V)
draw contour lines at the values specified in sequence V, which must be in increasing order.
tricontourf(..., Z, V)
fill the (len(V)-1) regions between the values in V, which must be in increasing order.
tricontour(Z, **kwargs)
Use keyword args to control colors, linewidth, origin, cmap ... see below for more details.
C = tricontour(...) returns a TriContourSet object.
Optional keyword arguments:
colors: [ None | string | (mpl_colors) ]
If None, the colormap specified by cmap will be used.
If a string, like ?r? or ?red?, all levels will be plotted in this color.
If a tuple of matplotlib color args (string, float, rgb, etc), different levels will be plotted in different colors in the order specified.
alpha: float
The alpha blending value
cmap: [ None | Colormap ]
A cm Colormap instance or None. If cmap is None and colors is None, a default Colormap is used.
norm: [ None | Normalize ]
A matplotlib.colors.Normalize instance for scaling data values to colors. If norm is None and colors is None, the default linear scaling is used.
levels [level0, level1, ..., leveln]
A list of floating point numbers indicating the level curves to draw, in increasing order; e.g., to draw just the zero contour pass levels=[0]
origin: [ None | ?upper? | ?lower? | ?image? ]
If None, the first value of Z will correspond to the lower left corner, location (0,0). If ?image?, the rc value for image.origin will be used.
This keyword is not active if X and Y are specified in the call to contour.
extent: [ None | (x0,x1,y0,y1) ]
If origin is not None, then extent is interpreted as in matplotlib.pyplot.imshow(): it gives the outer pixel boundaries. In this case, the position of Z[0,0] is the center of the pixel, not a
corner. If origin is None, then (x0, y0) is the position of Z[0,0], and (x1, y1) is the position of Z[-1,-1].
This keyword is not active if X and Y are specified in the call to contour.
locator: [ None | ticker.Locator subclass ]
If locator is None, the default MaxNLocator is used. The locator is used to determine the contour levels if they are not given explicitly via the V argument.
extend: [ ?neither? | ?both? | ?min? | ?max? ]
Unless this is ?neither?, contour levels are automatically added to one or both ends of the range so that all data are included. These added ranges are then mapped to the special colormap
values which default to the ends of the colormap range, but can be set via matplotlib.colors.Colormap.set_under() and matplotlib.colors.Colormap.set_over() methods.
xunits, yunits: [ None | registered units ]
Override axis units by specifying an instance of a matplotlib.units.ConversionInterface.
tricontour-only keyword arguments:
linewidths: [ None | number | tuple of numbers ]
If linewidths is None, the default width in lines.linewidth in matplotlibrc is used.
If a number, all levels will be plotted with this linewidth.
If a tuple, different levels will be plotted with different linewidths in the order specified
linestyles: [ None | ?solid? | ?dashed? | ?dashdot? | ?dotted? ]
If linestyles is None, the ?solid? is used.
linestyles can also be an iterable of the above strings specifying a set of linestyles to be used. If this iterable is shorter than the number of contour levels it will be repeated as
If contour is using a monochrome colormap and the contour level is less than 0, then the linestyle specified in contour.negative_linestyle in matplotlibrc will be used.
tricontourf-only keyword arguments:
antialiased: [ True | False ]
enable antialiasing
Note: tricontourf fills intervals that are closed at the top; that is, for boundaries z1 and z2, the filled region is:
z1 < z <= z2
There is one exception: if the lowest boundary coincides with the minimum value of the z array, then that minimum value will be included in the lowest interval.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.tripcolor(*args, **kwargs)
Create a pseudocolor plot of an unstructured triangular grid.
The triangulation can be specified in one of two ways; either:
tripcolor(triangulation, ...)
where triangulation is a matplotlib.tri.Triangulation object, or
tripcolor(x, y, ...)
tripcolor(x, y, triangles, ...)
tripcolor(x, y, triangles=triangles, ...)
tripcolor(x, y, mask=mask, ...)
tripcolor(x, y, triangles, mask=mask, ...)
in which case a Triangulation object will be created. See Triangulation for a explanation of these possibilities.
The next argument must be C, the array of color values, either one per point in the triangulation if color values are defined at points, or one per triangle in the triangulation if color values
are defined at triangles. If there are the same number of points and triangles in the triangulation it is assumed that color values are defined at points; to force the use of color values at
triangles use the kwarg facecolors*=C instead of just *C.
shading may be ?flat? (the default) or ?gouraud?. If shading is ?flat? and C values are defined at points, the color values used for each triangle are from the mean C of the triangle?s three
points. If shading is ?gouraud? then color values must be defined at points.
The remaining kwargs are the same as for pcolor().
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.triplot(*args, **kwargs)
Draw a unstructured triangular grid as lines and/or markers.
The triangulation to plot can be specified in one of two ways; either:
triplot(triangulation, ...)
where triangulation is a matplotlib.tri.Triangulation object, or
triplot(x, y, ...)
triplot(x, y, triangles, ...)
triplot(x, y, triangles=triangles, ...)
triplot(x, y, mask=mask, ...)
triplot(x, y, triangles, mask=mask, ...)
in which case a Triangulation object will be created. See Triangulation for a explanation of these possibilities.
The remaining args and kwargs are the same as for plot().
Return a list of 2 Line2D containing respectively:
□ the lines plotted for triangles edges
□ the markers plotted for triangles nodes
Additional kwargs: hold = [True|False] overrides default hold state
Make a second axes that shares the x-axis. The new axes will overlay ax (or the current axes if ax is None). The ticks for ax2 will be placed on the right, and the ax2 instance is returned.
See also
For an example
Make a second axes that shares the y-axis. The new axis will overlay ax (or the current axes if ax is None). The ticks for ax2 will be placed on the top, and the ax2 instance is returned.
Uninstalls the matplotlib display hook.
matplotlib.pyplot.violinplot(dataset, positions=None, vert=True, widths=0.5, showmeans=False, showextrema=True, showmedians=False, points=100, bw_method=None, hold=None, data=None)
Make a violin plot.
Call signature:
violinplot(dataset, positions=None, vert=True, widths=0.5,
showmeans=False, showextrema=True, showmedians=False,
points=100, bw_method=None):
Make a violin plot for each column of dataset or each vector in sequence dataset. Each filled area extends to represent the entire data range, with optional lines at the mean, the median, the
minimum, and the maximum.
dataset : Array or a sequence of vectors.
The input data.
: array-like, default = [1, 2, ..., n]
Sets the positions of the violins. The ticks and limits are automatically set to match the positions.
: bool, default = True.
If true, creates a vertical violin plot. Otherwise, creates a horizontal violin plot.
: array-like, default = 0.5
Either a scalar or a vector that sets the maximal width of each violin. The default is 0.5, which uses about half of the available horizontal space.
: bool, default = False
If True, will toggle rendering of the means.
: bool, default = True
If True, will toggle rendering of the extrema.
: bool, default = False
If True, will toggle rendering of the medians.
: scalar, default = 100
Defines the number of points to evaluate each of the gaussian kernel density estimations at.
: str, scalar or callable, optional
The method used to calculate the estimator bandwidth. This can be ?scott?, ?silverman?, a scalar constant or a callable. If a scalar, this will be used directly as kde.factor. If a callable, it
should take a GaussianKDE instance as its only parameter and return a scalar. If None (default), ?scott? is used.
result : dict
A dictionary mapping each component of the violinplot to a list of the corresponding collection instances created. The dictionary has the following keys:
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
• All arguments with the following names: ?dataset?.
Additional kwargs: hold = [True|False] overrides default hold state
set the default colormap to viridis and apply to current image if any. See help(colormaps) for more information
matplotlib.pyplot.vlines(x, ymin, ymax, colors='k', linestyles='solid', label='', hold=None, data=None, **kwargs)
Plot vertical lines.
Plot vertical lines at each x from ymin to ymax.
x : scalar or 1D array_like
x-indexes where to plot the lines.
ymin, ymax : scalar or 1D array_like
Parameters: Respective beginning and end of each line. If scalars are provided, all lines will have same length.
colors : array_like of colors, optional, default: ?k?
linestyles : [?solid? | ?dashed? | ?dashdot? | ?dotted?], optional
label : string, optional, default: ??
Returns: lines : LineCollection
Other Parameters:
kwargs : LineCollection properties.
See also
horizontal lines
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?ymin?, ?x?, ?colors?, ?ymax?.
Additional kwargs: hold = [True|False] overrides default hold state
(Source code, png, hires.png, pdf)
matplotlib.pyplot.waitforbuttonpress(*args, **kwargs)
Call signature:
waitforbuttonpress(self, timeout=-1)
Blocking call to interact with the figure.
This will return True is a key was pressed, False if a mouse button was pressed and None if timeout was reached without either being pressed.
If timeout is negative, does not timeout.
set the default colormap to winter and apply to current image if any. See help(colormaps) for more information
matplotlib.pyplot.xcorr(x, y, normed=True, detrend=, usevlines=True, maxlags=10, hold=None, data=None, **kwargs)
Plot the cross correlation between x and y.
x : sequence of scalars of length n
y : sequence of scalars of length n
hold : boolean, optional, default: True
detrend : callable, optional, default: mlab.detrend_none
x is detrended by the detrend callable. Default is no normalization.
Parameters: normed : boolean, optional, default: True
if True, normalize the data by the autocorrelation at the 0-th lag.
usevlines : boolean, optional, default: True
if True, Axes.vlines is used to plot the vertical lines from the origin to the acorr. Otherwise, Axes.plot is used.
maxlags : integer, optional, default: 10
number of lags to show. If None, will return all 2 * len(x) - 1 lags.
(lags, c, line, b) : where:
• lags are a length 2`maxlags+1 lag vector.
Returns: • c is the 2`maxlags+1 auto correlation vectorI
• line is a Line2D instance returned by plot.
• b is the x-axis (none, if plot is used).
Other Parameters:
linestyle : Line2D prop, optional, default: None
Only used if usevlines is False.
marker : string, optional, default: ?o?
In addition to the above described arguments, this function can take a data keyword argument. If such a data argument is given, the following arguments are replaced by data[<arg>]:
□ All arguments with the following names: ?y?, ?x?.
Additional kwargs: hold = [True|False] overrides default hold state
matplotlib.pyplot.xkcd(scale=1, length=100, randomness=2)
Turns on xkcd sketch-style drawing mode. This will only have effect on things drawn after this function is called.
For best results, the ?Humor Sans? font should be installed: it is not included with matplotlib.
scale : float, optional
The amplitude of the wiggle perpendicular to the source line.
length : float, optional
The length of the wiggle along the line.
randomness : float, optional
The scale factor by which the length is shrunken or expanded.
This function works by a number of rcParams, so it will probably override others you have set before.
If you want the effects of this function to be temporary, it can be used as a context manager, for example:
with plt.xkcd():
# This figure will be in XKCD-style
fig1 = plt.figure()
# ...
# This figure will be in regular style
fig2 = plt.figure()
matplotlib.pyplot.xlabel(s, *args, **kwargs)
Set the x axis label of the current axis.
Default override is:
override = {
'fontsize' : 'small',
'verticalalignment' : 'top',
'horizontalalignment' : 'center'
See also
For information on how override and the optional args work
matplotlib.pyplot.xlim(*args, **kwargs)
Get or set the x limits of the current axes.
xmin, xmax = xlim() # return the current xlim
xlim( (xmin, xmax) ) # set the xlim to xmin, xmax
xlim( xmin, xmax ) # set the xlim to xmin, xmax
If you do not specify args, you can pass the xmin and xmax as kwargs, e.g.:
xlim(xmax=3) # adjust the max leaving min unchanged
xlim(xmin=1) # adjust the min leaving max unchanged
Setting limits turns autoscaling off for the x-axis.
The new axis limits are returned as a length 2 tuple.
matplotlib.pyplot.xscale(*args, **kwargs)
Set the scaling of the x-axis.
call signature:
xscale(scale, **kwargs)
The available scales are: ?linear? | ?log? | ?logit? | ?symlog?
Different keywords may be accepted, depending on the scale:
The base of the logarithm
nonposx/nonposy: [?mask? | ?clip? ]
non-positive values in x or y can be masked as invalid, or clipped to a very small positive number
Where to place the subticks between each major tick. Should be a sequence of integers. For example, in a log10 scale: [2, 3, 4, 5, 6, 7, 8, 9]
will place 8 logarithmically spaced minor ticks between each major tick.
nonpos: [?mask? | ?clip? ]
values beyond ]0, 1[ can be masked as invalid, or clipped to a number very close to 0 or 1
The base of the logarithm
The range (-x, x) within which the plot is linear (to avoid having the plot go to infinity around zero).
Where to place the subticks between each major tick. Should be a sequence of integers. For example, in a log10 scale: [2, 3, 4, 5, 6, 7, 8, 9]
will place 8 logarithmically spaced minor ticks between each major tick.
This allows the linear range (-linthresh to linthresh) to be stretched relative to the logarithmic range. Its value is the number of decades to use for each half of the linear range. For
example, when linscale == 1.0 (the default), the space used for the positive and negative halves of the linear range will be equal to one decade in the logarithmic range.
matplotlib.pyplot.xticks(*args, **kwargs)
Get or set the x-limits of the current tick locations and labels.
# return locs, labels where locs is an array of tick locations and
# labels is an array of tick labels.
locs, labels = xticks()
# set the locations of the xticks
xticks( arange(6) )
# set the locations and labels of the xticks
xticks( arange(5), ('Tom', 'Dick', 'Harry', 'Sally', 'Sue') )
The keyword args, if any, are Text properties. For example, to rotate long labels:
xticks( arange(12), calendar.month_name[1:13], rotation=17 )
matplotlib.pyplot.ylabel(s, *args, **kwargs)
Set the y axis label of the current axis.
Defaults override is:
override = {
'fontsize' : 'small',
'verticalalignment' : 'center',
'horizontalalignment' : 'right',
'rotation'='vertical' : }
See also
For information on how override and the optional args work.
matplotlib.pyplot.ylim(*args, **kwargs)
Get or set the y-limits of the current axes.
ymin, ymax = ylim() # return the current ylim
ylim( (ymin, ymax) ) # set the ylim to ymin, ymax
ylim( ymin, ymax ) # set the ylim to ymin, ymax
If you do not specify args, you can pass the ymin and ymax as kwargs, e.g.:
ylim(ymax=3) # adjust the max leaving min unchanged
ylim(ymin=1) # adjust the min leaving max unchanged
Setting limits turns autoscaling off for the y-axis.
The new axis limits are returned as a length 2 tuple.
matplotlib.pyplot.yscale(*args, **kwargs)
Set the scaling of the y-axis.
call signature:
yscale(scale, **kwargs)
The available scales are: ?linear? | ?log? | ?logit? | ?symlog?
Different keywords may be accepted, depending on the scale:
The base of the logarithm
nonposx/nonposy: [?mask? | ?clip? ]
non-positive values in x or y can be masked as invalid, or clipped to a very small positive number
Where to place the subticks between each major tick. Should be a sequence of integers. For example, in a log10 scale: [2, 3, 4, 5, 6, 7, 8, 9]
will place 8 logarithmically spaced minor ticks between each major tick.
nonpos: [?mask? | ?clip? ]
values beyond ]0, 1[ can be masked as invalid, or clipped to a number very close to 0 or 1
The base of the logarithm
The range (-x, x) within which the plot is linear (to avoid having the plot go to infinity around zero).
Where to place the subticks between each major tick. Should be a sequence of integers. For example, in a log10 scale: [2, 3, 4, 5, 6, 7, 8, 9]
will place 8 logarithmically spaced minor ticks between each major tick.
This allows the linear range (-linthresh to linthresh) to be stretched relative to the logarithmic range. Its value is the number of decades to use for each half of the linear range. For
example, when linscale == 1.0 (the default), the space used for the positive and negative halves of the linear range will be equal to one decade in the logarithmic range.
matplotlib.pyplot.yticks(*args, **kwargs)
Get or set the y-limits of the current tick locations and labels.
# return locs, labels where locs is an array of tick locations and
# labels is an array of tick labels.
locs, labels = yticks()
# set the locations of the yticks
yticks( arange(6) )
# set the locations and labels of the yticks
yticks( arange(5), ('Tom', 'Dick', 'Harry', 'Sally', 'Sue') )
The keyword args, if any, are Text properties. For example, to rotate long labels:
yticks( arange(12), calendar.month_name[1:13], rotation=45 )
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Cochin to Krishnagiri distance
Distance in KM
The distance from Cochin to Krishnagiri is 460.757 Km
Distance in Mile
The distance from Cochin to Krishnagiri is 286.3 Mile
Distance in Straight KM
The Straight distance from Cochin to Krishnagiri is 357.6 KM
Distance in Straight Mile
The Straight distance from Cochin to Krishnagiri is 222.2 Mile
Travel Time
Travel Time 9 Hrs and 10 Mins
Cochin Latitud and Longitude
Latitud 9.9312763 Longitude 76.26731169999994
Krishnagiri Latitud and Longitude
Latitud 12.5186353 Longitude 78.2137702 | {"url":"http://www.distancebetween.org/cochin-to-krishnagiri","timestamp":"2024-11-04T04:10:18Z","content_type":"text/html","content_length":"1758","record_id":"<urn:uuid:aac1aa04-0c4c-4b42-acc3-6d553480511b>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00882.warc.gz"} |
Learn an Integral Result from Parseval's Theorem
Learn an Integral Result from Parseval’s Theorem
Estimated Read Time: 3 minute(s)
Common Topics: result, integral, theorem, known, dx
In this Insight article, Parseval’s theorem will be applied to a sinusoidal signal that lasts a finite period of time. It will be shown that it necessarily follows that ## (\frac{\sin(2 x_o)}{x_o})
( \frac{\pi}{2})=\int\limits_{-\infty}^{+\infty} \frac{\sin(x-x_o) \sin(x+x_o)}{(x-x_o)(x+x_o)} \, dx ##.
Note: This integral result was computed by this author in 2009. I have not done a complete literature search, but I anticipate this result is probably already known.
Parseval’s Theorem:
Parseval’s theorem says ## \int\limits_{-\infty}^{+\infty} V^2(t) \, dt=\frac{1}{2 \pi} \int\limits_{-\infty}^{+\infty} |\tilde{V}(\omega)|^2 \, d \omega ##.
Let ## V(t)=\sin(\omega_o t) ## for ## 0 \leq t \leq T ##, (where ## T ## is some completely arbitrary time interval), and ## V(t)=0 ## otherwise.
Let’s compute the Fourier transform ## \tilde{V}(\omega) ## :
## \tilde{V}(\omega)=\int\limits_{-\infty}^{+\infty} V(t)e^{-i \omega t} \, dt ##.
## \tilde{V}(\omega)=\int\limits_{0}^{T} \frac{1}{2i} (e^{i \omega_o t}-e^{-i \omega_o t})e^{-i \omega t} \, dt ##.
## \tilde{V}(\omega)=\frac{1}{2i} \int\limits_{0}^{T} [e^{-i (\omega-\omega_o)t}-e^{-i (\omega+\omega_o)t}] \, dt ##.
## \tilde{V} (\omega)=\frac{1}{2} (\frac{(e^{-i(\omega-\omega_o)T}-1)}{\omega-\omega_o}-\frac{(e^{-i(\omega+\omega_o)T}-1)}{\omega+\omega_o} ) ##.
The expression for ## \tilde{V}^*(\omega) \tilde{V} (\omega) ## is quite lengthy, and parts of the expansion separates into terms whose integrals are readily evaluated. The result is ## \int\
limits_{-\infty}^{+\infty} |\tilde{V}(\omega)|^2 \, d \omega= \int\limits_{-\infty}^{+\infty} \tilde{V}^*(\omega)\tilde{V}(\omega) \, d \omega=\pi T-\cos(\omega_o T) \int\limits_{-\infty}^{+\infty} \
frac{2 \sin[(\omega-\omega_o)T/2]\sin[(\omega+\omega_o)T/2]}{(\omega-\omega_o)(\omega+\omega_o)} \, d \omega ##, where the result for the integral on the right side still needs to be determined.
(Note: The numerator of the integral resulted from a term of the form ## \cos(\omega T)-\cos(\omega_o T) ##).
Meanwhile ## \int\limits_{-\infty}^{+\infty} V(t)^2 \, dt=\int\limits_{o}^{T} \sin^2(\omega_o t) \, dt=\frac{T}{2}-\frac{\sin(2 \omega_o T)}{4 \omega_o} ##.
Using Parseval’s theorem,
## \int\limits_{-\infty}^{+\infty} V^2(t) \, dt=\frac{1}{2 \pi} \int\limits_{-\infty}^{+\infty} |V(\omega)|^2 \, d \omega ##, and using the identity ## \sin(2 \omega_o T)=2 \sin(\omega_o T) \cos(\
omega_o T) ##, the result is that the following equality must hold:
## (\frac{\sin(\omega_o T)}{\omega_o})(\frac{\pi}{2})=\int\limits_{-\infty}^{+\infty} \frac{\sin[(\omega-\omega_o)T/2] \sin[(\omega+\omega_o)T/2]}{(\omega-\omega_o)(\omega+\omega_o)} \, d \omega ##.
Parseval’s Theorem thereby gives this Integral Result:
If we let ## T=2 ##, and change ## \omega ## to ## x ##, we get the integral result stated at the beginning of the article:
## (\frac{\sin(2 x_o)}{x_o})(\frac{\pi}{2})=\int\limits_{-\infty}^{+\infty} \frac{\sin(x-x_o) \sin(x+x_o)}{(x-x_o)(x+x_o)} \, dx ##.
Comparing to another somewhat well-known integral:
In the limit that ## x_o \rightarrow 0 ##, this gives the somewhat well-known result that
## \pi=\int\limits_{-\infty}^{+\infty} \frac{\sin^2(x)}{x^2} \, dx ##.
It is not known whether this integral has any immediate applications, but it is hoped the reader finds the result of interest. Parseval’s theorem turned out to be quite useful for generating this
Additional remarks:
Perhaps there is a way to get this same result for this integral by an application of the residue theorem or some other similar technique. Any feedback is welcome, and if there is another way to
get this same result, this author would welcome seeing the computation.
B.S. Physics with High Departmental Distinction= University of Illinois at Urbana-Champaign 1977. M.S. Physics UCLA 1979. Worked for 25+ years as a physicist doing electro-optic research at
Northrop-Grumman in Rolling Meadows, Illinois.
https://www.physicsforums.com/insights/wp-content/uploads/2017/07/Parsevalstheorem.png 135 240 Charles Link https://www.physicsforums.com/insights/wp-content/uploads/2019/02/
Physics_Forums_Insights_logo.png Charles Link2017-07-12 14:15:332020-12-25 18:30:07Learn an Integral Result from Parseval’s Theorem
5 replies
1. Delta² says:
Just a small addition regarding the statement of Parseval's theorem:
If ##V(t)## is real-valued (and in this article it is )then it is correct as it is stated
if ##V(t)## is complex-valued then in the statement of theorem it has to be put inside norm, that is ##int_{-infty}^{infty}|V(t)|^2dt=int_{-infty}^{infty}|tilde{V}(omega)|^2domega##
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2. Charles Link says:
I believe I have now also succeeded at evaluating this integral by use of the residue theorem: ## \ ## The integral can be rewritten as ## I=-frac{1}{2} int Re[ frac{e^{i2z}[1-e^{-i2(z-x_o)}]}
{(z-x_o)(z+x_o)}] , dz ##. A Taylor expansion shows the only pole is at ## z=-x_o ## (from the ## (z+x_o) ## term in the denominator). The contour will go along the x-axis with an infinitesimal
hemisphere loop over the pole and will be closed in the upper half complex plane. (I needed to consult my complex variables book for this next part:). The infinitesimal hemisphere loop in the
clockwise direction results in ## -B_o pi i ## where ## B_o ## is the residue of the function at ## z=-x_o ##. This results in ## -pi frac{sin(2x_o)}{2x_o} ## for the portion containing the
infinitesimal hemisphere loop over the pole (which is not part of the integration of the function along the x-axis, where the entire function is considered to be well-behaved.). Since the contour
doesn't enclose any poles, the complete integral around it must be zero, so the functional part along the x-axis must be equal to ## pi frac{sin(2x_o)}{2x_o} ##, which is the result that we
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3. Charles Link says:
Greg Bernhardt
Thanks Charles! Some day I will understand this :)Hi @Greg Bernhardt
I will be very pleased if someone comes up with an alternative method to evaluate this integral. I am pretty sure I computed it correctly, but I would really enjoy seeing an alternative solution.
:) :)
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4. TheAdmin says:
Thanks Charles! Some day I will understand this :)
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5. Charles Link says:
Additional comments: The result for this integral is similar to the type of result that would be obtained from residue theory if the integrand gets evaluated at the "pole" at ## x=x_o ##.
Alternatively, with the transformations ## y=x-x_o ## and ## y=pi u ##, in the limit ## T rightarrow +infty ##, ## frac{sin(pi u T)}{pi u }=delta(u) ##, but this only gives a result for large ##
T ## for an evaluation of the integral. So far, the Parseval method is the only way I have of solving it.
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Pipe Stress Budget
The Pipe Stress Budget calculator computes the pressure that a pipe can withstand based on the allowable stress, wall thickness and outside diameter.
INSTRUCTION: Choose units and enter the following:
• (S) Allowable Stress
• (t) Wall Thickness
• (D) Outside Diameter of Pipe
Pressure Budget (P): The calculator returns the pressure in pounds per square inch (psi). However, this can be automatically converted to compatible units (e.g., pascals) via the pull-down menu.
The Math / Science
The most common diameters of gas pipes, typically measured in inches, vary based on the application (residential, commercial, or industrial) and the type of gas system. For residential natural gas
supply systems, the following diameters are common:
1. ½ inch
2. ¾ inch
3. 1 inch
For larger commercial or industrial applications, pipes with diameters of 1¼ inch, 1½ inch, and 2 inches or more might be used. The size needed depends on the volume of gas required, the length of
the pipe run, and local building codes.
The thickness of a 1-inch pipe depends on its schedule, which indicates the wall thickness. Pipe schedules vary based on the pipe's application and material. Below are common schedules and
corresponding wall thicknesses for a 1-inch pipe:
1. Schedule 40 (Standard):
□ Wall thickness: 0.133 inches (3.38 mm)
2. Schedule 80 (Extra strong):
□ Wall thickness: 0.179 inches (4.55 mm)
3. Schedule 160 (High pressure):
□ Wall thickness: 0.250 inches (6.35 mm)
The thicker the pipe (higher schedule), the higher its pressure rating, but this also decreases the inner diameter.
This calculator is based on Barlow's formula which relates the internal pressure that a pipe can withstand to its dimensions and the strength of its material.
The Barlow formula is:
This formula figures prominently in the design of autoclaves and other pressure vessels.
Wikipedia (https://en.wikipedia.org/wiki/Barlow%27s_formula) | {"url":"https://www.vcalc.com/wiki/pipe-stress-budget","timestamp":"2024-11-01T22:22:42Z","content_type":"text/html","content_length":"49610","record_id":"<urn:uuid:ad1714fa-7dcd-4bc2-93e2-9c9a7f304b3c>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00266.warc.gz"} |
doc/develop/env_vars.rst - third_party/github/zephyrproject-rtos/zephyr - Git at Google
.. _env_vars:
Environment Variables
Various pages in this documentation refer to setting Zephyr-specific
environment variables. This page describes how.
Setting Variables
Option 1: Just Once
To set the environment variable :envvar:`MY_VARIABLE` to ``foo`` for the
lifetime of your current terminal window:
.. tabs::
.. group-tab:: Linux/macOS
.. code-block:: console
export MY_VARIABLE=foo
.. group-tab:: Windows
.. code-block:: console
set MY_VARIABLE=foo
.. warning::
This is best for experimentation. If you close your terminal window, use
another terminal window or tab, restart your computer, etc., this setting
will be lost forever.
Using options 2 or 3 is recommended if you want to keep using the setting.
Option 2: In all Terminals
.. tabs::
.. group-tab:: Linux/macOS
Add the ``export MY_VARIABLE=foo`` line to your shell's startup script in
your home directory. For Bash, this is usually :file:`~/.bashrc` on Linux
or :file:`~/.bash_profile` on macOS. Changes in these startup scripts
don't affect shell instances already started; try opening a new terminal
window to get the new settings.
.. group-tab:: Windows
You can use the ``setx`` program in ``cmd.exe`` or the third-party RapidEE
To use ``setx``, type this command, then close the terminal window. Any
new ``cmd.exe`` windows will have :envvar:`MY_VARIABLE` set to ``foo``.
.. code-block:: console
setx MY_VARIABLE foo
To install RapidEE, a freeware graphical environment variable editor,
`using Chocolatey`_ in an Administrator command prompt:
.. code-block:: console
choco install rapidee
You can then run ``rapidee`` from your terminal to launch the program and set
environment variables. Make sure to use the "User" environment variables area
-- otherwise, you have to run RapidEE as administrator. Also make sure to save
your changes by clicking the Save button at top left before exiting.Settings
you make in RapidEE will be available whenever you open a new terminal window.
.. _env_vars_zephyrrc:
Option 3: Using ``zephyrrc`` files
Choose this option if you don't want to make the variable's setting available
to all of your terminals, but still want to save the value for loading into
your environment when you are using Zephyr.
.. tabs::
.. group-tab:: Linux/macOS
Create a file named :file:`~/.zephyrrc` if it doesn't exist, then add this
line to it:
.. code-block:: console
export MY_VARIABLE=foo
To get this value back into your current terminal environment, **you must
run** ``source zephyr-env.sh`` from the main ``zephyr`` repository. Among
other things, this script sources :file:`~/.zephyrrc`.
The value will be lost if you close the window, etc.; run ``source
zephyr-env.sh`` again to get it back.
.. group-tab:: Windows
Add the line ``set MY_VARIABLE=foo`` to the file
:file:`%userprofile%\\zephyrrc.cmd` using a text editor such as Notepad to
save the value.
To get this value back into your current terminal environment, **you must
run** ``zephyr-env.cmd`` in a ``cmd.exe`` window after changing directory
to the main ``zephyr`` repository. Among other things, this script runs
The value will be lost if you close the window, etc.; run
``zephyr-env.cmd`` again to get it back.
These scripts:
- set :envvar:`ZEPHYR_BASE` (see below) to the location of the zephyr
- adds some Zephyr-specific locations (such as zephyr's :file:`scripts`
directory) to your :envvar:`PATH` environment variable
- loads any settings from the ``zephyrrc`` files described above in
You can thus use them any time you need any of these settings.
.. _zephyr-env:
Zephyr Environment Scripts
You can use the zephyr repository scripts ``zephyr-env.sh`` (for macOS and
Linux) and ``zephyr-env.cmd`` (for Windows) to load Zephyr-specific settings
into your current terminal's environment. To do so, run this command from the
zephyr repository:
.. tabs::
.. group-tab:: Linux/macOS
.. code-block:: console
source zephyr-env.sh
.. group-tab:: Windows
.. code-block:: console
These scripts:
- set :envvar:`ZEPHYR_BASE` (see below) to the location of the zephyr
- adds some Zephyr-specific locations (such as zephyr's :file:`scripts`
directory) to your :envvar:`PATH` environment variable
- loads any settings from the ``zephyrrc`` files described above in
You can thus use them any time you need any of these settings.
.. _env_vars_important:
Important Environment Variables
Some :ref:`important-build-vars` can also be set in the environment. Here
is a description of some of these important environment variables. This is not
a comprehensive list.
- :envvar:`BOARD`
- :envvar:`CONF_FILE`
- :envvar:`SHIELD`
- :envvar:`ZEPHYR_BASE`
- :envvar:`ZEPHYR_EXTRA_MODULES`
- :envvar:`ZEPHYR_MODULES`
The following additional environment variables are significant when configuring
the :ref:`toolchain <gs_toolchain>` used to build Zephyr applications.
- :envvar:`ZEPHYR_TOOLCHAIN_VARIANT`: the name of the toolchain to use
- :envvar:`<TOOLCHAIN>_TOOLCHAIN_PATH`: path to the toolchain specified by
:envvar:`ZEPHYR_TOOLCHAIN_VARIANT`. For example, if
``ZEPHYR_TOOLCHAIN_VARIANT=llvm``, use :envvar:`LLVM_TOOLCHAIN_PATH`. (Note
the capitalization when forming the environment variable name.)
You might need to update some of these variables when you
:ref:`update the Zephyr SDK toolchain <gs_toolchain_update>`.
Emulators and boards may also depend on additional programs. The build system
will try to locate those programs automatically, but may rely on additional
CMake or environment variables to do so. Please consult your emulator's or
board's documentation for more information.
.. _using Chocolatey: https://chocolatey.org/packages/RapidEE | {"url":"https://pigweed.googlesource.com/third_party/github/zephyrproject-rtos/zephyr/+/a0ea971e9f283271f7a3221f8b7d565e57b7c386/doc/develop/env_vars.rst","timestamp":"2024-11-06T06:24:59Z","content_type":"text/html","content_length":"65776","record_id":"<urn:uuid:5a94cbe1-3053-4e29-b464-b43e9c955df3>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00897.warc.gz"} |
Front Matter
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AIPP Books
Magnetic Field Effects on Quantum Wells
Sujaul Chowdhury;
Department of Physics,
Shahjalal University of Science and Technology
, Sylhet 3114,
Sujaul Chowdhury, Ph.D., is a Professor at the Department of Physics, Shahjalal University of Science and Technology (SUST), Bangladesh. He was a Humboldt Research Fellow at The Max Planck Institute,
Stuttgart, Germany.
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Chowdhury Shadman Awsaf;
Department of Physics,
Shahjalal University of Science and Technology
, Sylhet 3114,
Chowdhury Shadman Awsaf was a top M.S. student of the Department of Physics at Shahjalal University of Science and Technology (SUST). Awsaf is now studying Physics at Freie University, Berlin,
Search for other works by this author on:
Ponkog Kumar Das
Department of Physics,
Shahjalal University of Science and Technology
, Sylhet 3114,
Ponkog Kumar Das is an Assistant Professor at Shahjalal University of Science and Technology. He has published several journal articles and conference proceedings. He is a two-time Vice Chancellor
Award winner at SUST and was awarded the Prime Minister Gold Medal from the University Grants Commission (UGC), Bangladesh in 2016.
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ISBN electronic:
ISBN print:
Book Chapter
Front Matter
Sujaul Chowdhury
Department of Physics,
Shahjalal University of Science and Technology
, Sylhet 3114,
Search for other works by this author on:
Chowdhury Shadman Awsaf
Department of Physics,
Shahjalal University of Science and Technology
, Sylhet 3114,
Search for other works by this author on:
Ponkog Kumar Das
Department of Physics,
Shahjalal University of Science and Technology
, Sylhet 3114,
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Sujaul Chowdhury, Chowdhury Shadman Awsaf, Ponkog Kumar Das, "Front Matter", Magnetic Field Effects on Quantum Wells, Sujaul Chowdhury, Chowdhury Shadman Awsaf, Ponkog Kumar Das
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Sujaul Chowdhury;
Department of Physics,
Shahjalal University of Science and Technology
, Sylhet 3114,
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Chowdhury Shadman Awsaf;
Department of Physics,
Shahjalal University of Science and Technology
, Sylhet 3114,
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Ponkog Kumar Das
Department of Physics,
Shahjalal University of Science and Technology
, Sylhet 3114,
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Copyright © 2021 AIP Publishing LLC
AIP Publishing
This book is an analysis of a semiconductor nanostructure called isolated GaAs-AlGaAs quantum well (QW). It details the connectivity between quantum mechanics and semiconductor physics and is the
first comprehensive book on this topic. It provides a detailed analysis of the application of quantum mechanics of semiconductor nanostructures and electron transport under the influence of magnetic
field. Electronics, lasers, atomic clocks, and magnetic resonance scanners all fundamentally depend on our understanding of the quantum nature of light and matter explored in this book.
Magnetic Field Effects on Quantum Wells is a highly technical treatment featuring:
• Calculated parametric variations of transmission coefficient of the QW in non-tunneling regime in absence and in presence of magnetic field applied perpendicular to GaAs-AlGaAs interfaces
• Explanations of magnetic field dependence of the parametric var iations in a quantitatively exact manner
• Presentations of background material on microelectronics, nanostructure physics, and classical mechanics
This is an invaluable resource for researchers, scientists, industry professionals, faculty, and graduate students working with quantum mechanics. It is an excellent text for post-graduates
conducting research in semiconductor and condensed matter.
This book, Magnetic Field Effects on Quantum Wells, is an in-depth analysis of the topic and the first of its kind.
The scope and intended focus of the book are as follows:
1. The book covers the physics of a semiconductor nanostructure.
2. The book contains a comprehensive theoretical account of the topic.
3. The book is aimed at providing academics and scientists complete information on the physics of a non-tunneling regime of a semiconductor nanostructure in a magnetic field.
4. This is the first comprehensive account of the topic.
The book deals with a semiconductor nanostructure called an isolated GaAs–AlGaAs quantum well (QW). The parametric variations of the transmission coefficient of the QW in a non-tunneling regime are
calculated in the absence and presence of a magnetic field applied perpendicular to the GaAs–AlGaAs interfaces. Both analytical and numerical investigations are reported. The magnetic field
dependence of the parametric variations is explained in a quantitatively exact manner. The book also contains a comprehensive theoretical account of the topic. Background information about
microelectronics, nanostructure physics, and classical mechanics is also covered to enable readers to understand the book.
Unique features of this book include:
1. Complete calculations
2. Analytical and numerical accounts
3. Magnetic field effects brought out and explained in a quantitatively exact manner
4. A comprehensive theoretical account of the topic
Benefits of this book for the reader include:
1. Backgrounds on microelectronics, nanostructure physics, and classical mechanics
2. Complete acquaintance with the physics of the topic
Audiences of the book are graduate students and academics of physics and electrical and electronic engineering.
The organization of chapters is as follows. After a necessary introduction to the backgrounds of microelectronics and nanostructure physics in the first two chapters, Chap. 3 solves the 1D problem in
zero magnetic field and reveals the analytical expression for the transmission coefficient. In Chap. 4, the classical Hamiltonian function of a charged particle in an electric and a magnetic field is
calculated, which is used in Chap. 5 to obtain the corresponding Hamiltonian operator. In Chap. 5, the problem is resolved by reducing the 3D problem to a 1D problem. In Chap. 6, the analytical
expression of the magnetic-field-dependent transmission coefficient is obtained. Chapter 7 carries out numerical investigations to reveal the effects of the magnetic field on the transmission
coefficient. We find that a larger magnetic field reduces the effective depth of the quantum well. | {"url":"https://pubs.aip.org/books/monograph/49/chapter/20675084/Front-Matter","timestamp":"2024-11-03T13:51:30Z","content_type":"text/html","content_length":"98002","record_id":"<urn:uuid:10cd3883-d859-43bb-8519-6b80e04adb07>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00232.warc.gz"} |
Joint-significance test for simple mediation (wide-format input) — mdt_within_wide
Given a data frame, a predictor (IV), an outcome (DV), a mediator (M), and a grouping variable (group) conducts a joint-significant test for within-participant mediation (see Yzerbyt, Muller,
Batailler, & Judd, 2018).
mdt_within_wide(data, DV_A, DV_B, M_A, M_B)
a data frame containing the variables in the model.
an unquoted numeric variable in the data frame which will be used as the dependent variable value for the "A" independent variable condition.
an unquoted numeric variable in the data frame which will be used as the dependent variable value for the "B" independent variable condition.
an unquoted numeric variable in the data frame which will be used as the mediatior variable value for the "A" independent variable condition.
an unquoted numeric variable in the data frame which will be used as the mediatior variable value for the "b" independent variable condition.
Returns an object of class "mediation_model".
An object of class "mediation_model" is a list containing at least the components:
A character string containing the type of model that has been conducted (e.g., "simple mediation").
A character string containing the approach that has been used to conduct the mediation analysis (usually "joint significance").
A named list of character strings describing the variables used in the model.
A named list containing information on each relevant path of the mediation model.
A boolean indicating whether an indirect effect index has been computed or not. Defaults to FALSE. See add_index to compute mediation index.
(Optional) An object of class "indirect_index". Appears when one applies add_index to an object of class "mediation_model".
A list of objects of class "lm". Contains every model relevant to joint-significance testing.
The original data frame that has been passed through data argument.
With within-participant mediation analysis, one tests whether the effect of \(X\) on \(Y\) goes through a third variable \(M\). The specificity of within-participant mediation analysis lies in the
repeated measures design it relies on. With such a design, each sampled unit (e.g., participant) is measured on the dependent variable \(Y\) and the mediator \(M\) in the two conditions of \(X\). The
hypothesis behind this test is that \(X\) has an effect on \(M\) (\(a\)) which has an effect on \(Y\) (\(b\)), meaning that \(X\) has an indirect effect on \(Y\) through \(M\).
As with simple mediation, the total effect of \(X\) on \(Y\) can be conceptually described as follows:
$$c = c' + ab$$
with \(c\) the total effect of \(X\) on \(Y\), \(c'\) the direct of \(X\) on \(Y\), and \(ab\) the indirect effect of \(X\) on \(Y\) through \(M\) (see Models section).
To assess whether the indirect effect is different from the null, one has to assess the significance against the null for both \(a\) (the effect of \(X\) on \(M\)) and \(b\) (effect of \(M\) on \(Y\)
controlling for the effect of \(X\)). Both \(a\) and \(b\) need to be simultaneously significant for an indirect effect to be claimed (Judd, Kenny, & McClelland, 2001; Montoya & Hayes, 2011).
Data formatting
To be consistent with other mdt_* family functions, mdt_within takes a long-format data frame as data argument. With this kind of format, each sampled unit has two rows, one for the first
within-participant condition and one for the second within-participant condition. In addition, each row has one observation for the outcome and one observation for the mediator (see dohle_siegrist
for an example.
Because such formatting is not the most common among social scientists interested in within-participant mediation, JSmediation contains the mdt_within_wide function which handles wide-formatted data
input (but is syntax-inconsistent with other mdt_* family functions).
Variable coding
Models underlying within-participant mediation use difference scores as DV (see Models section). mdt_within_wide uses M_A \(-\) M_B and DV_A \(-\) DV_B in these models.
For within-participant mediation, three models will be fitted:
• \(Y_{2i} - Y_{1i} = c_{11}\)
• \(M_{2i} - M_{1i} = a_{21}\)
• \(Y_{2i} - Y_{1i} = c'_{31} + b_{32}(M_{2i} - M_{1i}) + d_{33}[0.5(M_{1i} + M_{2i}) - 0.5(\overline{M_{1} + M_{2}})]\)
with \(Y_{2i} - Y_{1i}\) the difference score between DV conditions for the outcome variable for the ith observation, \(M_{2i} - M_{1i}\) the difference score between DV conditions for the mediator
variable for the ith observation, \(M_{1i} + M_{2i}\) the sum of mediator variables values for DV conditions for the ith observation, and \(\overline{M_{1} + M_{2}}\) the mean sum of mediator
variables values for DV conditions across observations (see Montoya & Hayes, 2011).
Coefficients associated with \(a\), \(b\), \(c\), and \(c'\) paths are respectively \(a_{21}\), \(b_{32}\), \(c_{11}\), and \(c'_{31}\).
Judd, C. M., Kenny, D. A., & McClelland, G. H. (2001). Estimating and testing mediation and moderation in within-subject designs. Psychological Methods, 6(2), 115-134. doi: 10.1037//1082-989X.6.2.115
Montoya, A. K., & Hayes, A. F. (2017). Two-condition within-participant statistical mediation analysis: A path-analytic framework. Psychological Methods, 22(1), 6-27. doi: 10.1037/met0000086
Yzerbyt, V., Muller, D., Batailler, C., & Judd, C. M. (2018). New recommendations for testing indirect effects in mediational models: The need to report and test component paths. Journal of
Personality and Social Psychology, 115(6), 929–943. doi: 10.1037/pspa0000132 | {"url":"https://jsmediation.cedricbatailler.me/reference/mdt_within_wide","timestamp":"2024-11-04T16:52:02Z","content_type":"text/html","content_length":"16233","record_id":"<urn:uuid:4d182a4b-9628-448b-accd-cfa40d43640a>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00887.warc.gz"} |
Program to generate the n Fibonacci numbers using recursion - Functions in C Programming | Video Summary and Q&A | Glasp
Program to generate the n Fibonacci numbers using recursion - Functions in C Programming | Summary and Q&A
1.2K views
April 3, 2022
Program to generate the n Fibonacci numbers using recursion - Functions in C Programming
Learn how to generate Fibonacci numbers using recursion in a step-by-step manner.
Key Insights
• 🍳 Fibonacci numbers can be generated using recursion by breaking down the problem into smaller sub-problems.
• 🏛️ By defining base conditions for the starting values of Fibonacci numbers, recursion can be used to build upon these values.
• 🤙 Recursion involves calling the function on smaller values of n and adding the results together to find the Fibonacci number for n.
• #️⃣ The program provided in the content demonstrates the implementation of the recursion method to generate Fibonacci numbers.
• 💐 Understanding the tree diagram and following the program's flow can help in comprehending the recursion process for Fibonacci generation.
• 🔄 The variable "c" is used as a counter in the program, but it can be replaced with another variable like "i" if desired.
• 👻 Recursion allows for a more concise and elegant solution to generating Fibonacci numbers compared to traditional loops.
hello friends let us deal with one important program on recursion that is to generate fibonacci numbers using recursion we have seen fibonacci numbers just using a simple while loop or a value or
even a for loop in that case now this time i'll be trying it using recursion how do i get this values i'm supposed to design a base condition now if i kee... Read More
Questions & Answers
Q: What is the basic concept behind generating Fibonacci numbers using recursion?
The concept involves breaking down the problem into smaller sub-problems and using a base condition to determine the values at the starting points. Recursion is used to build upon the previously
calculated Fibonacci numbers.
Q: How is the base condition defined in this program?
For Fibonacci numbers 0 and 1, the base condition states that their respective values are 0 and 1.
Q: How does recursion help in generating Fibonacci numbers?
If the input value is greater than 1, the program recursively calls itself for the values of n-1 and n-2, and then adds them together to obtain the Fibonacci number for n.
Q: Can you explain the flow of the program using an example?
Suppose we want to find the Fibonacci number for n=4. The program would recursively call itself for n=3 and n=2 to obtain fib(3) and fib(2). The program then adds these two values together to find
the Fibonacci number for n=4.
Summary & Key Takeaways
• Fibonacci numbers are typically generated using loops, but this program demonstrates how to use recursion to achieve the same result.
• Recursion is achieved by breaking down the problem into smaller sub-problems and solving them sequentially.
• The program uses a base condition to determine the values of Fibonacci numbers at their starting points and builds upon them using recursion.
