Simulation-based optimization

Simulation-based optimization (also known as simply simulation optimization) integrates optimization techniques into simulation modeling and analysis. Because of the complexity of the simulation, the objective function may become difficult and expensive to evaluate. Usually, the underlying simulation model is stochastic, so that the objective function must be estimated using statistical estimation techniques (called output analysis in simulation methodology).
Once a system is mathematically modeled, computer-based simulations provide information about its behavior. Parametric simulation methods can be used to improve the performance of a system. In this method, the input of each variable is varied with other parameters remaining constant and the effect on the design objective is observed. This is a time-consuming method and improves the performance partially. To obtain the optimal solution with minimum computation and time, the problem is solved iteratively where in each iteration the solution moves closer to the optimum solution. Such methods are known as ‘numerical optimization’, ‘simulation-based optimization’ or 'simulation-based multi-objective optimization' used when more than one objective is involved.
In simulation experiment, the goal is to evaluate the effect of different values of input variables on a system. However, the interest is sometimes in finding the optimal value for input variables in terms of the system outcomes. One way could be running simulation experiments for all possible input variables. However, this approach is not always practical due to several possible situations and it just makes it intractable to run experiments for each scenario. For example, there might be too many possible values for input variables, or the simulation model might be too complicated and expensive to run for a large set of input variable values. In these cases, the goal is to iterative find optimal values for the input variables rather than trying all possible values. This process is called simulation optimization.
Specific simulation–based optimization methods can be chosen according to Figure 1 based on the decision variable types.
Optimization exists in two main branches of operations research:
"Optimization parametric (static)" – The objective is to find the values of the parameters, which are “static” for all states, with the goal of maximizing or minimizing a function. In this case, one can use mathematical programming, such as linear programming. In this scenario, simulation helps when the parameters contain noise or the evaluation of the problem would demand excessive computer time, due to its complexity.
"Optimization control (dynamic)" – This is used largely in computer science and electrical engineering. The optimal control is per state and the results change in each of them. One can use mathematical programming, as well as dynamic programming. In this scenario, simulation can generate random samples and solve complex and large-scale problems.
Simulation-based optimization methods.
Some important approaches in simulation optimization are discussed below.
Statistical ranking and selection methods (R/S).
Ranking and selection methods are designed for problems where the alternatives are fixed and known, and simulation is used to estimate the system performance.
In the simulation optimization setting, applicable methods include indifference zone approaches, optimal computing budget allocation, and knowledge gradient algorithms.
Response surface methodology (RSM).
In response surface methodology, the objective is to find the relationship between the input variables and the response variables. The process starts from trying to fit a linear regression model. If the P-value turns out to be low, then a higher degree polynomial regression, which is usually quadratic, will be implemented. The process of finding a good relationship between input and response variables will be done for each simulation test. In simulation optimization, response surface method can be used to find the best input variables that produce desired outcomes in terms of response variables.
Heuristic methods.
Heuristic methods change accuracy by speed. Their goal is to find a good solution faster than the traditional methods, when they are too slow or fail in solving the problem. Usually they find local optimal instead of the optimal value; however, the values are considered close enough of the final solution. Examples of these kinds of methods include tabu search and genetic algorithms.
Metamodels enable researchers to obtain reliable approximate model outputs without running expensive and time-consuming computer simulations. Therefore, the process of model optimization can take less computation time and cost.
Stochastic approximation.
Stochastic approximation is used when the function cannot be computed directly, only estimated via noisy observations. In these scenarios, this method (or family of methods) looks for the extrema of these function. The objective function would be:
Derivative-free optimization methods.
Derivative-free optimization is a subject of mathematical optimization. This method is applied to a certain optimization problem when its derivatives are unavailable or unreliable. Derivative-free methods establish a model based on sample function values or directly draw a sample set of function values without exploiting a detailed model. Since it needs no derivatives, it cannot be compared to derivative-based methods.
For unconstrained optimization problems, it has the form:
The limitations of derivative-free optimization:
1. Some methods cannot handle optimization problems with more than a few variables; the results are usually not so accurate. However, there are numerous practical cases where derivative-free methods have been successful in non-trivial simulation optimization problems that include randomness manifesting as "noise" in the objective function. See, for example, the following
2. When confronted with minimizing non-convex functions, it will show its limitation.
3. Derivative-free optimization methods are relatively simple and easy, but, like most optimization methods, some care is required in practical implementation (e.g., in choosing the algorithm parameters).
Dynamic programming and neuro-dynamic programming.
Dynamic programming.
Dynamic programming deals with situations where decisions are made in stages. The key to this kind of problem is to trade off the present and future costs.
One dynamic basic model has two features:
1) It has a discrete time dynamic system.
2) The cost function is additive over time.
For discrete features, dynamic programming has the form:
For the cost function, it has the form:
formula_14 is the cost at the end of the process.
As the cost cannot be optimized meaningfully, it can be used the expect value:
Neuro-dynamic programming.
Neuro-dynamic programming is the same as dynamic programming except that the former has the concept of approximation architectures. It combines artificial intelligence, simulation-base algorithms, and functional approach techniques. “Neuro” in this term origins from artificial intelligence community. It means learning how to make improved decisions for the future via built-in mechanism based on the current behavior. The most important part of neuro-dynamic programming is to build a trained neuro network for the optimal problem.
Limitations.
Simulation-based optimization has some limitations, such as the difficulty of creating a model that imitates the dynamic behavior of a system in a way that is considered good enough for its representation. Another problem is complexity in the determining uncontrollable parameters of both real-world system and simulation. Moreover, only a statistical estimation of real values can be obtained. It is not easy to determine the objective function, since it is a result of measurements, which can be harmful to the solutions.