Explore More Summaries from Ekeeda 📚 | {"url":"https://glasp.co/youtube/p/program-to-generate-the-n-fibonacci-numbers-using-recursion-functions-in-c-programming","timestamp":"2024-11-04T08:58:40Z","content_type":"text/html","content_length":"358995","record_id":"<urn:uuid:f5eaf661-9ffc-4214-b815-c72cb9f33f3e>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00002.warc.gz"} |
The inapproximability of lattice and coding problems with
We prove that the closest vector problem with preprocessing (CVPP) is NP-hard to approximate within any factor less than sqrt{5/3}. More specifically, we show that there exists a reduction from an
NP-hard problem to the approximate closest vector problem such that the lattice depends only on the size of the original problem, and the specific instance is encoded solely in the target vector. It
follows that there are lattices for which the closest vector problem cannot be approximated within factors gamma < sqrt{5/3} in polynomial time, no matter how the lattice is represented, unless NP is
equal to P (or NP is contained in P/poly, in case of nonuniform sequences of lattices). The result easily extends to any L[p] norm, for p >= 1, showing that CVPP in the L[p] norm is hard to
approximate within any factor gamma < {5/3}^{1/p}. As an intermediate step, we establish analogous results for the nearest codeword problem with preprocessing (NCPP), proving that for any finite
field GF(q), NCPP over GF(q) is NP-hard to approximate within any factor less than 5/3. | {"url":"https://cseweb.ucsd.edu/~daniele/papers/GapCVPP.xml","timestamp":"2024-11-07T15:54:58Z","content_type":"application/xml","content_length":"2527","record_id":"<urn:uuid:3c81a503-9be6-409a-b71f-f964a4a9dcb7>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00596.warc.gz"} |
Jennifer A Wolfe
• Associate Professor, Mathematics
• Member of the Graduate Faculty
Jennifer A. Wolfe (she/her) is a biracial Asian American cisgender woman, daughter of a Thai immigrant, first generation college graduate, and has been in mathematics teacher education for over 20
years. While her primary focus is on grades 6-12 mathematics teacher preparation, she has taught a variety of undergraduate and graduate mathematics and mathematics education courses, mentored grades
6-12 student teachers, and facilitated professional development workshops for in-service teachers in rural Appalachia and across the country. She also has extensive experience in working with K-16
students both within and outside the classroom setting.
As an Associate Professor of Mathematics Education at the University of Arizona, her work focuses on the use of equitable teaching practices through Complex Instruction, a research-based form of
collaborative learning that centers on dismantling hierarchies of competence and shifting power and resources to historical and presently excluded and marginalized folx of the global majority. She
supports pre-service and in-service secondary mathematics teachers in learning to co-create identity affirming spaces that center student voice, collaboration, community, equity, radical love, and
She is a recipient of the 2015 UArizona AAU Undergraduate STEM Teaching Excellence Award, the 2022 College of Science Innovation in Teaching Award, and the 2022 Department of Mathematics David
Lovelock Innovation in Education Award for effectively implementing research-based equitable teaching practices for engaging students in collaborative groupwork. She was awarded the Leicester &
Kathryn Sherrill Creative Teaching Award and recognized at the 2024 UA Awards of Distinction. This event honors the recipients of the University of Arizona's highest faculty teaching honors.
She has served as Co-PI/Senior Personnel on two NSF grants aimed at the (1) recruiting, supporting, and retaining secondary mathematics teachers and (2) investigating preservice and beginning
teachers views on equity and classroom implementation of equitable mathematics teaching practices.
She has served in leadership roles for Association for Mathematics Teachers Educators (AMTE), the National Council of Teachers of Mathematics (NCTM), and the Psychology of Mathematics Education-North
American Chapter (PMENA). She served as co-editor for the Informing Practice Department of the NCTM Mathematics Teaching in the Middle School journal and for the Ear to the Ground Department of the
NCTM Mathematics Teacher: Learning and Teaching PK-12 journal. She is a former Fellow of The Center for Inquiry and Equity in Mathematics, The Teaching Well, and the Institute for Teachers of Color
Committed to Racial Justice.
She is currently a co-host of the AMTE Teaching Math Teaching podcast, a mentor for the AMTE Service, Teaching, & Research (STaR) fellowship program, an early career induction program for faculty in
mathematics education, and serves on the editorial panel for the Mathematics Teacher Educator journal. Dr. Wolfe holds a Bachelor of Science and a Master of Arts in Mathematics, and a PhD in
Mathematics Education from the University of Kentucky.
• Ph.D. Mathematics Education
□ University of Kentucky, Lexington, Kentucky, United States of America
□ An exploratory mixed methods study of prospective middle grades teachers' mathematical connections while completing investigative tasks in geometry
• M.A. Mathematics
□ University of Kentucky, Lexington, Kentucky, United States of America
• B.S. Mathematics
□ University of Kentucky, Lexington, Kentucky, United States of America
• University Distinguished Faculty Award UArizona Foundation Leicester & Kathryn Sherrill Creative Teaching Award
□ The University of Arizona, Spring 2023
• Innovation in Teaching Award
□ The University of Arizona College of Science, Fall 2022
• Racial Healing for Educators, Pan-Asian, Pacific Islander, & North African (PAPINA) Racial Healing Affinity Group Fellow
□ The Teaching Well, Fall 2022
• David Lovelock Award for Innovation in Education
□ The University of Arizona Department of Mathematics, Spring 2022
• Fellowship
□ Institute for Teachers of Color Committed to Racial Justice (ITOC), Fellow 2020-2021, Summer 2020
□ The Center for Inquiry and Equity in Mathematics, Fellow, 2019-2020, Spring 2019
• Mentor/Fellow
□ NSF UArizona S-STEM Bridge Program, Fellow/Mentor, 2020-2022, Summer 2020
• Undergraduate STEM Education Teaching Excellence Award
□ The University of Arizona AAU, Spring 2015
• International Congress on Mathematics Education Travel Grant
□ National Science Foundation/National Council of Teachers of Mathematics, Summer 2012
• College of Science Distinguished Early Career Teaching Award
□ The University of Arizona, Spring 2012 (Award Nominee)
• National Science Foundation Service Teaching and Research STaR Fellow
□ National Science Foundation, Summer 2010
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2024-25 Courses
• Meth Tch Math Sec School
MATH 406B (Fall 2024)
2023-24 Courses
• Curr+Assmnt Sec Sch Math
MATH 406A (Spring 2024)
• Meth Tch Math Sec School
MATH 406B (Fall 2023)
2021-22 Courses
• Curr+Assmnt Sec Sch Math
MATH 406A (Spring 2022)
• Meth Tch Math Sec School
MATH 406B (Fall 2021)
2020-21 Courses
• Sec Math Teach Practicum
MATH 494C (Spring 2021)
• Understand Ele Math - B
MATH 302B (Spring 2021)
• Meth Tch Math Sec School
MATH 406B (Fall 2020)
2019-20 Courses
• Independent Study
MATH 599 (Spring 2020)
• Understand Ele Math - B
MATH 302B (Spring 2020)
• Meth Tch Math Sec School
MATH 406B (Fall 2019)
2018-19 Courses
• Directed Research
MATH 392 (Spring 2019)
• Sec Math Teach Practicum
MATH 494C (Spring 2019)
• Meth Tch Math Sec School
MATH 406B (Fall 2018)
2017-18 Courses
• Sec Math Teach Practicum
MATH 494C (Spring 2018)
• Understand Ele Math - B
MATH 302B (Fall 2017)
2016-17 Courses
• Understand Ele Math - B
MATH 302B (Fall 2016)
2015-16 Courses
• Understand Ele Math - B
MATH 302B (Spring 2016)
Scholarly Contributions
• Wood, M. B., Turner, E. E., Civil, M., & Eli, J. A. (2016). Proceedings of the 38th Annual North American Chapter of the International Group for the Psychology of Mathematics Education. Tucson,
AZ: The University of Arizona.
• Wolfe, J. A. (2022). Building Caring Communities in Math Methods. In Care after COVID: Reconstructing understanding of care in teacher education(pp 104-114). Routledge. doi:https://doi.org/
More info
This chapter grounds the need for care against systemic issues of inequity made hyper-visible during the COVID-19 pandemic. The author considers how mathematics teacher education can shift
towards enacting caring practices that will foster the development of positive mathematics identities before addressing how mathematics educators can cultivate collaborative learning environments
that center on building authentic, caring relationships. The chapter shares the author’s identities in relation to building caring communities before next considering what guides the author’s
understanding of care and how her enacted care shifted during and after the first year of COVID-19. The chapter concludes with examples of classroom practices for creating and sustaining caring
learning environments in mathematics teacher education.
• Wolfe, J. A. (2022). Building caring communities in math methods: COVID and classrooms in teacher education.. In M. Shoffner & A. Webb (Eds.) Care after COVID: Reconstructing Understandings of
Care in Teacher Education(pp 104-114). Routledge.
• Wolfe, J. A., & Safi, F. (2022). Lesson 7.3 Majority and Power: The Role of Mathematics in Making Sense of Representations. In B. Conway, L. Id-Deen, M. C. Raygoza, A Ruiz, J. W. Staley, & E.
Thanheiser (Eds.).Middle School Mathematics Lessons to Explore, Understand, and Respond to Social Injustice(pp 141-151). Corwin.
• Jessup, N. A., Wolfe, J. A., & Kalinec-Craig, C. (2021). Rehumanizing Mathematics Education and Building Community for Online Learning. In Online Learning in Mathematics Education. Research in
Mathematics Education.(pp 95-113). Springer. doi:10.1007/978-3-030-80230-1_5
• Jessup, N., Wolfe, J. A., & Kalinec-Craig, C. (2021). Rehumanizing mathematics education and building community in online spaces. In K. Hollenbrands, R. Anderson, & K. Oliver (Eds.), Online
Learning in Mathematics Education(pp 95-113). Brill Sense. doi:https://doi.org/10.1007/978-3-030-80230-1_5
• Eli, J. A., Civil, M., Mcgraw, R. H., Anhalt, C. O., Anhalt, C. O., Mcgraw, R. H., Civil, M., & Eli, J. A. (2019). Stronger together: The Arizona Mathematics Teaching (MaTh) Noyce Program’s
Collaborative Model for Secondary Teacher Preparation. In Recruiting, retaining, and preparing STEM teachers for a global generation. Rotterdam, Netherlands: Sense.
More info
Eli, J., McGraw R., Anhalt, C., & Civil, M. (2019). Stronger Together: The AZ Mathematics Teaching (MaTh) Noyce Program’s Collaborative Model for Secondary Teacher Preparation. Book chapter in J.
Leonard, A. Burrows, and R. Kitchen (Eds.), Recruiting, Preparing, and Retaining STEM Teachers for a Global Generation. Sense Publishers, The Netherlands.
• Eli, J. A. (2016). Broadening perspectives through purposeful reflection: A commentary on Strickland's case. In Cases for Teacher Educators: Facilitating Conversations about Inequities in
Mathematics Classrooms..
• Anhalt, C. O., & Eli, J. A. (2012). Building connections from whole number to polynomial long division: Teaching English language learners. In Beyond good teaching: Advancing Mathematics
Education for ELLs(pp 96-103). Reston, VA: National Council of Teachers of Mathematics.
More info
English language learners share a basic need: to engage, and be engaged, in meaningful mathematics. Through guiding principles and instructional tools, together with classroom vignettes and video
clips, this book shows how to go beyond good teaching to support ELLs in learning challenging mathematics while developing language skill. Position your students to share the valuable knowledge
that they bring to the classroom as they actively build and communicate their understanding. The design of this book is interactive and requires the reader to move back and forth between the
chapters and online resources at www.nctm.org/more4u. Occasionally, the reader is asked to stop and reflect before reading further in a chapter. At other times, the reader is asked to view video
clips of teaching practices for ELLs or to refer to graphic organizers, observation and analysis protocols, links to resources, and other supplementary materials. The authors encourage the reader
to use this resource in professional development.
• McGraw, R., Fernandes, A., Wolfe, J. A., & Jarnutowski, B. (2024). Unpacking mathematics preservice teachers’ conceptions of equity. Mathematics Education Research Journal, 36, 645-670.
• Jansen, A., & Center for Inquiry & Equity in Mathematics, . (2023). Entangling and Disentangling Inquiry and Equity: Voices of Mathematics Education and Mathematics Professors. Journal of Urban
Mathematics Education, 16(1), 10-39. doi:https://doi.org/10.21423/jumev16ia473
• Marshall, A., Sword, S., Applegate, M., Greenstein, S., Pendleton, T., Yong, K., Young, M., Wolfe, J. A., Chao, T., & Harris, P. (2023). “I Got You”: Centering Identities and Humanness in
Collaborations Between Mathematics Educators and Mathematicians. Journal for Humanistic Mathematics, 13(2), 309-337. doi:10.5642/jhummath.zuxw1688
• Wolfe, J. A. (2021). Teaching Is a Journey: A Journey in Becoming. Mathematics Teacher: Learning and Teaching PK–12, 114(3), 258-261. doi:10.5951/mtlt.2020.0378
More info
This department provides a space for current and past PK–12 teachers of mathematics to connect with other teachers of mathematics through their stories that lend personal and professional
• Jessup, N., Wolfe, J. A., Udiani, O., & Kalinec-Craig, C. (2020). Issues of equity when teaching and learning mathematics in a pandemic. Mathematics Teacher: Learning and Teaching PK-12, 113(10).
• Eli, J. A., Roy, G., & Hendrix, L. (2018). Using history to model with mathematics: The German tank problem.. Mathematics Teaching in the Middle School, 23(7), 370-377.
• Gonzalez, G., & Eli, J. A. (2017). Pre-service and in-service teachers' perspectives about launch a problem. Journal of Mathematics Teacher Education, 20(2), 159-201. doi:10.1007/
More info
Abstract: Launching a problem is critical in a problem-based lesson. We investigated teachers’ perspectives on the use of a problem that was analogous to the one provided during a launch. Our
goal was to identify teachers’ underlying assumptions regarding what should constitute a launch as elements of the practical rationality of mathematics teaching. We analyzed data from four focus
groups that consisted of pre-service (PST) and in-service (IST) teachers who viewed animated vignettes of classroom instruction. We applied Toulmin’s scheme to model the arguments that were
evident in the transcriptions of the discussions. We identified nine claims and 13 justifications for those claims, the majority of which were offered by the ISTs. ISTs’ assumptions focused on
reviewing, providing hints, and not confusing students, whereas PSTs’ assumptions focused on motivation and student engagement. Overall, the assumptions were contradictory and supported different
strategies. The assumptions also illustrated different stances regarding how to consider students’ prior knowledge during a launch. We identified a tension between ensuring that students could
begin a problem by relying on the launch and allowing them to struggle with the problem by limiting the information provided in the launch. This study has implications for teacher education
because it identifies how teachers’ underlying assumptions may affect their decisions to enable students to engage in productive struggle and exercise conceptual agency.
• Eli, J. A., Wood, M. B., Turner, E. E., Eli, J. A., & Civil, M. (2016). Sin Fronteras: Questioning Borders with(in) Mathematics Education. Proceedings of the Annual Meeting of the North American
Chapter of the International Group for the Psychology of Mathematics Education (38th, Tucson, Arizona, November 3-6, 2016).. International Group for the Psychology of Mathematics Education.
• Yow, J., Eli, J. A., Beisiegel, M., Welder, R., & McCloskey, A. (2016). Challenging transitions and crossing borders: Preparing novice mathematics teacher educators to support novice K-12
teachers. Mathematics Teacher Education and Development, 18(1), 52-69.
More info
This article shares findings from a survey of 69 recently graduated doctoral students in mathematics education. Similarities were found between the experiences of these novice mathematics teacher
educators and the experiences documented in the literature on novice K-12 mathematics teachers. Using Giroux’s framework of border crossing (1992/2005), findings showed that novice mathematics
teacher educators needed more teaching experiences during their doctoral preparation programs as well as more mentoring during their initial years as professors. These findings are consistent
with research findings on the experiences of novice K-12 mathematics teachers. The article then discusses how these findings impact the teaching and learning of mathematics across K-12 and
university settings and offers suggestions for improving the transition for mathematics teacher educators into their academic roles as novice professors.
• Orrill, C. H., Kim, O., Peters, S. A., Lischka, A. E., Jong, C., Sanchez, W. B., & Eli, J. A. (2015). Challenges and strategies for assessing mathematical knowledge for teaching. Mathematics
Teacher Education and Development, 17(1), 12-29.
More info
Abstract: Developing and writing assessment items that measure teachers' knowledge is an intricate and complex undertaking. In this paper, we begin with an overview of what is known about
measuring teacher knowledge. We then highlight the challenges inherent in creating assessment items that focus specifically on measuring teachers’ specialised knowledge for teaching. We offer
insights into three practices we have found valuable towards overcoming challenges in our own cross-disciplinary work to create assessment items for measuring teachers' knowledge for teaching.
• Eli, J. A., Mohr-Schroeder, M. J., & Lee, C. W. (2013). Mathematical connections and their relationship to mathematics knowledge for teaching geometry. School Science and Mathematics, 113(3),
More info
Abstract: Effective competition in a rapidly growing global economy places demands on a society to produce individuals capable of higher-order critical thinking, creative problem solving,
connection making, and innovation. We must look to our teacher education programs to help prospective middle grades teachers build the mathematical habits of mind that promote a conceptually
indexed, broad-based foundation of mathematics knowledge for teaching which encompasses the establishment and strengthening of mathematical connections. The purpose of this concurrent exploratory
mixed methods study was to examine prospective middle grades teachers’ mathematics knowledge for teaching geometry and the connections made while completing open and closed card sort tasks meant
to probe mathematical connections. Although prospective middle grades teachers’ mathematics knowledge for teaching geometry was below average, they were able to make over 280 mathematical
connections during the card sort tasks. Curricular connections made had a statistically significant positive impact on mathematics knowledge for teaching geometry.
• Royal, K. D., & Eli, J. A. (2013). Developing a psychometric ruler: An alternative presentation of rasch measurement output. Journal of Applied Quantitative Methods, 8(3), 1-10.
More info
Rasch measurement is one of the most popular analytical techniques available in the field of psychometrics. Despite the advantages of Rasch measurement, many researchers and consumers of
information have noted that interpreting Rasch output can be an arduous task. The purpose of this paper is to respond to this problem by presenting an alternative method for reporting results
that is arguably more user-friendly and easily interpretable by consumers of research.
• Eli, J. A., Mohr-Schroeder, M. J., & Lee, C. W. (2011). Exploring mathematical connections of prospective middle-grades teachers through card-sorting tasks. Mathematics Education Research
Journal, 23(3), 297-319.
More info
Abstract:Prospective teachers are expected to construct, emphasise, integrate, and make use of mathematical connections; in doing so, they acquire an understanding of mathematics that is fluid,
supple, and interconnected (Evitts Dissertation Abstracts International, 65(12), 4500, 2005). Given the importance of mathematical connection making, an exploratory study was conducted to
consider the ability of prospective middle-grades teachers to make mathematical connections while engaging in card-sorting activities. Twenty-eight prospective middle-grades teachers participated
in both an open and closed card sort. Data were analysed using constant comparative methods to extract meta themes to describe the types of connections made. Findings indicate that these
prospective teachers tended to make more procedural- and categorical-type mathematical connections and far fewer derivational or curricular mathematical connections. © 2011 Mathematics Education
Research Group of Australasia, Inc.
• Royal, K. D., Eli, J. A., & Bradley, K. D. (2010). Exploring community college faculty perceptions of student outcomes: Findings of a pilot study. Community College Journal of Research and
Practice, 34(7), 523-540.
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Abstract:This study explored the paradigmatic differences in perceptions of community college faculty employed at select Virginia and West Virginia community colleges collected via a web-based
survey. The study is framed within the faculty self-classification along the "hard" and "social/behavior" science paradigm continuum. Given the paradigmatic continuum, faculty perceptions' of
student outcomes were examined. Faculty respondents consistently reported the importance of intellectual growth; however, differences in relative importance of outcomes tied to emotional,
cultural, and social growth exist. The potential implications of these perceptions on student experiences and outcomes are considered. © Taylor & Francis Group, LLC.
• Bradley, K. D., Royal, K. D., Cunningham, J. D., Weber, J., & Eli, J. A. (2008). What constitutes good educational research? A consideration of ethics, methods, and theory.. Mid-Western
Educational Researcher, 21(1), 26-35.
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Abstract: The question of what constitutes good educational research has received much attention in recent years. To offer an empirical framework, a web-based survey was administered to faculty
and graduate students in the College of Education at a southeastern university to obtain their perceptions about characteristics related to high-quality educational research. Results suggest that
survey items connected to ethics and theory were relatively easy to endorse, while items connected to methodology illustrated variation in perception. This study provides a foundation for
dialogue regarding the quality of educational research.
Proceedings Publications
• McGraw, R., Fernandes, A., Wolfe, J. A., & Janutowski, B. (2022). Preservice and beginning teachers’ perspectives on equity.. In Proceedings of the 44th Annual North American Chapter of the
International Group for the Psychology of Mathematics Education, 1217-1221.
• Jansen, A., & Center for Inquiry & Equity in Mathematics, . (2021). Entangling and disentangling inquiry and equity: Voices of mathematics education and mathematics professors . In Proceedings of
the 43rd Annual North American Chapter of the International Group for the Psychology of Mathematics Education, 222.
• Wieman, R., Tyminski, A. M., Trocki, A., Perry, J., Johnson, K., Gonzalez, G., & Eli, J. A. (2019). Developing Theory, Research, and Tools for Effective Launching: Developing a Launch Framework..
In International Group for the Psychology of Mathematics Education.
• Weiman, R., Perry, J., Tyminski, A., Kalemanik, G., Gonzalez, G., Trocki, A., & Eli, J. A. (2018, November). Developing theory, research, and tools for effective launching.. In In T. E. Hodges ,
G. J. Roy, & A. M. Tyminski (Eds.) Proceedings of the 40th Annual North American Chapter of the International Group for the Psychology of Mathematics, 1497-1506.
• Eli, J. A. (2016, Fall 2016). Supporting secondary mathematics teacher preparation: A collaborative tetrad model.. In Psychology of Mathematics Education North American Chapter.
• Eli, J. A., Civil, M., Turner, E. E., & Wood, M. B. (2016).
Sin Fronteras: Questioning Borders with(in) Mathematics Education. Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics
Education (38th, Tucson, Arizona, November 3-6, 2016).
. In PME-NA (North American Chapter of the International Group for the Psychology of Mathematics Education).
• Wood, M. B., Eli, J. A., Eli, J. A., & Wood, M. B. (2016). Learning to facilitate groupwork through complex instruction.. In Proceedings of the 38th Annual North American Chapter of the
International Group for the Psychology of Mathematics Education., 933.
• Beisiegel, M., & Eli, J. A. (2013, Summer). An investigation of novice mathematics teacher educators' experiences in transition from doctoral student to faculty member.. In The 37th Conference of
the International Group for the Psychology of Mathematics Education, 5, 209.
More info
In A.M. Lindmeier, & A. Heinze. (Eds). Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education
• McCloskey, A., Beisiegel, M., Eli, J. A., Welder, R., & Yow, J. (2012, Fall). Parallel transitions: Challenges faced by new mathematics teachers and new mathematics teacher educators.. In The
34th Annual North American Chapter of the International Group for the Psychology of Mathematics Education, 737-740.
More info
In L.R. Van Zoest, J.J. Lo, & J.L. Kratky (Eds.), Proceedings of the 34th Annual North American Chapter of the International Group for the Psychology of Mathematics Education
• Beisiegel, M., Eli, J. A., McCloskey, A., Welder, R., & Yow, J. (2011, Fall). Transforming practices by understanding the connections between new mathematics teacher educators and new K-12
mathematics Teachers. In The 33rd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 1912.
More info
In L.R. Weist, & T. Lamberg (Eds.), Proceedings of the 33rd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.
• Eli, J. A., Mohr-Schroeder, M. J., & Lee, C. W. (2010, January). Prospective middle grades teachers' mathematical connections and its relationship to their mathematics knowledge for teaching.. In
The 8th annual Hawaii International Conference on Education, 1367-1418.
More info
With the implementation of No Child Left Behind legislation and a push for reform curricula, prospective teachers must be prepared to facilitate learning at a conceptual level. To address these
concerns, an exploratory mixed methods investigation of twenty-eight prospective middle grades teachers’ mathematics knowledge for teaching geometry and mathematical connection-making was
conducted at a large public southeastern university. Participants completed a diagnostic assessment in mathematics with a focus on geometry and measurement (CRMSTD, 2007), a mathematical
connections evaluation, and a card sort activity. Mixed methods data analysis revealed prospective middle grades teachers’ mathematics knowledge for teaching geometry was underdeveloped and the
mathematical connections made by prospective middle grades teachers were more procedural than conceptual in nature. Proceedings of the 8th annual Hawaii International Conference on Education
• McGraw, R., Jarnutowski, B., Fernandes, A., & Wolfe, J. A. (2023, February). Mathematics teacher education program students' perspectives on equity and mathematics teaching. Association of
Mathematics Teacher Educators. New Orleans: Association of Mathematics Teacher Educators.
• Yeh, C., Kokka, K., Jong, C., Chao, T., & Wolfe, J. A. (2023, February). Not your model minority: Interrogating the phrase "Asian American" in mathematics education.. Association of Mathematics
Teacher Educators. New Orleans, Louisana: Association of Mathematics Teacher Educators.
• Fernandes, A., Wolfe, J. A., Janutowski, B., & McGraw, R. (2022, November). Preservice and beginning teachers' perspectives on equity.. 44th International Group for the Psychology of Mathematics
Education-North American Chapter Conference. Nashville, TN: PMENA.
• Safi, F., Roy, G., & Wolfe, J. A. (2022, Fall). Five Equity-Based Practices: Focusing on Intentional Teacher Actions & Valuing Student Identity. National Council of Teachers of Mathematics Annual
Conference. Los Angeles: NCTM.
• Wolfe, J. A. (2022, October). Building caring communities in mathematics teacher preparation: Unpacking the hidden curriculum of teaching. [Invited Featured Speaker].. Association of Mathematics
Teacher Educators Virtual Professional Development Event: Unpacking the Hidden Curriculum of Being a Mathematics Teacher Educator.Association of Mathematics Teacher Educators.
• Wolfe, J. A., Fernandes, A., Janutowski, B., & McGraw, R. (2022, September). Preservice secondary mathematics teachers' perspectives of equity. National Council of Teachers of Mathematics
Research Conference. Los Angeles, California: NCTM.
• Jansen, A., & Center for Inquiry & Equity in Mathematics, . (2021). Entangling and Disentangling Inquiry and Equity: Voices of Mathematics Education and Mathematics Professors. Proceedings of the
43rd Annual North American Chapter of the International Group for the Psychology of Mathematics Education. Philadelphia, PA: Psychology of Mathematics Education-North American Chapter.
• Wolfe, J. A. (2021, Fall). Building Caring Communities in Mathematics Teacher Preprartion: Learning to Listen and Listening to Learn [Keynote]. Association of Mathematics Teacher Educators-TX.
San Antonio, Texas: Association of Mathematics Teacher Educators.
More info
I served as the invited Keynote Speaker for this conference which was held virtually. Please see the attached letter from the conference organizer and president of AMTE-TX
• Wolfe, J. A. (2021, October/Fall). Dismantling hierarchies of status: Empowering students through equitable teaching and groupwork. National Council of Teachers of Mathematics Annual Conference.
St. Louis, MO: National Council of Teachers of Mathematics.
• Wolfe, J. A. (2021, Spring). Dismantling Hierarchies of Status: Building Student Agency Through Equitable and Inclusive Teaching. National Council of Teachers of Mathematics Virtual Conference.
Virtual: National Council of Teachers of Mathematics.
More info
I was an invited speaker for this conference.
• Wolfe, J. A. (2021, Spring). Dismantling hierarchies of status: Building student agency through equitable and inclusive teaching. National Council of Teachers of Mathematics Virtual Conference.
Dallas, TX (Virtual): National Council of Teachers of Mathematics.
• Wolfe, J. A. (2021, Spring). How Has Preparing Teachers Online Shifted Teacher Educators' Priorities & Practices. Virtual Conversations: Teacher Education in Response to Today's Demands. [Invited
Panelist].. Collaboratory for Teacher Education at the University of Pennsylvaina Graduate School of Education. https://collaboratory.gse.upenn.edu/: The University of Pennsylvania Graudate
School of Education; The Collaboratory for Teacher Education at Penn GSE.
More info
I was an invited panelist for the virtual presentation and conversation.
• Hackett, M., & Wolfe, J. A. (2020, Spring). Lab School: Learn the work by doing the work.. Mathematics Educators Appreciation Day (MEAD) conference. Tucson, AZ: Center for Recruitment &
• Wolfe, J. A., & Salcido, M. (2020, Spring). Emphasizing Groupwork: A Focus on Status and Participation. Mathematics Educator Appreciate Day (MEAD) conference. Tucson, AZ: Center for Recruitment &
• Wolfe, J. A., Kalinec-Craig, C., Jessup, N., & Udiani, O. (2020, Spring). Online teaching and learning in the time of a pandemic: Issues of equity and access.. The Center for Inquiry and Equity
in Mathematics Webinar.
• Eli, J. A. (2019, Fall). Emphazing the group in groupwork: Empowering students through equitable teaching practices. [Invited Featured Speaker].. National Council of Teachers of Mathematics
Conference. Boston, MA: National Council of Teachers of Mathematics.
• Hackett, M., Wolfe, J. A., Salcido, M., & Quihuis, G. (2019, January/Spring). From Summer Labschool to the School Year: Implementing Access and Equity Practices in the Middle School Classroom..
Mathematics Educators Appreciation Day Conference. Tucson, AZ: Center for Recruitment and Retention of Teachers.
• Wolfe, J. A., & Hackett, M. (2019, February/Spring). Complex instruction lab school: A middle school mathematics professional development model. Western Regional Noyce Conference. Tucson AZ:
National Science Foundation.
• Yeh, C., Kokka, K., Louis, N., Jong, C., Eli, J. A., Chao, T., & Adiredja, A. (2019, Spring). Growing against the grain: Counterstories of Asian American mathematics education scholars. American
Education Research Association. Toronto, CA: American Education Research Association.
• Chao, T., Eli, J. A., Kokka, K., & Cathery, Y. (2018, April 2018). Critical Issues in Working with Asian American Students [Invited Panelist].. National Council of Supervisors of Mathematics.
Washington DC: National Council of Supervisors of Mathematics.
• Eli, J. A., & Safi, F. (2018, April 2018). We got this! Engaging students in equitable group collaboration with challenging mathematics.. National Council of Teachers of Mathematics Annual
Conference. Washington, D. C: National Council of Teachers of Mathematics.
• Eli, J. A., & Safi, F. (2018, April 2018). Working together: Building mathematical knowledge for teaching through equitable teaching practices. National Council of Teachers of Mathematics.
Washington DC: National Council of Teachers of Mathematics.
• Safi, F., & Eli, J. A. (2018, Spring). Working Together: Building Mathematical Knowledge For Teaching Through Equitable Teaching Practices. National Council of Supervisors of Mathematics.
Washington, D.C: National Council of Supervisors of Teachers.
• Weiman, R., Perry, J., Tyminski, A., Kelemanik, G., Gonzalez, G., Trocki, A., & Eli, J. A. (2018, Fall). Developing theory, research, and tools for effective launching. North American Chapter of
the International Group for the Psychology of Mathematics Education. Columbia, SC: Psychology of Mathematics Education-North American Chapter.
• Wood, M. B., & Eli, J. A. (2018, Fall). Mathematical Superheroes: Create a justification league in your classroom.. National Council of Teachers of Mathematics Conference. Hartford, CT: National
Council of Teachers of Mathematics.
• Eli, J. A. (2016, January 2016). Learning to Facilitate Groupwork Through Complex Instruction. Association of Mathematics Teacher Educators. Irvine, CA.
• Parks, A., Felton-Koestler, M., Eli, J. A., Wood, M. B., Wagner, A., & Crespo, S. (2016, January). Designing Complex Instruction Tasks to Support Prospective Teacher Learning in Elementary
Content and Methods Courses. 20th Annual Association of Mathematics Teacher Educators Conference. Irvine, CA: Association of Mathematics Teacher Educators.
• Eli, J. A. (2015, February). Secondary mathematics teacher preparation: A collaborative tetrad model. Research Council on Mathematics Learning. Las Vegas, NV: Research Council on Mathematics
More info
Abstract: Student teaching is often described as the most influential part of teacher preparation. During student teaching, pre-service teachers are expected to put into practice the integration
of content and pedagogy under the mentorship of knowledgeable others in a classroom setting. The traditional model of student teaching supervision involves daily interaction with an in-service
teacher coupled with periodic visits by a university supervisor, usually a mathematics educator. Although university mathematicians are responsible for significant portions of teacher preparation
prior to student teaching, they are often absent during this crucial period. In this session, I propose a new model of collaboration for supporting mathematics teacher preparation that includes
both mathematicians and mathematics educators in the student teaching semester. I will discuss preliminary findings from the implementation of the tetrad model with a focus on the professional
noticing of all tetrad members.
• Eli, J. A. (2015, January). Card Sorting: A Tool for Assessing Students' mathematical Connections. Mathematics Educators Appreciation Day Conference. Tucson High School: Center for Recruitment &
Retention of Teachers.
More info
Session Description: In this session, I will begin by introducing card sorting as a tool for exploring and assessing students' mathematical connections and will present participants with sample
card sort activities. Participants will then engage in and create their own card sort activities for assessing their students' mathematical connections on a topic of choice.
• Menendez, J. A., & Eli, J. A. (2015, November). Using Complex Instruction in content courses for prospective teachers. American Mathematical Association of Two Year Colleges Annual Conference.
New Orleans, LA: American Mathematical Association of Two Year Colleges.
• Eli, J. A. (2014, January). Using blender to help students visualize and model solids of revolution. Mathematics Educators Appreciation Day Conference. Tucson High School: Center for Recruitment
& Retention of Teachers.
• Shirley-Akers, K., Eli, J. A., & Royal, K. D. (2014, April). Evaluating faculty research productivity and collaborations with social network analysis. American Educational Research Association
Annual Conference. Philadelphia, PA: American Educational Research Association.
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Abstract: Research productivity and collaborations are essential aspects of advancing academia. Publishing is a critical mechanism in higher education to allow faculty members to share new
information in all disciplinary fields. Due to its importance, scholarly work is often heavily considered for promotion, tenure, compensation, and other merit decisions. Social network analysis
(SNA) offers a useful methodology for evaluating productivity and collaborations within research networks. The purpose of this paper is to present a primer of SNA and demonstrate how SNA can be
used to discern faculty research productivity and collaborations in highly integrated and complex research networks
• Beisiegel, M., & Eli, J. A. (2013, July). An investigation of novice mathematics teacher educators' experiences in transition from doctoral student to faculty member. International Group for the
Psychology of Mathematics Education. Kiel, Germany: International Group for the Psychology of Mathematics Education.
• Eli, J. A. (2013, January). A pre-service secondary mathematics teachers' implementation of a solids of revolution task: Learning to launch. Association of Mathematics Teacher Educators Annual
Conference. Orlando, FL: Association of Mathematics Teacher Educators.
• Eli, J. A. (2013, October). Developing a collaborative model for mentoring secondary mathematics student teachers. American Mathematical Society Sectional Meeting; Special Session. Louisville,
Ky: American Mathematical Society.
• Gonzalez, G., & Eli, J. A. (2013, April). Teaching tensions when scaffolding students' work in a problem-based lesson. American Educational Research Association Annual Conference. San Francisco,
CA: American Educational Research Association.
• McGraw, R., Eli, J. A., & Felton, M. D. (2013, February). Mathematics education research: What, who, and how?. The University of Arizona Mathematics Department Research Tutorial Groups. The
University of Arizona: Department of Mathematics.
• Beisiegel, M., Eli, J. A., & McCloskey, A. (2012, April). New mathematics educators' preparation for academic careers: An exploratory study. National Council of Mathematics Teachers Research
Pre-Session. Philadelphia, PA: National Council of Teachers of Mathematics.
• Eli, J. A., Anhalt, C., Fernandez, M., & Wilson, P. (2012, February). Building reflective secondary pre-service teachers: A co-constructed triad of collaboration. Association of Mathematics
Teacher Educators Annual Conference. Fort Worth, TX: Association of Mathematics Teacher Educators.
• Eli, J. A., Beisiegel, M., McCloskey, A., Welder, R., & Yow, J. (2012, April). Becoming a mathematics teacher educator: Novice faculty members' perceptions of the impact of doctoral program
experiences. American Educational Research Association Annual Conference. Vancouver, Canada: American Educational Research Association.
• Eli, J. A., Yow, J., & Welder, R. (2012, April). Similarities among new teacher educators and new K-12 mathematics teachers. National Council of Teachers of Mathematics Research Pre-Session.
Philadelphia, PA: National Council of Teachers of Mathematics.
• McCloskey, A., Eli, J. A., Beisiegel, M., Welder, R., & Yow, J. (2012, November). Parallel transitions: Challenges faced by new mathematics teachers and new mathematics teacher educators.
International Group for the Psychology of Mathematics Education North American Chapter Annual Conference. Kalamazoo, MI: International Group for the Psychology of Mathematics Education North
American Chapter.
• Thomas, M., Lozano, G., Patterson, C., & Eli, J. A. (2012, January). Developing a protocol for analyzing the quality of classroom interactions in an undergraduate calculus course. AMS-MAA-MER
Annual Joint Conference. Boston, MA: American Mathematical Society; Mathematical Association of America.
• Welder, R., Yow, J., Beisiegel, M., & Eli, J. A. (2012, February). It's crazy; they told me it would be: Experiences of new mathematics teacher educators. Association of Mathematics Teacher
Educators. Fort Worth, TX: Association of Mathematics Teacher Educators.
• Anhalt, C., Eli, J. A., & Cuprak, J. (2011, April). Mentoring student teachers: Co-constructing a professional relationship. National Council of Teachers of Mathematics Annual Conference.
Indianapolis, IN: National Council of Teachers of Mathematics.
• Beisiegel, M., Eli, J. A., McCloskey, A., Welder, R., & Yow, J. (2011, January). How do the experiences of new mathematics teachers and new mathematics educators compare?. Association of
Mathematics Teacher Educators Pre-conference STaR Fellows Teacher Preparation Research Interest Group. Irvine, CA: Association of Mathematics Teacher Educators.
• Beisiegel, M., Eli, J. A., McCloskey, A., Welder, R., & Yow, J. (2011, November). Transforming our practices by understanding the connections and comparing the challenges faced by new mathematics
teachers and new mathematics teacher educators. International Group for the Psychology of Mathematics Education North American Chapter Annual Conference. Reno, NV: International Group for the
Psychology of Mathematics Education North American Chapter.
• Beisigel, M., Eli, J. A., McCloskey, A., Welder, R., & Yow, J. (2011, October). Mathematics teacher educators: A missing link in the improvement and scholarship of mathematics teacher education.
International Society for the Scholarship of Teaching and Learning Conference. Milwaukee, WI: International Society for the Scholarship of Teaching and Learning.
• Eli, J. A., Mohr-Schroeder, M. J., & Lee, C. W. (2011, January). Exploring mathematical connections through the use of card sort activities. Association of Mathematics Teacher Educators. Irvine,
CA: Association of Mathematics Teacher Educators.
• Royal, K. D., & Eli, J. A. (2011, April). Using rasch measurement to measure factors affecting the frequency of academic misconduct. American Educational Research Association Annual Conference.
New Orleans, LA: American Educational Research Association.
• Anhalt, C., Cuprak, J., & Eli, J. A. (2010, April). Making connections: Long division of whole numbers and algebraic expressions. National Council of Teachers of Mathematics Annual Conference.
San Diego, CA: National Council of Teachers of Mathematics.
• Eli, J. A. (2010, March). Dissect, rearrange, and motivate: Area explorations of two-dimensional figures. The University of Arizona Institute for Mathematics and Education Middle School Teachers'
Circle. The University of Arizona: Institute for Mathematics and Education.
• Eli, J. A., Mohr-Schroeder, M. J., & Lee, C. W. (2010, January). Prospective middle grades teachers' mathematical connections and its relationship to their mathematics knowledge for teaching. The
8th annual Hawaii International Conference on Education. Honolulu, HI: Hawaii International Conference on Education.
• Bradley, K. D., Cunningham, J. D., & Eli, J. A. (2009, February). The relationship between primary methodology and perceptions of good educational research. American Association of Colleges for
Teacher Education. Chicago, IL: American Association of Colleges for Teacher Education.
• Eli, J. A. (2009, October). Exploring the web of connections. The University of Arizona Department of Mathematics Instructional Colloquium. The University of Arizona: Department of Mathematics.
• Eli, J. A., & Mohr-Schroeder, M. J. (2009, April). An exploratory study of prospective teachers' mathematical connections while completing tasks in geometry. American Educational Research
Association Annual Conference. San Diego, CA: American Educational Research Association.
Poster Presentations
• Eli, J. A., & Gonzalez, G. (2014, April). Middle grades teachers' knowledge for teaching solids of revolution. National Council of Teachers of Mathematics Research Conference. New Orleans, LA:
National Council of Teachers of Mathematics.
• Eli, J. A., Beisiegel, M., McCloskey, A., Welder, R., & Yow, J. (2012, July). Transitioning to faculty: new mathematics teacher educators' perceptions of successful doctoral program preparation.
International Congress on Mathematical Education Conference. Seoul, South Korea.
• Eli, J. A., & Mohr-Schroeder, M. J. (2009, April). Prospective middle grades teachers' mathematical connections in geometry. National Council of Teachers of Mathematics Research Pre-Session.
Washington, DC: National Council of Teachers of Mathematics.
• Wolfe, J. A. (2022, May). Facilitating mathematics groupwork through complex instruction. [Year 4]. Professional Development Workshop. UArizona Algebra Academy.. UArizona Early Academic Outreach.
• Wolfe, J. A., & Hackett, M. (2022, June). Sunnyside Lab School for Teachers (S-Lab). [Year 5]. Professional Development.. Sunnyside Unified School District Professional Development Summer
• Wolfe, J. A. (2021, May). Facilitating mathematics groupwork through complex instruction. [Year 3]. Professional Development. UArizona Algebra Academy.. UArizona Early Academic Outreach.
• Wolfe, J. A., & Hackett, M. (2021, June). Sunnyside Lab School for Teachers (S-Lab). [Year 4]. Professional Development.. Sunnyside Unified School District Professional Development Summer
• Wolfe, J. A., & Hackett, M. (2020, July). Sunnyside Lab School for Teachers (S-Lab). [Year 3]. Professional Development.. Sunnyside Unified School District Professional Development Summer
More info
Created, developed, and lead a 2-week professional development (labschool) for Sunnyside Unified School District. This particular labschool was modified for live online remote teaching with a
focus on equity and interactive collaboration for the google meets online learning environment.
• Eli, J. A. (2019, May). Facilitating mathematics groupwork through complex instruction. [Year 2]. Professional Development. UArizona Algebra Academy.. UArizona Early Academic Outreach.
• Wolfe, J. A., & Hackett, M. (2019, June). Sunnyside Lab School for Teachers (S-Lab). [Year 2]. Professional Development.. Sunnyside Unified School District Professional Development Summer
• Wolfe, J. A. (2018, May). Engaging students in equitable group collaboration: An introduction to complex instruction. [Year 1]. Professional Development. UArizona Algebra Academy.. UArizona Early
Academic Outreach.
• Eli, J. A. (2016, December). Who's Doing the Math? Learning to Facilitate Groupwork Through Complex Instruction. The University of Arizona College of Science, Department of Mathematics
Newsletter. http://w3.math.arizona.edu/files/newsletter/2016/Mathematics_NewsFall2016-web.pdf
• Torre, A. J., & Eli, J. A. (2016, March). Math education research: A complete non-scary and actually interesting area of math. Arizona Daily Wildcat.
More info
Please note that this was an article written by Alexa Jane Torres where I was interviewed along with my colleague Dr. Rebecca McGraw | {"url":"https://profiles.arizona.edu/person/jwolfe628","timestamp":"2024-11-07T10:27:40Z","content_type":"text/html","content_length":"113826","record_id":"<urn:uuid:e9afd666-d152-4bbb-b4c0-da37392cf57b>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00672.warc.gz"} |
Data Science Tutorials
How to perform the MANOVA test in R?. when there are several response variables, a multivariate analysis of variance can be used to examine them all at once (MANOVA). This article explains how to use
R to compute manova. For example, we might run an experiment in which we give two groups of mice two… | {"url":"https://datasciencetut.com/page/27/","timestamp":"2024-11-06T20:42:52Z","content_type":"text/html","content_length":"150078","record_id":"<urn:uuid:a2d70327-daf8-42c6-8733-072662b7ed5c>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00869.warc.gz"} |
Geometry: Parametric curves
Parametric curves
So far we have we have not yet been able to describe geometric shapes different from a circle, triangle or a line. The theory of parametric equations allows us to do describe shapes of various kinds.
We restrict our attention to parametric curves.
A parametric curve #\orange{C}# is a figure in the plane which is described by two equations
\[\orange C \colon \phantom{x}\begin{cases}\blue{x}&=\blue{x(t)}\\ \green{y}&=\green{y(t)} \end{cases}\]
where #t# varies over a specified #\text{interval}#. These equations are called parametric equations. The curve #\orange{C}# consists of all points #\ivcc{\blue{x(t)}}{\green{y(t)}}# for #t# in the
The parametric equations \[\begin{array}{rcl}\blue{x(t)}&=&\blue{t^2}\\\green{y(t)}&=&\green{2-t},\end{array}\]with #t# in the interval #\ivcc{-4}{2}# define a parametric curve.
In the figure you see the curve in the example defined by the equations \[\begin{array}{rcl}\blue{x(t)}&=&\blue{t^2}\\\green{y(t)}&=&\green{2-t},\end{array}\]
on the interval #[-4,2]#.
Two parametric curves with the same parametric equations can look very different if the intervals are different. In the example, the curve #\orange{C}# was defined by \[\begin{array}{rcl}\blue{x(t)}&
=&\blue{t^2}\\\green{y(t)}&=&\green{2-t},\end{array}\] with #t# ranging from #-4# to #2#.
When we change the interval to #t# ranging from #0# to #6#, we get the following figure:
More dimensions
One can also define a parametric equation with more variables and using more equations. For example, the parametric equation given by a scaled version of \[\begin{array}{rcl}x(t)&=&\cos(2t)+t\\y(t)&=
&t+t^2-4\\z(t)&=&\cos(2t) + \sin(3t) +0.2t,\end{array}\]where #t# ranges over the interval #\ivcc{-10}{10}# defines a curve in three dimensional space.
We will not study these kinds of curves in this course.
Such parametric curves can be used to describe the orbit of a certain point #\orange P#. If such an orbit is described by parametrical equations \[\begin{cases}\blue x &= \blue{x(t)},\\ \green y & =
\green{y(t)}. \end{cases}\] then one often writes #\orange{P_{t}}# for the 'location' of #\orange P# at time #t# - i.e. \[\orange{P_{t_0}}=\ivcc{\blue{x(t_0)}}{\green{y(t_0)}}\]
where #t_0# is a number in the interval. In the figure is an example where \[\ivcc{ \blue{x(t)}}{ \green{y(t)}} = \ivcc{ \blue{\sin(t) + \cos(t)}}{ \green{\cos(t)}}\]
On a parametric curve there is a notion of direction. Informally the direction of the curve is determined by the direction the curve has when #t# increases. One can reverse the direction of a
parametric curve by "moving in the other direction". For example a parametric curve on an interval #[a,b]# \[\begin{cases}\blue{x}&=\blue{x(t)}\\ \green{y}&=\green{y(t)} \end{cases}\] can be reversed
with the following curve: \[\begin{cases}\blue{x}&=\blue{x(-t)}\\ \green{y}&=\green{y(-t)} \end{cases}\]
on the interval #[-b, -a]#.
These notions can be made precise using the derivative. We will do so later.
The intersections of the curve with the axes of the #x,y#-plane can be computed. To compute the intersection of the curve with the #y#-axis, one should solve #\blue{x(t)}=0#. The intersections with
the #x#-axis are computed by solving #\green{y(t)}=0#.
Parametric curves are very useful to describe motion in physics.
Throwing a ball
The curve
\[\orange P \colon \phantom{x}\begin{cases}\blue{x(t)}&=10 \cdot \cos( \theta ) t\\ \green{y(t)}&= 10 \cdot \sin(\theta) t - \frac{1}{2}g \cdot t^2 \end{cases}\]
describes the orbit of an object being thrown from the origin with a varying angle #\theta# and a speed of #10 \text{ } m/s# with a gravitational constant #g#. On most places on earth the
gravitational constant is around #9.8 \text{ } m /s^2#.
It is possible to adjust the values in the picture by using the sliders. Note that the distance thrown is maximal when the angle is #45# degrees and that the value of #\theta# in the slider is in
Friction In the example we disregard friction for simplicity.
Maximal height The maximal height of a parametric curve is attained whenever #\green{y(t)}# is maximal within the specified interval. If we want to know the maximal height of the curve in the
example, we should find the maximal value of
\[\green{y(t)}= 10 \cdot \sin(\theta)\cdot t - \frac{1}{2}g \cdot t^2 \]
The time at which the maximum value is attained can be found by setting the derivative to zero:
\[\frac{\partial y(t)}{\partial t}=10\cdot \sin(\theta) - g\cdot t = 0\quad\text{ so }\quad t= \frac{10\cdot \sin(\theta)}{g}\]
The maximal height of the curve is therefore the height that is attained at time #t_h= \frac{10\cdot \sin(\theta)}{g}#, so we substitute this value:
\[\begin{array}{rcl}\green{y(t_h)}&=& 10 \cdot \sin(\theta)\cdot t_h - \frac{1}{2}g \cdot t_h^2\\ &=& 10\cdot \sin(\theta)\cdot \frac{10\cdot \sin(\theta)}{g} - \frac{1}{2}g\cdot \left(\frac{10\cdot
\sin(\theta)}{g}\right)^2\\ &=& 100\cdot \frac{\sin^2(\theta)}{g}- 50\cdot \frac{\sin^2(\theta)}{g} \\ &=&50\cdot \frac{\sin^2(\theta)}{g} \end{array}\]
This gives a maximal height of # 50\cdot \frac{\sin^2(\theta)}{g}#.
Every graph of a function can be described with a parametric curve. In this sense, the theory of parametric curves is a broader theory than the theory of functions and graphs.
If #f(x)# is a function on a domain #\ivcc{a}{b}# then we can define a parametric curve #\orange{C}# as follows
\[\begin{array}{rcl}\blue{x(t)}&=&\blue{t}\\\green{y(t)}&=&\green{f(t)},\end{array}\] where #t# ranges over the interval #\ivcc{a}{b}#. This curve coincides with the graph of the function # f(x)#.
Technicality Using the definition of a parametric curve as above, it is actually not true that every graph of a function can be described with the parametric equation as in the example. There is a
technicality that the domain of a function need not be an interval whereas the variable #t# in a parametric curve should always vary over an interval. The precise statement therefore is: Every graph
of a function defined on an interval can be described with a parametric curve.
Another way to get rid of this technicality is relaxing the definition of a parametric equation such that #t# is also allowed to vary over any subset of the real numbers.
Converse statement Not every parametric curve can be described as the graph of a single function. Take for example the unit circle, described with the parametric equation \[\begin{array}{rcl}\green{x
(t)}&=&\green{\cos (t)}\\ \blue{y(t)}&=&\blue{\sin(t)}.\end{array}\]
This describes the circle with radius #1# centered around the origin.
Since a function can only have a single #y#-value corresponding to every #x#-value, we need at least two functions to describe the unit circle. These would be one for the top half and one for the
bottom half of the circle.
Sketching a parametric curve Given parametric equations #\ivcc{\blue{x(t)}}{\green{y(t)}}# and an interval #\ivcc{a}{b}# for #t# it can be very useful to make a sketch of the curve. This can be done
by picking some explicit values for #t# in #\ivcc{a}{b}# and substituting these in the parametric equations #\blue{x(t)}# and #\green{y(t)}#. After drawing these in the plane it will, in most cases,
be clear what the corresponding curve should be.
Consider the curve #\orange C# given by #\ivcc{ \blue{x(t)}}{ \green{y(t)} }= \ivcc{ \blue{\frac{3t}{1+t^3}}}{ \green{\frac{3t^2}{1+3t^3}}}# defined for #t \neq -1#. We picked some values for #t# and
plotted the point #\orange{P_t}#. The dotted line is the curve #\orange C# itself.
Determine the maximal height #h# of the parametric curve given by equations
x(t) &= \left(t+3\right)\cdot \left(t+6\right)+1, \\
y(t) &= 5\cdot \cos \left(\sqrt{t^2-4}\right)-1.
where #t# lies in #\left[ -9 , 9 \right] #.
The maximal height is #4#.
The maximal height is attained whenever #y(t) = 5\cdot \cos \left(\sqrt{t^2-4}\right)-1# is maximal. The cosine is periodic and takes maximal value #1#. This happens whenever #\sqrt{t^2-4} = 0#.
Squaring the equation gives us #t^2-4 = 0#. We solve this by factorizing. We get #\left(t-2\right)\cdot \left(t+2\right) = 0# and find solutions #\left[ t=-2 , t=2 \right] #. These lie in the desired
interval. Consequently, the maximal height is given by substituting one of these values in #y(t)#. We see that the height is given by #4#.
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clump history command
clump history <name sid > keyword ...
Adds a history of a clump value. A history is a table of floating point values assigned by sampling a model attribute periodically during a simulation. Review the common c history commands for
details regarding the manipulation of histories.
The history can be assigned a name for later reference with the optional name keyword, if not the history will be given a default name based on its internally assigned id number.
Following one of the keywords given below, either 1) an additional id keyword followed by an integer id, or 2) an additional position keyword followed by a position vector v must be given to
define the specific clump.
displacement [clumphistoryblock]
Magnitude of the accumulated clump displacement vector as a result of cycling.
displacement-x [clumphistoryblock]
The \(x\)-component of accumulated displacement as a result of cycling.
displacement-y [clumphistoryblock]
The \(y\)-component of accumulated displacement as a result of cycling.
displacement-z (3D ONLY) [clumphistoryblock]
The \(z\)-component of accumulated displacement as a result of cycling.
euler-x (3D ONLY) [clumphistoryblock]
The \(x\)-euler angle (in degrees) of the current clump orientation. The orientation is updated only when orientation tracking has been enabled (see the model orientation-tracking command).
See the euler keyword of the clump attribute command.
euler-y (3D ONLY) [clumphistoryblock]
The \(y\)-euler angle (in degrees) of the current clump orientation. The orientation is updated only when orientation tracking has been enabled (see the model orientation-tracking command).
See the euler keyword of the clump attribute command.
euler-z (3D ONLY) [clumphistoryblock]
The \(z\)-euler angle (in degrees) of the current clump orientation. The orientation is updated only when orientation tracking has been enabled (see the model orientation-tracking command).
See the euler keyword of the clump attribute command.
extra [clumphistoryblock]
Take a history of an extra variable. Use index to specify the extra index and type to specify the type.
force-contact [clumphistoryblock]
The magnitude of the force resulting from contacts.
force-contact-x [clumphistoryblock]
The \(x\)-component of the contact force.
force-contact-y [clumphistoryblock]
The \(y\)-component of the contact force.
force-contact-z (3D ONLY) [clumphistoryblock]
The \(z\)-component of the contact force.
force-unbalanced [clumphistoryblock]
The magnitude of the unbalanced force (sum of the applied, contact and gravitational forces).
force-unbalanced-x [clumphistoryblock]
The \(x\)-component of the unbalanced force.
force-unbalanced-y [clumphistoryblock]
The \(y\)-component of the unbalanced force.
force-unbalanced-z (3D ONLY) [clumphistoryblock]
The \(z\)-component of the unbalanced force.
moment-contact [clumphistoryblock]
{Value in 2D; Magnitude in 3D} of the sum of the contact moments.
moment-contact-x (3D ONLY) [clumphistoryblock]
The \(x\)-component of the contact moment.
moment-contact-y (3D ONLY) [clumphistoryblock]
The \(y\)-component of the contact moment.
moment-contact-z (3D ONLY) [clumphistoryblock]
The \(z\)-component of the contact moment.
moment-unbalanced [clumphistoryblock]
{Value in 2D; Magnitude in 3D} of the sum of the contact and applied moments.
moment-unbalanced-x (3D ONLY) [clumphistoryblock]
The \(x\)-component of the unbalanced moment.
moment-unbalanced-y (3D ONLY) [clumphistoryblock]
The \(y\)-component of the unbalanced moment.
moment-unbalanced-z (3D ONLY) [clumphistoryblock]
The \(z\)-component of the unbalanced moment.
position-x [clumphistoryblock]
The \(x\)-component of the location of clump centroid.
position-y [clumphistoryblock]
The \(y\)-component of the location of clump centroid.
position-z (3D ONLY) [clumphistoryblock]
The \(z\)-component of the location of clump centroid.
rotation (2D ONLY) [clumphistoryblock]
Current clump orientation. The orientation is updated only when orientation tracking has been enabled (see model orientation-tracking command).
spin [clumphistoryblock]
{Value in 2D; Magnitude in 3D} of the clump angular velocity in radians per second.
spin-x (3D ONLY) [clumphistoryblock]
The \(x\)-component of the clump angular velocity in radians per second.
spin-y (3D ONLY) [clumphistoryblock]
The \(y\)-component of the clump angular velocity in radians per second.
spin-z (3D ONLY) [clumphistoryblock]
The \(z\)-component of the clump angular velocity in radians per second.
stress [clumphistoryblock]
The clump stress. Use the quantity keyword to specify the quantity to be recorded.
velocity [clumphistoryblock]
Magnitude of the clump translational velocity vector.
velocity-x [clumphistoryblock]
The \(x\)-component of the clump velocity.
velocity-y [clumphistoryblock]
The \(y\)-component of the clump velocity.
velocity-z (3D ONLY) [clumphistoryblock]
The \(z\)-component of the clump velocity.
clump history Keyword Block
The following modifiers are available to specify addition information for the clump history keywords. They will not modify history values to which they are not applicable. In other words, the
component keyword will not modify scalar or tensor values. displacement, displacement-x, displacement-y, displacement-z, euler-x, euler-y, euler-z, extra, force-contact, force-contact-x,
force-contact-y, force-contact-z, force-unbalanced, force-unbalanced-x, force-unbalanced-y, force-unbalanced-z, moment-contact, moment-contact-x, moment-contact-y, moment-contact-z, moment-unbalanced
, moment-unbalanced-x, moment-unbalanced-y, moment-unbalanced-z, position-x, position-y, position-z, rotation, spin, spin-x, spin-y, spin-z, stress, velocity, velocity-x, velocity-y and velocity-z.
component keyword
This keyword selects which scalar to retrieve from a vector type value, such as velocity or displacement. If the value type is not a vector, this setting is ignored. The available options
Record the \(x\)-component of the vector.
Record the \(y\)-component of the vector.
z (3D ONLY)
Record the \(z\)-component of the vector.
Record the vector magnitude.
id i
Record the history of clump with ID i.
index i
For keywords that require it (most notably extra), this specifies the index that should be used.
log b
If on, the returned number is the base 10 log of the absolute value of the original value. The default is off.
position v
Record the history of clump either containing or closest to position v.
quantity keyword
This keyword selects which scalar to retrieve from a symmetric tensor type value, such as stress or strain. If the value type is not a tensor, this setting is ignored. The available options
intermediate (3D ONLY)
Record the intermediate principal value of the tensor.
Record the maximum (most positive) principal value of the tensor. Note that compressive stresses are negative.
Record the mean value, defined as the trace of the tensor divided by { 2 in 2D; 3 in 3D}. For stresses this is most often referred to as the pressure.
Record the minimum (most negative) principal value of the tensor. Note that compressive stresses are negative.
Record the norm of the full tensor. This is defined as the sum of each tensor component squared.
octahedral (3D ONLY)
Record the octahedral measure of the tensor.
Record the maximum shear value.
total-measure (3D ONLY)
Record the distance of the tensor value to the origin in principal space.
Record the volumetric change, or trace of the tensor.
von-mises (3D ONLY)
Record the Von Mises measure.
Record the xx-component of the tensor.
Record the xy-component of the tensor.
xz (3D ONLY)
Record the xz-component of the tensor.
Record the yy-component of the tensor.
yz (3D ONLY)
Record the yz-component of the tensor.
zz (3D ONLY)
Record the zz-component of the tensor.
type keyword
In certain cases the type (scalar, vector, or tensor) of the value cannot necessarily be determined ahead of time. Extra variables, for example, can hold values of all three types. This
keyword allows one to specify which type it is assumed to be. If the original value type does not match, 0.0 is returned.
Specify a scalar float type.
Specify a tensor type.
Specify a vector type.
Usage Examples
Record the \(z\)-component of position of clump with ID 1. The history is named 10.
clump history name '10' position-z id 1
Record the \(y\)-component of position of clump with ID 2.
clump history position-y id 2
CS: BEEFUP. ‘history’ for balls, clumps, etc is pretty thin. Greater use in examples seems advisable.
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Probability, Symmetry, Linearity (2/6)
I plan six lectures on possible directions of modification/generalization of the probability theory, both concerning mathematical foundations and applications within and without pure mathematics.
Specifically, I will address two issues.
1. Enhancement of stochastic symmetry by linearization and Hilbertization of set-theoretic categories.
2. Non-symmetric probability theory in heterogeneous environments of molecular biology and of linguistics. | {"url":"https://indico.math.cnrs.fr/event/574/","timestamp":"2024-11-02T18:33:05Z","content_type":"text/html","content_length":"97344","record_id":"<urn:uuid:9725ac69-faf3-45ea-9951-042e04d6bacd>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00829.warc.gz"} |
Breakthrough in Quantum Computing: Google's Quantum Processor Surpasses Supercomputers
The realm of quantum computing has witnessed a monumental milestone as Google's quantum processor, known as Sycamore, has surpassed the capabilities of the world's most powerful supercomputers in
performing a specific computational task. This breakthrough marks a pivotal moment in the evolution of quantum computing, paving the way for transformative applications in various fields.
Quantum computing operates on the principles of quantum mechanics, a branch of physics that governs the behavior of subatomic particles. Unlike classical computers, which process information in bits
that can be either 0 or 1, quantum computers utilize qubits, which possess the remarkable property of superposition, allowing them to exist in multiple states simultaneously. This unique attribute
enables quantum computers to perform certain calculations exponentially faster than conventional supercomputers.
Google's Sycamore Quantum Processor:
Google's Sycamore quantum processor consists of 53 superconducting qubits, arranged in a grid-like structure. By carefully controlling the interactions between these qubits, researchers were able to
create a quantum circuit that performed a specific task with unprecedented speed.
Computational Task:
The task assigned to Sycamore was to sample from a random quantum circuit, which involves generating a series of random quantum operations and measuring the resulting quantum state. This task is
computationally intensive for classical computers because it requires determining the probability of all possible outcomes, which grows exponentially with the number of qubits.
Surpassing Supercomputers:
Sycamore completed the sampling task in approximately 200 seconds. In contrast, the world's fastest supercomputer, Summit, would have required approximately 10,000 years to perform the same
computation. This remarkable acceleration represents a significant advantage for quantum computing over classical approaches.
Significance of the Breakthrough:
The achievement of Sycamore heralds a new era in quantum computing. It demonstrates the practical viability of quantum processors for solving certain computational problems that are intractable for
classical computers. This breakthrough has profound implications for a wide range of fields, including:
• Drug Discovery: Quantum computers can accelerate the simulation of molecular systems, enabling more efficient development of new drugs.
• Materials Science: They can facilitate the design and optimization of novel materials with enhanced properties.
• Financial Modeling: Quantum algorithms can improve the accuracy and efficiency of complex financial models.
• Artificial Intelligence: Quantum processors can enhance the performance of machine learning algorithms.
Challenges and Future Prospects:
While Sycamore's breakthrough is a major milestone, quantum computing technology is still in its infancy. Challenges remain in scaling up quantum processors to larger numbers of qubits, reducing
errors, and developing practical applications. However, ongoing research and advancements promise continued progress in the field.
Google's Sycamore quantum processor has achieved a groundbreaking accomplishment by surpassing supercomputers in performing a specific computational task. This breakthrough marks a pivotal moment in
the evolution of quantum computing, opening doors to solve complex problems and drive transformative applications across diverse fields. As the technology continues to mature, it has the potential to
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Mock CCC '19 Contest 1 S3 - Pusheen Eats Tuna Sashimi and Tuna Nigiri
View as PDF
Pusheen has been dreaming about tuna sashimi! She has decided that she needs to eat more tuna in her life, so she decides to visit her favourite restaurant to eat tuna sashimi and tuna nigiri.
Pusheen decides to take her helicopter to her favourite restaurant. Due to various regulations, Pusheen is required to fix the helicopter at a specific altitude if she's not taking off or landing, so
the helicopter can be effectively treated as traveling in a 2D plane. Due to fuel restrictions, Pusheen will travel only to points with nonnegative coordinates further subject to the constraint that
and .
The helicopter is a bit finicky to maneuver, so in a single second, Pusheen can only change either the velocity of the helicopter in the x-direction or the y-direction by one unit. Pusheen is not
required to change the velocity of the helicopter within a second. Note that the change in velocity happens before the helicopter moves within that second - this means that at the end of every
second, Pusheen is guaranteed to be at a lattice point. Pusheen can only board/disembark from the helicopter when it is not moving.
More formally, if Pusheen's location at the beginning of second is and her helicopter's velocity is , and Pusheen decides to change the velocity at the start of that second by , then Pusheen's
location at the beginning of second is and her helicopter's velocity is . Pusheen can only board/disembark when . Pusheen's helicopter will fly through all points on the line segment from to in that
It's also a windy day! Some lattice points have high winds that Pusheen must not fly through at any point in time. Help Pusheen figure out if she can get to her favourite restaurant, and figure out
how quickly she can do so!
Pusheen's starting and ending locations will not have wind.
All lattice points with wind are distinct.
In tests worth 1 mark, and .
In tests worth 2 additional marks, and .
In tests worth 3 additional marks, .
Input Specification
The first line contains three nonnegative integers, , , and .
The next line contains two nonnegative integers, and , representing Pusheen's starting location.
The next line contains two nonnegative integers, and , representing the location of Pusheen's favourite restaurant.
The next lines each contain two space-separated integers, and , representing a lattice point with wind.
Output Specification
If Pusheen cannot make it to her favourite restaurant, output -1. Otherwise, output the minimum number of seconds needed for Pusheen to be able to make it to her favourite restaurant.
Sample Input 1
Sample Output 1
Sample Input 2
Sample Output 2
Sample Input 3
Sample Output 3
Sample Input 4
Sample Output 4
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History, Progress and New Results in Synthetic Passive Element Design Employing CFTAs
After the presentation of the Current Follower Transconductance Amplifier (CFTA) active element, it has found a numerous application possibilities while designing linear and non-linear analog
function blocks. This paper gives a short review of the CFTA and mainly focuses on the synthetic floating and grounded passive element design, which can also be electronically controllable. Except
the design of synthetic inductors, also possible realizations of floating and grounded capacitors and resistors are described, where the value of these passive elements can be adjusted by means of
active elements’ parameters. For the design of the corresponding circuit realizations, the Mason-Coates signal flow graph approach is used. The performance of some discussed synthetic elements is
verified and evaluated by Spice simulations on simple analog frequency filters.
K. C. Smith, A. Sedra, “The current conveyor: a new circuit building block,” IEEE Proc., vol. 56, pp. 1368–1369, 1968.
A. Sedra, K. C. Smith, “A second-generation current conveyor and its application,” IEEE Trans. Circuit Theory, vol. 17, pp. 132–134, 1970.
A. Fabre, “Third-generation current conveyor: a new helpful active element,” Electronics Letters, vol. 31, no. 5, pp. 338–339, 1995.
T. Dostal, J. Pospisil, “Current and voltage conveyors - a family of three port immittance converters,” Proc. ISCAS, Roma, pp. 419–422, 1982.
C. Acar, S. Ozoguz, “A new versatile building block: current differencing buffered amplifier suitable for analog signal-processing filters,” Microelectronics Journal, vol. 30, no. 2, pp. 157-160,
J. Koton, K. Vrba, N. Herencsar, “Tuneable filter using voltage conveyors and current active elements,” Int. J. Electronics, vol. 96, no. 8, pp. 787–794, 2009.
D. Biolek, “CDTA - Building Block for Current-Mode Analog Signal Processing,” in Proc. Int. Conf. ECCTD03 Krakow, Poland, vol. III, pp. 397–400, 2003.
R. Prokop, V. Musil, “CCTA-a new modern circuit block and its internal realization,” in Proc. Int. Conf. Electronic Devices and Systems, pp. 89-93, Brno, Czech Republic, 2005.
D. Biolek, V. Biolkova, “CTTA Current-mode filters based on current dividers,” in Proc. 11th Electronic Devices and Systems Conference, pp. 2–7, 2004.
N. Herencsar, J. Koton, I. Lattenberg, K. Vrba, “Signal-flow Graphs for Current-Mode Universal Filter Design Using Current Follower Transconductance Amplifiers (CFTAs),” in Proc. Int. Conf. Applied
Electronics - APPEL, Pilsen, Czech Republic, pp. 69–72, 2008.
C. Hou, Y. Wu, S. Liu, “New configuration for single-CCII first-order and biquadratic current mode filters,” Int. J. Electronics, vol. 71, no. 4, pp. 637–644, 1991.
K. Vrba, J. Cajka, V. Zeman, “New RC-Active Network Using Current Conveyors,” Radioengineering, vol. 6, no. 2, pp. 18–21, 1997.
G. W. Roberts, A. S. Sedra, “All Current-mode Frequency Selective Filters,” Electronics Letters, vol. 25, no. 12, pp. 759–760, 1989.
Y.-S. Hwang, P.-T. Hung, W. Chen, S.-I. Liu, “CCII-based linear transformation elliptic filters,” Int. J. Electronics, vol. 89, no. 2, pp. 123–133, 2002.
J. Koton, K. Vrba, P. Ushakov, J. Misurec, “Designing Electronically Tunable Frequency Filters Using the Signal Flow Graph Theory,” in Proc. 31th Int. Conf. Telecommunications and Signal Processing -
TSP, pp. 41–43, 2008.
A. Uygur, H. Kuntman, A. Zeki, “Multi-input multi-output CDTA-based KHN filter,” in Proc. 4th Int. Conf. Electrical and Electronics, Bursa, Turkey, 2005.
A. U. Keskin, D. Biolek, E. Hancioglu, V. Biolkova, “Current-mode KHN filter employing current differencing transconductance amplifiers,” Int. J. Electronics and Communications, vol. 60, no. 6, pp.
443-446, 2006.
N. A. Shah, M. Quadri, S. Z. Iqbal, “Realization of CDTA based current mode universal filter,” Indian Journal of Pure and Applied Physics, vol. 46, no. 4, pp. 283-285, 2008.
D. Biolek, V. Biolkova, Z. Kolka, “Current-mode biquad employing single CDTA,” Indian Journal of Pure and Applied Physics, vol. 47, no. 7, pp. 535-537, 2009.
A. Lahiri, “New current-mode quadrature oscillators using CDTA,” IEICE Electronics Express, vol. 6, no. 3, pp. 135-140, 2009.
D. Prasad, D. R. Bhaskar, A. K. Singh, “New grounded and floating simulated inductance circuits using current differencing transconductance amplifiers,” Radioengineering, vol. 19, no. 1, pp. 194-198,
W. Tangsrirat, T. Pukkalanun, P. Mongkolwai, and W. Surakampontorn, “Simple current-mode analog multiplier, divider, square-rooter and squarer based on CDTAs,” Int. J. Electronics and Communications,
vol. 65, no. 3, pp. 198-203, 2011.
L. Tan, K. Liu, Y. Bai, J. Teng, “Construction of CDBA and CDTA behavioral models and the applications in symbolic circuits analysis,” Analog. Integr. Circ. Sig. Process., vol 75., no. 3, pp.
517–523, 2013.
R. L. Geiger, E. Snchez-Sinencio, “Active Filter Design Using Operational Transconductance Amplifiers: A Tutorial,” IEEE Circuits and Devices Magazine, Vol. 1, pp. 20–32, 1985.
N. Herencsar, J. Koton, K. Vrba, A. Lahiri, O. Cicekoglu, “Current-Controlled CFTA-Based Current-Mode SITO Universal Filter and Quadrature Oscillator,” in Proc. Int. Conf. Applied Electronics -
APPEL, Pilsen, Czech Republic, pp. 121–124, 2010.
N. Herencsar, J. Koton, K. Vrba, O. Cicekoglu, “New Active-C Grounded Positive Inductance Simulator Based on CFTAs,” in Proc. Int. Conf. Telecommunications and Signal Processing - TSP, pp. 35–37,
P. Suwanjan, W. Jaikla, “CFTA Based MISO Current-mode Biquad Filter,” in Proc. Recent Researches in Circuits, Systems, Multimedia and Automatic Control, Rovaniemi, Finland, pp. 93-97, 2012.
W. Tangsrirat, “Active-C Realization of nth-order Current-Mode Allpole Lowpass Filters Using CFTAs,” in Proc. Int. Multiconf. Engineers and Computer Scientists - IMECS, vol. II, pp. 1–4, 2012.
Datasheet AD844: 60 MHz 2000 V/ s Monolithic Op Amp, Analog Devices, Rev. 7, 2009.
Datasheet OPA861: Wide Bandwidth Operational Transconductance Amplifier (OTA), Texas Instruments, SBOS338G, 2013.
Datasheet MAX435/MAX436: Wideband Transconductance Amplifiers, MAXIM, Rev. 1, 1993.
N. Herencsar, J. Koton, K. Vrba, J. Misurec, “A Novel Current-Mode SIMO Type Universal Filter Using CFTAs,” Contemporary Engineering Sciences, vol 2, no. 2, pp. 59–66, 2009.
Datasheet UCC-N1B: Universal Current Conveyor (UCC) and Second-Generation Current Conveyor (CCII+/-), ON Semiconductor & Brno University of Technology, Rev. 1, 2010.
N. Herencsar, J. Koton, K. Vrba, I. Lattenberg, J. Misurec, “Generalized Design Method for Voltage-Controlled Current- Mode Multifunction Filters,” in Proc. 16th Telecommunications Forum TELFOR 2008,
Belgrade, Serbia, pp. 400–403, 2008.
D. Biolek, R. Senani, V. Biolkova, Z. Kolka, “Active Elements for Analog Signal Processing: Classification, Review, and New Proposals,” Radioengineering, vol. 17. no. 4, 2008.
J. Sirirat, D. Prasertsom, W. Tangsrirat, “High-Output-Impedance Current-Mode Electronically Tunable Universal Filter Using Single CFTA,” in Proc. Int. Symp. Communications and Information
Technologies - ISCIT, Tokyo, Japan, pp. 200–203, 2010.
N. Herencsar, J. Koton, K. Vrba, “Realization of Current-Mode KHN Equivalent Biquad Using Current Follower Transconductance Amplifiers (CFTAs),” IEICE Transactions on Fundamentals of Electronics
Communications and Computer Sciences, vol. E93, pp. 1816–1819, 2010.
W. Tangsrirat, “Novel current-mode and voltage-mode universal biquad filters using single CFTA,” Indian J. Engineering and Materials Science, vol. 17, pp. 99–104, 2010.
J. Sirirat, W. Tangsrirat, W. Surakampontorn, “Voltage-mode electronically tunable universal filter employing single CFTA,” in Proc. Int. Conf. Electrical Engineering/Electronics Computer
Telecommunications and Information Technology - ECTI-CON, Chaing Mai, pp. 759–763, 2010.
W. Tangsrirat, “Single-input three-output electronically tunable universal current-mode filter using current follower transconductance amplifiers,” Int . J. Electronics and Communications - AEU, vol.
65, pp. 783–787, 2011.
J. Satansup, T. Pukkalanun, W. Tangsrirat, “Current-Mode KHN Biquad Filter Using Modified CFTAs and Grounded Capacitors,” in Proc. Int. MultiConf. of Engineers and Computer Scientists - IMECS, Hong
Kong, vol. II, 2011.
D. Duangmalai, A. Noppakarn, W. Jaikla, “Electronically Tunable Low-Component-Count Current-Mode Biquadratic Filter Using CFTAs,” in Proc. Int. Conf. Information and Electronics Engineering - IPCSIT,
Singapore, vol 6, pp. 263–267, 2011.
J. Satansup, W. Tansrirat, “Realization of current-mode KHN-equivalent biquad filter using ZC-CFTAs and grounded capacitors,” Indian J. Pure and Applied Physics, vol. 49, pp. 841–846, 2011.
J. Satansup, W. Tansrirat, “Single-Input Five-Output Electronically Tunable Current-Mode Biquad Consisting of Only ZC-CFTAs and Grounded Capacitors,” Radioengineering, vol. 20, no. 3, pp. 650–655,
W. Jaikla, S. Lawanwisut, M. Siriprucyanun, P. Prommee, “A Four-Inputs Single-Output Current-Mode Biquad Filter Using a Minimum Number of Active and Passive Components,” in Proc. Int. Conf.
Telecommunications and Signal Processing - TSP, Prague, Czech Republic, pp. 378–381, 2012.
W. Tangsrirat, “Active-C Realization on nth-order Current-Mode Allpole Lowpass Filters Using CFTAs,” in Proc. Int. MultiConf. of Engineers and Computer Scientists - IMECS, Hong Kong, vol. II, 2012.
P. Suwanjan, W. Jaikla, “CFTA Based MISO Current-mode Biquad Filter,” in Proc. Int. Conf. Recent Trends in Circuits, Systems, Multimedia and Automatic Control, pp. 93–97, 2012.
M. Kumngern, “Electronically tunable current-mode universal biquadratic filter using a single CCCFTA,” in Proc. Int. Symp. Circuits and Systems - ISCAS, Seoul, South Korea, pp. 1175–1178, 2012.
B. Singh, A.K. Singh, R. Senani, “New Universal Current-mode Biquad Using Only Three ZC-CFTAs,” Radioengineering, vol 21, no. 1, pp. 273–280, 2012.
S. Lawanwisut, M. Siripruchyanun, “A Current-mode Multifunction Biquadratic Filter Using CFTAs,” J. of KMUTNB, vol. 22, no. 3, pp. 479–485, 2012.
W. Tangsrirat, P. Mongkolwai, T. Pukkalanun, “Current-mode high-Q bandpass filter and mixed-mode quadrature oscillator using ZC-CFTAs and grounded capacitors,” Indian J Pure and Applied Physics, vol.
50, pp. 600–607, 2012.
X. Nie, Z. Pan, “Multiple-input single-output low-input and high-output impedance current-mode biquadratic filter employing five modified CFTAs and only two grounded capacitors,” Microelectronics
Journal, vol. 44, no. 9, pp. 802–806, 2013.
W. Tangsrirat, “Gm-Realization of Controlled-Gain Current Follower Transconductance Amplifier,” The Scientific World Journal, vol. 2013, pp. 1–8, doi:10.1155/2013/201565, 2013.
R.S. Tomar, C. Chauhan, S.V. Singh, D.S. Chauhan, “Current-Mode Active-C Biquiad Filter Using Single MO-CCCFTA,” in Proc. Int. Conf. Recent Trends in Computing and Communication Engineering - RTCCE,
pp. 259–262, 2013.
W. Tangsrirat, S. Unhavanich, “Signal flow graph realization of single input five-output current-mode universal biquad using current follower transconductance amplifiers,” Rev. Roum. Sci. Techn. -
Electrotechn. et Energ., vol. 59, no. 2, pp. 183–191, 2014.
N. Herencsar, J. Koton, K. Vrba, “Electronically Tunable Phase Shifter Employing Current-Controlled Current Follower Transconductance Amplifiers (CCCFTAs),” in Proc. 32nd Int. Cont.
Telecommunications and Signal Processing - TSP, Budapest, Hungary, pp. 54–57, 2009.
P. Mongokolwai, T. Dumawipata, W. Tangsrirat, “Current-Mode Quadrature Oscillator Employing ZC-CFTA Based First-Order Allpass Sections,” in Int. Conf. Modeling and Simulation Technology, Tokyo,
Japan, pp. 472–475, 2011.
T. Nakyoy, W. Jaikla, “Resistorless First-Order Current-Mode Allpass Filter Using Only Single CFTA and Its Application,” in Proc. IEEE Int. Symp. Electronic Design, Test and Application - DELTA,
Queenstown, doi: 10.1109/DELTA.2011.28, pp. 105–109, 2011.
K. Intawichai, W. Tangsrirat, “Signal flow graph realization of nth-order current-mode allpass filters using CFTAs,” in Proc. Int. Conf. Electrical Engineering/Electronics Computer Telecommunications
and Information Technology - ECTI-CON, Krabi, doi:10.1109/ECTICon.2013.6559519, pp. 1–6, 2013.
A. Iamarejin, S. Maneewan, P. Suwanjan, W. Jaikla, “Current-mode variable current gain first-order allpass filter employing CFTAs,” Przeglad Elektrotechniczny, vol. 89, no. 2a, pp. 238–241, 2013.
A. Lahiri, “Resistor-less mixed-mode quadrature sinusoidal oscillator,” Int. J. Computer and Electrical Engineering, vol. 2, no. 1, pp. 63–66, 2010.
N. Herencsar, K. Vrba, J. Koton, A. Lahiri, “Realizations of single-resistance-controlled quadrature oscillators using a generalized current follower transconductance amplifier and a unity-gain
voltagefollower,” Int. J. Electronics, vol. 97, no. 8, pp. 897–906, 2010.
S. Maneewan, B. Sreewirote, W. Jaikla, “A Current-mode Quadrature Oscillator Using a Minimum Number of Active and Passive Components,” in Proc. IEEE Int. Conf. Vehicular Electronics and Safety -
ICVES, Beijing, doi:10.1109/ICVES.2011.5983835, pp. 312–315, 2011.
M. Kumngern, U. Torteanchai, “A Current-Mode Four-Phase Third-Order Quadrature Oscillator Using a MCCCFTA,” in Proc. IEEE Int. Conf. Cyber Technology in Automation, Control and Intelligent Systems,
Bangkok, Thailand, pp. 156–159, 2012.
D. Prasertsom, W. Tangsrirat, “Current Gain Controlled CFTA and Its Application to Resistorless Quadrature Oscillator,” in Proc. Int. Conf. Electrical Engineering/Electronics Computer
Telecommunications and Information Technology - ECTI-CON, Phetchaburi, doi: 10.1109/ECTICon. 2012.6254271, pp. 1–4, 2012.
P. Uttaphut, “Realization of Electronically Tunable Current-Mode Multiphase Sinusoidal Oscillators using CFTAs,” World Academy of Science, Engineering and Technology, vol. 69, pp. 719–722, 2012.
Y.A. Li, “Electronically tunable current-mode biquadratic filter and fourphase quadrature oscillator,” Microelectronics Journal, vol. 45, no. 3, pp. 330–335, 2014.
P. Mongkolwai, W. Tangsrirat, “CFTA-based current multiplier/divider circuit,” in Proc. IEEE Int. Symp. Intelligent Signal Processing and Communications Systems - ISPACS, Chiang Mai, doi: 10.1109/
ISPACS.2011.6146074, pp. 1–4, 2011.
P. Silapan, C. Chanapromma, “Multiple Ouput CFTAs (MO-CFTAs)-based Wide-range Linearly/Electronically Controllable Current-mode Square-rooting Circuit,” in Proc. IEEE Int. Symp. Intelligent Signal
Processing and Communications Systems - ISPACS, Chiang Mai, doi: 10.1109/ISPACS.2011.6146152, pp. 1–4, 2011.
P. Silapan, C. Chanapromma, T. Worachak, “Realization of Electronically Controllable Current-mode Square-rooting Circuit Based on MOCFTA,” World Academy of Science, Engineering and Technology, vol.
58, pp. 493–496, 2011.
W. Kongnun, A. Aurasopon, “A Novel Electronically Controllable of Current-Mode Level Shifted Multicarrier PWM Based on MOCFTA,” Radioengineering, vol 22, no. 3, pp. 907–915, 2013.
W. Kongnun, P. Silapan, “A Single MO-CFTA Based Electronically/Temperature Insensitive Current-mode Half-wave and Full-wave Rectifiers,” Advances in Electrical and Electronic Engineering, vol. 11,
no. 4, pp. 275–283, 2013.
N. Herencsar, J. Koton, K. Vrba, A. Lahiri, “Floating Simulators Based on Current Follower Transconductance Amplifiers (CFTAs),” Advances in Communications, Computers, Systems, Circuits and Devices,
pp. 23–26, 2010.
N. Herencsar, J. Koton, K. Vrba, O. Cicekoglu, “New Active-C Grounded Positive Inductance Simulator Based on CFTAs,” in Proc. 33th Int. Conf. Telecommunications and Signal Processing - TSP, Baden bei
Wien, Austria, pp. 35–37, 2010.
M. Fakhfakh, M. Pierzchala, B. Rodanski, “An Improved Design of VCCS-Based Active Inductors,” in Proc. Int. Conf. on Synthesis, Modeling, Analysis and Simulation Methods and Applications to Circuit
Design - SMACD, Seville, pp. 101–104, doi: 10.1109/SMACD.2012.6339427, 2012.
N. Herencsar, A. Lahiri, J. Koton, K. Vrba, R. Sotner, “New Floating Lossless Inductance Simulator Using Z-copy Current Follower Transconductance Amplifier,” in Proc. 22nd Int. Conf.
Radioelektronika, Brno, Czech Republic, pp. 93–96, 2012.
D. Siriphot, S. Maneewan, W. Jaikla, “Single Active Element Based Electronically Controllable Grounded Inductance Simulator,” in Proc. IEEE Int. Conf. Biomedical Engineering - BMEiCON, Amphur Muang,
pp. 1–4, doi: 10.1109/BMEiCon.2013.6687724, 2013.
Y.-A. Li, “A series of new circuits based on CFTAs,” Int. J. Electronics and Communications - AEU, vol 66, pp. 587–592, 2012.
M. Venclovsky, Transform-based filter design technique based on passive structures, Master’s thesis, Brno University of Technology, 2009.
W.-K. Chen, The Circuits and Filters Handbook, New York, CRC Press, 2003, 2nd edition, ISBN 0-8493-0912-3.
T. Deliyannis, Y. Sun, J.K. Fidler, Continuous-Time Active Filter Design, New York, CRC Press, 1999, ISBN 0-8493-2573-0.
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Getting Started
Getting Started: SEQDESIGN Procedure
This section illustrates a clinical study design that uses a two-sided O’Brien-Fleming design (O’Brien and Fleming 1979) to stop the trial early for ethical concerns about possible harm or for
unexpectedly strong efficacy of the new drug.
Suppose that a pharmaceutical company is conducting a clinical trial to test the efficacy of a new cholesterol-lowering drug. The primary focus is low-density lipoprotein (LDL), the so-called bad
cholesterol, which is a risk factor for coronary heart disease. LDL is measured in
The trial consists of two groups of equally allocated patients with elevated LDL levels: an experimental group given the new drug and a placebo control group. Suppose the changes in LDL level after
the treatment for individuals in the experimental and control groups are normally distributed with means
For a fixed-sample design with a total sample size
Following the derivation in the section Test for the Difference between Two Normal Means, the statistic
Thus, under the null hypothesis
With a Type I error probability
Also suppose that for the trial, the alternative reference
ods graphics on;
proc seqdesign altref=-10
TwoSidedOBrienFleming: design nstages=4
samplesize model=twosamplemean(stddev=20);
ods output Boundary=Bnd_LDL;
ods graphics off;
The ALTREF= option specifies the alternative reference, and the actual maximum information is derived in the SEQDESIGN procedure. With the specified ODS GRAPHICS ON statement, the PLOTS=BOUNDARY
option displays a boundary plot with the rejection and acceptance regions.
In the DESIGN statement, the label TwoSidedOBrienFleming identifies the design in the output tables. By default (or equivalently if you specify ALT=TWOSIDED and STOP=REJECT in the DESIGN statement),
the design has a two-sided alternative hypothesis in which early stopping in the interim stages occurs to reject the null hypothesis. That is, at each interim stage, the trial either is stopped to
reject the null hypothesis or continues to the next stage.
The NSTAGES=
For a two-sided design with early stopping to reject the null hypothesis, there are two boundaries for the design: an upper Figure 78.7.
A property of the boundaries constructed with the O’Brien-Fleming design is that the null hypothesis is more difficult to reject in the early stages than in the later stages. That is, the trial is
rejected in the early stages only with overwhelming evidence, because in these stages there might not be a sufficient number of responses for a reliable estimate of the treatment effect.
The SAMPLESIZE statement with the MODEL=TWOSAMPLEMEAN option uses the derived maximum information to compute required sample sizes for a two-sample test for mean difference. The ODS OUTPUT statement
with the BOUNDARY=BND_LDL option creates an output data set named BND_LDL which contains the resulting boundary information.
In a clinical trial, the amount of information about an unknown parameter available from the data can be measured by the Fisher information. For a maximum likelihood statistic, the information level
is the inverse of its variance. See the section Maximum Likelihood Estimator for a detailed description of Fisher information. At each stage of the trial, data are collected and analyzed with a
statistical procedure, and a test statistic and its corresponding information level are computed.
In this example, you can use the REG procedure to compute the maximum likelihood estimate BOUND_LDL data set. At each subsequent stage, you can use the SEQTEST procedure to compare the test statistic
with adjusted boundaries derived from the boundary information stored in the test information table created by the SEQTEST procedure at the previous stage. The test information tables are structured
for input to the SEQTEST procedure.
At each interim stage, the trial will either be stopped to reject the null hypothesis or continue to the next stage. At the final stage, the null hypothesis is either rejected or accepted.
By default (or equivalently if you specify INFO=EQUAL in the DESIGN statement), the SEQDESIGN procedure derives boundary values with equally spaced information levels for all stages—that is, the same
information increment between successive stages. The "Design Information," "Method Information," and "Boundary Information" tables are displayed by default, as shown in Figure 78.4, Figure 78.5, and
Figure 78.6, respectively.
The "Design Information" table in Figure 78.4 displays design specifications and four derived statistics: the actual maximum information, the maximum information, the average sample number under the
null hypothesis (Null Ref ASN), and the average sample number under the alternative hypothesis (Alt Ref ASN). Except for the actual maximum information, each statistic is expressed as a percentage of
the identical statistic for the corresponding fixed-sample information. The average sample number is the expected sample size (for nonsurvival data) or expected number of events (for survival data).
Note that for a symmetric two-sided design, the ALTREF=
The SEQDESIGN Procedure
Design: TwoSidedOBrienFleming
Standardized Z
Reject Null
The maximum information is the information level at the final stage of the group sequential trial. The Max Information (Percent Fixed-Sample) is the maximum information for the sequential design
expressed as a percentage of the information for the corresponding fixed-sample design. In Figure 78.4, the Max Information (Percent Fixed-Sample) is
The Null Ref ASN (Percent Fixed-Sample) is the average sample number (expected sample size) required under the null hypothesis for the group sequential design expressed as a percentage of the sample
size for the corresponding fixed-sample design. In Figure 78.4, the Null Ref ASN is
Similarly, the Alt Ref ASN (Percent Fixed-Sample) is the average sample number (expected sample size) required under the alternative hypothesis for the group sequential design expressed as a
percentage of the sample size for the corresponding fixed-sample design. In Figure 78.4, the Alt Ref ASN is
In this example, the O’Brien-Fleming design requires only a slight increase in sample size if the trial proceeds to the final stage. On the other hand, if the alternative hypothesis is correct, this
design provides a substantial saving in sample size on average.
The "Method Information" table in Figure 78.5 displays the computed Type I and Type II error probabilities
With the zero null reference, the drift parameter is the standardized alternative reference at the final stage Specified and Derived Parameters for a detailed description of the drift parameter. The
drift parameters for the design are derived in the SEQDESIGN procedure even if the alternative reference is not specified or derived in the procedure.
The O’Brien-Fleming method belongs to the unified family of designs, which is parameterized by two parameters, Table 78.3 for parameter values of commonly used methods in the unified family. The
"Method Information" table in Figure 78.5 displays the values of Unified Family Methods.
The "Boundary Information" table in Figure 78.6 displays the information level, including the proportion, actual level, and corresponding sample size (N) at each stage. The table also displays the
lower and upper alternative references, and the lower and upper boundary values at each stage.
1 0.2500 0.026851 42.96116 -1.63862 1.63862 -4.04859 4.04859
2 0.5000 0.053701 85.92233 -2.31736 2.31736 -2.86278 2.86278
3 0.7500 0.080552 128.8835 -2.83817 2.83817 -2.33745 2.33745
4 1.0000 0.107403 171.8447 -3.27724 3.27724 -2.02429 2.02429
The information proportion is the proportion of maximum information available at each stage and N is the corresponding sample size. By default (or equivalently if you specify BOUNDARYSCALE=STDZ), the
procedure displays boundary values with the standardized
In this example, a standardized
Note that in a typical trial, the actual information levels do not match the information levels specified in the design. The SEQTEST procedure modifies the boundary values stored in the BOUND_LDL
data set to adjust for these new information levels.
With the specified ODS GRAPHICS ON statement, a detailed boundary plot with the rejection and acceptance regions is displayed, as shown in Figure 78.7. This plot displays the boundary values in the
"Boundary Information" table in Figure 78.6. The stages are indicated by vertical lines with accompanying stage numbers. The horizontal axis indicates the sample sizes for the stages. Note that
comparing with a fixed-sample design, only a small increase in sample size is needed for the O’Brien-Fleming design, as shown in Figure 78.7.
If a test statistic at an interim stage is in the rejection region (shaded area), the trial stops and the null hypothesis is rejected. If the statistic is not in any rejection region, the trial
continues to the next stage.
The boundary plot also displays critical values for the corresponding fixed-sample design. The symbol "
When you specify the SAMPLESIZE statement, the maximum information (either explicitly specified or derived in the SEQDESIGN procedure) is used to compute the required sample sizes for the study. The
MODEL=TWOSAMPLEMEAN(STDDEV=20) option specifies the test for the difference between two normal means. See the section Test for the Difference between Two Normal Means for a detailed derivation of
these required sample sizes.
The "Sample Size Summary" table in Figure 78.8 displays the parameters for the sample size computation and the resulting maximum and expected sample sizes.
The "Sample Sizes (N)" table in Figure 78.9 displays the required sample sizes at each stage for the trial, in both fractional and integer numbers. The derived fractional sample sizes are displayed
under the heading "Fractional N." These sample sizes are rounded up to integers under the heading "Ceiling N." By default (or equivalently if you specify WEIGHT=1 in the MODEL=TWOSAMPLEMEAN option),
the sample sizes for the two groups are equal for the two-sample test.