Balanced clustering

Balanced clustering is a special case of clustering where, in the strictest sense, cluster sizes are constrained to formula_1 or formula_2, where formula_3 is the number of points and formula_4 is the number of clusters. A typical algorithm is balanced k-means, which minimizes mean square error (MSE). Another type of balanced clustering called balance-driven clustering has a two-objective cost function that minimizes both the imbalance and the MSE. Typical cost functions are ratio cut and Ncut. Balanced clustering can be used for example in scenarios where freight has to be delivered to formula_3 locations with formula_4 cars. It is then preferred that each car delivers to an equal number of locations.
Software.
There exists implementations for balanced k-means and Ncut


Autonomous convergence theorem

In mathematics, an autonomous convergence theorem is one of a family of related theorems which specify conditions guaranteeing global asymptotic stability of a continuous autonomous dynamical system.
History.
The Markus–Yamabe conjecture was formulated as an attempt to give conditions for global stability of continuous dynamical systems in two dimensions. However, the Markus–Yamabe conjecture does not hold for dimensions higher than two, a problem which autonomous convergence theorems attempt to address. The first autonomous convergence theorem was constructed by Russell Smith. This theorem was later refined by Michael Li and James Muldowney.
An example autonomous convergence theorem.
A comparatively simple autonomous convergence theorem is as follows:
This autonomous convergence theorem is very closely related to the Banach fixed-point theorem.
How autonomous convergence works.
Note: this is an intuitive description of how autonomous convergence theorems guarantee stability, not a strictly mathematical description.
The key point in the example theorem given above is the existence of a negative logarithmic norm, which is derived from a vector norm. The vector norm effectively measures the distance between points in the vector space on which the differential equation is defined, and the negative logarithmic norm means that distances between points, as measured by the corresponding vector norm, are decreasing with time under the action of formula_5. So long as the trajectories of all points in the phase space are bounded, all trajectories must therefore eventually converge to the same point.
The autonomous convergence theorems by Russell Smith, Michael Li and James Muldowney work in a similar manner, but they rely on showing that the area of two-dimensional shapes in phase space decrease with time. This means that no periodic orbits can exist, as all closed loops must shrink to a point. If the system is bounded, then according to Pugh's closing lemma there can be no chaotic behaviour either, so all trajectories must eventually reach an equilibrium.
Michael Li has also developed an extended autonomous convergence theorem which is applicable to dynamical systems containing an invariant manifold.


Kernel embedding of distributions

In machine learning, the kernel embedding of distributions (also called the kernel mean or mean map) comprises a class of nonparametric methods in which a probability distribution is represented as an element of a reproducing kernel Hilbert space (RKHS). A generalization of the individual data-point feature mapping done in classical kernel methods, the embedding of distributions into infinite-dimensional feature spaces can preserve all of the statistical features of arbitrary distributions, while allowing one to compare and manipulate distributions using Hilbert space operations such as inner products, distances, projections, linear transformations, and spectral analysis. This learning framework is very general and can be applied to distributions over any space formula_1 on which a sensible kernel function (measuring similarity between elements of formula_1) may be defined. For example, various kernels have been proposed for learning from data which are: vectors in formula_3, discrete classes/categories, strings, graphs/networks, images, time series, manifolds, dynamical systems, and other structured objects. The theory behind kernel embeddings of distributions has been primarily developed by Alex Smola, Le Song , Arthur Gretton, and Bernhard Schölkopf. A review of recent works on kernel embedding of distributions can be found in.
The analysis of distributions is fundamental in machine learning and statistics, and many algorithms in these fields rely on information theoretic approaches such as entropy, mutual information, or Kullback–Leibler divergence. However, to estimate these quantities, one must first either perform density estimation, or employ sophisticated space-partitioning/bias-correction strategies which are typically infeasible for high-dimensional data. Commonly, methods for modeling complex distributions rely on parametric assumptions that may be unfounded or computationally challenging (e.g. Gaussian mixture models), while nonparametric methods like kernel density estimation (Note: the smoothing kernels in this context have a different interpretation than the kernels discussed here) or characteristic function representation (via the Fourier transform of the distribution) break down in high-dimensional settings.
Methods based on the kernel embedding of distributions sidestep these problems and also possess the following advantages:
Thus, learning via the kernel embedding of distributions offers a principled drop-in replacement for information theoretic approaches and is a framework which not only subsumes many popular methods in machine learning and statistics as special cases, but also can lead to entirely new learning algorithms.
Definitions.
Let formula_4 denote a random variable with domain formula_5 and distribution formula_6 Given a kernel formula_7 on formula_8 the Moore–Aronszajn theorem asserts the existence of a RKHS formula_9 (a Hilbert space of functions formula_10 equipped with an inner product formula_11 and a norm formula_12) in which the element formula_13 satisfies the reproducing property
One may alternatively consider formula_13 an implicit feature mapping formula_16 from formula_5 to formula_18 (which is therefore also called the feature space), so that formula_19 can be viewed as a measure of similarity between points formula_20 While the similarity measure is linear in the feature space, it may be highly nonlinear in the original space depending on the choice of kernel.
Kernel embedding.
The kernel embedding of the distribution formula_21 in formula_18 (also called the kernel mean or mean map) is given by:
If formula_21 allows a square integrable density formula_25, then formula_26, where formula_27 is the Hilbert–Schmidt integral operator. A kernel is "characteristic" if the mean embedding formula_28 is injective. Each distribution can thus be uniquely represented in the RKHS and all statistical features of distributions are preserved by the kernel embedding if a characteristic kernel is used.
Empirical kernel embedding.
Given formula_29 training examples formula_30 drawn independently and identically distributed (i.i.d.) from formula_31 the kernel embedding of formula_21 can be empirically estimated as
Joint distribution embedding.
If formula_34 denotes another random variable (for simplicity, assume the co-domain of formula_34 is also formula_5 with the same kernel formula_7 which satisfies formula_38), then the joint distribution formula_39 can be mapped into a tensor product feature space formula_40 via
By the equivalence between a tensor and a linear map, this joint embedding may be interpreted as an uncentered cross-covariance operator formula_42 from which the cross-covariance of functions formula_43 can be computed as
Given formula_29 pairs of training examples formula_46 drawn i.i.d. from formula_21, we can also empirically estimate the joint distribution kernel embedding via
Conditional distribution embedding.
Given a conditional distribution formula_49 one can define the corresponding RKHS embedding as
Note that the embedding of formula_51 thus defines a family of points in the RKHS indexed by the values formula_52 taken by conditioning variable formula_4. By fixing formula_4 to a particular value, we obtain a single element in formula_9, and thus it is natural to define the operator
which given the feature mapping of formula_52 outputs the conditional embedding of formula_34 given formula_59 Assuming that for all formula_60 it can be shown that
This assumption is always true for finite domains with characteristic kernels, but may not necessarily hold for continuous domains. Nevertheless, even in cases where the assumption fails, formula_62 may still be used to approximate the conditional kernel embedding formula_63 and in practice, the inversion operator is replaced with a regularized version of itself formula_64 (where formula_65 denotes the identity matrix).
Given training examples formula_66 the empirical kernel conditional embedding operator may be estimated as