1 42.96 21.48 21.48 0.0269 44 22 22 0.0275
2 85.92 42.96 42.96 0.0537 86 43 43 0.0538
3 128.88 64.44 64.44 0.0806 130 65 65 0.0812
4 171.84 85.92 85.92 0.1074 172 86 86 0.1075
In practice, integer sample sizes are used in the trial, and the resulting information levels increase slightly. Thus, | {"url":"http://support.sas.com/documentation/cdl/en/statug/63347/HTML/default/statug_seqdesign_sect004.htm","timestamp":"2024-11-05T13:41:24Z","content_type":"application/xhtml+xml","content_length":"55518","record_id":"<urn:uuid:6d0dd6a4-4777-48ea-8a14-d58832b1133e>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00418.warc.gz"} |
Solving modular systems of equation
Solving modular systems of equation
I am trying to solve a system of equations mod $p$, for $p$ prime. I have build matrices $A$ and $b$ and I am using the function solve_right() to solve $Ax = b$. But I need to solve $Ax = b$ mod $p$.
Is there any functions that could do that ? thanks !
1 Answer
Sort by ยป oldest newest most voted
If you define your matrices containing elements of the field $\mathbb Z/p\mathbb Z$, solve_right does what you need.
sage: A = matrix(GF(17), [[6,3], [3,8]])
sage: b = matrix(GF(17), [1, 2]).transpose()
sage: A.solve_right(b)
If for some reason you need matrices over (say) $\mathbb Z$ but still need to solve modulo some prime $p$, you can use change_ring.
sage: A = matrix(ZZ, [[1,2], [3,4]])
sage: b = matrix(ZZ, [5, 6]).transpose()
sage: A.change_ring(GF(17)).solve_right(b.change_ring(GF(17)))
edit flag offensive delete link more
great, thanks. I'll try the last option.
Sasha-dpt ( 2017-04-02 22:27:03 +0100 )edit | {"url":"https://ask.sagemath.org/question/37146/solving-modular-systems-of-equation/?sort=oldest","timestamp":"2024-11-11T17:03:57Z","content_type":"application/xhtml+xml","content_length":"53205","record_id":"<urn:uuid:e5e8b3e6-ec28-4aa3-8ef9-7790aeff4b2b>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00550.warc.gz"} |
Differential geometric approach to quantum mechanics
30642 views
Plenty of books/papers have been written about differential geometry in relation with general relativity, string theory, classical/quantum/gauge field theory and classical mechanics (Mathematical
Methods of Classical Mechanics by V. I. Arnold comes to mind). I was wondering if people have investigated non-relativistic quantum mechanics using differential geometry and if this approach has been
fruitful? Any good resources/books that discuss this method would be greatly appreciated.
Yeah, the approach basically uses Kahler manifolds --- for a review of this "geometric description", see Ashtekar and Schilling "Geometrical Formulation of Quantum Mechanics" arXiv:gr-qc/9706069.
For a geometrical perspective on quantum information theory, and just quantum theory and quantum states in general, look up The Geometry of Quantum States.
I do not understand well the question. Are you asking whether the issue of (first) quantization on manifolds has been investigated? If this is the question the answer is positive. Several researchers
focused on that problem. The point is that the standard procedure based on Stone-von Neumann theorem generally does not hold for manifolds different form ${\mathbb R}^n$. The algebra of elementary
observables (that generated by position and momentum in $\mathbb R^n$ is very difficult to define). Naive approaches face the technical problem of symmetric operators which should represent
observables but are not (essentially) self adjoint. The situation becomes simpler when the space is homogeneous, i.e. when there is a (at least topological) group acting transitively on the space.
The action can be defined either in terms of isometries, conformal transformations or diffeomorphisms. In this case there are procedures, in particular due to Isham and Landsman and collaborators
(Letters in Mathemattical Physics 20:11-18, 1990, Nuclear Physics B365 (1991) 121-160) leading to a *-algebra of essentially self adjoint operators defining observables in a suitable Hilbert space
associated with the manifold. These procedures include (generalizations of) anyons theory and Aharonov Bohm quantization as well as the standard quantization procedure in $\mathbb R^n$.
Thanks! I have only recently starting to recognize how important differential geometry appears to be in physics, and I'm getting more and more interested in it. I am basically trying to understand in
what level of details we can describe quantum mechanics using differential geometry. So "(first) quantization on manifolds" sounds really interested, although I will be lying if I say I understood
most of your message ;) (this is due to my lack of knowledge). But if I have some spare time then I will try to look up some of the terms you have mentioned.
I'd say the answer to this is clearly: geometric quantization.
I'd argue that this is the theory of quantization, everything else is an approximation to it (in particular the popular algebraic deformation quantization is an approximation to full non-perturbative
geometric quantization). And it is all based on differential geometry -- on higher differential geometry, actually, if one includes quantization of local field theory -- I have some technical details
on this here.
For an exposition of how differential geometry and Lie theory yield the theory of quantization see here.
Thanks! I've saved your pdf file on my computer and hope to be able to tackle it after I have learnt more differential geometry. I'm currently reading "Geometry, Topology and Physics" by Mikio
Nakahara, but I have the feeling I may need to buy a more advanced book on diff. geometry before I can better appreciate your paper ;).
@Hunter, my remark on field theory was for completeness, to drive home the point that this really eventually gives the full story, not just some fragment. But I gather you want to and probably should
start with just quantum mechanics for the time being, hence with just standard geometric quantization. A fairly comprehensive and commented list of literature on that is here. That lists
introductions, and textbooks, and original articles and recent developments. Dig around a bit and see what suits your needs. | {"url":"https://www.physicsoverflow.org/17343/differential-geometric-approach-to-quantum-mechanics","timestamp":"2024-11-09T19:02:47Z","content_type":"text/html","content_length":"180035","record_id":"<urn:uuid:cea3de16-ea61-4978-a94b-3d1a055c5503>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00024.warc.gz"} |
What size heater do I need for a 50 gallon aquarium?
Finding the Right Aquarium Heater Size
Aquarium Heater Size Guide
40 Gallon/150 Liter 100 watt 150 watt
50 Gallon/200 Liter 150 watt 200 watt
65 Gallon/250 Liter 200 watt 250 watt
75 Gallon/300 Liter 250 watt 300 watt
What size heater do I need for a 55 gallon fish tank?
The size of your tank is vital for choosing an appropriate heater for your tank. As a rough estimate, you use between 2.5 and 5 watts per gallon of water. For a 55-gallon tank, I’d recommend using
between a 150-watt and 300-watt heater.
How long does it take to heat up a 50 gallon fish tank?
it might take between 24 and 48 hours and get to the temperature. When I threw my heater in my tank (about 2ft, 20 gallons), it took about 30-36 hours to get to the right temp. Give it a little time,
and first chance you get, go and snag a little thermometer from your local FS.
How many watts is 50 gallons?
A typical 50 gallon electric water heater runs at 4500 watts. In an electric circuit of 240 volts, 4500 watts is equivalent to 18.75 amps.
Is it better to have 1 or 2 heaters in an aquarium?
In large aquariums, it may require two heaters. For example, a 100 gallon aquarium would require 500 Watts of heat, but it would be better to use a 300 Watt heater at each end of the tank, rather
than one 500 Watt heater, to keep the water temperature consistent throughout the aquarium.
How do I choose an aquarium heater?
A good rule of thumb for aquarium heaters is 5 watts per gallon for aquariums 55 gallons or smaller, and 3 watts per gallon for those over 60 gallons. Use a larger size or a second heater if your
aquarium is in an especially cold room or is located on an exterior wall or near an outside door.
Where should I put my aquarium heater?
In your home aquarium, the best location for placing a heater is near the maximum water flow, such as the outlet (or inlet) from the filter, or in the stream of a powerhead. Having water flowing
directly past the heater is what quickly and evenly disperses heated water throughout the tank.
Is a higher watt heater better?
With our products, heat output is measured in wattage. That doesn’t necessarily mean more is better. Just because you can get a 2,000-watt heater for the same price as a 750-watt one, doesn’t mean
you should. Too much heat for the room will cause the heater to fail.
How many watts do you need for a 55 gallon aquarium?
A standard 55 gallon aquarium will require a 165-watt heater at a minimum. A heater this size will be sufficient if the difference between the aquariums temperature and the ambient room temperature
is around 5 degrees Fahrenheit.
How many watts does a 60 gallon water heater use?
4500 Watt
GSW 60 Gallon 240 Volt 4500 Watt Electric Water Heater.
How many gallons can a 50 watt heater heat?
The general rule of thumb is to have a capacity of approximately 5 watts per gallon of water. Therefore, a 10 gallon aquarium will need a 50 watt heater. As the tank size increases, the larger water
volume is able to retain the heat better….Aquarium Heater Size.
Aquarium Size Heater Capacity
75-100 Gallons 250-300 Watts
What is the best water heater for a fish tank?
Hanging/Immersible Heater. An immersible heater is also known as a hanging heater.
Submersible Heater. Submersible heaters are ones that sit under the water.
Substrate Heater. These are typically used as an addition to other heaters and come in the form of wires which are fixed onto the base of the aquarium.
In-line Heater.
Filter Heater.
Do all fish tanks need a heater?
The lighting in a fish tank can make the tank warmer. If you have sufficient light in your fish tank that the water temperature is habitable for your fish, you will not need a heater. Another factor
related to the tank’s lighting system is exposure to sunlight.
What are some good fish for a 55 gallon tank?
Gourami. A pair of three-spot gouramis are a beautiful addition to a 55-gallon tank,and many species grow to around 6-inches in length.
Angelfish. One of the most popular tropical fish for sure,but in my opinion shouldn’t be kept in a tank smaller than 55 gallons.
Neons and Other Tetras.
Cherry Barbs.
Should I get a heater for my fish tank?
It’s better to buy a high-quality heater once. Heaters are the kind of equipment where quality really counts.
Read the instructions. Too often,we just go ahead and do something without reading the instructions.
Check the temperature of your tank with a thermometer.
Consider installing a temperature controller. | {"url":"https://locke-movie.com/2022/08/25/what-size-heater-do-i-need-for-a-50-gallon-aquarium/","timestamp":"2024-11-11T13:51:20Z","content_type":"text/html","content_length":"46208","record_id":"<urn:uuid:4c5fe0c8-0c04-43ca-9a8a-549f9846bab1>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00251.warc.gz"} |
from Montana to 3430 Boychinovtsi
16 Mins - Total Flight Time from Montana to 3430 Boychinovtsi
Plane takes off from Montana, BG and lands in 3430 Boychinovtsi, BG.
Current Time in Boychinovtsi: Friday November 8th 6:42am.
Estimated Arrival Time: If you were to fly from Boychinovtsi now, your arrival time would be Friday November 8th 6:58am (based on Boychinovtsi time zone).
* Flight duration has been calculated using an average speed of 435 knots per hour. 15 minutes has been added due to takeoff and landing time. Note that this time varies based on runway traffic.
Other factors such as taxing and not being able to reach or maintain a speed of 435 knots per hour has not been taken into account.
Flight Time Summary
Your in air flight time starts at Montana and ends at 3430 Boychinovtsi.
Estimated arrival time: Friday November 8th 6:58am (based on destination time zone).
You can see why your trip to Montana takes 16 mins by taking a look at how far of a distance you would need to travel. You may do so by checking the flight distance between Montana and 3430
After seeing how far Montana is from 3430 Boychinovtsi by plane, you may also want to get information on route elevation from Montana to 3430 Boychinovtsi.
Did you know that 3430 Boychinovtsi can be reached by car? If you'd like to drive there, you can check the travel time from Montana to 3430 Boychinovtsi.
To see how far your destination is by car, you can check the distance from Montana to 3430 Boychinovtsi.
If you need a road map so that you can get a better understanding of the route to 3430 Boychinovtsi, you may want to check the road map from Montana to 3430 Boychinovtsi.
If you're now considering driving, you may want to take a look at the driving directions from Montana to 3430 Boychinovtsi.
Whether the trip is worth the drive can also be calculated by figuring out the fuel cost from Montana to 3430 Boychinovtsi.
Recent Flight Times Calculations for Montana BG:
Flight Time from Montana to Kriva Palanka
Flight Time from Montana to Calafat
Flight Time from Montana to Pirot
Flight Time from Montana to 3640 Yakimovo
Flight Time from Montana to 3670 Medkovets
« RSS Flight Times Calculations for Montana BG » | {"url":"https://www.distancesto.com/flight-time/bg/montana-to-3430-boychinovtsi/history/91906.html","timestamp":"2024-11-08T04:42:27Z","content_type":"text/html","content_length":"49279","record_id":"<urn:uuid:c49b27fd-fcb0-4b0b-9619-1f9da2c429bf>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00558.warc.gz"} |
mikrocontroller code?
Potential drop (difference) over L with DC when current is stationary: constant current * ohmic resistance.
Don't you agree?
Yes, wait, if only an inductor is connected directly on a 5v dc powersource, there is always 5v difference
Pulsed LR circuit with T1 =T2 = tau, left is voltage over R right is voltage over L. Left have the same peak voltages, V/div aren't the same. | {"url":"https://www.ionizationx.com/index.php/topic,948.16.html?PHPSESSID=l821n1qsekc9k3lft10328kqb7","timestamp":"2024-11-01T21:06:46Z","content_type":"application/xhtml+xml","content_length":"27666","record_id":"<urn:uuid:48808c1b-f521-435d-8f66-d6184de46d25>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00666.warc.gz"} |
New library functions
Next: New library packages Up: New features that have Previous: Arithmetic over finite
• ellipsoid: surface area of an ellipsoid
• evalr: evaluation over a range(s)
• fnormal: floating point normalization
• galois: compute the galois group of a (univariate) polynomial (up to degree 7)
• history: a history mechanism (an alternative to Maple "s)
• iratrecon: rational reconstruction
• mellin: Mellin transform
• poisson: Poisson series expansion
• irreduc: irreducibility test of a (multivariate) polynomial
• roots: compute the roots of a (univariate) polynomial
• shake: real interval arithmetic (to Digits precision)
• sturm: use the Sturm sequence to find the number of real roots in an interval
• sturmseq: compute the Sturm sequence of a polynomial over Q or R
• thiele: continued fraction interpolation
• userinfo: generate user information
• ztrans, invztrans: the Z-transform and its inverse
• C: Maple to C language translator
• FFT: Fast Fourier transform
• Hermite: compute the Hermite normal form of a matrix over a finite field
• Irreduc: irreducibility test of a (univariate) polynomial mod p
• MOLS: generates mutually orthogonal Latin squares
• Nullspace: compute the nullspace of a matrix over a finite field
• Primitive: tests if a univariate polynomial over a finite field is primitive
• Randpoly: random univariate polynomial of degree
• Randprime: random monic irreducible univariate polynomial of degree
• Smith: compute the Smith normal form of a matrix over a finite field | {"url":"http://thproxy.jinr.ru/Documents/MapleV/summary/section3_3.html","timestamp":"2024-11-02T14:47:25Z","content_type":"text/html","content_length":"2975","record_id":"<urn:uuid:7efe3c88-bf97-4e89-b085-d05a7d82c5e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00377.warc.gz"} |
Run a simulation model with correlations between distributions and compute SPC (process capability) indicators
This tutorial will help you set up and run a simulation model with correlations between distributions and compute SPC process capability indices in Excel using XLSTAT.
What is a simulation model
Simulation models allow to obtain information, such as mean or median or confidence intervals, on variables that do not have an exact value, but for which we either know or assume a distribution. If
some “result” variables depend on these “distributed” variables by the way of known or assumed formulae, then the “result” variables will also have a distribution. Sim allows you to define the
distributions, and then to obtain, through simulations, an empirical distribution of the input and output variables as well as the corresponding statistics.
Simulation models are used in many areas such as finance and insurance, medicine, oil and gas prospecting, accounting, or sales prediction.
Four elements are involved in the construction of a simulation model:
1. Distributions are associated to random variables. XLSTAT gives a choice of more than 30 distributions to describe the uncertainty on the values that a variable can take. For example, you can
choose a triangular distribution if you have a quantity for which you know it can vary between two bounds, but with a value that is more likely (a mode). At each iteration of the computation of
the simulation model, a random draw is performed for each distribution that has been defined.
2. Scenario variables allow to include in the simulation model a quantity that is fixed in the model, except during the tornado analysis where it can vary between two bounds.
3. Result variables correspond to outputs of the model. They depend either directly or indirectly, through one or more Excel formulae, on the random variables to which distributions have been
associated and, if available, on the scenario variables. The goal of computing the simulation model is to obtain the distribution of the result variables.
4. Statistics allow to track a given statistic for a result variable. For example, we might want to monitor the standard deviation of a result variable.
A correct model should comprise at least one distribution and one result variable. Models can contain any number of these four elements. A model can be limited to a single Excel sheet or can use a
whole Excel folder.
Dataset for running a simulation model integrating correlations between distributions
In this tutorial we add to the model of the first tutorial a correlation matrix and an SPC analysis. Our simulation model is based on sales and costs of a shop. The benefit is simply the difference
between sales and costs in this simple case. Based on historical data for costs and sales that were analyzed with the tool “distribution fitting” we found out that the costs follow a normal
distribution (mu=120, sigma=10) and the sales a normal distribution (mu=80, sigma=20) (see Histograms and distribution fitting tutorial in Excel or more details).
We suppose that the costs and the sales are correlated with a spearman correlation coefficient of 0.8. This is shown in the correlation matrix. The lower triangle is sufficient. It is important that
the headers of the rows and columns are the same as the names given to the the distribution variables when they were defined.
Additionally an SPC analysis for the three model variables is carried out. During the planning for the running year we defined upper and lower specification limits and a target value that is
identical to the static model.
Note: The SPC analysis is only available, if a valid SPC license is present.
This model can be found in the Model sheet.
Running the simulation model integrating correlations between distributions and computing the SPC indicators
To start the simulation run, select the XLSTAT / Sim / Simulation - Run command, or click the corresponding button of the Sim toolbar.
The Simulation - Run dialog box appears.
Set the number of simulations to 1000. Activate the Correlation/covariance matrix and select the correlation matrix including the row and column headers.
In the Charts - Sensitivity tab, enter the parameters of the tornado and spider analysis. Select the standard cell value as default value. Choose 10 data points in the interval from -10% up to +10%
of the value deviation:
In the SPC tab, if a valid SPC license is present, we activate the calculate process capabilities option.
Choose for the selection field LSL the three Excel cells below the cell LSL, because the column label is not necessary. Fill in USL in the same way.
In the field Name select the 3 cells with the names of the model elements for which the SPC analysis should be carried out: "sales", "costs" and "benefit" located to the left of the model elements.
Last, activate the target option to calculate SPC values that need a target value.
The computations begin once you have clicked OK.
Interpreting the results of a simulation model integrating correlations between distributions and SPC indicators
In the model summary the proximity matrix is displayed:
After the following tables that contain the details of the distribution and result variables, additional results of the SPC analysis are displayed:
Finally the correlation matrix of the different distribution and result variables is displayed. We see that the spearman correlation between costs and sales is close to 0.8. If the number of
iterations during the simulation was bigger, then this correlation will be even closer to 0.8. | {"url":"https://help.xlstat.com/6644-run-simulation-model-correlations-between","timestamp":"2024-11-07T20:03:16Z","content_type":"text/html","content_length":"30754","record_id":"<urn:uuid:719b57b4-71f0-4bfe-8be0-f9036514a659>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00734.warc.gz"} |
12. The number $sqrt2$ - Avidemia
Let us now return for a moment to the particular irrational number which we discussed in §§ 4-5. We there constructed a section by means of the inequalities \(x^{2} < 2\), \(x^{2} > 2\). This was a
section of the positive rational numbers only; but we replace it (as was explained in § 8) by a section of all the rational numbers. We denote the section or number thus defined by the symbol \(\sqrt
The classes by means of which the product of \(\sqrt{2}\) by itself is defined are (i) \((aa’)\), where \(a\) and \(a’\) are positive rational numbers whose squares are less than \(2\), (ii) \((AA’)
\), where \(A\) and \(A’\) are positive rational numbers whose squares are greater than \(2\). These classes exhaust all positive rational numbers save one, which can only be \(2\) itself. Thus \[(\
sqrt{2})^{2} = \sqrt{2}\sqrt{2} = 2.\]
Again \[(-\sqrt{2})^{2} = (-\sqrt{2})(-\sqrt{2}) = \sqrt{2}\sqrt{2} = (\sqrt{2})^{2} = 2.\] Thus the equation \(x^{2} = 2\) has the two roots \(\sqrt{2}\) and \(-\sqrt{2}\). Similarly we could
discuss the equations \(x^{2} = 3\), \(x^{3} = 7, \dots\) and the corresponding irrational numbers \(\sqrt{3}\), \(-\sqrt{3}\), \(\sqrt[3]{7}, \dots\).
$\leftarrow$ 10-11. Algebraical operations with real numbers Main Page 13-14. Quadratic surds $\rightarrow$ | {"url":"https://avidemia.com/pure-mathematics/the-number-sqrt2/","timestamp":"2024-11-10T05:17:56Z","content_type":"text/html","content_length":"76698","record_id":"<urn:uuid:f84a6331-7203-468b-a7ff-593863d7efcb>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00187.warc.gz"} |
F# Math - Numerical computing and F# PowerPack
This article is the first article of a series where I'll explain some of the F# features that are useful for numeric computing as well as some functionality from the F# PowerPack library. Most of the
content was originally written for the Numerical Computing in F# chapter on MSDN (that I announced earlier), but then we decided to focus on using F# with third party libraries that provide more
efficient implementation and richer set of standard numeric functionality that's needed when implementing machine learning and probabilistic algorithms or performing statistical analysis. If you're
interested in these topics, then the last section (below) gives links to the important MSDN articles.
However, F# PowerPack still contains some useful functionality. It includes two additional numeric types and an implementation of matrix that integrates nicely with F#. The series also demonstrates
how to use features of the F# language (and core libraries) to write numeric code elegantly. In particular, we'll use the following aspects:
These are just a few of the F# language features that are useful when writing numeric code, but there are many others. The usual F# development style using interactive tools, type safety that
prevents common errors, units of measure as well the expressivity of F# make it a great tool for writing numeric code. For more information, take a look at the MSDN overview article Writing Succinct
and Correct Numerical Computations with F#.
Numerical computing and F# PowerPack
If you're looking for information about other PowerPack components, then Daniel Mohl (@@dmohl) wrote a series that covers numerical types and modules (Part 1), asynchronous extensions (Part 2),
additional collection types (Part 3), as well as lexing, parsing and SI units (Part 4).
In this article series, we look at most of the numerical computing features provided by the F# PowerPack library. The following list shows the upcoming articles of the series:
Numerical Computing in F# (MSDN)
As already mentioned, F# PowerPack provides only basic implementation of matrices, which is not suitable for tasks that require efficient matrix multiplication. The library also does not implement
any advanced operations such matrix decomposition or any support for working with probability distributions or statistical analysis.
If you're interested in writing highly-efficient numeric code in F#, then it is recommended to use a third-party library (such as open-source Math.NET Numerics) that provide more efficient
implementation and wider range of features. The MSDN section Introduction to Numerical Computing in F# provides more information including a review of existing libraries. It also includes examples
that demonstrate the following three options:
References & Links | {"url":"https://tomasp.net/blog/powerpack-introduction.aspx/","timestamp":"2024-11-12T00:03:14Z","content_type":"text/html","content_length":"26691","record_id":"<urn:uuid:786a4b6c-5937-49c7-853d-b5f9920f9837>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00733.warc.gz"} |
Machine Learning to Balance the Load in Parallel Branch-and-Bound
We describe in this paper a new approach to parallelize branch-and-bound on a certain number of processors. We propose to split the optimization of the original problem into the optimization of
several subproblems that can be optimized separately with the goal that the amount of work that each processor carries out is balanced between the processors, while achieving interesting speedups.
The main innovation of our approach consists in the use of machine learning to create a function able to estimate the difficulty (number of nodes) of a subproblem of the original problem. We also
present a set of features that we developed in order to characterize the encountered subproblems. These features are used as input of the function learned with machine learning in order to estimate
the difficulty of a subproblem. The estimates of the numbers of nodes are then used to decide how to partition the original optimization tree into a given number of subproblems, and to decide how to
distribute them among the available processors. The experiments that we carry out show that our approach succeeds in balancing the amount of work between the processors, and that interesting speedups
can be achieved with little effort.
Department of Electrical Engineering and Computer Science, University of Liege, Liege, Belgium, March 2015
View Machine Learning to Balance the Load in Parallel Branch-and-Bound | {"url":"https://optimization-online.org/2015/03/4832/","timestamp":"2024-11-10T09:39:39Z","content_type":"text/html","content_length":"84419","record_id":"<urn:uuid:46881734-9d55-4f5b-acc1-c9d0fb3d90c7>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00279.warc.gz"} |
How to Determine CMRR When Measuring a 40 uOhm (2000 Amp) PDN with a 2-Port Probe | Signal Edge Solutions
Benjamin Dannan and Steve Sandler
This article is the first part of a 3-part series:
Most of us are aware of the ground loop in the 2-port measurement. Most of us are also aware that we need to introduce a ground loop isolator to correct the error. If not, we’ve published plenty on
the subject. But how much CMRR do you need to add? How does the use of a probe impact this requirement?
Rest assured, this measurement can be successfully performed, as shown in Figure 1, and we’ll show you how to determine much CMRR it takes in this blog.
Figure 1 – Depiction of 37 µΩ Known DUT 2-port Shunt Through Impedance Measurement.
Companies like Intel, Nvidia, AMD, Qualcomm, and Broadcom are putting more cores on a single die. Ampere is an excellent example where they have designed a cloud processor solution with 192 cores on
a single die. While 2000 Amps may seem excessive, the power demands of data centers, supercomputing, and AI have already surpassed this number. Without adding other considerations on the thermal
design or PDN validation, let’s focus on what is required to measure a 2000 Amp PDN.
Measuring a 2000 Amp PDN with a 2-port probe or even with two 1-port probes is not intuitive or even trivial. To effectively measure a 2000 Amp PDN, it is important first to understand the desired
impedance to be measured to support 2000 Amps. With reference to EQ(1), if it is assumed that a 0.8V power domain has an 80 mV peak-to-peak ripple spec, then with a 2000 Amp step load, the target
impedance (ZTGT) is 40 µΩ.
The Ground Loop Error
With the understanding of the ZTGT, efforts can be made to understand the requirements to be able to make a measurement a the desired of 40 µΩ. It quickly becomes clear that the common mode rejection
ratio (CMRR) becomes an important figure of merit to mitigate the ground loop for successful low impedance measurements into the micro-ohms associated with the two-port shunt through measurement.
This method is well-published, so this discussion will not include details.
Figure 1 depicts a 2-port measurement setup using 1-meter PDN cables for a 40 µΩ Device Under Test (DUT). The ground loop error is clearly seen, and the error from this is shown in Figure 2.
Figure 2 – 40 µΩ DUT Impedance Measurement Setup with 1-meter PDN Cables.
Figure 3 - 40 µΩ DUT Impedance Measurement Setup with 1-meter PDN Cables.
The ground loop resistance is the parallel equivalent of the two cables connecting the VNA to the DUT. In this case, the two cables are identical as are the probe ground pins. So, the resistance of
a single cable and probe pin is twice this measured value.
The return resistance is the sum of the cable shield and pin resistance.
Solving for the measured value and adding the DUT results in
This makes clear that the ground loop comprises two parts, the cable and the pin. Each contributes a resistance to the ground loop error.
A common error is to use calibration to remove the ground loop error. The calibration will subtract the constant value of R_measured from the result, which theoretically provides the correct
This method doesn’t work because the value of K is a constant, but the value of the cable and pin resistance is not. Bending the cable or adding compression to the pin will change the value of the
ground resistance by a small amount. The ground resistance of BNC connectors, used on many VNA’s, is also not constant. However, this small change in value is of the same order of magnitude as the
DUT, while the cable + pin resistance is many orders of magnitude larger than the DUT.
This method attempts to take the difference between two very large numbers to obtain a very small result. This is often unsuccessful.
Introducing a Ground Isolator
The introduction of a ground isolator changes the equation to
In this case, the ground resistance is first divided by CMRR of the isolator, significantly reducing the sensitivity to small changes in the ground loop resistance.
A transformer isolator limits this CMRR at low frequency, while solid-state isolators do not. On the other hand, transformers typically work to a higher frequency than a solid-state isolator, so
this is a tradeoff. But the question is how much isolation is required for a reasonable measurement.
Assuming a willingness to tolerate a 10% ground loop error in our 40uOhm measurement, the error term has to be less than 40µΩ.
This allows us to solve for the required isolator CMRR to perform this measurement.
The shield resistance of a 1-meter PDN cable is approximately 15mΩ. The resistance of the Picotest P2104A pin is also typically about 15mΩ, though contact pressure can vary quite a bit, and there is
a significant tolerance.
The minimum CMRR to make the measurement is 77.5dB at low frequency.
Reducing the Ground Loop Error with CMRR
To demonstrate the CMRR measurement requirements, an ideal, parameterized, Op Amp is inserted into the simulation setup shown in Figure 1 and is depicted in Figure 3. The results for a 2-port shunt
measurement with 1-meter PDN cables and an ideal op amp is shown in Figure 4. The ideal op amp represents the ground loop isolator. The CMRR for an op amp with 57dB and 77.5dB is shown versus a
result with no CMRR. The result with 57dB of CMRR demonstrates another Picotest product, the J2113A.
Figure 4 – 40 µΩ Impedance Measurement Setup with 1-meter PDN cables and Ideal Op Amp.
Figure 5 - 40 uΩ DUT Impedance Measurement Setup with 1-meter PDN Cables with CMRR Variation.
As shown by Figure 4, an uncalibrated error of 7.4% for our 40 µΩ DUT is achieved when CMRR equals 77.5dB. However, as shown in Figure 4, when CMRR is equal to 57dB, the uncalibrated error is almost
100%. This again emphasizes the earlier point that measuring a 2000 Amp PDN at 40 µΩ is not easy or intuitive. Calibration can then be used to further improve this accuracy.
Follow this LINK to check out part 2 of this blog discussion, demonstrating how to measure a 40 uΩ DUT using a 2-port probe. | {"url":"https://www.signaledgesolutions.com/post/the-challenge-of-measuring-a-40-uohm-2000-amp-pdn-with-a-2-port-probe-how-much-cmrr-is-needed","timestamp":"2024-11-04T18:04:23Z","content_type":"text/html","content_length":"1050498","record_id":"<urn:uuid:9497ffbc-d107-46e5-9bd6-086d29c8ce75>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00164.warc.gz"} |
Multifractal Detrended Fluctuation Analysis of Streamflow in the Yellow River Basin, China
College of Water Resources and Architectural Engineering, Northwest A&F University, Yangling, Shaanxi 712100, China
Institute of Soil and Water Conservation, Northwest A&F University, Yangling, Shaanxi 712100, China
Institute of Soil and Water Conservation, Chinese Academy of Science and Ministry of Water Resources, Yangling, Shaanxi 712100, China
Authors to whom correspondence should be addressed.
Submission received: 21 November 2014 / Revised: 3 April 2015 / Accepted: 8 April 2015 / Published: 17 April 2015
Multifractal detrended fluctuation analysis (MFDFA) can provide information about inner regularity, randomness and long-range correlation of time series, promoting the knowledge of their evolution
regularity. The MFDFA are applied to detect long-range correlations and multifractal behavior of streamflow series at four hydrological stations (Toudaoguai, Longmen, Huangfu and Ganguyi) in the main
channel and tributaries of the Yellow River. The results showed that there was one crossover point in the log−log curve of the fluctuation function F[q](s) versus s. The location for the crossover
point is approximately one year, implying an unchanged annual periodicity within the streamflow variations. The annual periodical feature of streamflow was removed by using seasonal trend
decomposition based on locally weighted regression (STL). All the decomposed streamflow series were characterized by long-term persistence in the study areas. Strong dependence of the generalized
Hurst exponent h(q) on q exhibited multifractal behavior in streamflow time series at four stations in the Yellow River basin. The reduction of dependence of h(q) on q for shuffled time series showed
that the multifractality of streamflow series was responsible for the correlation properties, as well as the probability density function of the streamflow series.
1. Introduction
Streamflow is an important component of water circulation, which carries large amounts of information on this dynamic mechanism. Water circulation and hydrological processes are greatly affected by
climate change and human activities such as agricultural irrigation, exploitation of water resources and construction of reservoirs, and these factors have caused evident changes in the last century.
Some studies showed that climate change led to changes in streamflow. Labat
et al
. [
] showed that global streamflow increased with the rise of global temperature by approximately 4% from only a 1 °C global temperature rise. Zhao
et al
. [
] analyzed the annual and seasonal trends of streamflow and the correlations between streamflow and climatic variables in the Poyang Lake basin, indicating that both annual and seasonal streamflow
showed increasing trends, and streamflow is more sensitive to changes in precipitation than potential evapotranspiration. Some studies paid close attention to the effects of human activities on
streamflow changes. Mu
et al
. [
] showed that soil conservation measures had a great impact on the streamflow regime in the Loess Plateau. Baker and Miller [
] employed the soil and water assessment tool (SWAT) model to assess the relative impact of land cover changes on hydrological processes in Kenya’s Rift Valley, and the results showed that land use
changes led to corresponding increases in surface runoff and decreases in groundwater recharge. These studies suggest that streamflow variation is complex as a result of the interaction of numerous
related factors. Thus, the inherent law of streamflow series, which is important to water resources management and ecological conservation of river basins, is worth investigation.
Streamflow time series represent complex nonlinear and non-stationary characteristics. The long-range correlation in streamflow time series is helpful for prediction. The earliest research on
long-range correlation can be found in Hurst [
]. Hurst’s finding is recognized as the first example of self-affine fractal behavior in empirical time series [
]. Since then, many researchers have shown great interest in detecting the long-range correlation of streamflow [
]. In these studies, detrended fluctuation analysis (DFA) was commonly applied to investigate the long-range correlation of streamflow. The DFA was introduced by Peng
et al
. [
], and the method is proven to be efficient for long-range correlation detection. DFA can decompose the changing trends of the time series at different timescales. Labat
et al
. [
] applied DFA to investigate streamflow series of two karstic watersheds in the south of France, suggesting that the correlation properties exist in small scales and anti-correlated properties exist
in large scales. Hirpa
et al
. [
] analyzed and compared the long-range correlations of river flow fluctuations from 14 stations in the Flint River Basin in the state of Georgia in the southeastern United States. They found that the
basin area is an important factor in long-range correlation studies of streamflow. To detect multifractal properties of the time series, Kantelhardt
et al
. [
] developed the multifractal detrended fluctuation analysis (MFDFA) by considering the
th-order statistical moment of the fluctuation to detect the scaling behavior of time series with a different density. The MFDFA is a development of DFA and can be applied to non-stationary
streamflow time series. Kantelhardt
et al
. [
] found that daily streamflow records were long-range correlated, which is related to the spatial behavior of rainfall above a crossover timescale of several weeks. Zhang
et al
. [
] found that streamflow time series were characterized by short-term correlations on shorter time scale in the Pearl River basin, China, and the streamflow variations were mainly the result of
climate changes, especially precipitation variations. Rego
et al
. [
] applied the MFDFA to detect the complex water fluctuations of 12 principal Brazilian rivers, and the result indicated the presence of multifractality and long-range correlations for all the
stations after eliminating the climatic periodicity. Rybski
et al
. [
] investigated the scaling behavior of long daily river discharge at 42 hydrological stations around the world by using the MFDFA, suggesting that daily streamflow is characterized by a power-law
decay of the autocorrelation function above some crossover time that is several weeks in most cases. Below the crossover time, pronounced short-term correlations occurred. The above analyses
indicated that the hydrological system was a complex dynamic system that displays self-similar and exhibits self-fractal behaviors over a certain range of time scales. However, it should be noted
that some period or quasi-period may lead to spurious appearance of crossovers when we detect the long-range correlations through the MFDFA method [
]. Better understanding of the scaling properties of streamflow removing periodicity will be important for prediction of the evolution of streamflow.
The Yellow River is the second longest river in China. The annual streamflow is only approximately 2% of China’s total, but it directly supports 12% of the national population (mostly farmers and
rural people), feeds 15% of the irrigation area, and contributes to 9% of China’s GDP. The Yellow River suffered from serious water shortages and significant reduction (
< 0.05) in streamflow in the past decades [
]. It is important to understand the streamflow processes in terms of the streamflow time series reflecting the effects of the climate, topographical features, and human activities. Exploring the
scaling behaviors and multifractal characteristics of streamflow series of the Yellow River is important to understand the future evolution trend of streamflow and to make decisions for water
Thus, the objectives of this study are: to analyze the statistical characteristics of streamflow series in the main channel and tributaries of the Yellow River; to remove the periodicity of
streamflow for investigating the scaling behavior; and to detect multifractal characteristics in the streamflow series.
2. Study Area
The Yellow River basin covers an area of approximately 75.2 × 10
, located between 96° ~ 119° E and 32° ~ 42° N. The river originates from the Qinghai−Tibetan Plateau in western China and flows eastward through the Loess Plateau and the North China Plain before
entering into the Bohai Sea, for a total length of 5464 km (
Figure 1
). From the river source to Toudaoguai, its upper reaches, it is 3472 km long with an area of 38.6 × 10
. The streamflow in the upper reaches accounts for approximately 61% of the whole basin. The upper reaches belongs to an arid climate with an annual average precipitation of 396 mm. Longmen station
lies in the middle reaches of the Yellow River. The region between Toudaoguai and Longmen is suffering from severe soil erosion. The climate is semi-arid and arid with an annual average precipitation
of 516 mm.
The Huangfuchuan watershed is located in a region crisscrossed by wind and water erosion at 110°18' ~ 111°12' E and 39°12' ~ 39°54' N, with a catchment area of 3246 km
Figure 1
). The Huangfuchuan River is a first-order tributary in midstream Yellow River with a river length of 137 km and average channel slope of 2.7%. The watershed is located in the transitional belt of
warm temperate and mesothermal zones with an average precipitation of 350–450 mm, more than 80% of which occurs between June and September [
]. The Huangfuchuan watershed is one of the most severe soil erosion areas and brings approximately 0.15 billion t sediment into the Yellow River each year. The Yanhe watershed is located to the
south of the Huangfuchuan River with a drainage area of 7687 km
Figure 1
). The Yanhe River is also a first-order tributary in the midstream of the Yellow River. The Yanhe watershed is dominated by a typical warm temperate continental monsoonal climate. Annual
precipitation is approximately 500 mm and the average annual temperature ranges from 8.8 °C to 10.2 °C. The Yanhe watershed is covered by forests in the south and steppe grassland in the north, and
arable land is mostly distributed in the alluvial plains and gentle slope hills. In the Yanhe watershed, the loess hilly-gully region accounts for 90% of the basin, and the slope in most areas is >
15°, where soil erosion is very serious [
3. Data and Method
3.1. Data
Monthly streamflow at the Toudaoguai and Longmen stations and daily streamflow at the Huangfu and Ganguyi stations are available for this study (
Figure 1
). The detailed information of the streamflow series is listed in
Table 1
. All the streamflow data are collected from the Hydrological Yearbook and the Yellow River Conservancy Commission. The data in this study have been checked by corresponding agencies to guarantee
their consistency.
Table 1. Data information at the four stations in the Yellow River.
River Station Area (km^2) Series Length Time Interval Annual Streamflow (10^8 m^3)
Yellow River Toudaoguai 367,898 January 1919–December 2009 Monthly 228.01
Yellow River Longmen 497,552 January 1919–December 2009 Monthly 285.69
Huangfuchuan Huangfu 3,175 1 January 1960–31 December 2009 Daily 1.20
Yanhe Ganguyi 5,891 1 January 1960–31 December 2009 Daily 1.97
3.2. Method
3.2.1. Seasonal Trend Decomposition Based on Locally Weighted Regression
The seasonal trend decomposition based on locally weighted regression (STL) method can be applied for detecting nonlinear trends and seasonal components in long-term time series. The STL method is a
filtering procedure for decomposing a time series into three components: seasonality, trend, and remainder. A complete description and details about the model can be found in Cleveland
et al
. [
]. The STL method was applied to the streamflow time series with a view of analyzing trends and seasonal components over time, since some components of streamflow time series produce distortions
impeding our understanding of their long-range correlations. The streamflow time series (
) was regarded as an additive form of three components: Trend (
), Seasonality (
) and Remainder (
In this study, streamflow time series were divided into the components of trend, seasonality and remainder by the STL. Seasonality was removed and the remaining components formed a new time series
for MFDFA analysis.
3.2.2. Multifractal Detrended Fluctuation Analysis
The multifractal detrended fluctuation analysis (MFDFA) can be used to detect the scaling behaviors and multifractal properties of nonstationary time series such as hydrological series. For a
streamflow time series
is the length of the time series. The detailed computational procedure can be found as follows [
Step 1
. The accumulated deviation of the series can be calculated as:
$Y ( i ) = ∑ k = 1 i [ x k − x ¯ ] i = 1 , … , N$
$x ¯$
is the mean of
$x k$
Step 2. The accumulated deviation Y(i) can be divided into non-overlapping intervals by the equal length s. There are N[s] = int(N/s) segments and N[s] is the nearest integer part of N/s. Since the
length N of the series may not be an integer multiple of the timescale s, some data may remain at the end of the series Y(i). In order not to ignore the rest of the series, the same computation
procedure is repeated from the end to the start of the series. Thus, 2N[s] segments can be obtained.
Step 3
. A least squares fit method is applied to calculate the trend of each 2
segments, and the variance is determined by:
$F 2 ( s , v ) = 1 s ∑ i = 1 s { Y [ ( v − 1 ) s + i ] − y v ( i ) } 2$
for each segment
= 1,…,
, and
$F 2 ( s , v ) = 1 s ∑ i = 1 s { Y [ N − ( v − N s ) s + i ] − y v ( i ) } 2$
+ 1,…, 2
Here, y[v](i) is the fitted piecewise polynomial trend function in segment v, and any order polynomial can be calculated, such as linear, quadratic, or cubic.
Step 4
. For the 2
segments, the
th-order fluctuation function is determined by:
$F q ( s ) = { 1 2 N s ∑ v = 1 2 N s [ F 2 ( s , v ) ] q / 2 } 1 / q$
$q ≠ 0$
$s ≥ m + 2$
. In this study, we make m = 1 and
Step 5
. There is a power−law relationship between the fluctuation function
) and
For each order of
, the scaling behavior of the fluctuation functions can be determined by the logarithmic chart of
versus s
. The slope of log
) and log
is the generalized Hurst exponent
). For stationary time series, the exponent
(2) is identical to the well-known Hurst exponent
, and
(2) varies between 0 and 1 [
]. The exponent
(2) can be used to analyze correlations in time series. The scaling exponent
= 0.5 means that the time series are uncorrelated; 0.5 <
< 1 implies long-term persistence and 0 <
< 0.5 implies short-term persistence [
In addition, when the time series is monofractal,
) will be a constant coefficient, independent of
. When the time series is multifractal, there is a significant dependence of
) on
> 0,
) depicts the scaling behavior of the segments with large fluctuations and
< 0,
) depicts the scaling behavior of the segments with small fluctuations [
The relationship between the generalized Hurst exponent
) and the scaling exponent
$τ ( q )$
is as follows:
$τ ( q ) = q h ( q ) − 1$
The singularity spectrum
(α) is another index used to describe a multifractal time series, which can be obtained from
$τ ( q )$
via the first-order Legendre transformation:
$α = τ ′ ( q ) , f ( α ) = q α − τ ( q )$
where α is the Hölder exponent. Using Equation (6), we can obtain the relationship as follows:
$α = h ( q ) + q h ′ ( q ) , f ( α ) = q [ α − h ( q ) ] + 1$
The width of the spectrum f(α) reflects the strength of multifractality effects in the time series. For monofractal data, the spectrum f(α) will be a single point, and both functions $τ ( q )$ and h(
q) are linear.
4. Results
4.1. Statistical Characteristics
Temporal changes of streamflow at the four stations are shown in
Figure 2
. The streamflow shows fluctuating changes and starts to decrease in 1985 at the Toudaoguai and Longmen stations. This may be due to the construction of Longyangxia Reservoir upstream of the Yellow
River. The variation of streamflow at the Ganguyi station is relatively stable, while the streamflow at the Huangfu station varies greatly with many flood peaks and low zero flow values.
Figure 2. Temporal variation of streamflow at the Toudaoguai, Longmen, Huangfu and Ganguyi stations of the Yellow River.
Table 2
shows basic statistical characteristic of streamflow in the main channel and tributary stations of the Yellow River. The average monthly streamflow at Toudaoguai and Longmen stations are 18.9 × 10
and 23.6 × 10
, respectively. The standard deviation of streamflow at the Longmen station is higher than that at Toudaoguai, indicating that streamflow fluctuates greatly at the Longmen station. The skewness and
kurtosis are greater than 0, suggesting that the streamflow distribution is not subject to the random normal distribution. The daily streamflow at Ganguyi, with more precipitation and extensive
vegetation, is relatively higher than that in Huangfuchuan watershed, which has less precipitation and poor vegetation. The standard deviation of streamflow at Huangfu is greater than that at Ganguyi
due to the different geomorphological types. The Huangfuchuan watershed is covered by bare weathered rocks, and low permeability and storage capacity improve runoff yield. In addition, the
Huangfuchuan watershed is a rainstorm center, leading to changes of large magnitude in streamflow. The streamflow distributions at both Ganguyi and Huangfu stations are non-normal distribution due to
skewness and kurtosis greater than 0.
Table 2. Characteristics of streamflow for the main river and tributaries of the Yellow River basin.
Table 2. Characteristics of streamflow for the main river and tributaries of the Yellow River basin.
Hydrological Station Mean (10^8 m^3) Max (10^8 m^3) Min (10^8 m^3) Standard Deviation Skewness Kurtosis
Toudaoguai 18.9 112.0 1.31 15.3 1.92 4.37
Longmen 23.6 126.1 2.81 17.7 1.76 3.33
Ganguyi 0.005 1.296 0 0.019 26.5 1246.7
Huangfu 0.003 1.305 0 0.024 26.9 1036.2
4.2. Multifractal Detrended Fluctuation Analysis
Figure 3
shows the scaling behavior of streamflow at four hydrological stations. Clearly, one crossover point can be found for the four log−log plots of
versus s
of the streamflow series. The crossover points occur at approximately 12 months at the Toudaoguai and Longmen stations, and after approximately 372 days at Ganguyi and Huangfu stations. The location
of the crossover point indicates annual periodicity of streamflow, implying a strong relationship between precipitation and streamflow.
The crossover points have been detected by the MFDFA method in many studies, suggesting that the scaling behavior of the time series is more complicated, and different parts have various scaling
exponents [
]. The crossover point often comes from a change in the correlation properties of the series at different scales of time or space, or from seasonal component in the time series [
]. The STL method was employed to eliminate the effect of annual periodicity of streamflow.
Figure 4
illustrates different components decomposed by the STL method for the monthly streamflow at Toudaoguai station. The originally observed time series was decomposed into seasonality, trend and residue.
The seasonal component shows regular variations in monthly time series, with high flows in flood season and low flows in dry season being a consistent pattern, as a result of annual climate patterns.
The trend component shows fluctuations before 1985, but there exists an obvious decreasing trend after 1985 which is possibly caused by the construction of Longyangxia Reservoir.