where formula_68 are implicitly formed feature matrices, formula_69 is the Gram matrix for samples of formula_4, and formula_71 is a regularization parameter needed to avoid overfitting.
Thus, the empirical estimate of the kernel conditional embedding is given by a weighted sum of samples of formula_34 in the feature space:
where formula_74 and formula_75
Rules of probability as operations in the RKHS.
This section illustrates how basic probabilistic rules may be reformulated as (multi)linear algebraic operations in the kernel embedding framework and is primarily based on the work of Song et al. The following notation is adopted:
In practice, all embeddings are empirically estimated from data formula_132 and it assumed that a set of samples formula_133 may be used to estimate the kernel embedding of the prior distribution formula_134.
Kernel sum rule.
In probability theory, the marginal distribution of formula_4 can be computed by integrating out formula_125 from the joint density (including the prior distribution on formula_34)
The analog of this rule in the kernel embedding framework states that formula_139 the RKHS embedding of formula_140, can be computed via
where formula_142 is the kernel embedding of formula_143 In practical implementations, the kernel sum rule takes the following form
where
is the empirical kernel embedding of the prior distribution, formula_146 formula_147, and formula_148 are Gram matrices with entries formula_149 respectively.
Kernel chain rule.
In probability theory, a joint distribution can be factorized into a product between conditional and marginal distributions
The analog of this rule in the kernel embedding framework states that formula_151 the joint embedding of formula_152 can be factorized as a composition of conditional embedding operator with the auto-covariance operator associated with formula_153
where
In practical implementations, the kernel chain rule takes the following form
Kernel Bayes' rule.
In probability theory, a posterior distribution can be expressed in terms of a prior distribution and a likelihood function as
The analog of this rule in the kernel embedding framework expresses the kernel embedding of the conditional distribution in terms of conditional embedding operators which are modified by the prior distribution
where from the chain rule:
In practical implementations, the kernel Bayes' rule takes the following form
where
Two regularization parameters are used in this framework: formula_164 for the estimation of formula_165 and formula_166 for the estimation of the final conditional embedding operator
The latter regularization is done on square of formula_168 because formula_169 may not be positive definite.
Applications.
Measuring distance between distributions.
The maximum mean discrepancy (MMD) is a distance-measure between distributions formula_170 and formula_171 which is defined as the squared distance between their embeddings in the RKHS
While most distance-measures between distributions such as the widely used Kullback–Leibler divergence either require density estimation (either parametrically or nonparametrically) or space partitioning/bias correction strategies, the MMD is easily estimated as an empirical mean which is concentrated around the true value of the MMD. The characterization of this distance as the "maximum mean discrepancy" refers to the fact that computing the MMD is equivalent to finding the RKHS function that maximizes the difference in expectations between the two probability distributions
Kernel two-sample test.
Given "n" training examples from formula_170 and "m" samples from formula_171, one can formulate a test statistic based on the empirical estimate of the MMD
to obtain a two-sample test of the null hypothesis that both samples stem from the same distribution (i.e. formula_177) against the broad alternative formula_178.
Density estimation via kernel embeddings.
Although learning algorithms in the kernel embedding framework circumvent the need for intermediate density estimation, one may nonetheless use the empirical embedding to perform density estimation based on "n" samples drawn from an underlying distribution formula_179. This can be done by solving the following optimization problem
where the maximization is done over the entire space of distributions on formula_182 Here, formula_183 is the kernel embedding of the proposed density formula_184 and formula_185 is an entropy-like quantity (e.g. Entropy, KL divergence, Bregman divergence). The distribution which solves this optimization may be interpreted as a compromise between fitting the empirical kernel means of the samples well, while still allocating a substantial portion of the probability mass to all regions of the probability space (much of which may not be represented in the training examples). In practice, a good approximate solution of the difficult optimization may be found by restricting the space of candidate densities to a mixture of "M" candidate distributions with regularized mixing proportions. Connections between the ideas underlying Gaussian processes and conditional random fields may be drawn with the estimation of conditional probability distributions in this fashion, if one views the feature mappings associated with the kernel as sufficient statistics in generalized (possibly infinite-dimensional) exponential families.
Measuring dependence of random variables.
A measure of the statistical dependence between random variables formula_4 and formula_34 (from any domains on which sensible kernels can be defined) can be formulated based on the Hilbert–Schmidt Independence Criterion