Figure 3. The fluctuation function F[2](s) versus time scale s in double logarithmic plots for the streamflow time series at the Toudaoguai, Longmen, Ganguyi and Huangfu stations of the Yellow River.
Figure 4. STL decomposition for streamflow at Toudaoguai station. The plots show the original measured streamflow, the seasonal component, trend component and the remainder from top to bottom,
As mentioned above, the STL method can easily extract periodicity of streamflow time series. The periodicity of the streamflow at the four stations are therefore removed by the STL method. New
streamflow time series including trend and remainder noise are constructed to detect the long-term correlations.
Figure 5
shows the scaling behavior of streamflow time series removing the periodicity at the four stations by the MFDFA method. It can be clearly found that the crossover point disappears due to annual
periodicity in the log−log plots of
versus s
. The
(2) values of streamflow are respectively 0.8257 and 0.8231 at Toudaoguai and Longmen stations, suggesting long-term persistence of the streamflow time series. It implies that scaling properties of
the streamflow series in the upper and middle reaches of the Yellow River are similar. At the Ganguyi and Huangfu stations, streamflow fluctuations are also characterized by long-term persistence, as
the values of
(2) are 0.5742 and 0.5882, respectively. This implies that the changing trends may remain stable for the next few years. The scaling properties of streamflow series at the four stations showed
similar characteristics over a scale. This may imply universal scaling properties within the Yellow River basin because they have similar climate patterns.
Figure 5. The fluctuation function F[2](s) versus time scale s in double logarithmic plots for the streamflow time series removing the periodicity at the Toudaoguai, Longmen, Ganguyi and Huangfu
stations of the Yellow River.
Figure 6
a–c show the dependence of
) and τ(
) on
, and the relationship between singularity spectrum
(α) and singularity exponent
. It shows that
) varies with the changes in
, which suggests that the streamflow series at the four stations in the Yellow River are characterized by multifractality. There are different behaviors for
< 0 and
> 0 between the scaling exponent τ(
) and
at the four stations. The slopes of τ(
versus q
are summarized in
Table 3
Figure 6. The relationships of (a) the generalized Hurst exponent h(q) and q; (b) the mass exponent function τ(q) and q; and (c) the singularity spectrum f(α) and singularity exponent α for the
streamflow series at the Toudaoguai, Longmen, Ganguyi and Huangfu stations of the Yellow River.
Table 3. Slopes of the mass exponent function τ(q) for the streamflow series at Toudaoguai, Longmen, Ganguyi and Huangfu stations of the Yellow River.
Table 3. Slopes of the mass
exponent function τ(q) for the
streamflow series at Toudaoguai,
Longmen, Ganguyi and Huangfu
stations of the Yellow River.
Study Area Slopes
−10 < q < 0 0 < q < 10
Toudaoguai 1.4070 0.6501
Longmen 1.3361 0.6191
Ganguyi 1.2538 0.2738
Huangfu 1.5456 0.2358
The nonlinear τ(
) means multiple scaling, and the degree of nonlinearity of the τ(
) function can reflect the degree of multifractality [
]. The slopes of τ(
) at the Toudaoguai and Longmen stations are similar, indicating a similar degree of multifractality. The distinct values of slopes between
< 0 and
> 0 of τ(
) are 0.7569, 0.7170, 0.980 and 1.3098, respectively, at Toudaoguai, Longmen, Ganguyi, and Huangfu stations. This may suggest that the streamflow time series at the Huangfu station have the highest
degree of multifractality in scaling properties among these stations.
Figure 6
c shows the multifractal spectra
(α) of four streamflow time series. All of them exhibit the shape of a parabolic curve, indicating the multifractal structure of the streamflow time series. The multifractal spectra at Toudaoguai,
Longmen, and Ganguyi stations are not symmetrical and all have left truncation, whereas the multifractal spectrum has a right truncation at the Huangfu station. This is because streamflow time series
at Huangfu station have a multifractal structure that is sensitive to large magnitudes of local fluctuations. Ihlen [
] concluded that the time series had a long left tail when it was insensitive to local fluctuations with small magnitudes, and would have a long right tail when it was insensitive to the larger
magnitudes of local fluctuations. The widest range (
$Δ α$
) of the multifractal spectrum is similar between Toudaoguai and Longmen stations, and the values are 0.86 and 0.88, respectively, indicating similar multifractal characteristics. The widest range of
the multifractal spectrum at Ganguyi and Huangfu stations are 1.10 and 1.43, respectively, suggesting that the strength and complexities of streamflow fluctuations at Huangfu are higher than those at
the Ganguyi station.
5. Discussion
There are two causes that lead to the multifractality of the time series [
]. One is that a broadening of the probability density function (PDF) of the time series can result in multifractality. Another is different fluctuations of correlations in small and large scales. To
distinguish the two types of multifractality, we can shuffle the streamflow time series at the Toudaoguai, Longmen, Huangfu and Ganguyi stations. The method generates two random integers,
, in the interval from 1 to
is the length of the streamflow time series) and then swaps the position of
) and
). The process is repeated 10
times in this study and the shuffled time series formed. The shuffled time series have the same distribution as the original time series, but they do not have the fluctuations of correlations and
will exhibit a random behavior with
) = 0.5. If the shuffled series show weaker multifractality characteristics than the original one, it means that both of the two reasons mentioned above together lead to the multifractality in the
streamflow time series.
Figure 7
shows that the
) is dependent on
and the curves of
versus q
are lower for the shuffled time series than the original one, behaving as a monotonically decreasing function of
for all hydrological stations, which indicates that the multifractality is due to the correlation properties and the PDF of the hydrological series.
The streamflow time series in the Yellow River present multifractal characteristics, which is related to precipitation, different topographical features, the size of the drainage area, land use/
cover, the hydrodynamic system of self-organized behavior, and the exploitation of water resources. Koscielny-Bunde
et al
. [
] examined the temporal correlations and multifractal properties of long-term streamflow records from 41 hydrological stations throughout the world and showed that there were no universal scaling
behaviors in the streamflow series, and the widest range of the multifractal spectrum was slightly decreasing with increasing basin area. Özger
et al
. [
] also showed that the larger the drainage area, the smaller the multifractality and the higher the persistency was. The widest ranges of the multifractal spectrum are 1.10 (at Ganguyi, 5891 km
) and 1.43 (at Huangfu, 3246 km
), respectively. The results in this study are consistent with Koscielny-Bunde
et al
. [
] and Özger
et al
. [
]. Kantelhardt
et al
. [
] found that the persistence of streamflow was related to storage processes occurring in the soil and the highly intermittent spatial behavior of rainfall. The
(2) values and multifractal spectra
(α) of the streamflow series at the Toudaoguai and Longmen stations are similar, but the values
(2) and multifractal spectra
(α) are different at the Ganguyi and Huangfu stations. For the mainstream of the Yellow River, most of the streamflow comes from the upper reaches of the Yellow River. Fluctuations in streamflow may
be similar at Toudaoguai and Longmen stations, resulting in similar scaling behaviors, but the Ganguyi and Huangfu stations are located in different regions which have uneven spatial and temporal
distribution of precipitation, various land covers and different soil types. The average annual precipitation are 364.3 mm and 504.1 mm in Huangfuchuan and Yanhe watershed, respectively [
]. There are many rainstorms in the Huangfuchuan basin, which has complex geomorphological types including a weathered sandstone hilly-gully region, a loess hilly-gully region and a sandy loess
hilly-gully region. By contrast, the loess hilly-gully region is the dominant geomorphological type, accounting for 90% of the Yanhe basin. Human activities, such as exploitation, utilization of
water resources, and the construction of check dams can heavily affect streamflow variation. The number of check dams had reached 567, and the dam-controlled area was 2216.47 km
in 2010, accounting for 68.3% of the Huangfuchuan basin area [
]. Yang
et al
. [
] illustrated that hydrological processes downstream of dams were closely associated with the regulating activities of reservoirs, and dam construction had significant influence on hydrological
alterations. The interaction of all these complicated factors leads to the different multifractality of the streamflow series at the Huangfu and Ganguyi stations.
Figure 7. Generalized Hurst exponent h(q) as a function of q for original and shuffled streamflow series of the Toudaoguai, Longmen, Ganguyi, and Huangfu stations of the Yellow River.
The long-range correlations and multifractal behavior of streamflow time series are detected by MFDFA method. By contrast, the traditional methods such as rescaled range analysis and spectral
analysis may produce spurious results of long-range correlation when the time series have changing trends. These trends are systematic deviations from the average streamflow that are caused by human
activities, the seasonal cycle, or a changing climate [
]. The MFDFA has been employed to detect long-range correlation of time series with the superposition of a non-stationary trend. This method can deal with non-stationary time series and also avoid
the spurious detection of long-range correlations [
]. The annual periodicity of streamflow time series resulted in the existence of a crossover point using MFDFA method (
Figure 3
). The STL method provides a graphical view for describing nonlinear trends with seasonal periodicity by using the generalized additive modeling approach [
]. This method can be applied to addressing the changes in timing, amplitude, and variance that occur in the seasonal cycle. The STL method decomposes the time series into three components: seasonal,
trend, and remainder. Periodical components can be extracted and removed from the original time series [
]. Afterwards, the MFDFA method can be used to detect the long-range correlation and multifractal behavior of streamflow time series.
6. Conclusions
This study applied the MFDFA and STL methods to investigate long-term correlations and multifractal behaviors of the streamflow series at the Toudaoguai station and Longmen stations in the main river
and at the Huangfu and Ganguyi stations in tributaries of the Yellow River. The results can be summarized as follows:
One crossover point can be found for the four log−log plots of F[q](s) versus s in the streamflow series. The crossover point occurred in approximately 12 months at the Toudaoguai and Longmen
stations, and in approximately 372 days at Ganguyi and Huangfu station, which is attributed to a one-year periodicity of streamflow.
The scaling properties of the decomposed streamflow series at the four stations showed long-range correlations. This may imply universal scaling properties within Yellow River basin.
The q dependence of h(q) and τ(q) indicated that streamflow time series have multifractal behavior, which is due to the correlation properties, as well as to the PDF of the hydrological series.
Comparing with Yanhe, streamflow time series at Huangfuchuan have a multifractal structure that is sensitive to large magnitudes of local fluctuations. Different precipitation−geomorphological
types−runoff relationships at Yanhe and Huangfuchuan watershed may be the major effect factors inducing different multifractal behaviors.
The study is financially supported by the Major Programs of the Chinese Academy of Sciences (KZZD-EW-04-03), the National Natural Science Foundation of China (Grant Nos.: 41271295, 41201266) and the
West Light Foundation of the Chinese Academy of Science.
Author Contributions
Xingmin Mu and Guangju Zhao conceived of and designed the research framework. Erhui Li, Guangju Zhao and Peng Gao collected and arranged the data. Erhui Li analyzed the data and wrote the paper.
Guangju Zhao and Peng Gao revised the paper.
Conflicts of Interest
The authors declare no conflict of interest.
© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://
Share and Cite
MDPI and ACS Style
Li, E.; Mu, X.; Zhao, G.; Gao, P. Multifractal Detrended Fluctuation Analysis of Streamflow in the Yellow River Basin, China. Water 2015, 7, 1670-1686. https://doi.org/10.3390/w7041670
AMA Style
Li E, Mu X, Zhao G, Gao P. Multifractal Detrended Fluctuation Analysis of Streamflow in the Yellow River Basin, China. Water. 2015; 7(4):1670-1686. https://doi.org/10.3390/w7041670
Chicago/Turabian Style
Li, Erhui, Xingmin Mu, Guangju Zhao, and Peng Gao. 2015. "Multifractal Detrended Fluctuation Analysis of Streamflow in the Yellow River Basin, China" Water 7, no. 4: 1670-1686. https://doi.org/
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In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. This is in contrast to a variable, which is not fixed.
Unspecified constants[ ]
The most widely mentioned sort of constant is a fixed, but possibly unspecified, number. Usually the term constant is used in connection with mathematical functions of one or more variable arguments.
These arguments, or other variables, are often called x, y, or z, using lower-case letters from the end of the English alphabet. Constants are by convention usually denoted by lower-case letters from
the beginning of the English alphabet, such as a, b, and c.
Specified constants[ ]
Of course, some constants have special symbols, because they are specified, such as 1 or π.
A special case of this may be found in physics, chemistry, and related fields, where certain features of the natural world that are described by numbers are found to have the same value at all times
and places.
For example, in Albert Einstein's special theory of relativity, we have the formula
Here, the letter c stands for the speed of light in a vacuum, which is the same in all physical situations (to the best of current knowledge). In contrast, the letter m stands for the mass of an
object, which could be anything, so it is a variable. E stands for the object's rest energy, another variable, and the formula defines a function that gives rest energy in terms of mass.
(See mathematical constant and physical constant.)
Constant term[ ]
A constant term is a number that appear as an addend in a formula, such as
${\displaystyle f(x) = \sin x + c.}$
Here the constant c is the constant term of the function f. The value of c has not been specified in this formula, but it must be a specific value for f to be a specific function.
The constant term may depend on how the formula is written. For example
${\displaystyle f(x) = x^3+(\sin x)^2 + 4}$
${\displaystyle g(x) = x^3-(\cos x)^2 + 5}$
are formulae for the same function.
In a polynomial (or a generalisation of a polynomial, such as a Taylor series or Fourier expansion), the constant term is associated to the exponent zero. Note that the constant term may be zero,
however. In a sense, any formula has a constant term, if you allow the constant term to be zero.
For some purposes, the constant is taken to be the value of f(0), but this depends on the function being defined at 0; it would not work for f(x)=1-1/x.
Constants vs variables[ ]
A number that is constant in one place may be a variable in another. Consider the example above, with a function f defined by
f(x) = sin x + c.
Now consider a functional F, a function whose argument is itself another function, defined by
F(g) = g(π/2).
Then for the function f given above, we have
F(f) = c + 1.
In the formula for f(x), we are fixing c and varying x, so c is a constant. But in the formula for F(f), we are varying both c and f, so c is a variable. Even this statement might be false in the
presence of some larger context that gives yet another point of view.
Thus, there is no precise definition of "constant" in mathematics; only phrases such as "constant function" or "constant term of a polynomial" can be defined.
There is a mathematicians' joke to the effect that "variables don't; constants aren't." That is, the term variable is frequently used to mean a value that is fixed in a given equation, albeit
unknown; while the term constant is used to mean an arbitrary quantity which may assume any value, as in the constant of integration.
See also[ ]
• Astronomical constant
• Mathematical constant
• Physical constant
bg:Константа da:Konstant et:Konstant es:Constante (física) eo:Konstanto lt:Konstanta nl:Constant (eigenschap) pt:Constante ru:Константа simple:Constant sv:Konstant zh:常数 {{enWP|Constant | {"url":"https://psychology.fandom.com/wiki/Constant","timestamp":"2024-11-11T00:38:04Z","content_type":"text/html","content_length":"168592","record_id":"<urn:uuid:452eca47-93b4-403f-b9e1-acd2978b2f02>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00383.warc.gz"} |
Operations On Rational Numbers Worksheet With Answers
Operations On Rational Numbers Worksheet With Answers work as foundational tools in the world of maths, offering an organized yet versatile system for students to check out and master numerical
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Time Constant - (Intro to Engineering) - Vocab, Definition, Explanations | Fiveable
Time Constant
from class:
Intro to Engineering
The time constant is a measure of the time required for a system to respond to a change in its state, specifically in the context of charging or discharging capacitors and inductors. It indicates how
quickly the voltage or current reaches approximately 63.2% of its final value after a sudden change, which is crucial in understanding transient responses in circuits involving capacitance and
congrats on reading the definition of Time Constant. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The time constant, denoted as \(\tau\), is calculated using the formula \(\tau = R \times C\) for RC circuits, where \(R\) is resistance and \(C\) is capacitance.
2. In RL circuits, the time constant is given by \(\tau = \frac{L}{R}\), where \(L\) is inductance and \(R\) is resistance.
3. A shorter time constant indicates a faster response time to changes, meaning the system will reach its steady state more quickly.
4. In a charging capacitor, after one time constant, the capacitor will have charged to about 63.2% of its maximum voltage; after five time constants, it will be considered fully charged.
5. Time constants can vary significantly depending on the values of resistance and capacitance or inductance in a circuit, impacting how electronic devices perform in real-time applications.
Review Questions
• How does the time constant affect the transient response of capacitors and inductors?
□ The time constant directly influences how quickly capacitors and inductors can charge or discharge their stored energy. In capacitors, a smaller time constant means they will reach their
maximum voltage more quickly, while inductors with a shorter time constant will react faster to changes in current. Understanding this relationship helps engineers design circuits that
respond appropriately to changing signals.
• Describe the significance of the time constant in designing circuits that require precise timing and control.
□ The time constant plays a crucial role in circuit design, especially in applications where timing is critical, such as oscillators, timers, and filters. By carefully selecting resistance and
capacitance values to achieve the desired time constant, engineers can control how fast or slow the circuit responds to changes. This ensures that electronic devices perform reliably and
predictably under varying conditions.
• Evaluate the impact of varying resistance and capacitance on the time constant and how it alters circuit behavior during transients.
□ Varying resistance and capacitance directly affects the time constant, which in turn alters how quickly a circuit reaches its steady state during transients. For example, increasing
resistance results in a longer time constant, slowing down the response of both charging capacitors and inductors. Conversely, decreasing resistance leads to a quicker response. This
evaluation allows engineers to tailor circuits for specific applications by adjusting these parameters for optimal performance.
© 2024 Fiveable Inc. All rights reserved.
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What is the acceleration of a car that increases its velocity from 0?
What is the acceleration of a car that increases its velocity from 0?
Since acceleration = (change in velocity)/(time it takes), if the change in velocity = 0, the acceleration = 0, too.
What is the acceleration of a car that changes velocity from 20 km/hr to 50 km/hr in 4 seconds?
Answer: acceleration is 0.003 m/s^2 .
Which term is defined as a change in velocity?
Acceleration: The rate of change of velocity is acceleration. Like velocity, acceleration is a vector and has both magnitude and direction. For example, a car in straight-line motion is said to have
forward (positive) acceleration if it is speeding up and rearward (negative) acceleration if it is slowing down.
How do you find velocity from acceleration?
Multiply the acceleration by time to obtain the velocity change: velocity change = 6.95 * 4 = 27.8 m/s . Since the initial velocity was zero, the final velocity is equal to the change of speed.
Is acceleration the change in velocity over time?
Because acceleration has both a magnitude and a direction, it is a vector quantity. Velocity is also a vector quantity. Acceleration is defined as the change in the velocity vector in a time
interval, divided by the time interval.
Is acceleration a change in speed or velocity?
Acceleration is a vector quantity that is defined as the rate at which an object changes its velocity. An object is accelerating if it is changing its velocity.
What is acceleration time?
Acceleration (a) is the change in velocity (Δv) over the change in time (Δt), represented by the equation a = Δv/Δt. This allows you to measure how fast velocity changes in meters per second squared
(m/s^2). Acceleration is also a vector quantity, so it includes both magnitude and direction.
Is acceleration always m/s 2?
Because acceleration is velocity in m/s divided by time in s, the SI units for acceleration are m/s2, meters per second squared or meters per second per second, which literally means by how many
meters per second the velocity changes every second. The quicker you turn, the greater the acceleration.
What is called acceleration?
A point or an object moving in a straight line is accelerated if it speeds up or slows down. Motion on a circle is accelerated even if the speed is constant, because the direction is continually
changing. Because acceleration has both a magnitude and a direction, it is a vector quantity.
What is acceleration and velocity?
Velocity is the rate of change of displacement. Acceleration is the rate of change of velocity. Velocity is a vector quantity because it consists of both magnitude and direction. Acceleration is also
a vector quantity as it is just the rate of change of velocity.
Is acceleration the derivative of velocity?
To determine whether velocity is increasing or decreasing, we plug 1 into the acceleration function, because that will give us the rate of change of velocity, since acceleration is the derivative of
What happens to acceleration if velocity increases?
When an object is speeding up, the acceleration is in the same direction as the velocity. Thus, this object has a positive acceleration. In Example B, the object is moving in the negative direction
(i.e., has a negative velocity) and is slowing down. Thus, this object has a negative acceleration.
Acceleration (a) is the change in velocity (Δv) over the change in time (Δt), represented by the equation a = Δv/Δt. This allows you to measure how fast velocity changes in meters per second squared
(m/s^2). Acceleration is also a vector quantity, so it includes both magnitude and direction. Created by Sal Khan.
How is acceleration measured in meters per second?
Acceleration is also a vector quantity, so it includes both magnitude and direction. Acceleration (a) is the change in velocity (Δv) over the change in time (Δt), represented by the equation a = Δv/
Δt. This allows you to measure how fast velocity changes in meters per second squared (m/s^2).
Which is the medium for both velocity and acceleration?
Time is the medium for both velocity and acceleration to occur. In a certain amount of time, an object will have moved a certain distance- This distance is defined by the velocity or acceleration.
What is the equation for acceleration in Khan Academy?
Acceleration (video) | Khan Academy Acceleration (a) is the change in velocity (Δv) over the change in time (Δt), represented by the equation a = Δv/Δt. This allows you to measure how fast velocity
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Statistics Sunday: Different Means
While reading a history of mathematics book earlier this year, I was surprised to learn about the number of means one can compute to summarize values. I was familiar with the common
descriptive statistics
for central tendency - mean, median, and mode - and was aware that this particular mean is also called the arithmetic mean, so I suspected there were more kinds of means. But I wasn't exposed to any
of them beyond arithmetic mean in my statistics classes.
I've started to do some research into the different kinds of means to share here. Today, I'll start with the geometric mean.
The arithmetic mean, of course, is calculated by adding together all values then dividing by the number of values. But there are many cases where the arithmetic mean isn't really an appropriate
measure. If you're dealing with values that are serially correlated - there is shared variance between values of a variable over time - the arithmetic mean may not be the best descriptive statistic.
For instance, say you're tracking return on an investment over time. Those values will be correlated across time and you'll have compounding that must be taken into account. The geometric mean is
well-suited for this situation - in fact, it's frequently used among investment professionals.
The geometric mean is calculated by multiplying
values together, then taking the
th root. As you can imagine, for a few values, this could easily be calculated by hand. For instance, to demonstrate, let's say I have 5 values - return over the last 5 years:
Year 1 - 1%
Year 2 - 7%
Year 3 - -2%
Year 4 - 6%
Year 5 - 3%
For a $100 investment, the value over the 5 years would be:
Year 1 - $100 * 1.01 = $101.00
Year 2 - $101 * 1.07 = $108.07
Year 3 - $108.07 * 0.98 = $105.91
Year 4 - $105.91 * 1.06 = $112.26
Year 5 - $112.26 * 1.03 = $115.63
The arithmetc mean of these 5 return rates (1.01, 1.07, 0.98, 1.06, and 1.03) would be 1.03, or a 3.0% rate of return. The geometric mean would be the product of these 5 values (approximately 1.156)
taken the 5th root = 1.029 or 2.9%. Pretty close. If we had more values and/or more volatility in those values, we might see more of a discrepancy between the values. One thing to note, though, is
that the geometric mean will be less than or equal to the arithmetic mean; it won't be greater than it.
For large
s, you'll want to have a computer do this calculation for you. (The same could be said for the arithmetic mean, I suppose.)
Fortunately, many data analysis programs offer a geometric mean calculation. You can compute this in Excel for up to 255 values using the GEOMEAN function. SPSS offers geometric mean in the Analyze->
Compare Means->Means option. And the psych package for R offers a geometric.mean function.
Today, I'm looking back over my own data for the year - books read, blog posts written, writing accomplished, and so on - and generating some metrics to describe 2017 for me. I may not need to take
the geometric mean of anything, but it's always good to have different descriptive statistics in your back pocket. You never know when they might be useful. Look for a "measurement" post sometime
today or tomorrow.
Happy new year, everyone! Have a fun celebration tonight, stay safe, and I'll see you in 2018!
A few edits: 1) As Jay points out below, geometric mean isn't so great if a value is 0 or negative. Any value times 0 is of course 0, and any positive value times a negative value is negative. So
your result will be meaningless if you have 0s or negatives.
2) My friend, David, over at
The Daily Parker
let me know that this isn't how he learned to compute rate of return in business school. This is probably a good demonstration that there are many ways to summarize a set of values, and also a
demonstration that I don't really know a lot about investment or economics. I love numbers, but not so much numbers with $ signs in front of them. Mostly I wanted to share how to compute geometric
mean and I based my example of a few different examples I saw on the internet. (Yes, I know, just because it's on the internet doesn't mean it's correct.) So if the investment example is incorrect or
meaningless, I'm okay with that. But the geometric mean can be well-suited for other applications, as long as you watch out for #1 above.
Thank you both for your feedback!
1 comment:
1. A few things to note:
-The geometric mean is only meaningful for ratio-scale data. A value of 0 is an edge case where the GM will equal 0 regardless of all other values, but this is unlikely to be a desirable feature
of it.
-To compute the GM, you can take the natural logs of the data, compute the arithmetic mean of those, and then exponentiate. No need for a custom function. | {"url":"https://www.deeplytrivial.com/2017/12/statistics-sunday-different-means.html","timestamp":"2024-11-07T23:40:48Z","content_type":"text/html","content_length":"102244","record_id":"<urn:uuid:28c8470a-cac2-4ee9-9d22-ca1c4324372e>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00380.warc.gz"} |
Sublinear Time Low-Rank Approximation of Distance Matrices
Part of Advances in Neural Information Processing Systems 31 (NeurIPS 2018)
Ainesh Bakshi, David Woodruff
Let $\PP=\{ p_1, p_2, \ldots p_n \}$ and $\QQ = \{ q_1, q_2 \ldots q_m \}$ be two point sets in an arbitrary metric space. Let $\AA$ represent the $m\times n$ pairwise distance matrix with $\AA_{i,j}
= d(p_i, q_j)$. Such distance matrices are commonly computed in software packages and have applications to learning image manifolds, handwriting recognition, and multi-dimensional unfolding, among
other things. In an attempt to reduce their description size, we study low rank approximation of such matrices. Our main result is to show that for any underlying distance metric $d$, it is possible
to achieve an additive error low rank approximation in sublinear time. We note that it is provably impossible to achieve such a guarantee in sublinear time for arbitrary matrices $\AA$, and our proof
exploits special properties of distance matrices. We develop a recursive algorithm based on additive projection-cost preserving sampling. We then show that in general, relative error approximation in
sublinear time is impossible for distance matrices, even if one allows for bicriteria solutions. Additionally, we show that if $\PP = \QQ$ and $d$ is the squared Euclidean distance, which is not a
metric but rather the square of a metric, then a relative error bicriteria solution can be found in sublinear time. Finally, we empirically compare our algorithm with the SVD and input sparsity time
algorithms. Our algorithm is several hundred times faster than the SVD, and about $8$-$20$ times faster than input sparsity methods on real-world and and synthetic datasets of size $10^8$.
Accuracy-wise, our algorithm is only slightly worse than that of the SVD (optimal) and input-sparsity time algorithms. | {"url":"https://proceedings.nips.cc/paper_files/paper/2018/hash/c45008212f7bdf6eab6050c2a564435a-Abstract.html","timestamp":"2024-11-14T04:49:46Z","content_type":"text/html","content_length":"9645","record_id":"<urn:uuid:db7445b0-f9dc-45bd-9e64-c58230ce5bd1>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00474.warc.gz"} |
Properties of Information Flow
What are the properties of information flow? To establish the terminology with which I will pose properties to consider, I start off with the most basic of properties. If A carries the information
that B, then A carries the information that B.
$A \rightarrow B \vdash A \rightarrow B$
Here are two other straightforward properties of information:
$((A \rightarrow B) \wedge (A \rightarrow C)) \vdash (A \rightarrow (B \wedge C))$
$((A \rightarrow C) \wedge (B \rightarrow C)) \vdash ((A \vee B) \rightarrow C)$
What further properties are there to consider and which should be accepted and which should be rejected in developing an account of information flow?
Well, here are 3 other properties to consider. They are valid in Fred Dretske’s inverse conditional probability account of information flow [see an earlier post here] but come out invalid in standard
formal systems for counterfactuals (Jonathan Cohen and Aaron Meskin suggest a counterfactual theory of information).
1. $((A \rightarrow B) \wedge (B \rightarrow C)) \vdash (A \rightarrow C)$
2. $A \rightarrow B \vdash eg B \rightarrow eg A$
3. $A \rightarrow B \vdash (A \wedge C) \rightarrow B$
Property 1, that information flow is transitive, is a property that we would intuitively accept; if A carries the information that B and B carries the information that C, then A carries the
information that C. It is a mainstay of Dretske’s account, where it is named the Xerox principle. It is also adopted by Barwise and Seligman in Information Flow: The Logic of Distributed Systems. The
transitivity of information flow though has recently been challenged by Hilmi Demir [see http://sites.google.com/site/asilbenhilmi/Demir-Dissertation.pdf?attredirects=0, p. 84.].
What of the other two properties? I would say that Property 2 should be counted as a property of information flow. If A carries the information B, then a unique path can be traced from A back to B.
There is a many-to-one relationship between the set X and B, where $A \in X$. If $A \rightarrow B$, then every time A obtains B obtains, so if B does not obtain, then neither does A. This principle
provokes the realisation that information flows forwards and backwards. Not only can current events carry information about past events, but current events can also carry information about future
As for Property 3, at first I am inclined to say that this also should be counted as a property of information flow. In a strict sense, if A carries the information B, then A should say everything
regarding the obtaining of B. Consider this non-monotonic reasoning
• Ellie being a bird carries the information that Ellie flies
• Ellie being a bird and Ellie being an Emu does not carry the information that Ellie flies
In this case, we can simply say that Ellie being a bird does not carry the information that Ellie flies. There is insufficient information to imply that Ellie does or does not fly. However
• Ellie being an emu carries the information that Ellie does not fly
• Ellie being an eagle carries the information that Ellie does fly
So information flow would be monotonic if A carries the information B implies that A is truly a sufficient condition for B. Being a bird is not a sufficient condition for flying. Being a bird does
carry information about the probability of flying though.
However, one could take this non-monotonic reasoning further.
• Ellie being an eagle carries the information that Ellie flies
• Ellie being an eagle and Ellie having clipped wings does not carry the information that Ellie flies
An adherence to the monotonicity of information flow would force one to deny the first point here. Only the second point would hold.
But in many of these examples one could just keep on adding extra clauses, so that an apparent information carrying relationship could be invalidated. Perhaps then the best option is to deny the
monotonicity of information flow. In the above example, Ellie being an eagle does carry the information that Ellie flies. Maybe one could appeal to something like the Dretskean notion of relevant
alternatives in order to accommodate this?
1 thought on “Properties of Information Flow”
1. I hope you do not mind my commenting on an older blog post. Regarding non-monotonicity, I think we are indeed in a position where the background conditions for information flow cannot be known,
or fully specified, except perhaps within purely mathematical systems.
Jon Barwise and Jerry Seligman tackle this somewhat in their book. Their solution frames monotonicity as valid when reasoning about normal tokens of a local logic. Barwise has a nice paper:
Jon Barwise, “State spaces, local logics, and non-monotonicity,” in Logic, language and computation, vol. 2 (Center for the Study of Language and Information, 1999), 1-20, http://portal.acm.org/ | {"url":"http://theinformationalturn.net/philosophy_information/properties-of-information-flow/","timestamp":"2024-11-04T03:51:05Z","content_type":"text/html","content_length":"45883","record_id":"<urn:uuid:48302361-1c00-4589-8ca1-c0da76a78403>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00019.warc.gz"} |
Volts, Amps, Watts, Inverter & Solar power - Simplified
Okay. I think it's about time to tackle the big questions. Whats the difference between, volts, amps, watts and how much solar power do I need ?
First of all, lets start with a basic foundation to work from. Since you're reading this blog, you must already know that the primary topic and usage for power is on our teardrop trailers. All of our
teardrop trailer's are off the grid. We use a deep cycle marine battery to power fans,heating blankets, 12 volt coolers, lights and a whole pile of different accesories.
Since we began our custom teardrop trailer builds, we have come across a handful of people that still want to depend on the "Grid", and some people even want to still use regular kitchen appliances
when camping, such as blenders,toasters and yes even coffee makers. I must admit, I am guilty of wanting a good freshly brewed coffee first thing in the morning, no matter where I am camped out. So
let's get right to it then. Lets use one item at a time to decide how much power you'll need on your teardrop camping trip.
First we need to understand each definition.
Volts. The volt is a measure of electric potential. Electrical potential is a type of potential energy, and refers to the energy that could be released if electric current is allowed to flow. We use
a 12 volt deep cycle marine battery. Sometimes we connect 2 batteries together. Positive with positive and negative with negative. It still maintains 12 volts.
Amps. Amperes are used to express flow rate of electric charge. For any point experiencing a current, if the number of charged particles passing through it — or the charge on the particles passing
through it — is increased, the amperes of current at that point will proportionately increase.
Now lets break this down using a basic 12 volt fan as an example; We're using a 12 volt battery as it's source. Almost everything you purchase for a 12 volt system will tell you how many amps it
draws, or uses. let's say the fan we choose uses 3 amps. Okay, Let's stop there for a second and now go to the next definition.
Watts. One watt is also defined as the current flow of one ampere with voltage of one volt. How to convert watts to amps The current I in ampers (A) is equal to the power P in watts (W) divided by
the voltage V in volts (V): I(A) = P(W) / V(V)
Inverter. A power inverter, or inverter, is an electronic device or circuitry that changes direct current (DC) to alternating current (AC).[1] The input voltage, output voltage and frequency, and
overall power handling depend on the design of the specific device or circuitry. The inverter does not produce any power; the power is provided by the DC source. In our case, the deep cycle marine
Solar. Before you figure out what size solar panel you want, you need to figure out how much power you'll need throughout the day and night. We personally use 100W flexible panel and 2 deep cycle
marine batteries. If all you need is a little lighting at night, you'll be fine with a smaller 15W solar panel and a single deep cycle marine battery.
Okay, lets go back to that 12 volt fan. It does not require an inverter to work. It also does not need a solar panel to work. If you want to recharge your battery, you'll need a regular battery
charger,solar panel or a 7 pin round trickle charge from your car to trailer battery. If you want to use an appliance like a coffee maker or toaster you'll need an inverter. We use a 3000W inverter.
(remember- a coffee maker uses approx 1250W ) A 100W solar panel can produce approx 6 amps per hour on a sunny day. 5 hrs, gives you 30 amps returned back to your battery. That 12 volt fan draws
approx 3 amps. Your newly charged deep cycle marine battery should have approx 100 amp hours already in reserve.
A 100ah battery should provide 1 amp for 100 hours, 2 amps for 50 hours, 3 amps for 33 hours etc. It would be nice if this would work all the way up to 100 amps for 1 hour, but there are some limits
to the maximum rate of current draw, and how much of that 100amps you can actually use without destroying your battery.
There's a lot more to understanding a 12 volt system but I hope this write up is a begining in helping you to calculate what size system works best for you. Happy trails... | {"url":"https://www.theteardroptrailer.com/post/2016-1-17-volts-amps-watts-inverter-solar-power-simplified","timestamp":"2024-11-08T17:57:22Z","content_type":"text/html","content_length":"1051134","record_id":"<urn:uuid:7dc2d220-5d9e-405d-a898-358b6cce1051>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00459.warc.gz"} |
Multiple Regression
From Verify.Wiki
The extension of simple linear regression to multiple explanatory (or predictor variables) is known as multiple linear regression (or multivariable linear regression). Nearly all real-world
regression models involve multiple predictors, and basic descriptions of linear regression are often phrased in terms of the multiple regression model. Multiple linear regression is used for two main
1. To describe the linear dependence of the response ${\displaystyle Y}$ on a collection of explanatory variables ${\displaystyle X_{1},X_{2},...,X_{k}}$.
2. To predict values of the response ${\displaystyle Y}$ from values of the explanatory variable ${\displaystyle X_{1},X_{2},...,X_{k}}$, for which more data are available.
Any hyper-plane fitted through a cloud of data will deviate from each data point to greater or lesser degree. The vertical distance between a data point and the fitted line is termed a "residual".
This distance is a measure of prediction error, in the sense that it is the discrepancy between the actual value of the response variable and the value predicted by the hyper-plane. Linear regression
determines the best-fit hyper-plane through a scattering of data, such that the sum of squared residuals is minimized; equivalently, it minimizes the error variance. The fit is "best" in precisely
that sense: the sum of squared errors is as small as possible. That is why it is also termed "Ordinary Least Squares" regression. ^[1]
The Multiple Linear Regression Model
Formally, consider a collection of explanatory variables ${\displaystyle X_{1},X_{2},...,X_{k}}$ and a response variable ${\displaystyle Y}$ and suppose there are ${\displaystyle n}$ randomly
selected subjects in an experiment. With ${\displaystyle \epsilon _{i}}$ as unknown random errors and ${\displaystyle i=1,2,...,n}$, the multiple linear regression model is:
${\displaystyle y_{i}=\beta _{0}+\beta _{1}x_{1i}+\beta _{2}x_{2i}+...++\beta _{k}x_{ki}+\epsilon _{i}}$.
We call ${\displaystyle \beta _{0},\beta _{1},...,\beta _{k}}$ the unknown parameters and additionally, we assume ${\displaystyle \beta _{0},\beta _{1},...,\beta _{k}}$ to be constants (and not
random variables). Note that usually, multiple regression is written in terms of vectors and matrices. Specifically, it is usually written:
${\displaystyle {\mathbf {y}}={\mathbf {X}}{\mathbf {\beta }}+{\mathbf {\epsilon }}}$
where ${\displaystyle {\mathbf {y}}={\begin{bmatrix}y_{1}\\y_{2}\\\vdots \\y_{n}\end{bmatrix}},{\mathbf {X}}={\begin{bmatrix}1&x_{11}&\cdots &x_{1k}\\1&x_{21}&\cdots &x_{2k}\\\vdots &\vdots &\ddots &
\vdots \\1&x_{n1}&\cdots &x_{nk}\end{bmatrix}},{\mathbf {\beta }}={\begin{bmatrix}\beta _{1}\\\beta _{2}\\\vdots \\\beta _{n}\end{bmatrix}},and{\mathbf {\epsilon }}={\begin{bmatrix}\epsilon _{1}\\\
epsilon _{2}\\\vdots \\\epsilon _{n}\end{bmatrix}}}$.
1. The ${\displaystyle x_{ji}}$ are nonrandom and measured with negligible error. Note that the vectors ${\displaystyle {\mathbf {x}}}$ from the matrix ${\displaystyle {\mathbf {X}}}$ can also be
transformations of explanatory variables.
2. ${\displaystyle \epsilon }$ is a random vector.
3. For each ${\displaystyle i}$, ${\displaystyle E(\epsilon _{i})=0}$ where ${\displaystyle E}$ is the expected value. That is, the ${\displaystyle \epsilon _{i}}$ have mean equal to 0. This can
also be written as ${\displaystyle E(\epsilon )=0}$.
4. For each ${\displaystyle i}$, ${\displaystyle var(\epsilon _{i})=\sigma ^{2}}$ where ${\displaystyle var}$ is the variance. That is, the ${\displaystyle \epsilon _{i}}$ have homogeneous variance
${\displaystyle \sigma ^{2}}$. This can also be written as ${\displaystyle var(\epsilon )=\sigma ^{2}}$.
5. For each ${\displaystyle ieq j}$, ${\displaystyle E(\epsilon _{i}\epsilon _{j})=0}$. That is, the ${\displaystyle \epsilon _{i}}$ are uncorrelated random variables.
6. It is often assumed (although not necessary) that ${\displaystyle \epsilon }$ follows a multivariate normal distribution with mean ${\displaystyle {\mathbf {0}}}$ and variance ${\displaystyle \
sigma ^{2}{\mathbf {I}}}$ with ${\displaystyle {\mathbf {0}}}$ a vector of zeros with length ${\displaystyle n}$ and ${\displaystyle I}$ the ${\displaystyle n\times n}$ identity matrix.
Notice assumptions 4 and 5 can be written compactly as ${\displaystyle var({\mathbf {\epsilon }})=\sigma ^{2}{\mathbf {I}}}$.
Partitioning the total variability
From the about section, the goal of multiple regression is to estimate ${\displaystyle \mu (x_{i})=y_{i}}$ for all ${\displaystyle i}$. This is accomplished by using the estimates of ${\displaystyle
\beta }$ which are attained through a partitioning of the "total variability" of the observed response ${\displaystyle Y}$ where the total variability of ${\displaystyle Y}$ is the quantity ${\
displaystyle \sum _{i=1}^{n}(y_{i}-{\bar {Y}})^{2}}$. Denoting the least squares estimate of ${\displaystyle \beta }$ as ${\displaystyle {\hat {\beta }}}$, the process follows in two steps.
1. Estimate ${\displaystyle {\hat {\beta }}}$.
2. Calculate ${\displaystyle {\hat {y}}_{i}={\hat {\mu }}(x_{i})={\hat {\beta }}{\mathbf {X}}}$. The ${\displaystyle {\hat {y}}_{i}}$ are called the predicted values.
Note that the partitioning of the total variability of ${\displaystyle Y}$ is achieved by adding ${\displaystyle 0=-{\hat {y}}_{i}+{\hat {y}}_{i}}$ to the equation of total variability in the
following way: ${\displaystyle \sum _{i=1}^{n}(y_{i}-{\bar {Y}})^{2}=\sum _{i=1}^{n}(y_{i}-{\hat {y}}_{i}+{\hat {y}}_{i}-{\bar {Y}})^{2}=\sum _{i=1}^{n}(y_{i}-{\hat {y}}_{i})+\sum _{i=1}^{n}({\hat
{y}}_{i}-{\bar {Y}})^{2}}$,
and that the quantities from equation above are given special names:
1. The sum of squares total is ${\displaystyle SST=\sum _{i=1}^{n}(y_{i}-{\bar {Y}})^{2}}$,
2. The sum of squares regression is ${\displaystyle SSR=\sum _{i=1}^{n}({\hat {y}}_{i}-{\bar {Y}})^{2}}$, and
3. The sum of squares error is ${\displaystyle SSE=\sum _{i=1}^{n}(y_{i}-{\hat {y}}_{i})}$.
Note that some authors refer to the sum of squares regression as the The sum of squares model.
The ${\displaystyle i^{th}}$ residual is the difference between the observed ${\displaystyle y_{i}}$ and the predicted ${\displaystyle {\hat {y}}_{i}}$. This is written as ${\displaystyle e_{i}=y_{i}
-{\hat {y}}_{i}}$. Now, as ${\displaystyle {\hat {y}}_{i}=\beta _{0}+\beta _{1}x_{1i}+\beta _{2}x_{2i}+...+\beta _{k}x_{ki}}$, it follows that ${\displaystyle e_{i}=y_{i}-\beta _{0}-\beta _{1}x_{1i}-
\beta _{2}x_{2i}-...-\beta _{k}x_{ki}}$. Due to this, in order to attain quality predictions, ${\displaystyle {\hat {\beta }}_{0},{\hat {\beta }}_{1},\dots ,{\hat {\beta }}_{k}}$ are chosen to
minimize ${\displaystyle e_{i}}$ for each ${\displaystyle i}$.
Parameter Estimation With Ordinary Least Squares
One method for estimating the matrix of unknown parameters ${\displaystyle \beta }$ is through the use of Ordinary Least Squares (OLS). This is accomplished by finding values for ${\displaystyle {\
hat {\beta }}}$ that minimize the sum of the squared residuals: ${\displaystyle ({\mathbf {y}}-{\mathbf {X}}\beta )^{T}({\mathbf {y}}-{\mathbf {X}}\beta )}$ where ${\displaystyle T}$ is the matrix
OLS proceeds by taking the matrix calculus partial derivatives of ${\displaystyle ({\mathbf {y}}-{\mathbf {X}}\beta )^{T}({\mathbf {y}}-{\mathbf {X}}\beta )}$ with respect to ${\displaystyle \beta }$
. After algebra, the OLS estimates (which are also known as the normal equations) we have
${\displaystyle ({\mathbf {X}}^{T}{\mathbf {X}})\beta ={\mathbf {X}}^{T}{\mathbf {y}}}$
Therefore, assuming that the inverse of ${\displaystyle ({\mathbf {X}}^{T}{\mathbf {X}})}$ exists, the OLS predictions are attained with the equation ${\displaystyle \beta =({\mathbf {X}}^{T}{\mathbf
{X}})^{-1}{\mathbf {X}}^{T}{\mathbf {y}}}$. Note that ${\displaystyle {\hat {\beta }}}$ is unbiased estimator for ${\displaystyle \beta }$ as ${\displaystyle E({\hat {\beta }})=\beta }$. Note also
that the unbiased property of the parameter estimates follows from the assumption that ${\displaystyle E(\epsilon _{i})=0}$.
Degrees of Freedom
For each of the sum of squares equations (from the partitioning of total variability section), there are related degrees of freedom. For simple linear regression, we have:
1. ${\displaystyle df_{total}=n-1}$,
2. ${\displaystyle df_{regression}=p-1}$, and
3. ${\displaystyle df_{error}=n-p}$.
Note that ${\displaystyle df_{regression}}$ is the number of explanatory variables in the model minus 1 and ${\displaystyle df_{error}}$ is the n minus the number of explanatory variables ${\
displaystyle \beta }$ denoted ${\displaystyle p}$ in the model.
Mean Squares
The mean squares are the ratio of the sum of squares over the respective degrees of freedom. Therefore, for simple linear regression:
1. the mean square for the model is ${\displaystyle MSR={\dfrac {SSR}{df_{regression}}}={\dfrac {SSR}{p-1}}}$ and
2. the mean square error is ${\displaystyle MSE={\dfrac {SSE}{df_{error}}}={\dfrac {SSE}{n-p}}}$.
Note also that MSE is an unbiased estimate for the variance ${\displaystyle \sigma ^{2}}$. Specifically, ${\displaystyle {\hat {\sigma }}^{2}=MSE}$.
Coefficient of Determination
The coefficient of determination, denoted by ${\displaystyle R^{2}}$ is a measure of fit for the estimated model. Specifically, ${\displaystyle R^{2}}$ is a measure of the amount of variance (of Y)
explained by the explanatory variable ${\displaystyle X}$. For simple linear regression, the equation is:
${\displaystyle R^{2}={\dfrac {SSR}{SST}}=1-{\dfrac {SSE}{SST}}}$.
Note that ${\displaystyle R^{2}}$ is a number between 0 and 1. For example, ${\displaystyle R^{2}=1}$ implies that all points fall on a straight line while ${\displaystyle R^{2}=0}$ (or if ${\
displaystyle R^{2}}$ is close to 0) implies that the points are extremely scattered or that the points follow a non-linear pattern. In either case, the regression model is poor when ${\displaystyle R
^{2}}$ is close to 0 while an ${\displaystyle R^{2}}$ close to 1 indicates that the model produces quality predictions (i.e. the model is a good fit).
The main criticisms of multiple linear regression involve the required linearity (in the coefficients) of the model as well as the (optional) assumption of the normality in the response ${\
displaystyle Y}$.