and can be used as a principled replacement for mutual information, Pearson correlation or any other dependence measure used in learning algorithms. Most notably, HSIC can detect arbitrary dependencies (when a characteristic kernel is used in the embeddings, HSIC is zero if and only if the variables are independent), and can be used to measure dependence between different types of data (e.g. images and text captions). Given "n" i.i.d. samples of each random variable, a simple parameter-free unbiased estimator of HSIC which exhibits concentration about the true value can be computed in formula_189 time, where the Gram matrices of the two datasets are approximated using formula_190 with formula_191. The desirable properties of HSIC have led to the formulation of numerous algorithms which utilize this dependence measure for a variety of common machine learning tasks such as: feature selection (BAHSIC ), clustering (CLUHSIC ), and dimensionality reduction (MUHSIC ).
HSIC can be extended to measure the dependence of multiple random variables. The question of when HSIC captures independence in this case has recently been studied: for
more than two variables
Kernel belief propagation.
Belief propagation is a fundamental algorithm for inference in graphical models in which nodes repeatedly pass and receive messages corresponding to the evaluation of conditional expectations. In the kernel embedding framework, the messages may be represented as RKHS functions and the conditional distribution embeddings can be applied to efficiently compute message updates. Given "n" samples of random variables represented by nodes in a Markov random field, the incoming message to node "t" from node "u" can be expressed as
if it assumed to lie in the RKHS. The kernel belief propagation update message from "t" to node "s" is then given by
where formula_195 denotes the element-wise vector product, formula_196 is the set of nodes connected to "t" excluding node "s", formula_197, formula_198 are the Gram matrices of the samples from variables formula_199, respectively, and formula_200 is the feature matrix for the samples from formula_201.
Thus, if the incoming messages to node "t" are linear combinations of feature mapped samples from formula_202, then the outgoing message from this node is also a linear combination of feature mapped samples from formula_203. This RKHS function representation of message-passing updates therefore produces an efficient belief propagation algorithm in which the potentials are nonparametric functions inferred from the data so that arbitrary statistical relationships may be modeled.
Nonparametric filtering in hidden Markov models.
In the hidden Markov model (HMM), two key quantities of interest are the transition probabilities between hidden states formula_204 and the emission probabilities formula_205 for observations. Using the kernel conditional distribution embedding framework, these quantities may be expressed in terms of samples from the HMM. A serious limitation of the embedding methods in this domain is the need for training samples containing hidden states, as otherwise inference with arbitrary distributions in the HMM is not possible.
One common use of HMMs is filtering in which the goal is to estimate posterior distribution over the hidden state formula_206 at time step "t" given a history of previous observations formula_207 from the system. In filtering, a belief state formula_208 is recursively maintained via a prediction step (where updates formula_209 are computed by marginalizing out the previous hidden state) followed by a conditioning step (where updates formula_210 are computed by applying Bayes' rule to condition on a new observation). The RKHS embedding of the belief state at time "t+1" can be recursively expressed as
by computing the embeddings of the prediction step via the kernel sum rule and the embedding of the conditioning step via kernel Bayes' rule. Assuming a training sample formula_212 is given, one can in practice estimate
and filtering with kernel embeddings is thus implemented recursively using the following updates for the weights formula_214
where formula_217 denote the Gram matrices of formula_218 and formula_219 respectively, formula_220 is a transfer Gram matrix defined as formula_221 and formula_222
Support measure machines.
The support measure machine (SMM) is a generalization of the support vector machine (SVM) in which the training examples are probability distributions paired with labels formula_223. SMMs solve the standard SVM dual optimization problem using the following expected kernel
which is computable in closed form for many common specific distributions formula_225 (such as the Gaussian distribution) combined with popular embedding kernels formula_7 (e.g. the Gaussian kernel or polynomial kernel), or can be accurately empirically estimated from i.i.d. samples formula_227 via
Under certain choices of the embedding kernel formula_7, the SMM applied to training examples formula_230 is equivalent to a SVM trained on samples formula_231, and thus the SMM can be viewed as a "flexible" SVM in which a different data-dependent kernel (specified by the assumed form of the distribution formula_225) may be placed on each training point.
Domain adaptation under covariate, target, and conditional shift.
The goal of domain adaptation is the formulation of learning algorithms which generalize well when the training and test data have different distributions. Given training examples formula_233 and a test set formula_234 where the formula_235 are unknown, three types of differences are commonly assumed between the distribution of the training examples formula_236 and the test distribution formula_237:
By utilizing the kernel embedding of marginal and conditional distributions, practical approaches to deal with the presence of these types of differences between training and test domains can be formulated. Covariate shift may be accounted for by reweighting examples via estimates of the ratio formula_244 obtained directly from the kernel embeddings of the marginal distributions of formula_4 in each domain without any need for explicit estimation of the distributions. Target shift, which cannot be similarly dealt with since no samples from formula_34 are available in the test domain, is accounted for by weighting training examples using the vector formula_247 which solves the following optimization problem (where in practice, empirical approximations must be used)
To deal with location scale conditional shift, one can perform a LS transformation of the training points to obtain new transformed training data formula_250 (where formula_195 denotes the element-wise vector product). To ensure similar distributions between the new transformed training samples and the test data, formula_252 are estimated by minimizing the following empirical kernel embedding distance
In general, the kernel embedding methods for dealing with LS conditional shift and target shift may be combined to find a reweighted transformation of the training data which mimics the test distribution, and these methods may perform well even in the presence of conditional shifts other than location-scale changes.
Domain generalization via invariant feature representation.
Given "N" sets of training examples sampled i.i.d. from distributions formula_254, the goal of domain generalization is to formulate learning algorithms which perform well on test examples sampled from a previously unseen domain formula_255 where no data from the test domain is available at training time. If conditional distributions formula_256 are assumed to be relatively similar across all domains, then a learner capable of domain generalization must estimate a functional relationship between the variables which is robust to changes in the marginals formula_170. Based on kernel embeddings of these distributions, Domain Invariant Component Analysis (DICA) is a method which determines the transformation of the training data that minimizes the difference between marginal distributions while preserving a common conditional distribution shared between all training domains. DICA thus extracts "invariants", features that transfer across domains, and may be viewed as a generalization of many popular dimension-reduction methods such as kernel principal component analysis, transfer component analysis, and covariance operator inverse regression.
Defining a probability distribution formula_258 on the RKHS formula_9 with
DICA measures dissimilarity between domains via distributional variance which is computed as
where
so formula_263 is a formula_264 Gram matrix over the distributions from which the training data are sampled. Finding an orthogonal transform onto a low-dimensional subspace "B" (in the feature space) which minimizes the distributional variance, DICA simultaneously ensures that "B" aligns with the bases of a central subspace "C" for which formula_34 becomes independent of formula_4 given formula_267 across all domains. In the absence of target values formula_34, an unsupervised version of DICA may be formulated which finds a low-dimensional subspace that minimizes distributional variance while simultaneously maximizing the variance of formula_4 (in the feature space) across all domains (rather than preserving a central subspace).
Distribution regression.
In distribution regression, the goal is to regress from probability distributions to reals (or vectors). Many important machine learning and statistical tasks fit into this framework, including multi-instance learning, and point estimation problems without analytical solution (such as hyperparameter or entropy estimation). In practice only samples from sampled distributions are observable, and the estimates have to rely on similarities computed between "sets of points". Distribution regression has been successfully applied for example in supervised entropy learning, and aerosol prediction using multispectral satellite images.
Given formula_270 training data, where the formula_271 bag contains samples from a probability distribution formula_272 and the formula_273 output label is formula_274, one can tackle the distribution regression task by taking the embeddings of the distributions, and learning the regressor from the embeddings to the outputs. In other words, one can consider the following kernel ridge regression problem formula_275
where
with a formula_7 kernel on the domain of formula_272-s formula_280, formula_281 is a kernel on the embedded distributions, and formula_282 is the RKHS determined by formula_281. Examples for formula_281 include the linear kernel formula_285, the Gaussian kernel formula_286, the exponential kernel formula_287, the Cauchy kernel formula_288, the generalized t-student kernel formula_289, or the inverse multiquadrics kernel formula_290.
The prediction on a new distribution formula_291 takes the simple, analytical form
where formula_293, formula_294, formula_295, formula_296. Under mild regularity conditions this estimator can be shown to be consistent and it can achieve the one-stage sampled (as if one had access to the true formula_272-s) minimax optimal rate. In the formula_298 objective function formula_299-s are real numbers; the results can also be extended to the case when formula_299-s are formula_301-dimensional vectors, or more generally elements of a separable Hilbert space using operator-valued formula_281 kernels.
Example.
In this simple example, which is taken from Song et al., formula_303 are assumed to be discrete random variables which take values in the set formula_304 and the kernel is chosen to be the Kronecker delta function, so formula_305. The feature map corresponding to this kernel is the standard basis vector formula_306. The kernel embeddings of such a distributions are thus vectors of marginal probabilities while the embeddings of joint distributions in this setting are formula_307 matrices specifying joint probability tables, and the explicit form of these embeddings is
The conditional distribution embedding operator,
is in this setting a conditional probability table
and
Thus, the embeddings of the conditional distribution under a fixed value of formula_4 may be computed as
In this discrete-valued setting with the Kronecker delta kernel, the kernel sum rule becomes
The kernel chain rule in this case is given by