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1. ↑ http://seismo.berkeley.edu/~kirchner/eps_120/Toolkits/Toolkit_10.pdf | {"url":"https://verify.wiki/wiki/Multiple_Regression","timestamp":"2024-11-12T12:06:04Z","content_type":"text/html","content_length":"149551","record_id":"<urn:uuid:5e07cd95-fe1c-4578-8a3f-947a444fa830>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00143.warc.gz"} |
3.4: Binomial Distribution (Special Topic)
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Example \(\PageIndex{1}\): Shock Study
Suppose we randomly selected four individuals to participate in the "shock" study. What is the chance exactly one of them will be a success? Let's call the four people Allen (A), Brittany (B),
Caroline (C), and Damian (D) for convenience. Also, suppose 35% of people are successes as in the previous version of this example.
Let's consider a scenario where one person refuses:
\[ \begin{align*} P(A = \text{refuse}; B = \text{shock}; C = \text{shock}; D = \text{shock}) &= P(A = \text{refuse}) P(B = \text{shock}) P(C = \text{shock}) P(D = \text{shock}) \\[5pt] &= (0.35)
(0.65)(0.65)(0.65) \\[5pt] &= (0.35)^1(0.65)^3 \\[5pt] &= 0.096 \end{align*}\]
But there are three other scenarios: Brittany, Caroline, or Damian could have been the one to refuse. In each of these cases, the probability is again
\[P=(0.35)^1(0.65)^3. \nonumber\]
These four scenarios exhaust all the possible ways that exactly one of these four people could refuse to administer the most severe shock, so the total probability is
\[4 \times (0.35)^1(0.65)^3 = 0.38. \nonumber\]
Exercise \(\PageIndex{1}\)
Verify that the scenario where Brittany is the only one to refuse to give the most severe shock has probability \((0.35)^1(0.65)^3.\)
\[ \begin{align*} P(A = shock; B = refuse; C = shock; D = shock) &= (0.65)(0.35)(0.65)(0.65) \\[5pt] &= (0.35)^1(0.65)^3.\end{align*}\]
The Binomial Distribution
The scenario outlined in Example \(\PageIndex{1}\) is a special case of what is called the binomial distribution. The binomial distribution describes the probability of having exactly k successes in
n independent Bernoulli trials with probability of a success p (in Example \(\PageIndex{1}\), n = 4, k = 1, p = 0.35). We would like to determine the probabilities associated with the binomial
distribution more generally, i.e. we want a formula where we can use n, k, and p to obtain the probability. To do this, we reexamine each part of the example.
There were four individuals who could have been the one to refuse, and each of these four scenarios had the same probability. Thus, we could identify the nal probability as
\[ \text {[# of scenarios]} \times \text {P(single scenario)} \label {3.39}\]
The first component of this equation is the number of ways to arrange the k = 1 successes among the n = 4 trials. The second component is the probability of any of the four (equally probable)
Consider P(single scenario) under the general case of k successes and n-k failures in the n trials. In any such scenario, we apply the Multiplication Rule for independent events:
\[ p^k (1- p)^{n-k}\]
This is our general formula for P(single scenario).
Secondly, we introduce a general formula for the number of ways to choose k successes in n trials, i.e. arrange k successes and n - k failures:
\[ \binom {n}{k} = \dfrac {n!}{k!(n - k)!} \label{3.4.Y}\]
The quantity \( \binom {n}{k} \) is read n choose k.^30 The exclamation point notation (e.g. k!) denotes a factorial expression.
\[ 0! &= 1 \nonumber \\[5pt] 1! &= 1 \nonumber \\[5pt] 2! &= 2 \times 1 = 2 \nonumber \\[5pt] 3! &= 3 \times 2 \times 1 = 6 \nonumber \\[5pt] 4! &= 4 \times 3 \times 2 \times 1 = 24 \nonumber \\[5pt]
& \vdots \nonumber \\[5pt] n! &= n \times (n - 1) \times \dots \times 3 \times 2 \times 1 \label{eq3.4.X} \]
Substituting Equation \ref{eq3.4.X} into Equation \ref{3.4.Y}, we can compute the number of ways to choose \(k = 1\) successes in \(n = 4\) trials:
\[ \binom {4}{1} &= \dfrac {4!}{1! (4 - 1)!} \\[5pt] &= \dfrac {4!}{1! 3!} \\[5pt]&= \dfrac {4 \times 3 \times 2 \times 1}{(1)(3 \times 2 \times 1)} \\[5pt]&= 4 \]
This result is exactly what we found by carefully thinking of each possible scenario in Example \(\PageIndex{1}\).
Other notations
Other notation for n choose k includes \(nC_k\), \(C^k_n\), and \(C(n, k)\).
Substituting \(n\) choose \(k\) for the number of scenarios and \(p^k(1 - p)^{n-k}\) for the single scenario probability in Equation \ref{3.39} yields the general binomial formula (Equation \ref
Definition: Binomial distribution
Suppose the probability of a single trial being a success is p. Then the probability of observing exactly k successes in n independent trials is given by
\[ \binom {n}{k} p^k (1 - p)^{n - k} = \dfrac {n!}{k!(n - k)!} p^k (1 - p)^{n - k} \label {3.40}\]
Additionally, the mean, variance, and standard deviation of the number of observed successes are
\[\mu = np \sigma^2 = np(1 - p) \sigma = \sqrt {np(1- p)} \label{3.41}\]
TIP: Four conditions to check if it is binomial?
1. The trials independent.
2. The number of trials, \(n\), is fixed.
3. Each trial outcome can be classified as a success or failure.
4. The probability of a success, \(p\), is the same for each trial.
Example \(\PageIndex{2}\)
What is the probability that 3 of 8 randomly selected students will refuse to administer the worst shock, i.e. 5 of 8 will?
We would like to apply the binomial model, so we check our conditions. The number of trials is fixed (n = 8) (condition 2) and each trial outcome can be classi ed as a success or failure (condition
3). Because the sample is random, the trials are independent (condition 1) and the probability of a success is the same for each trial (condition 4).
In the outcome of interest, there are k = 3 successes in n = 8 trials, and the probability of a success is p = 0.35. So the probability that 3 of 8 will refuse is given by
\[ \begin{align*} \binom {8}{3} {(0.35)}^k (1 - 0.35)^{8 - 3} &= \dfrac {8!}{3!(8 - 3)!} {(0.35)}^k (1 - 0.35)^{8 - 3} \\[5pt] &= \dfrac {8!}{3! 5!} {(0.35)}^3 {(0.65)}^5 \end{align*}\]
Dealing with the factorial part:
\[ \begin{align*} \dfrac {8!}{3!5!} &= \dfrac {8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(3 \times 2 \times 1)( 5 \times 4 \times 3 \times 2 \times 1 )} \\[5pt] &= \dfrac {8 \
times 7 \times 6 }{ 3 \times 2 \times 1 } \\[5pt] & = 56 \end{align*}\]
Using \((0.35)^3(0.65)^5 \approx 0.005\), the final probability is about 56 * 0.005 = 0.28.
TIP: computing binomial probabilities
The rst step in using the binomial model is to check that the model is appropriate. The second step is to identify n, p, and k. The final step is to apply the formulas and interpret the results.
TIP: computing n choose k
In general, it is useful to do some cancelation in the factorials immediately. Alternatively, many computer programs and calculators have built in functions to compute n choose k, factorials, and
even entire binomial probabilities.
Exercise \(\PageIndex{2A}\)
If you ran a study and randomly sampled 40 students, how many would you expect to refuse to administer the worst shock? What is the standard deviation of the number of people who would refuse?
Equation \ref{3.41} may be useful.
We are asked to determine the expected number (the mean) and the standard deviation, both of which can be directly computed from the formulas in Equation \ref{3.41}:
\[\mu = np = 40 \times 0.35 = 14 \nonumber\]
\[\sigma = \sqrt{np(1 - p)} = \sqrt { 40 \times 0.35 \times 0.65} = 0.02. \nonumber\]
Because very roughly 95% of observations fall within 2 standard deviations of the mean (see Section 1.6.4), we would probably observe at least 8 but less than 20 individuals in our sample who
would refuse to administer the shock.
Exercise \(\PageIndex{2B}\)
The probability that a random smoker will develop a severe lung condition in his or her lifetime is about 0:3. If you have 4 friends who smoke, are the conditions for the binomial model satisfied?
One possible answer: if the friends know each other, then the independence assumption is probably not satis ed. For example, acquaintances may have similar smoking habits.
Example \(\PageIndex{3}\)
Suppose these four friends do not know each other and we can treat them as if they were a random sample from the population. Is the binomial model appropriate? What is the probability that
1. none of them will develop a severe lung condition?
2. One will develop a severe lung condition?
3. That no more than one will develop a severe lung condition?
To check if the binomial model is appropriate, we must verify the conditions. (i) Since we are supposing we can treat the friends as a random sample, they are independent. (ii) We have a fixed number
of trials (n = 4). (iii) Each outcome is a success or failure. (iv) The probability of a success is the same for each trials since the individuals are like a random sample (p = 0.3 if we say a
"success" is someone getting a lung condition, a morbid choice). Compute parts (a) and (b) from the binomial formula in Equation \ref{3.40}:
\[P(0) = \binom {4}{0}(0.3)^0(0.7)^4 = 1 \times 1 \times 0.7^4 = 0.2401 \nonumber\]
\[P(1) = \binom {4}{1}(0.3)^1(0.7)^3 = 0.4116. \nonumber\]
Note: 0! = 1.
Part (c) can be computed as the sum of parts (a) and (b):
\[P(0)+P(1) = 0.2401+0.4116 = 0.6517.\]
That is, there is about a 65% chance that no more than one of your four smoking friends will develop a severe lung condition.
Exercise \(\PageIndex{3A}\)
What is the probability that at least 2 of your 4 smoking friends will develop a severe lung condition in their lifetimes?
The complement (no more than one will develop a severe lung condition) as computed in Example \(\PageIndex{3}\) as 0.6517, so we compute one minus this value: 0.3483.
Exercise \(\PageIndex{3B}\)
Suppose you have 7 friends who are smokers and they can be treated as a random sample of smokers.
1. How many would you expect to develop a severe lung condition, i.e. what is the mean?
2. What is the probability that at most 2 of your 7 friends will develop a severe lung condition.
Answer a
\(\mu\) = 0.3 \times 7 = 2.1.
Answer b
P(0, 1, or 2 develop severe lung condition) = P(k = 0)+P(k = 1)+P(k = 2) = 0:6471.
Below we consider the first term in the binomial probability, n choose k under some special scenarios.
Exercise \(\PageIndex{3C}\)
Why is it true that \( \binom {n}{0} = 1\) and \( \binom {n}{n} = 1 \) for any number n?
Frame these expressions into words. How many different ways are there to arrange 0 successes and n failures in n trials? (1 way.) How many different ways are there to arrange n successes and 0
failures in n trials? (1 way.)
Exercise \(\PageIndex{3D}\)
How many ways can you arrange one success and n -1 failures in n trials? How many ways can you arrange n -1 successes and one failure in n trials?
One success and n - 1 failures: there are exactly n unique places we can put the success, so there are n ways to arrange one success and n - 1 failures. A similar argument is used for the second
question. Mathematically, we show these results by verifying the following two equations:
\[ \binom {n}{1} = n, \binom {n}{n - 1} = n \]
Normal Approximation to the Binomial Distribution
The binomial formula is cumbersome when the sample size (n) is large, particularly when we consider a range of observations. In some cases we may use the normal distribution as an easier and faster
way to estimate binomial probabilities.
Example \(\PageIndex{4}\)
Approximately 20% of the US population smokes cigarettes. A local government believed their community had a lower smoker rate and commissioned a survey of 400 randomly selected individuals. The
survey found that only 59 of the 400 participants smoke cigarettes. If the true proportion of smokers in the community was really 20%, what is the probability of observing 59 or fewer smokers in a
sample of 400 people?
We leave the usual verification that the four conditions for the binomial model are valid as an exercise.
The question posed is equivalent to asking, what is the probability of observing \(k = 0, 1, \dots, 58, or 59\) smokers in a sample of n = 400 when p = 0.20? We can compute these 60 different
probabilities and add them together to nd the answer:
\[P(k = 0 or k = 1 or \dots or k = 59)\]
\[= P(k = 0) + P(k = 1) + \dots + P(k = 59)\]
\[= 0.0041\]
If the true proportion of smokers in the community is p = 0.20, then the probability of observing 59 or fewer smokers in a sample of n = 400 is less than 0.0041. The computations in Example 3.50 are
tedious and long. In general, we should avoid such work if an alternative method exists that is faster, easier, and still accurate. Recall that calculating probabilities of a range of values is much
easier in the normal model. We might wonder, is it reasonable to use the normal model in place of the binomial distribution? Surprisingly, yes, if certain conditions are met.
Exercise \(\PageIndex{4}\)
Here we consider the binomial model when the probability of a success is p = 0.10. Figure 3.17 shows four hollow histograms for simulated samples from the binomial distribution using four different
sample sizes: n = 10, 30, 100, 300. What happens to the shape of the distributions as the sample size increases? What distribution does the last hollow histogram resemble?
The distribution is transformed from a blocky and skewed distribution into one that rather resembles the normal distribution in last hollow histogram
Figure 3.17: Hollow histograms of samples from the binomial model when p = 0.10. The sample sizes for the four plots are n = 10, 30, 100, and 300, respectively.
Normal approximation of the binomial distribution
The binomial distribution with probability of success p is nearly normal when the sample size n is sufficiently large that np and n(1 - p) are both at least 10. The approximate normal distribution
has parameters corresponding to the mean and standard deviation of the binomial distribution:
\[ \mu = np \sigma = \sqrt {np(1- p)}\]
The normal approximation may be used when computing the range of many possible successes. For instance, we may apply the normal distribution to the setting of Example 3.50.
Example \(\PageIndex{5}\)
How can we use the normal approximation to estimate the probability of observing 59 or fewer smokers in a sample of 400, if the true proportion of smokers is p = 0.20?
Showing that the binomial model is reasonable was a suggested exercise in Example 3.50. We also verify that both np and n(1- p) are at least 10:
\[np = 400 \times 0.20 = 80 n(1 - p) = 400 \times 0.8 = 320\]
With these conditions checked, we may use the normal approximation in place of the binomial distribution using the mean and standard deviation from the binomial model:
\[\mu = np = 80 \sigma = \sqrt {np(1 - p)} = 8\]
We want to find the probability of observing fewer than 59 smokers using this model.
Exercise \(\PageIndex{5}\)
Use the normal model N(\(\mu = 80, \sigma = 8\)) to estimate the probability of observing fewer than 59 smokers. Your answer should be approximately equal to the solution of Example 3.50: 0.0041.
Compute the Z score rst: Z = \( \dfrac {59-80}{8} = -2.63\). The corresponding left tail area is 0.0043.
Caution: The normal approximation may fail on small intervals
The normal approximation to the binomial distribution tends to perform poorly when estimating the probability of a small range of counts, even when the conditions are met.
The normal Approximation Breaks down on small intervals
Suppose we wanted to compute the probability of observing 69, 70, or 71 smokers in 400 when p = 0.20. With such a large sample, we might be tempted to apply the normal approximation and use the range
69 to 71. However, we would nd that the binomial solution and the normal approximation notably differ:
\[\text {Binomial}: 0.0703 \text {Normal}: 0.0476\]
We can identify the cause of this discrepancy using Figure 3.18, which shows the areas representing the binomial probability (outlined) and normal approximation (shaded). Notice that the width of the
area under the normal distribution is 0.5 units too slim on both sides of the interval.
Figure 3.18: A normal curve with the area between 69 and 71 shaded. The outlined area represents the exact binomial probability.
TIP: Improving the accuracy of the normal approximation to the binomial distribution
The normal approximation to the binomial distribution for intervals of values is usually improved if cutoff values are modified slightly. The cutoff values for the lower end of a shaded region should
be reduced by 0.5, and the cutoff value for the upper end should be increased by 0.5.
The tip to add extra area when applying the normal approximation is most often useful when examining a range of observations. While it is possible to apply it when computing a tail area, the benefit
of the modification usually disappears since the total interval is typically quite wide.
Contributors and Attributions
• David M Diez (Google/YouTube), Christopher D Barr (Harvard School of Public Health), Mine Çetinkaya-Rundel (Duke University) | {"url":"https://stats.libretexts.org/Bookshelves/Introductory_Statistics/OpenIntro_Statistics_(Diez_et_al)./03%3A_Distributions_of_Random_Variables/3.04%3A_Binomial_Distribution_(Special_Topic)","timestamp":"2024-11-05T18:27:59Z","content_type":"text/html","content_length":"149898","record_id":"<urn:uuid:c47b295a-fd09-440f-b847-54ce567f6d9e>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00569.warc.gz"} |
This package implements tropical numbers and tropical algebras in Julia. Tropical algebra is also known as the semiring algebra, which is a set $R$ equipped with two binary operations $\oplus$ and $\
otimes$, called addition and multiplication, such that:
• $(R, \oplus)$ is a monoid with identity element called $\mathbb{0}$;
• $(R, \otimes)$ is a monoid with identity element called $\mathbb{1}$;
• Addition is commutative;
• Multiplication by the additive identity $\mathbb{0}$ annihilates ;
• Multiplication left- and right-distributes over addition;
• Explicitly stated, $(R, \oplus)$ is a commutative monoid.
To install this package, press ] in Julia REPL to enter package mode, then type
A Topical algebra can be described as a tuple $(R, \oplus, \otimes, \mathbb{0}, \mathbb{1})$, where $R$ is the set, $\oplus$ and $\otimes$ are the opeartions and $\mathbb{0}$, $\mathbb{1}$ are their
identity element, respectively. In this package, the following tropical algebras are implemented:
• TropicalAndOr: $([T, F], \lor, \land, F, T)$;
• Tropical (TropicalMaxPlus): $(\mathbb{R}, \max, +, -\infty, 0)$;
• TropicalMinPlus: $(\mathbb{R}, \min, +, \infty, 0)$;
• TropicalMaxMul: $(\mathbb{R}^+, \max, \times, 0, 1)$.
julia> using TropicalNumbers
julia> Tropical(3) * Tropical(4)
julia> TropicalMaxMul(3) * TropicalMaxMul(4)
1. TropicalMaxPlus is an alias of Tropical.
2. TropicalMaxMul should not contain negative numbers. However, this package does not check the data validity. Not only for performance reason, but also for future GPU support.
Related packages include
These packages include unnecessary fields in its tropical numbers, such as isinf. However, Inf and -Inf can be used directly for floating point numbers, which is more memory efficient and
computationally cheap. TropicalNumbers is designed for high performance matrix multiplication on both CPU and GPU. | {"url":"https://juliapackages.com/p/tropicalnumbers","timestamp":"2024-11-04T01:51:18Z","content_type":"text/html","content_length":"85684","record_id":"<urn:uuid:44e4ff97-c1ed-4000-a80e-5067b7423882>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00261.warc.gz"} |
SPSS Tutorial 5
SPSS Tutorial 5 - Generating Crosstabs
When we have categorical data it can be interesting to begin the analysis by using crosstabs. For example if we have two variables; gender and attendance. We can assess whether there are differences
between the variables and also whether gender has an impact on the likelihood of attendance.
Cross tabulation is a way to examine the relationship between two variables. In SPSS, you can access cross tabs using the following:
Analyze > Descriptive Statistics > Crosstabs…
In other words, click on Analyze in the menu bar, select Descriptive Statistics, then Crosstabs…
This brings up the crosstab dialog box displayed in Figure 1 below:
Once we have opened the crosstab dialog box find and move the independent and dependent variable to Column(s): and Row(s): boxes, respectively. In this case, the independent variable [gender] goes to
Column(s): box and the dependent variable [attend] goes to Row(s): box. This is displayed in Figure 2:
In the Cell Display window, select the Column in the Percentages section. It allows you to have percentage value separately calculated for each category of the independent variable. Make sure that
Observed was selected in the Counts section although it should be checked by default. This is displayed in Figure 3:
Click on Continue and OK buttons to see the cross tabulation. This generates the output displayed in Figure 4. The output displays two tables, one for case processing summary and the other for cross
tabulation of the two variables. Notice that [sex] is on the column and [attend] is on the row of the cross tabulation as we assigned. | {"url":"http://www.justindoran.ie/spss-tutorial-5.html","timestamp":"2024-11-05T01:18:16Z","content_type":"text/html","content_length":"29033","record_id":"<urn:uuid:2cc67138-5b9e-4f73-8d91-98fee1d2b95a>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00195.warc.gz"} |
Empirical Formula And Molecular Formula Worksheet
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Show Work On A Separate Sheet Of Paper.
Related Post: | {"url":"https://communityconferencing.org/en/empirical-formula-and-molecular-formula-worksheet.html","timestamp":"2024-11-11T15:04:38Z","content_type":"text/html","content_length":"29245","record_id":"<urn:uuid:990fc883-b16c-46f9-9447-063269e5a7a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00686.warc.gz"} |
8.3-8.4 Quiz (Proportionality and Similarity)
Which two triangles are similar?
A and B
B and C
A and C
Find the value for x that makes these
triangles similar
Which theorem can you use to justify that
triangle VXW is similar to triangle ZXY?
Solve for x:
Triangle Angle Bisector theorem
Three parallel lines theorem
Triangle Proportionality theorem
Converse of the Triangle Proportionality theorem
Which theorem would you use to solve this?
Triangle Angle Bisector
Triangle Proportionality
Similar triangles
Converse of Triangle Proportionality
To prove that the line between the triangle is
parallel, you would be applying which theorem?
yes, it is parallel
no, it is not parallel
there is not information
Which construction shows a 2:3 ratio from A to P and A to B?
SU =
What is the measure of segment UZ?
UZ =
Find the measure of x
To solve for x,which theorem did you NOT use?
Triangle Proportionality
Triangle Angle Bisector
Similar triangles
Three Parallel Lines
Reason #3 of the proof would be:
Similar triangles theorem
AA theorem
Triangle proportionality theorem
SSS theorem
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Subsitution system essay
The substitute laws reinforced the perception that the war was "a rich man's war and a poor man's fight." Many soldiers earning scanty military pay simmered with anger over serving with the richly
rewarded substitutes, whom they considered little better than mercenaries.
Substitute Teaching - schools.nyc.gov Substitute Teachers are used by the New York City public schools, on an as-needed basis, to cover the classroom in the absence of the regular (full-time)
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University Instead of Frege's system, we shall present a streamlined system known as first-order logic or the predicate calculus. The predicate calculus dates from the 1910's and 1920's. It is basic
for all subsequent logical research. It is a very general system of logic which accurately expresses a huge variety of assertions and modes of reasoning.
Math · Algebra (all content) · System of equations · Solving systems of equations with substitution Solving linear systems by substitution (old) Solving systems of equations with substitution
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Systems of Equations by Substitution Worksheets
Solve the system of equations: The first equation has a coefficient of 1 on the y, so we'll solve the first equation for y to get. Now we can substitute for y in the equation 2y + 6x = -8: We
simplify to get:-6x – 8 + 6x = -8. Combining the x terms, we get -8 = -8. We know this statement is true, because we just lost $8 the other day, and now we're $8 poorer. Solving Systems of Linear
Equations by Substitution Solving Systems of Linear Equations by Substitution Graphing is a useful tool for solving systems of equations, but it can sometimes be time-consuming. A quicker way to
solve systems is to isolate one variable in one equation, and substitute the resulting expression for that variable in the other equation. Solving linear systems by substitution (old) (video
Calcasieu Parish Public Schools / Homepage
substitution method to solve system of linear equations ...
One Word Substitution list PDF BankExamsToday.Com Page 2 An act of no longer caring for, using, or doing something, failure to do one's job or duty - Dereliction An act of officially charging someone
with a crime - Indictment
EssaysForStudent.com - 89,000+ Free Term Papers and Essays Our communication and information exchange is not just limited to this site. We'd be glad to see you on our social network pages. Find
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Department of Mathematics - Adams, William
From his Obituary
On Thursday, February 15, 2024, William Wells ”Bill” Adams of Silver Spring, Maryland, died peacefully. Beloved husband of Elizabeth Shaw Adams and devoted father of Ruth and Sarah (Dwight Shank)
Adams, loving grandfather of Aaron, Hannah and Dylan, and dear brother of the late Carol Steele. Bill was a first generation college student who received his undergraduate degree from UCLA and his
Ph.D. from Columbia University. He devoted most of his career to teaching and research in Mathematics at the University of Maryland, writing many articles and a few books on Number Theory. Bill was
an avid birder who spent much of his free time looking to the trees in search of ever more elusive birds to add to his extensive life list (2,746 species). Bill shared his birding passion with his
wife, Liz, and daughter, Ruth, many friends, and his grandson, Dylan. When not birding, Bill could be found ushering at many theaters in the area, cheering on his grandchildren in all their
activities, solving Kenken, reading, playing cards, boating in Southold and cheering on the Washington Nationals and Maryland Terps.
Report on the Occasion of his Retirement Mathematical Research
William Adams received his Ph.D. from Columbia University in 1964 under Serge Lang. He then had positions at Berkeley, UCLA, and the Institute for Advanced Study. In 1966, he was hired as Associate
Professor by Maryland and he was promoted to Professor in 1971.
The most significant results of the first twenty years of Bill’s career were in the areas of Diophantine approximation and transcendence theory. The main purpose of Diophantine approximation is to
study the approximation of real numbers by rational numbers. An easy application of the box principle, or of continued fractions, shows that, for a given real number α, there are infinitely many
pairs of integers p, q such that |qα − p| < 1/q. For all α outside a set of measure zero, the number of solutions of this inequality with 1 ≤ q ≤ x is asymptotic to 2 log x as x → ∞. One of Bill’s
early, striking results is that the number of solutions to |qe − p| < 1/q with |q| ≤ x is asymptotic to a constant times (log x/ log log x)^3/2 as x → ∞. This shows that e does not behave like most
real numbers in this regard.
It is well-known that if α is a quadratic irrational, then |qα − p| < 1/(q√5) has infinitely many solutions, and that 1/√5 is the best constant that works for all α. Now suppose that K ⊂ R is a cubic
extension of Q and let {1, β[1], β[2]} be a basis of K as a Q-vector space. Let c[0] > 0 be the infimum of constants c such that
|qβ[1] − p[1]| < (c/q)1/2, |qβ[2] − p[2]| < (c/q)1/2
has infinitely many solutions in integers p[1], p[2], q, and let C[0] be the supremum of the c[0] as K and β[1], β[2] vary. Cassels and Davenport showed that 2/7 ≤ C[0] ≤ 46−1/4(more generally, for
any β[1], β[2] ∈ R such that {1, β[1], β[2]} is linearly independent over Q). In a series of papers, Bill completely settled the question in the cubic case by showing that in fact C[0] = 2/7.
In the area of transcendence theory, Bill developed a p-adic analogue of Gel'fond and Schneider’s theory of transcendence and algebraic differential equations. This gave new proofs of the
transcendency of the p-adic numbers eα and αβ, where α and β are p-adic numbers algebraic over Q (satisfying some standard conditions), and also gave the first p-adic transcendence measures for such
numbers. He also proved that if β has degree r ≥ 4 over Q, then the transcendence degree over Q of Q(αβ, αβ2, . . . , αβr−1) is at least 2.
In the late 1980s, Bill started working in computational ring theory, in partic ular the theory of Gröbner bases. Recall that a polynomial ring in one variable over of field is a principal ideal
domain and the generator of an ideal can be found from a set of generators, essentially by the Euclidean algorithm. This is no longer possible for polynomials in several variables. The Gröbner basis
algorithm, which is in some ways a cross between the Euclidean algorithm and the Gaussian reduction algorithm from linear algebra, is a replacement that finds generators of ideals in rings of
polynomials in several variables. The method, which found early success in Hironaka’s work on resolution of singularities, is now indispensable in many computer algebra calculations. One of Bill’s
most important contributions to the subject is his book An Introduction to Gröbner Bases, written jointly with Philippe Loustaunau. This is a well-written introduction to the subject and is described
by Math. Reviews as “excellent” (I have read the book and heartily agree). Bill has written many papers on Gröbner bases and their applications. As an example of how wide reaching the techniques are,
consider the following result from a paper written by Adams jointly with Berenstein, Loustaunau, Sabadini, and Struppa. Let K be a compact subset of n-dimensional quaternionic space H^n, with n > 1,
such that H^n \K is connected. If f is a regular function on H^n \K, then f extends to an entire function. This is of course the analogue of Hartogs’ theorem in several complex variables. It was
proved by Pertici, but the proof in the present paper is almost purely algebraic.
They consider the Cauchy–Fueter complex of differential operators whose solution sheaf is the sheaf of regular functions of several quaternionic variables, and they study a free resolution of this
complex. Gröbner basis techniques allow them to prove the vanishing of some of its Ext-modules and explicitly calculate the degrees of syzygies. It is quite surprising that these techniques work even
in this non-commutative setting. This and related papers are featured in the book Analysis of Dirac Systems and Computational Algebra by Colombo, Sabadini, Sommen, and Struppa (Birkhäuser, 2004).
Teaching Accomplishments
Bill had a distinguished career in teaching. On the graduate level, he directed 13 Ph.D. theses and 4 M.A. theses and has taught numerous graduate courses. On the undergraduate level, he excelled in
both large lectures and small sections, and he was the advisor for many undergraduates. He gave several talks to high school students, including one to the US Mathematical Olympiad Team. In the
summer of 1990, he and L. Washington co-mentored three students from the Research Science Institute (including future Fields Medalist Terry Tao).
Three times, for a total of 8 years, Bill served as Associate Chair for Un dergraduate Studies. He also served on numerous curriculum development and review committees.
Bill had an impressive service record at both the national and the local levels. He served for two and one half years as the Program Director for Algebra/Number Theory for the National Science
Foundation. He was the Editor for Number Theory for the Proceedings of the American Math. Society for 9 years, and was Associate Editor of the Journal of Symbolic Computation for 7 years.
In the early 1990’s, Bill was Project Director for the JPBM Committee on Professional Recognition and Rewards. This was a major committee sponsored by the AMS, MAA, and SIAM, and Bill personally
conducted 19 site visits at various universities during his directorship.
At the university level, Bill served on the Faculty Senate and many com mittees for the Faculty Senate, including chair of the Senate Faculty Affairs Committee, several program review committees, and
several search commit tees. At the department level, in addition to being Undergraduate Chair, he served on numerous committees.
Bill helped design and was the main organizer of the new Developmental Mathematics Program in the mathematics department. This very successful program was featured as the cover story in the December
2003 issue of Focus, a publication of the MAA.Finally, Bill deserves much credit for building the number theory group at the University of Maryland. He arrived at Maryland at the same time as
L. Goldstein and H. Jacquet. In the next few years, G. Cooke, T. Kubota, D. Garbanati, M. Razar, and S. Kudla were hired. Bill was one of the main organizers of the Special Year in Number Theory in
1977-1978, which featured talks by many of the top number theorists in the world and which resulted in the hiring of L. Washington and D. Zagier. Bill’s efforts over the years greatly enhanced the
university’s international reputation.
Bill played an unintended role in the history of the University. He openly opposed an action of the administration in response to some campus protests in the late 1960’s. In retaliation, the
administration turned down his promotion to full professor. The chair of the math department resigned in protest, and Kirwan (for whom our building is named) became acting chair, starting him on the
path that led to becoming chancellor of the Maryland system. During the controversy, one of the regents told Bill Adams, ”You’re too idealistic to work in a university.” No comment, except that Bill
was very idealistic and principled, and a wonderful person. | {"url":"https://www-math.umd.edu/people/in-memoriam/item/229-wwa.html","timestamp":"2024-11-11T03:30:07Z","content_type":"text/html","content_length":"182321","record_id":"<urn:uuid:fc8196ad-2136-4a88-a979-d9e1e38602c7>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00505.warc.gz"} |
More quantum gates and lattices
The previous post ended with unanswered questions about describing the Conway group, Co[0], in terms of quantum gates with dyadic rational coefficients. It turned out to be easier than expected,
although the construction is much more complicated than the counterpart in the previous post.
We’ll start with a construction for a much smaller simple group — the order-168 group PSL(2, 7) — which can be generated by three CNOT gates acting on the three lower qubits in the diagram below:
For the moment, ignore the two unused upper qubits; their purpose will become clear soon enough. The three lower qubits have eight computational basis states (000, 001, 010, 011, 100, 101, 110, and
111). Viewing each of these states as a three-dimensional vector over $\mathbb{F}_2$, the CNOT gates induce linear shearing maps (and together generate the full 168-element group of linear
Now we introduce another two CNOT gates:
These act only on the two upper qubits, and generate the symmetric group $S_3$ (freely permuting the states 01, 10, and 11). These two gates, together with the three gates in the previous diagram,
therefore generate the direct product $S_3 \times PSL(2, 7)$ of order 1008.
It is helpful to partition the 32 computational basis states into four 8-element sets:
• the set W consisting of all basis states of the form 00xyz;
• the set A consisting of all basis states of the form 01xyz;
• the set B consisting of all basis states of the form 10xyz;
• the set C consisting of all basis states of the form 11xyz;
where x, y, z are elements of {0, 1}. Then the two gates on the upper qubits are responsible for bodily permuting the three sets {A, B, C}; the three gates on the lower qubits induce the same linear
permutation in each of W, A, B, and C (viewed as 3-dimensional vector spaces over the field of two elements).
Note that the permutation group is transitive on the 8-element set W, and transitive on the 24-element set V (the complement of W, or equivalently the union of A, B, and C), but no elements of V are
ever interchanged with elements of W. This will remain the case as we continue to add further gates.
For the time being, we’ll therefore suspend thinking about the smaller 8-element set W, and concentrate on the larger 24-element set V.
We now introduce a sixth CNOT gate, bridging the upper and lower qubits:
This now expands the size of the group by a factor of 64, resulting in a 64512-element group called the trio group. The vector spaces A, B, and C are effectively upgraded into affine spaces which can
be semi-independently translated (the constraint is that the images of their ‘origins’ — 01000, 10000, and 11000 — always have a modulo-2 sum of zero).
The trio group, considered as a permutation group on the 24 basis states in V, is a maximal subgroup of the simple sporadic group M[24]. That means that adding a single further gate to break out of
the trio group will immediately upgrade us to the 244823040-element Mathieu group! Unfortunately, I wasn’t able to find a particularly simple choice of gate:
The effect of this complicated gate is to do the following:
• Within each of the sets W and A, apply a particular non-affine permutation which exchanges the eight elements by treating them as four ‘pairs’ and swapping each element with its partner:
□ 000 ⇔ 100;
□ 001 ⇔ 110;
□ 010 ⇔ 111;
□ 011 ⇔ 101;
• Swap four elements of B with four elements of C by means of the Fredkin gate on the far right.
In terms of its action on V, this is an element of M[24], but does not belong to the trio group. Interestingly, it belongs to many of the other important subgroups of M[24] — namely PSL(3, 4) (also
called M[21]), the ‘sextet group’, and the ‘octad group’. At this point, the group generated by these seven gates is now the direct product of the alternating group A[8] (acting on the set W) and the
Mathieu group M[24] (acting on the set V).
The elements of the Mathieu group are permutations on the set V, which can be viewed as 24-by-24 permutation matrices. Permutation matrices are matrices with entries in {0, 1}, where there’s exactly
one ‘1’ in each row and column. If we throw in a Z-gate acting on the uppermost qubit, we’ll expand this to a group 2^12:M[24] of ‘signed permutation matrices’ instead, where some of the ‘1’s are
replaced with ‘−1’s. (The sets of rows and columns containing negative entries must form codewords of the binary Golay code; this is why each permutation matrix has only 2^12 sign choices instead of
2^24.) This group is interesting inasmuch as it’s the group of permutations and bit-flips which preserve the binary Golay code. It’s also a maximal subgroup, called the monomial subgroup, of the
Conway group Co[0].
Instead of using this Z-gate, we’ll bypass the monomial subgroup and jump straight to the Conway group by adding a single additional three-qubit gate:
By a slight abuse of notation, we’ll reuse the symbols W and V to refer to the vector spaces spanned by first 8 basis states and final 24 basis states, respectively. After introducing this final
gate, the resulting group of 32-by-32 matrices is a direct product of:
• an order-348364800 group (the orientation-preserving index-2 subgroup of the Weyl group of E[8]) acting on the 8-dimensional vector space W;
• an order-8315553613086720000 group (the Conway group, Co[0]) acting on the 24-dimensional vector space V.
These are the groups of rotational symmetries of the two most remarkable* Euclidean lattices — the E[8] lattice and the Leech lattice, respectively. Indeed, if we take all linear combinations (with
integer coefficients!) of the images (under the group we’ve constructed) of a particular computational basis state, then we recover either the E[8] lattice or the Leech lattice (depending on whether
we used one of the basis states in W or in V).
* as well as being highly symmetric, they give the optimal packings of unit spheres in 8- and 24-dimensional space, as proved by Maryna Viazovska.
These two groups each have a centre of order 2 (consisting of the identity matrix and its negation), modulo which they’re simple groups:
• the quotient of the order-348364800 group by its centre is PSΩ(8, 2);
• the quotient of Co[0] by its centre is the sporadic simple group Co[1].
This process of quotienting by the centre is especially natural in this quantum gate formulation, as scalar multiples of a vector correspond to the same quantum state.
Further connections
If we take $n \geq 4$ qubits and the gates mentioned in the previous post, we remarked that the group generated is the full rotation group $SO(2^n)$. If instead we replace the Toffoli gate in our
arsenal with a CNOT gate, we get exactly the symmetry groups of the Barnes-Wall lattices! The orders of the groups are enumerated in sequence A014115 of the OEIS.
Thanks go to Conway and Sloane for their magnificent and thoroughly illuminating book, Sphere Packings, Lattices, and Groups. Also, the website jspaint.app (which is undoubtedly the most useful thing
to have ever been written in JavaScript) was helpful for creating the illustrations herein.
2 Responses to More quantum gates and lattices
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Measurement of the branching fractions for B+ -> D*+ D- K+, B+ -> D*- D+ K+, and B0 -> D*- D0 K+ decays
Charmed hadron spectroscopy is a quickly progressing field that has enjoyed a growing interest in recent years from both theorist and experimentalists. Open-charmed hadrons consist of a heavy $c$
quark bound to a light quark ($u$, $d$, $s$ quarks). These particles are in particular spin-parity states making for a very interesting quantum mechanical system allowing the study of the QCD
potential holding them together. Decays of type $B\rightarrow D^{}DK$ offer a rich environment to study charmed spectroscopy thanks to their abundance and low background. Using $pp$ collision data at
7, 8, and 13 TeV center-of-mass energies corresponding to an integrated luminosity of $9.1\mathrm{fb}^{-1}$ collected with the LHCb detector at CERN's Large Hadron Collider, this thesis presents the
most accurate measurement to-date of four branching fraction ratios of $B\rightarrow D^{}DK$ modes. These are measured to be
\frac{\mathcal{B} (B^+ \rightarrow D^{+}D^-K^+)}{\mathcal{B} (B^+ \rightarrow \overline{D}^{0}D^0K^+)} &= 0.590 \pm 0.017 (\mathrm{stat}) \pm 0.008 (\mathrm{syst}) \pm 0.013 (\mathcal{B}(D)) , \
\frac{\mathcal{B} (B^+ \rightarrow D^{-}D^+K^+)}{\mathcal{B} (B^+ \rightarrow \overline{D}^{0}D^0K^+)} &= 0.524 \pm 0.015 (\mathrm{stat}) \pm 0.007 (\mathrm{syst}) \pm 0.012 (\mathcal{B}(D)) , \
\frac{\mathcal{B} (B^0 \rightarrow D^{-}D^0K^+)}{\mathcal{B} (B^0 \rightarrow D^{-}D^0K^+} &= 1.767 \pm 0.028 (\mathrm{stat}) \pm 0.018 (\mathrm{syst}) \pm 0.036 (\mathcal{B}(D)) , \[10pt]
\frac{\mathcal{B} (B^+ \rightarrow D^{+}D^-K^+)}{\mathcal{B} (B^+ \rightarrow D^{-}D^+K^+)} &= 1.098 \pm 0.040 (\mathrm{stat}) \pm 0.017 (\mathrm{syst}),
where the first uncertainty is statistical, the second systematic, and the third is due to the uncertainties on the $D$ meson decay branching fractions measured by other experiments. Furthermore,
this thesis shows the Dalitz plots of the final states for these decay modes using data samples an order of magnitude larger than samples collected by other experiments. These measurements are an
important step towards a full amplitude analysis of $B\rightarrow D^{*}DK$ decays.
In addition, this thesis summarizes the installation and studies of the BCAM system used to monitor the position of the Inner Tracker detector of LHCb.
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I need help ith my homework, someone please? The first answer will be voted as the best for some points.=]?
A few days ago
I need help ith my homework, someone please? The first answer will be voted as the best for some points.=]?
Solve using the quadratic formula.
Top 1 Answers
A few days ago
Favorite Answer
first put the equations into standard form…
2x^2+3x=4 >>>>>>> 2x^2+3x-4=0
therfore a=2 , b=3, c=-4
3x^2=x-7>>>>>>>>> 3x^2-x+7=0
therefore a=3, b= -1, c=7
now use the above info for each question in the quadratic formula….. | {"url":"https://essaywriter.nyc/education-questions/homework-help/i-need-help-ith-my-homework-someone-please-the/","timestamp":"2024-11-05T04:08:05Z","content_type":"text/html","content_length":"119373","record_id":"<urn:uuid:cb496e75-d4c9-448d-a800-56678d330395>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00872.warc.gz"} |
Si/SiGe MODQW
nextnano^3 - Tutorial
next generation 3D nano device simulator
1D Tutorial
Si/SiGe MODQW (Modulation Doped Quantum Well)
Authors: Stefan Birner
==> 1DSiGe_Si_Schaeffler_SemicondSciTechnol1997_nn3.in - input file for the nextnano^3 software
==> 1DSiGe_Si_Schaeffler_SemicondSciTechnol1997_nnp.in - input file for the nextnano++ software
These input files are included in the latest version.
Si/SiGe MODQW (Modulation Doped Quantum Well)
This tutorial aims to reproduce Fig. 11 of
F. Schäffler
High-Mobility Si and Ge structures
Semiconductor Science and Technology 12, 1515 (1997)
Step 1: Layer sequence
width [nm] material strain doping
1 Schottky barrier 0.8 eV
2 15.0 Si cap strained w.r.t. Si[0.75]Ge[0.25]
3 22.5 Si[0.75]Ge[0.25] layer
4 15.0 Si[0.75]Ge[0.25] doping layer 2 x 10^18 cm^-3 (fully ionized)
5 10.0 Si[0.75]Ge[0.25] barrier (spacer)
6 18.0 Si channel strained w.r.t. Si[0.75]Ge[0.25]
7 69.5 Si[0.75]Ge[0.25] buffer layer
Step 2: Material parameters
The material parameters were taken from:
F. Schäffler
High-Mobility Si and Ge structures
Semiconductor Science and Technology 12, 1515 (1997)
The temperature was set to 0.1 Kelvin.
The Si layers are strained pseudomorphically with respect to a Si[0.75]Ge[0.25] substrate (buffer layer).
Step 3: Method
Self-consistent solution of the Schrödinger-Poisson equation within single-band effective-mass approximation (using ellipsoidal effective mass tensors) for both Delta conduction band edges.
Step 4: Results
• The following figure shows the self-consistently calculated conduction band profile and the lowest wave functions of an n-type Si/Si[0.75]Ge[0.25] modulation doped quantum well (MODQW) grown on a
relaxed Si[0.75]Ge[0.25] buffer layer.
The strain lifts the sixfold degeneracy of the lowest conduction band (Delta[6]) and leads to a splitting into a twofold (Delta[2]) and a fourfold (Delta[4]) degenerate conduction band edge.
• The following figure shows the lowest three wave functions (psi²) of the structure. Two eigenstates that have very similar energies and are occupied (i.e. they are below the Fermi level) whereas
the third eigenstate is not occupied at 0.1 K.
• The electron density (in units of 1 x 10^18 cm^-3) is plotted in this figure. The lowest states in each channel are occupied, i.e. are below the Fermi level.
The integrated electron densities are:
- in the parasiticSi[0.75]Ge[0.25] channel: 0.75 x 10^12 cm^-2
- in the strained Si channel: 0.66 x 10^12 cm^-2 | {"url":"https://www.nextnano.de/nextnano3/tutorial/1Dtutorial_SiSiGeMODQW.htm","timestamp":"2024-11-08T11:29:20Z","content_type":"text/html","content_length":"10048","record_id":"<urn:uuid:f6805fc1-7cff-40a7-8274-4846369a27a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00347.warc.gz"} |
CAT 2014 : Concepts, Shortcuts and More!
Today we look at a few questions that appeared in XAT 2014 exam. Note that XAT 2014 questions are also a good set of practice questions for CAT 2014.
The questions below have the question followed by a Video Solution!
1.Evaluate the following:
The solution to this problem is explained in the Video below:
2. Two numbers, 297
and 792
, belong to base B number system. If the first number is a factor of the second number then the value of B is
3. Aditya has a total of 18 red and blue marbles in two bags (each bag has marbles of both colors). A marble is randomly drawn from the first bag followed by another randomly drawn from the second
bag, the probability of both being red is 5/16. What is the probability of both marbles being blue?
4. The probability that a randomly chosen positive divisor of 10
is an integer multiple of 10
is a
, then 'b - a' would be:
5. The number of the possible values of X in the equation |X + 7| + |X - 8| = 16 is
6. A polynomial ax
+ bx
+ cx + d intersects x-axis at 1 and -1, and y-axis at 2. The value of b is:
Cannot be Determined
7. S = ax/(b+c+x) where all parameters are positive. If x is increased and all other terms are kept constant, S will
Increases and then decreases
Decreases and then increases
Cannot be determined
8. x, 17, 3x - y
- 2, and 3x + y
- 30, are four consecutive terms of an increasing arithmetic sequence. The sum of the four number is divisible by:
In our future posts, we will be adding more questions from XAT 2014, which should be also be a very good practice set for CAT 2014.
To register for Oliveboard's XAT course,
In our previous blog post, we had picked 3 questions from previous CAT paper. Today we will look at solving these questions. In the process, we will be doing a deeper dive into a couple of important
Question 1: If x = -0.5, which of the following has the smallest value?
2^1/x , 1/x, 1/x^2, 2^x, 1/√-x
Solution: In solving this problem, we will first need to understand a couple of important concepts on Powers of numbers for CAT 2014. The video below explains all the concepts and basics related to
Now that we have the concept clear, we can look at the solution to this particular problem.
Question 2: Which among the following is the largest?
2^1/2, 3^1/3, 4^1/4, 6^1/6, 12^1/12
Solution: The solution relies on another basic concept of Number systems to compare different numbers with different exponents . The concept and the solution are given below.
Question 3: How may 2 digit numbers increase by 18 when the digits are reversed?
Solution: Note that 2 digit numbers can be expressed as ab, where a,b are digits. So the original number is 10a + b, reversed number 10b + a.
The complete solution along with a few more basics for CAT 2014 can be viewed below.
We will be returning with our next post in a couple of days time! Happy Learning!
In Today's post, we try to look at 3 problems that have appeared in past CAT papers, and provide clues to solve them.
The complete solution will be available in our next post.
1. If x = -0.5, which of the following has the smallest value?
2^1/x , 1/x, 1/x^2, 2^x, 1/√-x
2. Which among the following is the largest?
2^1/2, 3^1/3, 4^1/4, 6^1/6, 12^1/12
3. How many 2 digit numbers increase by 18 when the digits are reversed?
1. A positive number raised to any real number will always be positive. For eg, 4^-1/10 will be greater than 0.
2. Convert all numbers to have the same exponent. Then compare the numbers.
3. Let the number have digits a,b. Original number is 10a+b, reversed is 10b+a. Take the difference and try to solve it :)
For comprehensive CAT Prep, click here.
In the previous post, we asked a few questions to help us understand the various concepts in number systems by examples.
Here are Video Solutions to all these problems :).
1. What happens when you multiply 3 even numbers? 2 even numbers and an odd number? 2 odd numbers and an even number? and 3 odd numbers?
2. What happens when you multiply 'N' even numbers? and 'N' odd numbers
3. Can product of 4 consecutive numbers be Odd?
The solution to all these 3 problems are explained in the video below.
We then asked the question: Whats the smallest number should be added to 156789 to make it divisible by 11?
To solve this problem, we need to understand divisibility tests by 11. This and the solution to the problem is explained in the video below.
Our Next question was: Whats the smallest number that should be added to 677 to make it divisible by 4, 5, and 11?
To solve this, its important to understand concepts of LCM. The question is solved with concepts below.
Our Next question was related to power cycles: What is the units digit of 729^99?
Detailed solution with basic power cycles concept is below.
Our Last question was in advanced Power cycle: What will be the last 2 digits when 25^625 is multiplied by 3^75?
Concept explanation with detailed solution is given below.
Do let us know what you think about these solutions.
We will be back with more a more detailed Number Systems Lesson in the next week.
In other news: TCS will be the official test partner for CAT the next 5 years. This was announced a few days back.
Number systems is an important topic for CAT and all other MBA Exams. The various topics under Number systems include :
1. Classification of Numbers : Natural numbers, Whole numbers, Fractions, Real Numbers, Complex numbers etc.
2. Rules on Number Operations : For eg, sum of 2 odd numbers is even etc.
3. Divisibility Rules.
4. Properties of Prime and Composite numbers, factorisation.
5. Power Cycle of Numbers
Today we will start by asking questions for you to ponder over, and answer each of them in the next set of blog posts with detailed video concepts.
1. What happens when you multiply 3 even numbers? 2 even numbers and an odd number? 2 odd numbers and an even number? and 3 odd numbers?
2. What happens when you multiply 'N' even numbers? and 'N' odd numbers
3. Can product of 4 consecutive numbers be Odd?
4. Whats the smallest number should be added to 156789 to make it divisible by 11?
5. Whats the smallest number that should be added to 677 to make it divisible by 4, 5, and 11?
6. What is the units digit of 729^99?
7. What will be the last 2 digits when 25^625 is multiplied by 3^75?
Through these problems, we hope to cover the complete Number systems concepts, and help you prepare better for CAT and other leading MBA Exams.
With less than 6 months to go for CAT, Oliveboard will be starting its concept blog today.
What to Expect:
1. Each topic with basic concepts.
2. Worked out examples for these topics.
3. Exercises
4. Video solutions to selected questions.
We are sure this will help you ace CAT 2014 and other MBA exams!
Register at Oliveboard for more.
Have a doubt? Leave a comment and we will get back to you at the earliest. Or you can email us at : support [at] oliveboard [dot] in | {"url":"http://concepts.oliveboard.in/search?updated-max=2014-06-02T02:33:00-07:00&max-results=7","timestamp":"2024-11-13T02:39:37Z","content_type":"application/xhtml+xml","content_length":"65928","record_id":"<urn:uuid:0598fb84-eebf-40a9-8c57-1d831e8c1ed9>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00603.warc.gz"} |
How to write multithreading matrix multiplication
Hello! I’m studying parallel programming in Julia, so i decided to write some basics matrix multiplication without using Linear Algebra package, because it’s already multi-threaded.
using Base.Threads
using BenchmarkTools
function multiplyMatrices_oneThreadLoop(A::Matrix{Float64}, B::Matrix{Float64}, N::Int64)
C = zeros(N, N)
Threads.@threads for i in 1:N
for j in 1:N
for k in 1:N
C[i, j]= A[i, k] * B[k, j]
return C
function multiplyMatrices_spawnExample(A::Matrix{Float64}, B::Matrix{Float64}, N::Int64)
C = zeros(N, N)
@sync Threads.@spawn for i in 1:N
for j in 1:N
for k in 1:N
C[i, j]= A[i, k] * B[k, j]
return C
function multiplyMatrices_default(A::Matrix{Float64}, B::Matrix{Float64}, N::Int64)
C = zeros(N,N)
for i in 1:N
for j in 1:N
for k in 1:N
C[i, j]= A[i, k] * B[k, j]
return C
N = 5000
A = rand(N, N);
B = rand(N, N);
println("multi-threaded loop 1st run")
@btime multiplyMatrices_oneThreadLoop(A, B, N)
println("using sync spawn 1st run")
@btime multiplyMatrices_spawnExample(A,B,N)
println("default multiplication 1st run")
@btime multiplyMatrices_default(A, B, N)
println("multi-threaded loop 2nd run")
@btime multiplyMatrices_oneThreadLoop(A, B, N)
println("using sync spawn 2nd run")
@btime multiplyMatrices_spawnExample(A,B,N)
println("default multiplication 2nd run")
@btime multiplyMatrices_default(A, B, N)
println("multi-threaded loop 3rd run")
@btime multiplyMatrices_oneThreadLoop(A, B, N)
println("using sync spawn 3rd run")
@btime multiplyMatrices_spawnExample(A,B,N)
println("default multiplication 3rd run")
@btime multiplyMatrices_default(A, B, N)
when i run this code with julia -t 8, i saw that
1. Performance of thread.@spawn function is slower than default multiplication
2. All functions have higher performance on first run than on repeated runs, why it happens?
Where i did mistakes and how to fix them?
1 Like
one mistake is to call @btime multiplyMatrices_oneThreadLoop(A, B, N) instead of @btime multiplyMatrices_oneThreadLoop($A, $B, $N). This avoids issues with global variables.
Further, you do assignments instead of += in the loop. So your code did not calculate a matmul.
This way of implementing a matmul is also the obvious one but unfortunately quite slow.
See also this comment:
This version of the code gives me the desired results, second runs are not faster or slower. I reduced N to have more reasonable runtimes.
using Base.Threads
using BenchmarkTools
function multiplyMatrices_oneThreadLoop(A::Matrix{Float64}, B::Matrix{Float64}, N::Int64)
C = zeros(N, N)
Threads.@threads for i in 1:N
for j in 1:N
for k in 1:N
C[i, j] += A[i, k] * B[k, j]
return C
function multiplyMatrices_spawnExample(A::Matrix{Float64}, B::Matrix{Float64}, N::Int64)
C = zeros(N, N)
@sync Threads.@spawn for i in 1:N
for j in 1:N
for k in 1:N
C[i, j] += A[i, k] * B[k, j]
return C
function multiplyMatrices_default(A::Matrix{Float64}, B::Matrix{Float64}, N::Int64)
C = zeros(N,N)
for i in 1:N
for j in 1:N
for k in 1:N
C[i, j] += A[i, k] * B[k, j]
return C
N = 100
A = rand(N, N);
B = rand(N, N);
println("multi-threaded loop 1st run")
@btime multiplyMatrices_oneThreadLoop(A, B, N)
println("using sync spawn 1st run")
@btime multiplyMatrices_spawnExample($A,$B,$N)
println("default multiplication 1st run")
@btime multiplyMatrices_default($A, $B, $N)
println("multi-threaded loop 2nd run")
@btime multiplyMatrices_oneThreadLoop($A, $B, $N)
println("using sync spawn 2nd run")
@btime multiplyMatrices_spawnExample($A,$B,$N)
println("default multiplication 2nd run")
@btime multiplyMatrices_default($A, $B, $N)
println("multi-threaded loop 3rd run")
@btime multiplyMatrices_oneThreadLoop($A, $B, $N)
println("using sync spawn 3rd run")
@btime multiplyMatrices_spawnExample($A,$B,$N)
println("default multiplication 3rd run")
@btime multiplyMatrices_default($A, $B, $N)
2 Likes
It’s extremely un-idiomatic and error prone to pass N as a variable. You should use something like
M,K = size(A)
KB,N = size(B)
K == KB || throw(DimensionMismatch("matrices have incompatible dimensions"))
C = zeros(M, N)
which also supports rectangular matrices. Or, if you only want the square case, insist that all the sizes are the same. Once you know that your arrays are all of the proper sizes, adding @inbounds
may also improve results.
Before multithreading, it’s important to have a good (and correct – be sure to fix the aforementioned += problem) single-threaded implementation. Your initial implementation suffers tremendously from
memory locality issues. Most of your time (single or multithreaded) is spent waiting on your RAM rather than doing useful work. Even just making your loops in order of j, k, i (outermost to
innermost) would probably be a big improvement.
But even with that, this code will still be limited by your memory’s speed. Writing a high-performance matrix-matrix multiply is not trivial. A good introduction to modern implementations is here. A
key trick is to compute the multiplication over small blocks (e.g., 4x4 in the linked example) to better utilize the cache. If you can break through the initial memory bottleneck, you’ll also see
better results from C[i,j] = muladd(A[i,k], B[k,j], C[i,j]) which can combine the * and + into a single operation on most computers.
Once you have a good serial calculation, you can try to parallelize to get better performance.
If your goal is merely to explore parallelism (not matrix multiplication in particular), I’d suggest you try a different application like sorting. There’s a great example of multithreaded sorting in
Julia in this blog post.
5 Likes
First I thought the same but in fact it’s not the case, threading with 24 cores gives a 10x speedup. Ofc, it still can be optimized much more compared to Julia’s *. But still, the speed-up is there:
julia> using Chairmarks
julia> function matmul(A, B)
C = similar(A, size(A, 1), size(B, 2))
fill!(C, 0)
for i in 1:size(A, 1)
for j in 1:size(B, 2)
for k in 1:size(A, 2)
C[i, j] += A[i, k] * B[k, j]
return C
matmul (generic function with 1 method)
julia> function matmul_threaded(A, B)
C = similar(A, size(A, 1), size(B, 2))
fill!(C, 0)
Threads.@threads for i in 1:size(A, 1)
for j in 1:size(B, 2)
for k in 1:size(A, 2)
C[i, j] += A[i, k] * B[k, j]
return C
matmul_threaded (generic function with 1 method)
julia> A = rand(1000, 700);
julia> B = rand(700, 1200);
julia> A * B ≈ matmul(A, B) ≈ matmul_threaded(A, B)
julia> @b matmul($A, $B)
615.996 ms (2 allocs: 9.155 MiB, without a warmup)
julia> @b matmul_threaded($A, $B)
45.856 ms (123 allocs: 9.168 MiB)
julia> @b $A * $B
4.209 ms (2 allocs: 9.155 MiB)
julia> Threads.nthreads()
julia> A = rand(2000, 3000);
julia> B = rand(3000, 4000);
julia> A * B ≈ matmul(A, B) ≈ matmul_threaded(A, B)
julia> @b matmul($A, $B)
43.364 s (2 allocs: 61.035 MiB, 0.08% gc time, without a warmup)
julia> @b matmul_threaded($A, $B)
3.911 s (123 allocs: 61.048 MiB, without a warmup)
julia> @b $A * $B
111.380 ms (2 allocs: 61.035 MiB)
1 Like
You should use @inbounds | {"url":"https://discourse.julialang.org/t/how-to-write-multithreading-matrix-multiplication/121865","timestamp":"2024-11-04T01:52:28Z","content_type":"text/html","content_length":"36261","record_id":"<urn:uuid:f492a2f0-f05d-4c4e-8176-94108ae41ff2>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00485.warc.gz"} |
neverhpfilter Package
In the working paper titled “Why You Should Never Use the Hodrick-Prescott Filter”, James D. Hamilton proposes a new alternative to economic time series filtering. The neverhpfilter package provides
functions and data for reproducing his solution. Hamilton (2017) <doi:10.3386/w23429>
Hamilton’s abstract offers an excellent introduction:
1. The HP filter produces series with spurious dynamic relations that have no basis in the underlying data-generating process. (2) Filtered values at the end of the sample are very different
from those in the middle, and are also characterized by spurious dynamics. (3) A statistical formalization of the problem typically produces values for the smoothing parameter vastly at odds
with common practice, e.g., a value for \(\lambda\) far below 1600 for quarterly data. (4) There’s a better alternative. A regression of the variable at date \(t + h\) on the four most recent
values as of date \(t\) offers a robust approach to detrending that achieves all the objectives sought by users of the HP filter with none of its drawbacks.
Getting Started
Install from CRAN on R version >= 3.5.0.
Or install from the Github master branch on R version >= 3.5.0.
Load the package
Package Documentation
The package consists of 2 estimation functions, 12 economic xts objects, an xts object containing Robert Shiller’s U.S. Stock Markets and CAPE Ratio data from 1871 to Present, and a data.frame
containing the original filter estimates found on table 2 of Hamilton (2017) <doi:10.3386/w23429>
Documentation for each can be found here:
Finally, a vignette recreating the estimates of the original work can be found in Reproducing Hamilton. | {"url":"https://cran.ma.ic.ac.uk/web/packages/neverhpfilter/readme/README.html","timestamp":"2024-11-03T12:24:29Z","content_type":"application/xhtml+xml","content_length":"6559","record_id":"<urn:uuid:f924543c-7822-40bc-b472-8b16c5762afc>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00819.warc.gz"} |
(zk-learning) elaboration on the quadratic residue ZKP
I just finished the first lecture of the Zero Knowledge Proofs MOOC titled Introduction and History of ZKP. The lecture described an interactive zero knowledge protocol to prove a number is a
quadratic residue $mod > N$. In other words, a prover $P$ wants to convince a verifier $V$ that it knows $x$ such that $y = x^2 > mod > N$, without revealing $x$ to $V$. The variables $(y, N)$ are
known to $P$ and $V$, but $x$ is only known to $P$.
The lecture did an excellent job explaining the protocol, but some parts weren’t explained explicitly. That was for brevity’s sake. Some parts are left for the viewers to think through themselves,
and that’s a good thing. It allows more content to be packed into the lecture and forces the viewers to think more deeply about the subject.
In this article I’ll give a general description of zero knowledge interactive protocols that “clicked” for me, and then apply it to the quadratic residue example. I’ll make some facts about the
protocol explicit that were left implicit in the lecture - only after I identified those implicit facts and made them explicit could I say with confidence that I understood why the protocol works.
And I hope by the end of the article you’ll also better understand how that the protocol works.
If you’re reading this then I assume you’ve seen the first 30 minutes of the lecture (up to and including the Second example section).
General Description
Here’s my general description that I’m going to apply to the quadratic residue protocol:
In each iteration of the protocol, verifier $V$ asks the prover $P$ to perform one of two tasks $A$ or $B$, but ahead of time $P$ does not know which task it will be. $P$ and $V$ have agreed on
these tasks before interaction starts.
The tasks are chosen such that if the claim is true (i.e. $P$ knows the secret), then $P$ can always perform both $A$ and $B$ correctly. But if the claim is false (i.e. $P$ doesn’t know the
secret), then at best $P$ can only “fake” one of the tasks at the risk of not being able to perform the other.
If $V$ picks the task at random, then a $P$ who doesn’t know the secret has a 50% chance of faking the wrong task, and therefore being unable to perform the task chosen by $V$. After successive
iterations the likelihood that $P$ doesn’t know the secret and correctly guesses which task to “fake” becomes increasingly small. After $n$ iterations, the likelihood that a $P$ who doesn’t know
the secret guessed was able to guess the right task to fake $n$ times in a row is $\frac{1}{2^n}$.
As the likelihood that $P$ doesn’t know the secret decreases exponentially as $n$ increases, $V$ can be convinced that $P$ does in fact know the secret for sufficient $n$.
Before we apply my general description to the quadratic residue protocol, let me remind you of how the protocol works:
1. $P$ chooses a random $r$ and sends $s=r^2 > mod > N$ to $V$.
2. $V$ sends one bit $b$ back to $P$, and flips a coin to determine its value. If the coin lands heads then $b=1$. If the coin lands tails then $b=0$.
3. $P$ receives $b$. If $b$ is 1 then it sends $z=r*x$ to $V$. If $b=0$ then it sends $z=r$ to $V$ (this is the opposite of what you see in the slides because the slides have a typo).
4. $V$ receives $z$ and computes the expected value of $z^2$ using its knowledge of $s$ and $b$. If $b=1$ then the expected value for $z^2$ is $(r*x)^2 = r^2 * x^2 = s * y$. If $b=0$ then the
expected value of $z^2$ is $(r)^2 = r^2$.
5. $V$ rejects the proof if $z^2$ doesn’t match the expected value. If $z^2$ matches the expected value then $V$ either accepts the proof if it has been sufficiently convinced, or continues for more
interations to increase its confidence in the proof.
What if $P$ doesn’t know the secret?
Let’s look at a scenario where $P$ does not know $x$ but tries to fake it.
$P$ fakes case $b=1$
Here’s how $P$ might convince $V$ in the $b=1$ case without actually knowing $x$.
1. $P$ generates a random $r$, but instead of sending $s=r^2$ to $V$, it sends $s=\frac{r^2}{y}$
2. $V$ sends $b=1$ to $P$
3. $P$ sends $z=r$ to $V$.
4. $V$ checks the expected value of $z^2$ against its actual value. The expected value for $z^2$ is $sy=y*r^2/y=r^2$ and the actual value for $z^2$ is $(r)^2=r^2$. The actual value matches the
expected value, so $V$ is convinced.
What if $P$ guessed wrong and sent $s=\frac{r^2}{y}$ to $V$, but $V$ sent $b=0$ back? $P$ could send $z = r$ to $V$, but then $V$ would compute the expected value as $z^2 = s = \frac{r^2}{y}$ and the
actual value sent by $P$ would be $z^2 = (r)^2 = r^2$, a mismatch! In fact, the only way $P$ could rectify the situation is if it actually knew $x$ and sent $z = \frac{r}{\sqrt{y}} = \frac{r}{x}$ to
As you can see, if $P$ doesn’t know $x$ and tries to “fake” the $b=1$ case, there’s a 50% chance $V$ sends $b=0$ and $P$ is unable to fulfill its role. Then $V$ can conclude that $P$ doesn’t know the
secret $x$.
$P$ fakes case $b=0$
Here’s how $P$ could convince $V$ in the $b=0$ case without actually knowing $x$: $P$ can simply follow the original procedure. There’s nothing to “fake” because $P$ doesn’t need to know $x$ to
compute a random $r$ and send $z=r$ to $V$.
But $P$ can still get caught by the $b=1$ case. If $b=1$, then $V$ expects to receive $z=r*x$, which $P$ cannot compute because it doesn’t know $x$.
What if P could predict the next $b$?
If $P$ knew $b$ in a given iteration before sending $s$ to $V$, then you saw in the previous sections how it could convince $V$ it knows $x$ without actually knowing $x$. Therefore, it’s critical
that $P$ cannot predict $b$. Either the method by which $V$ selects $b$ must be hidden from $P$, or it must be selected with randomness. A fair coin flip was chosen for the protocol because it’s the
simplest approach that satisfies those conditions.
What if P re-used $r$ between iterations?
$P$ should be careful to not re-use $r$ between iterations because that could accidentally give away the secret! Here is a scenario where $V$ could extract the secret if $P$ isn’t careful
1. $P$ generates $r$ and sends $s = r^2$ to $V$.
2. $V$ sends $b=0$ to $P$ and $P$ sends back $z_1=r$
3. $V$ verifies $z_1^2 = s$ and continues to the next iteration
4. $P$ uses the same $r$ and sends $s = r^2$ to $V$
5. $V$ sends $b=1$ to $P$ and $P$ sends back $z_2=r*x$
6. Now $V$ has both $z_1$ and $z_2$ and can compute the secret as $\frac{z_2}{z_1} = \frac{r*x}{r} = x$
What does the quadratic residue protocol achieve?
In the words of Professor Goldwasser, $P$ convinces $V$ not by proving the statement but by proving that it could prove the statement if it wanted to. I’m still trying to wrap my head around that
I hope this article made the quadratic residue example a bit less confusing. If, like me, this is your first foray into zero knowledge and your entire conception of what a proof is and what it means
to prove something is upending, buckle up. I have a feeling it just gets more mind-blowing from here.
Where I’m going from here
I’d like to do another exercise based on the first lecture to deepen my understanding. The first example showed how to prove to a blind verifier that a piece of paper was made of two different
colors. I’d like to apply the simulation paradigm to that example to formally show that it’s PZK (Perfect Zero Knowledge). Keep an eye out for that article.
Written on February 5, 2023 | {"url":"https://daltyboy11.github.io/quadratic-residue-zkp-expanded/","timestamp":"2024-11-14T18:09:02Z","content_type":"text/html","content_length":"12717","record_id":"<urn:uuid:0d36230b-ad8b-4df3-b8f1-73b5b90f97fb>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00797.warc.gz"} |
Stephanos Venakides, Professor and Director of Graduate Studies
Integrable Systems
Integrable systems mostly consist of families of nonlinear differential equations (ordinary and partial) that can be solved (integrated) in explicit ways through the general principle of the Lax
pair, named after its discoverer, Peter Lax. The process of solution has conceptual similarities with the method of the Fourier transform used in the solution of linear differential equations. As
in the Fourier transform, there is a spectral variable at hand. While the solution of linear equations is given by a Fourier integral in the spectral variable along a certain contour, the
nonlinear case is more complicated: The initial data are used to specify (a) an oriented contour on the plane of the complex spectral variable and (b) a square "jump" matrix at each point of the
contour. To find the solution to the differential equation, one has to derive a matrix that (a) is an anlytic function of the spectral variable off the contour, (b) jumps across the contour, the
left limit being equal to the right limit multiplied by the jump matrix, and (c) has a certain normalization at the infinity point of the spectral variable. Such a problem is known as a
Riemann-Hilbert problem (RHP). Solving such a problem in the general case is as difficult (indeed, much more so) as evaluating a general Fourier integral.
The full asymptotic expansion of general Fourier integrals in physically interesting asymptotic limits was made possible by the method of stationary phase/steepest descent, attributed to Lord
Kelvin. One encounters such asymptotic limits in calculations of long-time system behaviors, as well as semi-classical (large frequency or small Planck constant) calculations. The foundation of
this approach is that the main contribution from the integral arises from the neighborhood of points of the contour of integration where the fast growing exponent under the integral is
stationary. Properly restricted to these neighborhoods, the integral reduces asymptotically to a Gaussian integral, hence it is readily computable.
The situation is analogous in the nonlinear case. Through a procedure introduced by Deift and Zhou in the case of long time limits, factorization of the jump matrix coupled with contour
deformations allows the localization of the contour, the simplification of the jump matrix and the rigorous asymptotic reduction to a solvable RHP. The procedure is known as steepest descent for
RHP, arising from the "pushing" of parts of the contour to regions where it is exponentially close to the identity and can be thus neglected.
In dispersive equations involving oscillations, the method was readily applicable when the asymptotic oscillation was weakly nonlinear i.e. consisted of modulated plane wave solutions. In the
presence of fully nonlinear oscillations simply finding the stationary points of a scalar function was not appropriate. In collaboration with Deift and Zhou, (a) we found that the reduced or
"model" RHP, which determines the main contribution to the solution has as contour a union of intervals or arcs in the complex plane, (b) we introduced the "g-function mechanism", a procedure
that led to a system of transcendental equations and inequalities that the endpoints of the intervals satisfy and from which they are identified uniquely when they exist. (c) having identified
these points, we solved the reduced RHP through a Riemann theta function and established that the waveform is mostly a modulated quasi-periodic nonlinear wave. This work was done in the context
of the celebrated Korteweg de Vries equation (KdV). In joint work, with Deift, Kriecherbauer, McLaughlin (Ken) and Zhou, we implemented this approach to prove an important universality result in
the theory of random matrices of the unitary ensemble.
In collaboration with Tovbis and Zhou, we then tackled the problem of the nonlinear focusing Schroedinger (NLS) equation that is known to be modulationally unstable (KdV is stable) and thus
presented a further difficulty. We have succeeded in obtaining the global space-time solution to the initial value problem for special data that contain only radiation and the solution till the
second break in the presence of a soliton content. In both cases, it is analytic properties of the spectral data (jump matrix) that save us from the instability. Spectral data NLS calculations
are delicate when possible; it required special work in collaboration with Tovbis to calculate the data in the above cases. Again with Tovbis, we revealed the deeper structure of the modulation
equations by bringing them into a form that involves determinants. We also analyzed the limit of the inverse scattering transform in the asymptotic limit.
What one learns from these theories is that as waveforms evolve, they break into more complicated waveforms or relax to simpler ones. Multiple theta functions in the formulae describe the
evolution of multiphase modes. The analogue of caustics appears in space-time along the boundaries at which the number of participating modes jumps. We have already shown that, with our intial
data, there is only one break in the pure radiation case. The local asymptotic analysis of the first break was performed by Bertola and Tovbis.
In collaboration with former student Belov, we are working to understand the second breaking of the solution of the NLS equation in the presence of solitons, as well as possible subsequent
successive breaks that are suggested by numerics. The challenge is that, for a fixed spatial position, we reach a point in time, at which there is an obstacle to our systematic advance of the
solution in time. We have made a rigorous asymptotic calculation of the curve in space-time, along which this difficulty presents itself. We suspect that overcoming this obstacle involves a
transformational advance in the asymptotic method itself and we are working in this direction.
Wave Propagation in complex media
In earlier work with Bonilla and Higuera, we studied the breakdown of the stability of the steady state in a Gunn semiconductor, that leads to the generation of a time periodic pulse train that
is commonly used as a microwave source. With Bonilla Kindelan and Moscoso we analyzed the generation and propagation of traveling fronts in semiconductor superlattices.
More recently, in collaboration with Haider and Shipman, we studied the scattering of plane waves off a photonic crystal slab, composed of two dielectrics that are distributed periodically along
the slab with different refractive indices. We discovered anomalous transmission behavior, as the angle of incidence is varied from normal. With Shipman, we showed that the anomaly is mediated by
resonance in the system, in which the incident wave excites a mode along the slab, and we derived an asymptotic formula for the anomalous transmission near the resonant frequency. The formula has
very good agreement with the results of simulation. Significantly, the derived profile depends only on a small number of parameters. These few parameters encapsulate all the possible geometric
configurations of the photonic crystal.
Most of the materials used in practice, are either linear or weakly nonlinear. However, fully nonlinear phenomena occur near the above resonance, due to the large magnitude of the resonant
fields. With Shipman, we constructed and solved a fully nonlinear model displaying such phenomena. The model involves a linear transmission line in which an incident plane wave scatters off a
point defect, that is coupled to a nonlinear oscillator. As the coupling increases from zero, a frequency band emerges near the resonant frequency of the defect, in which three (as opposed to
one) harmonic solutions are possible. Three solutions also appear at all frequencies beyond a high frequency threshold. As the coupling constant is further increased (but is still quite small)
the band grows, the threshold frequency diminishes, until the two 3-solution regions touch and merge to one. It was pointed out to us by physicist and applied mathematician, S. Komineas, that our
line/oscillator model is of interest in the Bose- Einstein condensation community, for being a simplification of a model for the onset of vortex-antivortex pairs in polariton superfluids in the
optical parametric oscillator (OPO) regime. Our current effort, with Shipman and Komineas, is to develop the mathematical tools for solving this broader model.
Mathematical Biology
Several years ago, I joined the "laser group" of colleagues Dan Kiehart (group leader, Biology) and Glenn Edwards (Physics). The group studies the drosophila dorsal closure (see below) and
derives its name from experiments involving laser ablations of the drosophila embryo. The group includes postdocs and graduate students and works through weekly meetings. My interest is the
modeling of the closure of the dorsal opening of the drosophila embryo in the process of morphogenesis. The dorsal opening has the shape of a human eye and is only covered by an extra-embryonic,
epithelial tissue, the amnioserosa; during closure the opposite flanks are "zipped" together at the canthi ("eye" edges). The challenge is to understand the nature of the forces, how they affect
the kinetics and their biological and physical origin.
We developed a mathematical model that connects the empirical kinematic observations with contributing tissue forces, that affect the morphology of the dorsal surface and, in particular, the
movements of the purse string and of the canthi. We model the coordinated elastic and contractile motor forces, attributed to the action of actin and myosin that drive DC, by introducing a unit
that satisfies a law similar to the law derived by Hill in the early modeling of muscle dynamics. We model the zipping process through a phenomenological law that summarizes the complicated
processes of the canthus. Our model recapitulates the experimental observations of wild type native, laser perturbed and mutant native closure made in earlier work of the group (Hutson et.al.)
The current goal is a transformational extension of the model that (a) will allow deformations that are not restricted to ones that are traverse to the dorsal midline, (b) will introduce the
individuality of the amnioserosa cells and (c) will account for small time-scale oscillations observed in the amnioserosa. The scantness of the understanding of the underlying biological
mechanism makes this effort quite challenging. | {"url":"https://fds.duke.edu/db/aas/math/faculty/ven","timestamp":"2024-11-15T03:23:59Z","content_type":"text/html","content_length":"25450","record_id":"<urn:uuid:1248e242-518d-4098-bbf9-4ebde71e0fdd>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00267.warc.gz"} |
The flaw in the Scully pay/performance regession
The flaw in the Scully pay/performance regession
In 1974, Gerald Scully published an academic article called "Pay and Performance in Major League Baseball." (Here's a Google search that finds a copy on David Berri's site.) It's a very famous paper,
because it reached the conclusion that, in the pre-free-agent era, teams were paying players far, far less than the players were earning for their employers.
It was also one of the first academic papers to try to find a connection between individual performance and team wins. Unfortunately, Scully chose SLG and K/BB ratio as his measures of performance,
but, I suppose, those probably seemed like reasonable choices at the time. But it turns out there's a much more serious problem.
In his set of variables to use in predicting winning percentage, the set of variables that included SLG and K/BB, Scully also included dummy variables for how far the team was out of first place.
That is, in trying to predict how well a team did, Scully based his estimates partially on ... how well the team did!
That biases the results so much that I can't believe nobody's mentioned it before ... at least they haven't in all the mentions I've seen of this study.
If it's not obvious why that's the wrong thing to do, let me try to explain.
Suppose you predict that, this season, your favorite team will slug .390 and have a K/BB ratio of 1.5. What will its winning percentage be?
Well, if Scully had used only SLG and K/BB in his regression, it would be easy to figure out: you just take his regression equation, which would look something like
PCT = (a * SLG) + (b * K/BB) + c (if an NL team) + d
Plug in Scully's estimates for a, b, and c, plug in .390 and 1.5, and there you go -- your estimate.
But Scully's actual equation included those two extra terms
PCT =(.92 * SLG) + (.90 * K/BB) - 38.57 (if an NL team) + 37.24 + 43.78 (if the team finished within 5 games of first) - 75.64 (if the team finished more than 20 games out)
So now how do you calculate your team's expected PCT? You can't! Because you don't know whether to include those last two variables. After all, how can you predict in advance whether your team will
wind up having finished near the top or the bottom? You can't! If you could, you probably wouldn't need this regression in the first place!
Not only does the regression not make sense, but, more importantly, by including those two dummy variables, Scully's estimates of productivity wind up completely wrong. For instance: what is the
effect of raising your SLG by 10 points? Well, that depends. Keeping all the other variables constant, it's .92 * 10, or 9.2 points. That's .0092, or about 1.5 wins in a 162-game season.
But wait! Those dummy variables for standings position won't necessarily stay constant. What if those 1.5 wins lifted you from 21 games back to 19.5 games back? In that case, the equation would give
9.2 for the SLG, but an extra 75.7 for the change in the dummy, for a total of 86.9 points! And what if they lifted you from 6 games back to 4.5 games back? In that case, the equation would estimate
an extra 43.8 point bump, for a total of 53.0!
So what's the benefit of an extra 10 points slugging on the team's winning percentage?
--> 9.2 points -- for a team 22 or more games out
-> 75.7 points -- for a team 21.5 to 20 games out
--> 9.2 points -- for a team 19.5 to 7 games out
-> 43.8 points -- for a team 6.5 to 5 games out
--> 9.2 points -- for a team less than 5 games out
That makes no sense like that. You can't figure out how much the player's productivity is worth unless you know which of the five groups the team is in. But which group the team is in is exactly what
you're trying to predict!
In any case, it's obvious that using 9.2 points as the measure of the player's increased productivity is wrong. It's *at least* 9.2 points, but sometimes substantially more. You need to average all
five cases out, in proportion to how often they'd occur (and how can you know how often they occur without further study?). When you do that, you'll obviously get more than 9.2 points. But, as far as
I can tell, Scully just used the SLG coefficient as his measure -- the player only got credit for the 9.2 points! And so he *severely underestimated* how much a player's performance helps his team.
Here's an example that will make it clearer. Suppose a lottery gives you a 1 in a million chance of winning $500,000. Then, if you do a regression to predict winnings based on how many tickets you
buy, you'll probably get something close to
Winnings = 0.5 * tickets bought
Which makes sense: a 1 in a million chance of winning half a million dollars is worth 50 cents.
But now, suppose I add a term that says whether or not you won. Now, I'll get
Winnings = 0.0 * tickets bought + $500,000 (if you won)
That's true, but it completely hides the relationship between the ticket and the winnings. If you ignore the dummy variable, it looks like the ticket is worthless!
Same for Scully's regression. By including part of the "winnings" for having a good SLG or K/BB "ticket" in a different term, he underestimates the value of the "ticket".
[S:So, since Scully's conclusion was that players are underpaid for their productivity, and Scully himself had underestimated that productivity ... well, the conclusion is completely unjustified by
the results of the study. It may be true that players were underpaid -- I think it almost certainly is -- but this particular study, famous as it is, doesn't even come close to proving it.:S]
UPDATE: As commenter MattyD points out (thanks!), I got it backwards. For the previous paragraph, I should have said:
However, Scully's conclusion on pay and productivity still holds. The study underestimated player productivity, but, if it found that players are paid less than even that underestimated production,
it's certainly true that they are underpaid relative to their true production.
Labels: economics, payroll
7 Comments: | {"url":"http://blog.philbirnbaum.com/2010/04/flaw-in-scully-payperformance-regession.html","timestamp":"2024-11-02T11:52:39Z","content_type":"application/xhtml+xml","content_length":"40572","record_id":"<urn:uuid:469415d7-b1e5-498c-88ce-845f9fbc29cd>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00870.warc.gz"} |
Determinant of 4x4 Matrix | Examples and How to Find
Determinant of 4×4 Matrix
The determinant of a 4x4 matrix is a special number that you find using a specific formula. A matrix with n rows and n columns is called a square matrix. So, a 4x4 matrix is a square matrix with four
rows and four columns. If we have a square matrix A, the determinant of A is...
represedted as A.
A determinant is a number linked to a square matrix (a matrix with the same number of rows and columns) in math. It gives us important information about the matrix, like whether it can be inverted or
if it has special properties.
To find the determinant, you use different methods based on the matrix size. For small matrices like 2x2 and 3x3, you can calculate it directly. For bigger ones, like 4x4 or larger, you might use the
Laplace formula or other advanced methods.
In simple words, the determinant is like a "summary" of a square matrix that tells us useful things about it and how it relates to the surrounding space.
Symbol of Determinant
The determinant of a matrix is shown using "det" or the matrix's name with bars around it, like \( \det(A) \) or \( |A| \). Sometimes, it's also shown with a script "d", like \( dA \).
What is a Matrix?
A matrix is a grid of numbers arranged in rows and columns. The elements in a matrix can be single numbers, rows, columns, or even other matrices. Matrices are used in math and computer science for
many purposes, like solving systems of linear equations, showing transformations and rotations in graphics and physics, and organizing data in a neat way.
Formula to Find Determinant of a 4x4 Matrix
To find the determinant of a 4x4 matrix, you can use this formula:
det(A) = a11 * det(A11) - a12 * det(A12) + a13 * det(A13) - a14 * det(A14)
Here’s what the terms mean:
• aij is the element in the ith row and jth column of the matrix A.
• Aij is the smaller matrix you get by removing the ith row and jth column from A.
This formula, called the Laplace formula, breaks down the 4x4 matrix into smaller parts until you get to 2x2 matrices, where the determinant can be calculated directly.
How to Calculate the Determinant of a 4x4 Matrix
To find the determinant of a 4x4 matrix, use the Laplace formula. Here’s an example matrix A:
| a11 a12 a13 a14 |
| a21 a22 a23 a24 |
| a31 a32 a33 a34 |
| a41 a42 a43 a44 |
The determinant of A is:
det(A) = a11 * det(A11) - a12 * det(A12) + a13 * det(A13) - a14 * det(A14)
You keep breaking down the matrices until you get to 2x2 matrices, where you can find the determinant easily.
This is one way to find the determinant of a 4x4 matrix. There are other methods, depending on what you need for your problem.
Frequently Asked Questions on Determinant of 4×4 Matrix
To find the determinant of a 4x4 matrix, we can use the regular method. Start with the first element in the first row, multiply it by the determinant of the smaller 3x3 matrix that remains after
removing the row and column of that element, and repeat this process for the other elements.
The determinant of a 4×4 matrix is a specific number found using a certain formula. When a matrix has the form n x n, it is called a square matrix. Thus, a 4×4 matrix is a square matrix with four
rows and four columns. If we have a square matrix A, we show its determinant as |A|.
To find the minor of an element in a matrix, like aija_{ij}aij in matrix A, follow these steps:
1. Remove the row and column that contain aija_{ij}aij.
2. Calculate the determinant of the smaller matrix that is left.
Repeat this process for each element in the matrix to find all the minors.
The determinant is found with this formula: |A| = a(ei - fh) - b(di - fg) + c(dh - eg). In simple terms, this means you do 'a times (e times i minus f times h) minus b times (d times i minus f times
g) plus c times (d times h minus e times g)'. It might seem tricky, but if you look closely at the pattern, it's actually easy! | {"url":"https://www.home-tution.com/maths-topics/determinant-of-4x4-matrix","timestamp":"2024-11-02T04:46:00Z","content_type":"text/html","content_length":"114533","record_id":"<urn:uuid:880e85e7-af06-483c-a132-78560afeb14b>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00579.warc.gz"} |
Poisson point process Archives – H. Paul Keeler
One of the most important stochastic processes is Poisson stochastic process, often called simply the Poisson process. In a previous post I gave the definition of a stochastic process (also called a
random process) alongside some examples of this important random object, including counting processes. The Poisson (stochastic) process is a counting process. This continuous-time stochastic
process is a highly studied and used object. It plays a key role in different probability fields, particularly those focused on stochastic processes such as stochastic calculus (with jumps) and the
theories of Markov processes, queueing, point processes (on the real line), and Levy processes.
The points in time when a Poisson stochastic process increases form a Poisson point process on the real line. In this setting the stochastic process and the point process can be considered two
interpretations of the same random object. The Poisson point process is often just called the Poisson process, but a Poisson point process can be defined on more generals spaces. In some literature,
such as the theory of Lévy processes, a Poisson point process is called a Poisson random measure, differentiating the Poisson point process from the Poisson stochastic process. Due to the connection
with the Poisson distribution, the two mathematical objects are named after Simeon Poisson, but he never studied these random objects.
The other important stochastic process is the Wiener process or Brownian (motion process), which I cover in another post. The Wiener process is arguably the most important stochastic process. I have
written that post and the current one with the same structure and style, reflecting and emphasizing the similarities between these two fundamental stochastic process.
In this post I will give a definition of the homogenous Poisson process. I will also describe some of its key properties and importance. In future posts I will cover the history and generalizations
of this stochastic process.
In the stochastic processes literature there are different definitions of the Poisson process. These depend on the settings such as the level of mathematical rigour. I give a mathematical definition
which captures the main characteristics of this stochastic process.
Definition: Homogeneous Poisson (stochastic) process
An integer-valued stochastic process \(\{N_t:t\geq 0 \}\) defined on a probability space \((\Omega,\mathcal{A},\mathbb{P})\) is a homogeneous Poisson (stochastic) process if it has the following
1. The initial value of the stochastic process \(\{N_t:t\geq 0 \}\) is zero with probability one, meaning \(P(N_0=0)=1\).
2. The increment \(N_t-N_s\) is independent of the past, that is, \(N_u\), where \(0\leq u\leq s\).
3. The increment \(N_t-N_s\) is a Poisson variable with mean \(\lambda (t-s)\).
In some literature, the initial value of the stochastic process may not be given. Alternatively, it is simply stated as \(N_0=0\) instead of the more precise (probabilistic) statement given above.
Also, some definitions of this stochastic process include an extra property or two. For example, from the above definition, we can infer that increments of the homogeneous Poisson process are
stationary due to the properties of the Poisson distribution. But a definition may include something like the following property, which explicitly states that this stochastic process is stationary.
4. For \(0\leq u\leq s\), the increment \(N_t-N_s\) is equal in distribution to \(N_{t-s}\).
The definitions may also describe the continuity of the realizations of the stochastic process, known as sample paths, which we will cover in the next section.
It’s interesting to compare these defining properties with the corresponding ones of the standard Wiener stochastic process. Both stochastic processes build upon divisible probability distributions.
Using this property, Lévy processes generalize these two stochastic processes.
The definition of the Poisson (stochastic) process means that it has stationary and independent increments. These are arguably the most important properties as they lead to the great tractability of
this stochastic process. The increments are Poisson random variables, implying they can have only positive (integer) values.
The Poisson (stochastic) process exhibits closure properties, meaning you apply certain operations, you get another Poisson (stochastic) process. For example, if we sum two independent Poisson
processes \(X= \{X_t:t\geq 0 \}\) and \(Y= \{Y_t:t\geq 0 \}\), then the resulting stochastic process \(Z=Z+Y = \{N_t:t\geq 0 \}\) is also a Poisson (stochastic) process. Such properties are useful
for proving mathematical results.
A single realization of a (homogeneous) Poisson stochastic process, where the blue marks show where the process jumps to the next value. In any finite time interval, there are a finite number of
Properties such as independence and stationarity of the increments are so-called distributional properties. But the sample paths of this stochastic process are also interesting. A sample path of a
Poisson stochastic process is almost surely non-decreasing, being constant except for jumps of size one. (The term almost surely comes from measure theory, but it means with probability one.) There
are only finitely number of jumps in each finite time interval.
The homogeneous Poisson (stochastic) process has the Markov property, making it an example of a Markov process. The homogenous Poisson process \(N=\{ N_t\}_{t\geq 0}\)s not a martingale. But
interestingly, the stochastic process is \(\{ W_t – \lambda t\}_{t\geq 0}\) is a martingale. (Such relations have been used to study such stochastic processes with tools from martingale theory.)
Stochastic or point process?
The Poisson (stochastic) process is a discrete-valued stochastic process in continuous time. The relation these types of stochastic processes and point process is a subtle one. For example, David Cox
and Valerie Isham write on page 3 of their monograph:
The borderline between point processes and a number of other kinds of stochastic process is not sharply defined. In particular, any stochastic process in continuous time in which the sample
paths are step functions, and therefore any any process with a discrete state space, is associated with a point process, where a point is a time of transition or, more generally, a time of entry
into a pre-assigned state or set of states. Whether it is useful to look at a particular process in this way depends on the purpose of the analysis.
For the Poisson case, this association is presented in the diagram below. We can see the Poisson point process (in red) associated with the Poisson (stochastic) process (in blue) by simply looking at
the time points where jumps occur.
A single realization of a (homogeneous) Poisson stochastic process (in blue). The jumps of the process form a (homogeneous) Poisson point process (in red) on the real line representing time.
Playing a prominent role in the theory of probability, the Poisson (stochastic) process is a highly important and studied stochastic process. It has connections to other stochastic processes and is
central in queueing theory and random measures.
The Poisson process is a building block for more complex continuous-time Markov processes with discrete state spaces, which are used as mathematical models. It is also essential in the study of jump
processes and subordinators.
The Poisson (stochastic) process is a member of some important families of stochastic processes, including Markov processes, Lévy processes, and birth-death processes. This stochastic process also
has many applications. For example, it plays a central role in quantitative finance. It is also used in the physical sciences as well as some branches of social sciences, as a mathematical model for
various random phenomena.
Generalizations and modifications
For the Poisson (stochastic) process, the index set and state space are respectively the non-negative numbers and counting numbers, that is \(T=[0,\infty)\) and \(S=0, 1, \dots\), so it has a
continuous index set but a discrete state space. Consequently, changing the state space, index set, or both offers an ways for generalizing and modifying the Poisson (stochastic) process.
The defining properties of the Poisson stochastic process, namely independence and stationarity of increments, results in it being easy to simulate. The Poisson stochastic process can be simulated
provided random variables can be simulated or sampled according to a Poisson distributions, which I have covered in this and this post.
Simulating a Poisson stochastic process is similar to simulating a Poisson point process. (Basically, it is the same method in a one-dimensional setting.) But I will leave the details of sampling
this stochastic process for another post.
Further reading
Here are some related links:
A very quick history of Wiener process and the Poisson (point and stochastic) process is covered in this talk by me.
In terms of books, the Poisson process has not received as much attention as the Wiener process, which is typically just called the Brownian (motion) process. That said, any book covering queueing
theory will cover the Poisson (stochastic) process.
More advanced readers can read about the Poisson (stochastic) process, the Wiener (or Brownian (motion)) process, and other Lévy processes:
On this topic, I recommend the introductory article:
• 2004, Applebaum, Lévy Processes – From Probability to Finance and Quantum Groups.
This stochastic process is of course also covered in general books on stochastics process such as:
Binomial point process
The binomial point process is arguably the simplest point process. It consists of a non-random number of points scattered randomly and independently over some bounded region of space. In this post I
will describe the binomial point process, how it leads to the Poisson point process, and its historical role as stars in the sky.
The binomial point process is an important stepping stone in the theory of point process. But I stress that for mathematical models, I would always use a Poisson point process instead of a binomial
one. The only exception would be if you were developing a model for a small, non-random number of points.
Uniform binomial point process
We start with the simplest binomial point process, which has uniformly located points. (I described simulating this point process in an early post. The code is here.)
Consider some bounded (or more precisely, compact) region, say, \(W\), of the plane plane \(\mathbb{R}^2\), but the space can be more general. The uniform binomial point process is created by
scattering \(n\) points uniformly and independently across the set \(W\).
A single realization of a binomial point process with n=30 points. The points are uniformly and independently scattered across a unit square.
Consider a single point uniformly scattered in the region \(W\), giving a binomial point process with \(n=1\). We look at some region \(B\), which is a subset of \(W\), implying \(B\subseteq W\).
What is the probability that the single point \(X\) falls in region \(B\)?
First we write \(\nu(W)\) and \(\nu(B)\) to denote the respective areas (or more precisely, Lebesgue measures) of the regions \(W\) and \(B\), hence \(\nu(B)\leq \nu(W)\). Then this probability,
say, \(p\), is simply the ratio of the two areas, giving
$$p= P(X\in B)=\frac{\nu(B)}{\nu(W)}.$$
The event of a single point being found in the set \(B\) is a single Bernoulli trial, like flipping a single coin. But if there are \(n\) points, then there are \(n\) Bernoulli trials, which bring us
to the binomial distribution.