Best worst method

Best Worst Method (BWM) is a multi-criteria decision-making (MCDM) method that was proposed by Dr. Jafar Rezaei in 2015. The method is used to evaluate a set of alternatives with respect to a set of decision criteria. The BWM is based on pairwise comparisons of the decision criteria. That is, after identifying the decision criteria by the decision-maker (DM), two criteria are selected by the DM: the best criterion and the worst criterion. The best criterion is the one that has the most important role in making the decision, while the worst criterion has the opposite role. The DM then gives his/her preferences of the best criterion over all the other criteria and also his/her preferences of all the criteria over the worst criterion using a number from a predefined scale (e.g. 1 to 9). These two sets of pairwise comparisons are used as input for an optimization problem, the optimal results of which are the weights of the criteria. The salient feature of the BWM is that it uses a structured way to generate pairwise comparisons which leads to reliable results.


Dick Volz Award

The Dick Volz Best US PhD Thesis in Robotics and Automation is a yearly award to recognize outstanding Ph.D. thesis in the field of robotics and automation at any research institution in the United States of America. It is awarded with a four years delay as it is based both on thesis quality as well as post-graduation impact, hence, the 2007 Dick Volz Best US PhD Thesis Award was awarded in 2011. Its European counterpart is the Georges Giralt PhD Award.
The award is named after Professor Emeritus Richard A. Volz. He was a former Texas A&M Department of Computer Science department head and the president of the IEEE Robotics and Automation Society in 2006-2007. To honor his outstanding research on robotics and control as well as his mentoring, the Dick Volz Best US PhD Thesis in Robotics and Automation was established.
The list of award recipients includes the following:
The award committee includes Seth A. Hutchinson (University of Illinois Urbana-Champaign), John M. Hollerbach (University of Utah), Vijay Kumar (University of Pennsylvania), Gaurav Sukhatme (University of Southern California), and Henrik I. Christensen (Georgia Tech).


Flat pseudospectral method

The flat pseudospectral method is part of the family of the Ross–Fahroo pseudospectral methods introduced by Ross and Fahroo. The method combines the concept of differential flatness with pseudospectral optimal control to generate outputs in the so-called flat space.
Concept.
Because the differentiation matrix, formula_1, in a pseudospectral method is square, higher-order derivatives of any polynomial, formula_2, can be obtained by powers of formula_1,
where formula_5 is the pseudospectral variable and formula_6 is a finite positive integer.
By differential flatness, there exists functions formula_7 and formula_8 such that the state and control variables can be written as,
The combination of these concepts generates the flat pseudospectral method; that is, x and u are written as,
Thus, an optimal control problem can be quickly and easily transformed to a problem with just the Y pseudospectral variable.


Positive systems

Positive systems constitute a class of systems that has the important property that its state variables are never negative, given a positive initial state. These systems appear frequently in practical applications, as these variables represent physical quantities, with positive sign (levels, heights, concentrations, etc.).
The fact that a system is positive has important implications in the control system design. For instance, an asymptotically stable positive linear time-invariant system always admits a diagonal quadratic Lyapunov function, which makes these systems more numerical tractable in the context of Lyapunov analysis.
It is also important to take this positivity into account for state observer design, as standard observers (for example Luenberger observers) might give illogical negative values.
Conditions for positivity.
A continuous-time linear system formula_1 is positive if and only if A is a Metzler matrix.
A discrete-time linear system formula_2 is positive if and only if A is a nonnegative matrix.


Ignition switch

An ignition switch, starter switch or start switch is a switch in the control system of a motor vehicle that activates the main electrical systems for the vehicle, including "accessories" (radio, power windows, etc.). In vehicles powered by internal combustion engines, the switch provides power to the starter solenoid and the ignition system components (including the engine control unit and ignition coil), and is frequently combined with the starter switch which activates the starter motor.
Historically, ignition switches were key switches that requires the proper key to be inserted in order for the switch functions to be unlocked. These mechanical switches remain common in modern vehicles, further combined with an immobiliser to only activate the switch functions when a transponder signal in the key is detected. However, many new vehicles have been equipped with so-called "keyless" systems, which replace the key switch with a push button that also requires a transponder signal.
The ignition locking system may be sometimes bypassed by disconnecting the wiring to the switch and manipulating it directly; this is known as hotwiring.
Ignition switches are generally a simple repair that can be completed without much knowledge.
They are mainly vehicle specific and plug and play.