For a uniform binomial point process \(N_W\), the number of randomly located points being found in a region \(B\) is a binomial random variable, say, \(N_W(B)\), with probability parameter \(p=\nu(B)
/ \nu(W)\). The probability mass function of \(N_W(B)\) is
$$ P(N_W(B)=k)={n\choose k} p^k(1-p)^{n-k}. $$
We can write the expression more explicitly
$$ P(N_W(B)=k)={n\choose k} \left[\frac{\nu(B)}{ \nu(W)}\right]^k\left[1-\frac{\nu(B)}{\nu(W)}\right]^{n-k}. $$
Poisson limit
Poisson random variable
A standard exercise in introductory probability is deriving the Poisson distribution by taking the limit of the binomial distribution. This is done by sending \(n\) (the total number of Bernoulli
trials) to infinity while keeping the binomial mean \(\mu:=p n\) fixed, which sends the probability \(p=\mu/n\) to zero.
More precisely, for \(\mu\geq0\), setting \(p_n=\mu/n \) and keeping \(\mu :=p_n n\) fixed, we have the limit result
$$\lim_{n\to \infty} {n \choose k} p_n^k (1-p_n)^{n-k} = \frac{\mu^k}{k!}\, e^{-\mu}.$$
We can use, for example, Stirling’s approximation to prove this limit result.
We can make the same limit argument with the binomial point process.
Homogeneous Poisson point process
We consider the intensity of the uniform binomial point process, which is the average number of points in a unit area. For a binomial point process, this is simply
$$\lambda := \frac{n}{\nu(W)}.$$
For the Poisson limit, we expand the region \(W\) so it covers the whole plane \(\mathbb{R}^2\), while keeping the intensity \(\lambda = n/\nu(W)\) fixed. This means that the area \(\nu(W)\)
approaches infinity while the probability \(p=\nu(B)/\nu(W)\) goes to zero. Then in the limit we arrive at the homogeneous Poisson point process \(N\) with intensity \(\lambda\).
The number of points of \(N\) falling in the set \(B\) is a random variable \(N(B)\) with the probability mass function
$$ P(N(B)=k)=\frac{[\lambda \nu(B)]^k}{k!}\,e^{-\lambda \nu(B)}. $$
General binomial point process
Typically in point process literature, one first encounters the uniform binomial point process. But we can generalize it so the points are distributed according to some general distribution.
We write \(\Lambda\) to denote a non-negative Radon measure on \(W\), meaning \(\Lambda(W)< \infty\) and \(\Lambda(B)\geq 0\) for all (measurable) sets \(B\subseteq W\). We can also assume a more
general space for the underlying space such as a compact metric space, which is (Borel) measurable. But the intuition still works for compact region of the plane \(\mathbb{R}^2\).
For the \(n\) points, we assume each point is distributed according to the probability measure
$$\bar{\Lambda}= \frac{\Lambda}{\Lambda(W)}.$$
The resulting point process is a general binomial point process. The proofs for this point process remain essentially the same, replacing the Lebesgue measure \(\nu\), such as area or volume, with
the non-negative measure \(\Lambda\).
A typical example of the intensity measure \(\Lambda\) has the form
$$\Lambda(B)= \int_B f(x) dx\,,$$
where \(f\) is a non-negative density function on \(W\). Then the probability density of a single point is
$$ p(x) = \frac{1}{c}f(x),$$
where \(c\) is a normalization constant
$$c= \int_W f(x) dx\,.$$
On a set \(W \subseteq \mathbb{R}^2\) using Cartesian coordinates, a specific example of the density \(f\) is
$$ f(x_1,x_2) = \lambda e^{-(x_1^2+x_2^2)}.$$
Assuming a general binomial point process \(N_W\) on \(W\), we can use the previous arguments to obtain the binomial distribution
$$ P(N_W(B)=k)={n\choose k} \left[\frac{\Lambda(B)}{\Lambda(W)}\right]^k\left[1-\frac{\Lambda(B)}{\Lambda(W)}\right]^{n-k}. $$
General Poisson point process
We can easily adapt the Poisson limit arguments for the general binomial Poisson point process, which results in the general Poisson point process \(N\) with intensity measure \(\Lambda\). The number
of points of \(N\) falling in the set \(B\) is a random variable \(N(B)\) with the probability mass function
$$ P(N(B)=k)=\frac{[\Lambda(B)]^k}{k!}\, e^{-\Lambda(B)}. $$
History: Stars in the sky
The uniform binomial point process is an example of a spatial point process. With points being scattered uniformly and independently, its sheer simplicity makes it a natural choice for an early
spatial model. But which scattered objects?
Perhaps not surprisingly, it is trying to understand star locations where we find the earliest known example of somebody describing something like a random point process. In 1767 in England John
Michell wrote:
what it is probable would have been the least apparent distance of any two or more stars, any where in the whole heavens, upon the supposition that they had been scattered by mere chance, as it
might happen
As an example, Michelle studied the six brightest stars in the Pleiades star cluster. He concluded the stars were not scattered by mere chance. Of course “scattered by mere chance” is not very
precise in today’s probability language, but we can make the reasonable assumption that Michell meant the points were uniformly and independently scattered.
Years later in 1860 Simon Newcomb examined Michell’s problem, motivating him to derive the Poisson distribution as the limit of the binomial distribution. Newcomb also studied star locations.
Stephen Stigler considers this as the first example of applying the Poisson distribution to real data, pre-dating the famous work by Ladislaus Bortkiewicz who studied rare events such as deaths from
horse kicks. We also owe Bortkiewicz the terms Poisson distribution and stochastic (in the sense of random).
Here, on my repository, are some pieces of code that simulate a uniform binomial point process on a rectangle.
Further reading
For an introduction to spatial statistics, I suggest the lectures notes by Baddeley, which form Chapter 1 of these published lectures, edited by Baddeley, Bárány, Schneider, and Weil. The binomial
point process is covered in Section 1.3.
The binomial point process is also covered briefly in the classic text Stochastic Geometry and its Applications by Chiu, Stoyan, Kendall and Mecke; see Section 2.2. Similar material is covered in the
book’s previous edition by Stoyan, Kendall and Mecke.
Haenggi also wrote a readable introductory book called Stochastic Geometry for Wireless networks, where he gives the basics of point process theory. The binomial point process is covered in Section
For some history on point processes and the Poisson distribution, I suggest starting with the respective papers:
• Guttorp and Thorarinsdottir, What Happened to Discrete Chaos, the Quenouille Process, and the Sharp Markov Property?;
• Stigler, Poisson on the Poisson distribution.
Histories of the Poisson distribution and the Poisson point process are found in the books:
• Haight, Handbook of the Poisson Distribution, Chapter 9;
• Last and Penrose, Lectures on the Poisson process, Appendix C.
Summary: Poisson simulations
Here’s a lazy summary post where I list all the posts on various Poisson simulations. I’ve also linked to code, which is found on this online repository. The code is typically written in MATLAB and
Poisson point processes
Some simulations of Poisson point processes are also covered in this post on the Julia programming language:
Checking simulations
Poisson line process
Poisson random variables
Simulating a Poisson point process on a n-dimensional sphere
In the previous post I outlined how to simulate or sample a homogeneous Poisson point process on the surface of a sphere. Now I will consider a homogeneous Poisson point process on the \((n-1)-\)
sphere, which is the surface of the Euclidean ball in \(n\) dimensions.
This is a short post because it immediately builds off the previous post. For positioning the points uniformly, I will use Method 2 from that post, which uses normal random variables, as it
immediately gives a fast method in \(n\) dimensions.
I wrote this post and the code more for curiosity than any immediate application. But simulating a Poisson point process in this setting requires placing points uniformly on a sphere. And there are
applications in that, such as Monte Carlo integration methods, as mentioned in this post, which nicely details different sampling methods.
As is the case for other shapes, simulating a Poisson point process requires two steps.
Number of points
The number of points of a Poisson point process on the surface of a sphere of radius \(r>0\) is a Poisson random variable. The mean of this random variable is \(\lambda S_{n-1}\), where \(S_{n-1}\)
is the surface area of the sphere. For a ball embedded in \(n\) dimension, the area of the corresponding sphere is given by
$$S_{n-1} = \frac{2 \pi ^{n/2} }{\Gamma(n/2)} r^{n-1},$$
where \(\Gamma\) is the gamma function, which is a natural generalization of the factorial. In MATLAB, we can simply use the function gamma. In Python, we need to use the SciPy function
scipy.special. gamma.
Locations of points
For each point on the sphere, we generate \(n\) standard normal or Gaussian random variables, say, \(W_1, \dots, W_n\), which are independent of each other. These random variables are the Cartesian
components of the random point. We rescale the components by the Euclidean norm, then multiply by the radius \(r\).
For \(i=1,\dots, n\), we obtain
These are the Cartesian coordinates of a point uniformly scattered on a sphere with radius \(r\) and a centre at the origin.
How does it work?
In the post on the circle setting, I gave a more detailed outline of the proof, where I said the method is like the Box-Muller transform in reverse. The joint density of the normal random variables
is from a multivariate normal distribution with zero correlation. This joint density a function of the Cartesian equation for a sphere. This means the density is constant on the sphere, implying that
the angle of the point \((W_1,\dots, W_n)\) will be uniformly distributed.
The vector formed from the normal variables \((W_1,\dots,W_n)\) is a random variable with a chi distribution. But the final vector, which stretches from the origin to the point \((X_1,\dots,X_n)\),
has length one, because we rescaled it with the Euclidean norm.
The code for all my posts is located online here. For this post, the code in MATLAB and Python is here.
Further reading
I recommend this blog post, which discusses different methods for randomly placing points on spheres and inside spheres (or, rather, balls) in a uniform manner. (Embedded in two dimensions, a sphere
is a circle and a ball is a disk.)
Our Method 2 for positioning points uniformly, which uses normal variables, comes from the paper:
• 1959, Muller, A note on a method for generating points uniformly on n-dimensional spheres.
Two recent works on this approach are the papers:
• 2010, Harman and Lacko, On decompositional algorithms for uniform sampling from -spheres and -balls;
• 2017, Voelker, Gosman, Stewart, Efficiently sampling vectors and coordinates.
Simulating a Poisson point process on a sphere
In this post I’ll describe how to simulate or sample a homogeneous Poisson point process on the surface of a sphere. I have already simulated this point process on a rectangle, triangle disk, and
Of course, by sphere, I mean the everyday object that is the surface of a three-dimensional ball, where this two-dimensional object is often denoted by \(S^2\). Mathematically, this is a
generalization from a Poisson point process on a circle, which is slightly simpler than randomly positioning points on a disk. I recommend reading those two posts first, as a lot of the material
presented here builds off them.
I have not needed such a simulation in my own work, but I imagine there are many reasons why you would want to simulate a Poisson point process on a sphere. As some motivation, we can imagine these
points on a sphere representing, say, meteorites or lightning hitting the Earth.
The generating the number of points is not difficult. The trick is positioning the points on the sphere in a uniform way. As is often the case, there are various ways to do this, and I recommend
this post, which lists the main ones. I will use two methods that I consider the most natural and intuitive ones, namely using spherical coordinates and normal random variables, which is what I did
in the post on the circle.
Incidentally, a simple modification allows you to scatter the points uniformly inside the sphere, but you would typically say ball in mathematics, giving a Poisson point process inside a ball; see
below for details.
As always, simulating a Poisson point process requires two steps.
Number of points
The number of points of a Poisson point process on the surface of a sphere of radius \(r>0\) is a Poisson random variable with mean \(\lambda S_2\), where \(S_2=4\pi r^2\) is the surface area of the
sphere. (In this post I give some details for simulating or sampling Poisson random variables or, more accurately, variates.)
Locations of points
For any homogeneous Poisson point process, we need to position the points uniformly on the underlying space, which is in this case is the sphere. I will outline two different methods for positioning
the points randomly and uniformly on a sphere.
Method 1: Spherical coordinates
The first method is based on spherical coordinates \((\rho, \theta,\phi)\), where the radial coordinate \(\rho\geq 0\), and the angular coordinates \(0 \leq \theta\leq 2\pi\) and \(0\leq \phi \leq \
pi\). The change of coordinates gives \(x=\rho\sin(\theta)\cos(\phi)\), \(y=\rho\sin(\theta)\sin(\phi)\), and \(z=\rho\cos(\phi)\).
Now we use Proposition 1.1 in this paper. For each point, we generate two uniform variables \(V\) and \(\Theta\) on the respective intervals \((-1,1)\) and \((0,2\pi)\). Then we place the point with
the Cartesian coordinates
$$X = r \sqrt{1-V^2} \cos\Theta, $$
$$Y = r \sqrt{1-V^2}\sin\Theta, $$
$$ Z=r V. $$
This method places a uniform point on a sphere with a radius \(r\).
How does it work?
I’ll skip the precise details, but give some interpretation of this method. The random variable \(\Phi := \arccos V\) is the \(\phi\)-coordinate of the uniform point, which implies \(\sin \Phi=\sqrt
{1-V^2}\), due to basic trigonometric identities. The area element in polar coordinates is \(dA = \rho^2 \sin\phi d\phi d\theta \), which is constant with respect to \(\theta\). After integrating
with respect to \(\phi\), we see that the random variable \(V=\cos\Phi\) needs to be uniform (instead of \(\Phi\)) to ensure the point is uniformly located on the surface.
Method 2: Normal random variables
For each point, we generate three standard normal or Gaussian random variables, say, \(W_x\), \(W_y\), and \(W_z\), which are independent of each other. (The term standard here means the normal
random variables have mean \(\mu =0\) and standard deviation \(\sigma=1\).) The three random variables are the Cartesian components of the random point. We rescale the components by the Euclidean
norm, then multiply by the radius \(r\), giving
These are the Cartesian coordinates of a point uniformly scattered on a sphere with radius \(r\) and a centre at the origin.
How does it work?
The procedure is somewhat like the Box-Muller transform in reverse. In the post on the circle setting, I gave an outline of the proof, which I recommend reading. The joint density of the normal
random variables is from a multivariate normal distribution with zero correlation. This joint density is constant on the sphere, implying that the angle of the point \((W_x, W_y, W_z)\) will be
uniformly distributed.
The vector formed from the normal variables \((W_x, W_y,W_z)\) is a random variable with a chi distribution. We can see that the vector from the origin to the point \((X,Y,Z)\) has length one,
because we rescaled it with the Euclidean norm.
Depending on your plotting software, the points may more resemble points on an ellipsoid than a sphere due to the different scaling of the x, y and z axes. To fix this in MATLAB, run the command:
axis square. In Python, it’s not straightforward to do this, as it seems to lack an automatic function, so I have skipped it.
I have presented some results produced by code written in MATLAB and Python. The blue points are the Poisson points on the sphere. I have used a surface plot (with clear faces) to illustrate the
underling sphere in black.
Note: The aspect ratio in 3-D Python plots tends to squash the sphere slightly, but it is a sphere.
The code for all my posts is located online here. For this post, the code in MATLAB and Python is here. In Python I used the library mpl_toolkits for doing 3-D plots.
Poisson point process inside the sphere
Perhaps you want to simulate a Poisson point process inside the ball. There are different ways we can do this, but I will describe just one way, which builds off Method 1 for positioning the points
uniformly. (In a later post, I will modify Method 2, giving a way to uniformly position points inside the ball.)
For this simulation method, you need to make two simple modifications to the simulation procedure.
Number of points
The number of points of a Poisson point process inside a sphere of radius \(r>0\) is a Poisson random variable with mean \(\lambda V_3\), where \(V_3=4\pi r^3\) is the volume of the sphere.
Locations of points
We will modify Method 1 as outlined above. To sample the points uniformly in the sphere, you need to generate uniform variables on the unit interval \((0,1)\), take their cubic roots, and then,
multiply them by the radius \(r\). (This is akin to the step of taking the square root in the disk setting.) The random variables for the angular coordinates are generated as before.
Further reading
I recommend this blog post, which discusses different methods for randomly placing points on spheres and inside spheres (or, rather, balls) in a uniform manner. (Embedded in two dimensions, a sphere
is a circle and a ball is a disk.)
Our Method 2 for positioning points uniformly, which uses normal variables, comes from the paper:
• 1959, Muller, A note on a method for generating points uniformly on n-dimensional spheres.
This sampling method relies upon old observations that normal variables are connected to spheres and circles. I also found this post on a similar topic. Perhaps not surprisingly, the above paper is
written by the same Muller behind the Box-Muller method for sampling normal random variables.
Update: The connection between the normal distribution and rotational symmetry has been the subject of some recent 3Blue1Brown videos on YouTube.
Here is some sample Python code for creating a 3-D scatter plot.
Simulating a Poisson point process on a circle
In this post, I’ll take a break from the more theoretical posts. Instead I’ll describe how to simulate or sample a homogeneous Poisson point process on a circle. I have already simulated this point
process on a rectangle, triangle and disk. In some sense, I should have done this simulation method before the disk one, as it’s easier to simulate. I recommend reading that post first, as the
material presented here builds off it.
Sampling a homogeneous Poisson point process on a circle is rather straightforward. It just requires using a fixed radius and uniformly choose angles from interval \((0, 2\pi)\). But the circle
setting gives an opportunity to employ a different method for positioning points uniformly on circles and, more generally, spheres. This approach uses Gaussian random variables, and it becomes much
more efficient when the points are placed on high dimensional spheres.
Simulating a Poisson point process requires two steps: simulating the random number of points and then randomly positioning each point.
Number of points
The number of points of a Poisson point process on circle of radius \(r>0\) is a Poisson random variable with mean \(\lambda C\), where \(C=2\pi r\) is the circumference of the circle. You just need
to be able to need to produce (pseudo-)random numbers according to a Poisson distribution.
To generate Poisson variables in MATLAB, use the poissrnd function with the argument \(\lambda C\). In Python, use either the scipy.stats.poisson or numpy.random.poisson function from the SciPy or
NumPy libraries. (If you’re curious how Poisson simulation works, I suggest seeing this post for details on sampling Poisson random variables or, more accurately, variates.)
Locations of points
For a homogeneous Poisson point process, we need to uniformly position points on the underlying space, which is this case is a circle. We will look at two different ways to position points uniformly
on a circle. The first is arguably the most natural approach.
Method 1: Polar coordinates
We use polar coordinates due to the nature of the problem. To position all the points uniformly on a circle, we simple generate uniform numbers on the unit interval \((0,1)\). We then multiply these
random numbers by \(2\pi\).
We have generated polar coordinates for points uniformly located on the circle. To plot the points, we have to convert the coordinates back to Cartesian form by using the change of coordinates: \(x=
\rho\cos(\theta)\) and \(y=\rho\sin(\theta)\).
Method 2: Normal random variables
For each point, we generate two standard normal or Gaussian random variables, say, \(W_x\) and \(W_y\), which are independent of each other. (The term standard here means the normal random variables
have mean \(\mu =0\) and standard deviation \(\sigma=1\).) These two random variables are the Cartesian components of a random point. We then rescale the two values by the Euclidean norm, giving
These are the Cartesian coordinates of points uniformly scattered around a unit circle with centre at the origin. We multiply the two random values \(X\) and \(Y\) by the \(r>0\) for a circle with
radius \(r\).
How does it work?
The procedure is somewhat like the Box-Muller transform in reverse. I’ll give an outline of the proof. The joint density of the random variables \(W_x\) and \(W_y\) is that of the bivariate normal
distribution with zero correlation, meaning it has the joint density
$$ f(x,y)=\frac{1}{2\pi}e^{[-(x^2+y^2)/2]}.$$
We see that the function \(f\) is a constant when we trace around any line for which \((x^2+y^2)\) is a constant, which is simply the Cartesian equation for a circle (where the radius is the square
root of the aforementioned constant). This means that the angle of the point \((W_x, W_y)\) will be uniformly distributed.
Now we just need to look at the distance of the random point. The vector formed from the pair of normal variables \((W_x, W_y)\) is a Rayleigh random variable. We can see that the vector from the
origin to the point \((X,Y)\) has length one, because we rescaled it with the Euclidean norm.
I have presented some results produced by code written in MATLAB and Python. The blue points are the Poisson points on the sphere. I have used a surface plot (with clear faces) in black to illustrate
the underling sphere.
The code for all my posts is located online here. For this post, the code in MATLAB and Python is here.
Further reading
I recommend this blog post, which discusses different methods for randomly placing points on spheres and inside spheres (or, rather, balls) in a uniform manner. (Embedded in two dimensions, a sphere
is a circle and a ball is a disk.) A key paper on using normal variables is the following:
• 1959, Muller, A note on a method for generating points uniformly on n-dimensional spheres.
As I mentioned in the post on the disk, the third edition of the classic book Stochastic Geometry and its Applications by Chiu, Stoyan, Kendall and Mecke details on page 54 how to uniformly place
points on a disk. It just requires a small modification for the circle.
Cox point process
In previous posts I have often stressed the importance of the Poisson point process as a mathematical model. But it can be unsuitable for certain mathematical models. We can generalize it by first
considering a non-negative random measure, called a driving or directing measure. Then a Poisson point process, which is independent of the random driving measure, is generated by using the random
measure as its intensity or mean measure. This doubly stochastic construction gives what is called a Cox point process.
In practice we don’t typically observe the driving measure. This means that it’s impossible to distinguish a Cox point process from a Poisson point process if there’s only one realization available.
By conditioning on the random driving measure, we can use the properties of the Poisson point process to derive those of the resulting Cox point process.
By the way, Cox point processes are also known as doubly stochastic Poisson point processes. Guttorp and Thorarinsdottir argue that we should call them the Quenouille point processes, as Maurice
Quenouille introduced an example of it before Sir David Cox. But I opt for the more common name.
In this post I’ll cover a couple examples of Cox point processes. But first I will need to give a more precise mathematical definition.
We consider a point process defined on some underlying mathematical space \(\mathbb{S}\), which is sometimes called the carrier space or state space. The underlying space is often the real line \(\
mathbb{R}\), the plane \(\mathbb{R}^2\), or some other familiar mathematical space like a square lattice.
For the first definition, we use the concept of a random measure.
Let \(M\) be a non-negative random measure on \(\mathbb{S} \). Then a point process \(\Phi\) defined on some underlying space \(\mathbb{S}\) is a Cox point process driven by the intensity measure
\(M\) if the conditional distribution of \(\Phi\) is a Poisson point process with intensity function \(M\).
We can give a slightly less general definition of a Cox point process by using a random intensity function.
Let \(Z=\{Z(x):x\in\mathbb{S} \}\) be a non-negative random field such that with probability one, \(x\rightarrow Z(x)\) is a locally integrable function. Then a point process \(\Phi\) defined on
some underlying space \(\mathbb{S}\) is a Cox point process driven by \(Z\) if the conditional distribution of \(\Phi\) is a Poisson point process with intensity function \(Z\).
The random driving measure \(M\) is then simply the integral
M(B)=\int_B Z(x)\, dx , \quad B\subseteq S.
The random driving measures take different forms, giving different Cox point processes. But there is a general observation that can be made for all Cox point processes. For any region \(B \subseteq S
\), it can be shown that the number of points \(\Phi (B)\) adheres to the inequality
\mathbb{Var} [\Phi (B)] \geq \mathbb{E} [\Phi (B)],
where \(\mathbb{Var} [\Phi (B)] \) is the variance of the random variable \(\Phi (B)\). As a comparison, for a Poisson point process \(\Phi’\), the variance of \(\Phi’ (B)\) is simply \(\mathbb{Var}
[\Phi’ (B)] =\mathbb{E} [\Phi’ (B)]\). Due to its greater variance, the Cox point process is said to be over-dispersed compared to the Poisson point process.
Special cases
There is an virtually unlimited number of ways to define a random driving measure, where each one yields a different a Cox point process. But in general we are restricted by examining only tractable
and interesting Cox point processes. I will give some common examples, but I stress that the Cox point process family is very large.
Mixed Poisson point process
For the random driving measure \(M\), an obvious example is the product form \(M= Y \mu \), where \(Y\) is some independent non-negative random variable and \(\mu\) is the Lebesgue measure on \(\
mathbb{S}\). This driving measure gives the mixed Poisson point process. The random variable \(Y\) is the only source of randomness.
Log-Gaussian Cox point process
Instead of a random variable, we can use a non-negative random field to define a random driving measure. We then have the product \(M= Y \mu \), where \(Y\) is now some independent non-negative
random field. (A random field is a collection of random variables indexed by some set, which in this case is the underlying space \(\mathbb{S}\).)
Arguably the most tractable and used random field is the Gaussian random field. This random field, like Gaussian or normal random variables, takes both negative and positive values. But if we define
the random field such that its logarithm is a Gaussian field \(Z\), then we obtain the non-negative random driving measure \(M=\mu e^Z \), giving the log-Gaussian Cox point process.
This point process has found applications in spatial statistics.
Cox-Poisson line-point process
To construct this Cox point process, we first consider a Poisson line process, which I discussed previously. Given a Poisson line process, we then place an independent one-dimensional Poisson point
process on each line. We then obtain an example of a Cox point process, which we could call a Cox line-point process or a Cox-Poisson line-point process. (But I am not sure of the best name.)
Researchers have recently used this point process to study wireless communication networks in cities, where the streets correspond to Poisson lines. For example, see these two preprints:
Shot-noise Cox point process
We construct the next Cox point process by first considering a Poisson point process on the space \(\mathbb{S}\) to create a shot noise term. (Shot noise is just the sum of some function over all the
points of a point process.) We then use it as the driving measure of the Cox point process.
More specifically, we first introduce a kernel function \(k(\cdot,\cdot)\) on \(\mathbb{S}\), where \(k(x,\cdot)\) is a probability density function for all points \(x\in \mathbb{S}\). We then
consider a Poisson point process \(\Phi’\) on \(\mathbb{S}\times (0,\infty)\). We assume the Poisson point process \(\Phi’\) has a locally integrable intensity function \(\mu \).
(We can interpret the point process \(\Phi’\) as a spatially-dependent marked Poisson point process, where the unmarked Poisson point process is defined on \(\mathbb{S}\). We then assume each point \
(X\) of this unmarked point process has a mark \(T \in (0,\infty)\) with probability density \(\mu(X,t)\).)
The resulting shot noise
Z(x)= \sum_{(Y,T)\in \Phi’} T \, k(Y,x)\,,
gives the random field. We then use it as the random intensity function to drive the shot-noise Cox point process.
In previous posts, I have detailed how to simulate non-Poisson point processes such as the Matérn and Thomas cluster point processes. These are examples of a Neyman-Scott point process, which is a
special case of a shot noise Cox point process. All these point processes find applications in spatial statistics.
Unfortunately, there is no universal way to simulate all Cox point processes. (And even if there were one, it would not be the most optimal way for every Cox point process.) The simulation method
depends on how the Cox point process is constructed, which usually means how its directing or driving measure is defined.
In previous posts I have presented ways (with code) to simulate these Cox point processes:
• Matérn (cluster) point processes (code);
• Thomas (cluster) point processes (code);
• Cox-Poisson line-point process (code).
In addition to the Matérn and Thomas point processes, there are ways to simulate more general shot noise Cox point processes. I will cover that in another post.
Further reading
For general Cox point processes, I suggest: Chapter 6 in the monograph Poisson Processes by Kingman; Chapter 5 in Statistical Inference and Simulation for Spatial Point Processes by Møller and
Waagepetersen; and Section 5.2 in Stochastic Geometry and its Applications by Chiu, Stoyan, Kendall and Mecke. For a much more mathematical treatment, see Chapter 13 in Lectures on the Poisson
Process by Last and Penrose. Grandell wrote two detailed monographs titled Mixed Poisson Process and Doubly Stochastic Poisson Processes.
Motivated by applications in spatial statistics, Jesper Møller has (co)-written papers on specific Cox point processes. For example:
• 2001, Møller, Syversveen, and Waagepetersen, Log Gaussian Cox Processes;
• 2003, Møller, Shot noise Cox Processes;
• 2005, Møller and Torrisi,Generalised shot noise Cox processes.
I also suggest the survey article:
• 2003, Møller and Waagepetersen, Modern statistics for spatial point processes.
Signal strengths of a wireless network
In two previous posts, here and here, I discussed the importance of the quantity called the signal-to-interference ratio, which is usually abbreviated as SIR, for studying communication in wireless
networks. In everyday terms, for a listener to hear a certain speaker in a room full of people speaking, the ratio of the speaker’s volume to the sum of the volumes of everyone else heard by the
listener. The SIR is the communication bottleneck for any receiver and transmitter pair in a wireless network.
But the strengths (or power values) of the signals are of course also important. In this post I will detail how we can model them using a a simple network model with a single observer.
Propagation model
For a transmitter located at \(X_i\in \mathbb{R}^2\), researchers usually attempt to represent the received power of the signal \(P_i\) with a propagation model. Assuming the power is received at \(x
\in \mathbb{R}^2\), this mathematical model consists of a random and a deterministic component taking the general form
P_i(x)=F_i\,\ell(|X_i-x|) ,
where \(\ell(r)\) is a non-negative function in \(r>0\) and \(F_i\) is a non-negative random variable.
The function \(\ell(r)\) is called the pathloss function, and common choices include \(\ell(r)=(\kappa r)^{-\beta}\) and \(\ell(r)=\kappa e^{-\beta r}\), where \(\beta>0\) and \(\kappa>0\) are model
The random variables \(F_i\) represent signal phenomena such as multi-path fading and shadowing (also called shadow fading), caused by the signal interacting with the physical environment such as
buildings. It is often called fading or shadowing variables.
We assume the transmitters locations \(X_1,\dots,X_n\) are on the plane \(\mathbb{R}^2\). Researchers typically assume they form a random point process or, more precisely, the realization of a random
point process.
From two dimensions to one dimension
For studying wireless networks, a popular technique is to consider a wireless network from the perspective of a single observer or user. Researchers then consider the incoming or received signals
from the entire network at the location of this observer or user. They do this by considering the inverses of the signal strengths, namely
L_i(x): = \frac{1}{P_i}=\frac{1}{F_i \,\ell(|X_i-x|) }.
Mathematically, this random function is simply a mapping from the two-dimensional plane \(\mathbb{R}^2\) to the one-dimensional non-negative real line \(\mathbb{R}_0^+=[0,\infty)\).
If the transmitters are located according to a non-random point pattern or a random point process, this random mapping generates a random point process on the non-negative real line. The resulting
one-dimensional point process of the values \(L_1,L_2,\dots, \) has been called (independently) propagation (loss) process or path loss (with fading) process. More recently, my co-authors and I
decided to call it a projection process, but of course the precise name doesn’t mattter
Intensity measure of signal strengths
Assuming a continuous monotonic path loss function \(\ell\) and the fading variables \(F_1, F_2\dots\) are iid, if the transmitters form a stationary random point process with intensity \(\lambda\),
then the inverse signal strengths \(L_1,L_2,\dots \) form a random point process on the non-negative real line with the intensity measure \(M\).
M(t) =\lambda \pi \mathbb{E}( [\ell(t F)^{-1} ]^2)\,,
where \(\ell^{-1}\) is the generalized inverse of the function \(\ell\). This expression can be generalized for a non-stationary point process with general intensity measure \(\Lambda\).
The inverses \(1/L_1,1/L_2,\dots \), which are the signal strengths, forprocess with intensity measure
\bar{M}(s) =\lambda \pi \mathbb{E}( [\ell( F/s)^{-1} ]^2).
Poisson transmitters gives Poisson signal strengths
Assuming a continuous monotonic path loss function \(\ell\) and the fading variables \(F_1, F_2\dots\) are iid, if the transmitters form a Poisson point process with intensity \(\lambda\), then the
inverse signal strengths \(L_1,L_2,\dots \) form a Poisson point process on the non-negative real line with the intensity measure \(M\).
If \(L_1,L_2,\dots \) form a homogeneous Poisson point process, then the inverses \(1/L_1,1/L_2,\dots \) will also form a Poisson point process with intensity measure \(\bar{M}(s) =\lambda \pi \
mathbb{E}( [\ell( F/s)^{-1} ]^2). \)
Propagation invariance
For \(\ell(r)=(\kappa r)^{-\beta}\) , the expression for the intensity measure \(M\) reduces to
M(t) = \lambda \pi t^{-2/\beta} \mathbb{E}( F^{-2/\beta})/\kappa^2.
What’s striking here is that information of the fading variable \(F\) is captured simply by one moment \(\mathbb{E}( F^{-2/\beta}) \). This means that two different distributions will give the same
results as long as this moment is matching. My co-authors and I have been called this observation propagation invariance.
Some history
To study just the (inverse) signal strengths as a point process on the non-negative real line was a very useful insight. It was made independently in these two papers:
• 2008, Haenggi, A geometric interpretation of fading in wireless
networks: Theory and applications;
• 2010, Błaszczyszyn, Karray, and Klepper, Impact of the geometry, path-loss exponent and random shadowing on the mean interference factor in wireless cellular networks.
My co-authors and I presented a general expression for the intensity measure \(M\) in the paper:
• 2018, Keeler, Ross and Xia, When do wireless network signals appear Poisson?.
This paper is also contains examples of various network models.
Further reading
A good starting point on this topic is the Wikipedia article Stochastic geometry models of wireless networks. The paper that my co-authors and I wrote has details on the projection process.
With Bartek Błaszczyszyn, Sayan Mukherjee, and Martin Haenggi, I co-wrote a short monograph on SINR models called Stochastic Geometry Analysis of Cellular Networks, which is written at a slightly
more advanced level. The book puts an emphasis on studying the point process formed from inverse signal strengths, we call the projection process.
The Standard Model of wireless networks
In the previous post I discussed the signal-to-interference-plus ratio or SIR in wireless networks. If noise is included, then then signal-to-interference-plus-noise ratio or just SINR. But I will
just write about SIR, as most results that hold for SIR, will also hold for SINR without any great mathematical difficulty.
The SIR is an important quantity due to reasons coming from information theory. If you’re unfamiliar with it, I suggest reading the previous post.
In this post, I will describe a very popular mathematical model of the SIR, which I like to call the standard model. (This is not a term used in the literature as I have borrowed it from physics.)
Definition of SIR
To define the SIR, we consider a wireless network of \(n\) transmitters with positions located at \(X_1,\dots,X_n\) in some region of space. At some location \(x\), we write \(P_i(x)\) to denote the
power value of a signal received at \(x\) from transmitter \(X_i\). Then at location \(x\), the SIR with respect to transmitter \(X_i\) is
\text{SIR}(x,X_i) := \frac{P_i(x)}{\sum\limits_{j\neq i} P_j(x)} .
Researchers usually attempt to represent the received power of the signal \(P_i(x)\) with a propagation model. This mathematical model consists of a random and a deterministic component given by
P_i(x)=F_i\ell(|X_i-x|) ,
where \(\ell(r)\) is a non-negative function in \(r\geq 0\) and \(F_i\) is a non-negative random variable. The function \(\ell(r)\) is often called the path loss function. The random variables
represent random fading or shadowing.
Standard model
Based on the three model components of fading, path loss, and transmitter locations, there are many combinations possible. That said, researchers generally (I would guess, say, 90 percent or more)
use a single combination, which I call the standard model.
The three standard model assumptions are:
1. Singular power law path loss \(\ell(r)=(\kappa r)^{-\beta}\).
2. Exponential distribution for fading variables, which are independent and identically distributed (iid).
3. Poisson point process for transmitter locations.
Why these three? Well, in short, because they work very well together. Incredibly, it’s sometimes possible to get relatively a simple mathematical expression for, say, the coverage probability \(\
mathbb{P}[\text{SIR}(x,X_i)>\tau ]\), where \(\tau>0\).
I’ll now detail the reasons more specifically.
Path loss
The \(\ell(r)=(\kappa r)^{-\beta}\) is very simple, despite having a singularity at \(r=0\). This allows simple algebraic manipulation of equations.
Some, such as myself, are initially skeptical of this function as it gives an infinitely strong signal at the transmitter due to the singularity in the function \(\ell(r)=(\kappa r)^{-\beta}\). More
specifically, the path loss of the signal from transmitter \(X_i\) approaches infinity as \(x\) approaches \(X_i\) .
But apparently, overall, the singularity does not have a significant impact on most mathematical results, at least qualitatively. That said, one still observe consequences of this somewhat physically
unrealistic model assumption. And I strongly doubt enough care is taken by researchers to observe and note this.
Fading and shadowing variables
Interestingly, the original reason why exponential variables were used is because it allowed the SIR problem to be reformulated into a problem of a Laplace transform of a random variable, which for a
random variable \(Y\) is defined as
\mathcal{L}_Y(t)=\mathbb{E}(e^{- Y t}) \, .
where \(t\geq 0\). (This is essentially the moment-generating function with \(-t\) instead of \(t\).)
The reason for this connection is that the tail distribution of an exponential variable \(F\) with mean \(\mu\) is simply \(\mathbb{P}(F>t)= e^{-t/\mu}\). In short, with the exponential assumption,
various conditioning arguments eventually lead to Laplace transforms of random variables.
Transmitters locations
No prizes for guessing that researcher overwhelmingly use a (homogeneous) Poisson point process for the transmitter (or receiver) locations. When developing mathematical models with point processes,
if you can’t get any results with the Poisson point process, then abandon all hope.
It’s the easier to work with this point process due to its independence property, which leads to another useful property. For Poisson point process, the Palm distribution is known, which is the
distribution of a point process conditioned on a point (or collection of points) existing in a specific location of the underlying space on which the point process is defined. In general, the Palm
distribution is not known for many point processes.
Random propagation effects can lead to Poisson
A lesser known reason why researchers would use the Poisson point process is that, from the perspective of a single observer in the network, it can be used to capture the randomness in the signal
strengths. Poisson approximation results in probability imply that randomly perturbing the signal strengths can make signals appear more Poisson, by which I mean the signal strengths behave
stochastically or statistically as though they were created by a Poisson network of transmitters.
The end result is that a non-Poisson network can appear more Poisson, even if the transmitters do not resemble (the realization of) a Poisson point process. The source of randomness that makes a
non-Poisson network appear more Poisson is the random propagation effects of fading, shadowing, randomly varying antenna gains, and so on, or some combination of these.
Further reading
A good starting point on this topic is the Wikipedia article Stochastic geometry models of wireless networks. This paper is also good:
• 2009, Haenggi, Andrews, Baccelli, Dousse, Franceschetti, Stochastic Geometry and Random Graphs for the Analysis and Design of Wireless Networks.
This paper by my co-authors and I has some details on standard model and why a general network model behaving Poisson in terms of the signal strengths:
• 2018, Keeler, Ross and Xia, When do wireless network signals appear Poisson?.
Early books on the subject include the two-volume textbooks Stochastic Geometry and Wireless Networks by François Baccelli and Bartek Błaszczyszyn, where the first volume is on theory and the second
volume is on applications. Martin Haenggi wrote a very readable introductory book called Stochastic Geometry for Wireless networks.
Finally, I co-wrote with Bartek Błaszczyszyn, Sayan Mukherjee, and Martin Haenggi a short monograph on SINR models called Stochastic Geometry Analysis of Cellular Networks, which is written at a
slightly more advanced level. This book has a section on why signal strengths appear Poisson.
Signal-to-interference ratio in wireless networks
The fundamentals of information theory say that to successfully communicate across any potential communication link the signal strength of the communication must be stronger than that of the back
ground noise, which leads to the fundamental quantity known as signal-to-noise ratio. Information theory holds in very general (or, in mathematical speak, abstract) settings. The communication could
be, for example, a phone call on an old wired landline, two people talking in a bar, or a hand-written letter, for which the respective signals in these examples are the electrical current, speaker’s
voice, and the writing. (Respective examples of noise could be, for example, thermal noise in the wires, loud music, or coffee stains on the letter.)
In wireless networks, it’s possible for a receiver to simultaneously detect signals from multiple transmitters, but the receiver typically only wants to receive one signal. The other unwanted or
interfering signals form a type of noise, which is usually called interference, and the other (interfering) transmitters are called interferers. Consequently, researchers working on wireless networks
study the signal-to-interference ratio, which is usually abbreviated as SIR. Another name for the SIR is carrier-to-interference ratio.
If we also include background noise, which is coming not from the interferers, then the quantity becomes the signal-to-interference-plus-noise ratio or just SINR. But I will just write about SIR,
though jumping from SIR to SINR is usually not difficult mathematically.
The concept of SIR makes successful communication more difficult to model and predict, as it just doesn’t depend on the distance of the communication link. Putting the concept in everyday terms, for
a listener to hear a certain speaker in a room full of people all speaking to the listener, it is not simply the distance to the speaker, but rather the ratio of the speaker’s volume to the sum of
the volumes of everyone else heard by the listener. The SIR is the communication bottleneck for any receiver and transmitter pair in a wireless network.
In wireless network research, much work has been done to examine and understand communication success in terms of interference and SIR, which has led to a popular mathematical model that incorporates
how signals propagate and the locations of transmitters and receivers.
To define the SIR, we consider a wireless network of transmitters with positions located at \(X_1,\dots,X_n\) in some region of space. At some location \(x\), we write \(P_i(x)\) to denote the power
value of a signal received at \(x\) from transmitter \(X_i\). Then at location \(x\), the SIR with respect to transmitter \(X_i\) is
\text{SIR}(x,X_i) :=\frac{P_i(x)}{\sum\limits_{j\neq i} P_j(x)} =\frac{P_i(x)}{\sum\limits_{j=1}^{n} P_j(x)-P_i(x)} .
The numerator is the signal and the denominator is the interference. This ratio tells us that increasing the number of transmitters \(n\) decreases the original SIR values. But then, in exchange,
there is a greater number of transmitters for the receiver to connect to, some of which may have larger \(P_i(x)\) values and, subsequently, SIR values. This delicate trade-off makes it challenging
and interesting to mathematically analyze and design networks that deliver high SIR values.
Researchers usually assume that the SIR is random. A quantity of interest is the tail distribution of the SIR, namely
\mathbb{P}[\text{SIR}(x,X_i)>\tau ] := \frac{P_i(x)}{\sum\limits_{j\neq i} P_j(x)} \,,
where \(\tau>0\) is some parameter, sometimes called the SIR threshold. For a given value of \(\tau\), the probability \(\mathbb{P}[\text{SIR}(x,X_i)>\tau]\) is sometimes called the coverage
probability, which is simply the probability that a signal coming from \(X_i\) can be received successfully at location \(x\).
Mathematical models
Researchers usually attempt to represent the received power of the signal \(P_i(x)\) with a propagation model. This mathematical model consists of a random and a deterministic component taking the
general form
P_i(x)=F_i\ell(|X_i-x|) ,
where \(F_i\) is a non-negative random variable and \(\ell(r)\) is a non-negative function in \(r \geq 0\).
Path loss
The function \(\ell(r)\) is called the path loss function, and common choices include \(\ell(r)=(\kappa r)^{-\beta}\) and \(\ell(r)=\kappa e^{-\beta r}\), where \(\beta>0\) and \(\kappa>0\) are model
constants, which need to be fitted to (or estimated with) real world data.
Researchers generally assume that the so-called path loss function \(\ell(r)\) is decreasing in \(r\), but actual path loss (that is, the change in signal strength over a path travelled) typically
increases with distance \(r\). Researchers originally assumed path loss functions to be increasing, not decreasing, giving the alternative (but equivalent) propagation model
P_i(x)= F_i/\ell(|X_i-x|).
But nowadays researchers assume that the function \(\ell(r)\) is decreasing in \(r\). (Although, based on personal experience, there is still some disagreement on the convention.)
Fading and shadowing
With the random variable \(F_i\), researchers seek to represent signal phenomena such as multi-path fading and shadowing (also called shadow fading), caused by the signal interacting with the
physical environment such as buildings. These variables are often called fading or shadowing variables, depending on what physical phenomena they are representing.
Typical distributions for fading variables include the exponential and gamma distributions, while the log-normal distribution is usually used for shadowing. The entire collection of fading or
shadowing variables is nearly always assumed to be independent and identically distributed (iid), but very occasionally random fields are used to include a degree of statistical dependence between
Transmitters locations
In general, we assume the transmitters locations \(X_1,\dots,X_n\) are on the plane \(\mathbb{R}^2\). To model interference, researchers initially proposed non-random models, but they were considered
inaccurate and intractable. Now researchers typically use random point processes or, more precisely, the realizations of random point processes for the transmitter locations.
Not surprisingly, the first natural choice is the Poisson point process. Other point processes have been used such as Matérn and Thomas cluster point processes, and Matérn hard-core point processes,
as well as determinantal point processes, which I’ll discuss in another post.
Some history
Early random models of wireless networks go back to the 60s and 70s, but these were based simply on geometry: meaning a transmitter could communicate successfully to a receiver if they were closer
than some fixed distance. Edgar Gilbert created the field of continuum percolation with this significant paper:
• 1961, Gilbert, Random plane networks.
Interest in random geometrical models of wireless networks continued into the 70s and 80s. But there was no SIR in these models.
Motivated by understanding SIR, researchers in the late 1990s and early 2000s started tackling SIR problems by using a random model based on techniques from stochastic geometry and point processes.
Early papers include:
• 1997, Baccelli, Klein, Lebourges ,and Zuyev, Stochastic geometry and architecture of communication networks;
• 2003, Baccelli and Błaszczyszyn , On a coverage process ranging from the Boolean model to the Poisson Voronoi tessellation, with applications to wireless communications;
• 2006, Baccelli, Mühlethaler, and Błaszczyszyn, An Aloha protocol for multihop mobile wireless networks.
But they didn’t know that some of their results had already been discovered independently by researchers working on wireless networks in the early 1990s. These papers include:
• 1994, Pupolin and Zorzi, Outage probability in multiple access packet radio networks in the presence of fading;
• 1990, Sousa and Silvester, Optimum transmission ranges in a direct-sequence spread-spectrum multihop packet radio network.
The early work focused more on small-scale networks like wireless ad hoc networks. Then the focus shifted dramatically to mobile or cellular phone networks with the publication of the paper:
• 2011, Andrews, Baccelli, Ganti, A tractable approach to coverage and rate in cellular networks.
It’s can be said with confidence that this paper inspired much of the interest in using point processes to develop models of wireless networks. The work generally considers the SINR in the downlink
channel for which the incoming signals originate from the phone base stations.
Further reading
A good starting point on this topic is the Wikipedia article Stochastic geometry models of wireless networks. This paper is also good:
• 2009, Haenggi, Andrews, Baccelli, Dousse, Franceschetti, Stochastic Geometry and Random Graphs for the Analysis and Design of Wireless Networks.
Early books on the subject include the two-volume textbooks Stochastic Geometry and Wireless Networks by François Baccelli and Bartek Błaszczyszyn, where the first volume is on theory and the second
volume is on applications. Martin Haenggi wrote a very readable introductory book called Stochastic Geometry for Wireless networks.
Finally, Bartek Błaszczyszyn, Sayan Mukherjee, Martin Haenggi, and I wrote a short book on SINR models called Stochastic Geometry Analysis of Cellular Networks, which is written at a slightly more
advanced level. The book put an emphasis on studying the point process formed from inverse signal strengths, we call the projection process. | {"url":"https://hpaulkeeler.com/tag/poisson-point-process/","timestamp":"2024-11-02T01:05:16Z","content_type":"text/html","content_length":"178249","record_id":"<urn:uuid:8e65e8c3-a990-4791-94c6-58460aeecb15>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00408.warc.gz"} |
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The sum of two digits of a number is 15. if 9 is added to the number, the digit is reversed. The numbers are?Options8578778287
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