Impulse invariance

Impulse invariance is a technique for designing discrete-time infinite-impulse-response (IIR) filters from continuous-time filters in which the impulse response of the continuous-time system is sampled to produce the impulse response of the discrete-time system. The frequency response of the discrete-time system will be a sum of shifted copies of the frequency response of the continuous-time system; if the continuous-time system is approximately band-limited to a frequency less than the Nyquist frequency of the sampling, then the frequency response of the discrete-time system will be approximately equal to it for frequencies below the Nyquist frequency.
Discussion.
The continuous-time system's impulse response, formula_1, is sampled with sampling period formula_2 to produce the discrete-time system's impulse response, formula_3.
Thus, the frequency responses of the two systems are related by
If the continuous time filter is approximately band-limited (i.e. formula_6 when formula_7), then the frequency response of the discrete-time system will be approximately the continuous-time system's frequency response for frequencies below π radians per sample (below the Nyquist frequency 1/(2"T") Hz):
Comparison to the bilinear transform.
Note that aliasing will occur, including aliasing below the Nyquist frequency to the extent that the continuous-time filter's response is nonzero above that frequency. The bilinear transform is an alternative to impulse invariance that uses a different mapping that maps the continuous-time system's frequency response, out to infinite frequency, into the range of frequencies up to the Nyquist frequency in the discrete-time case, as opposed to mapping frequencies linearly with circular overlap as impulse invariance does.
Effect on poles in system function.
If the continuous poles at formula_10, the system function can be written in partial fraction expansion as
Thus, using the inverse Laplace transform, the impulse response is
The corresponding discrete-time system's impulse response is then defined as the following
Performing a z-transform on the discrete-time impulse response produces the following discrete-time system function
Thus the poles from the continuous-time system function are translated to poles at z = eskT. The zeros, if any, are not so simply mapped.
Poles and zeros.
If the system function has zeros as well as poles, they can be mapped the same way, but the result is no longer an impulse invariance result: the discrete-time impulse response is not equal simply to samples of the continuous-time impulse response. This method is known as the matched Z-transform method, or pole–zero mapping.
Stability and causality.
Since poles in the continuous-time system at "s" = "sk" transform to poles in the discrete-time system at z = exp("skT"), poles in the left half of the "s"-plane map to inside the unit circle in the "z"-plane; so if the continuous-time filter is causal and stable, then the discrete-time filter will be causal and stable as well.
Corrected formula.
When a causal continuous-time impulse response has a discontinuity at formula_16, the expressions above are not consistent.
This is because formula_17 has different right and left limits, and should really only contribute their average, half its right value formula_18, to formula_19.
Making this correction gives
Performing a z-transform on the discrete-time impulse response produces the following discrete-time system function
The second sum is zero for filters without a discontinuity, which is why ignoring it is often safe.


DataAdapter

In ADO.NET, a DataAdapter functions as a bridge between a data source, and a disconnected data class, such as a DataSet. At the simplest level it will specify SQL commands that provide elementary CRUD functionality. At a more advanced level it offers all the functions required in order to create Strongly Typed DataSets, including DataRelations. Data adapters are an integral part of ADO.NET managed providers, which are the set of objects used to communicate between a data source and a dataset. (In addition to adapters, managed providers include connection objects, data reader objects, and command objects.) Adapters are used to exchange data between a data source and a dataset. In many applications, this means reading data from a database into a dataset, and then writing changed data from the dataset back to the database. However, a data adapter can move data between any source and a dataset. For example, there could be an adapter that moves data between a Microsoft Exchange server and a dataset.
Sometimes the data you work with is primarily read-only and you rarely need to make changes to the underlying data source Some situations also call for caching data in memory to minimize the number of database calls for data that does not change. The data adapter makes it easy for you to accomplish these things by helping to manage data in a disconnected mode. The data adapter fills a DataSet object when reading the data and writes in a single batch when persisting changes back to the database. A data adapter contains a reference to the connection object and opens and closes the connection automatically when reading from or writing to the database. Additionally, the data adapter contains command object references for SELECT, INSERT, UPDATE, and DELETE operations on the data. You will have a data adapter defined for each table in a DataSet and it will take care of all communication with the database for you. All you need to do is tell the data adapter when to load from or write to the database.


Profile angle

The profile angle of a gear is the angle at a specified pitch point between a line tangent to a tooth surface and the line normal to the pitch surface (which is a radial line of a pitch circle). This definition is applicable to every type of gear for which a pitch surface can be defined. The profile angle gives the direction of the tangent to a tooth profile.
In spur gears and straight bevel gears, tooth profiles are considered only in a transverse plane, and the general terms profile angle and pressure angle are customarily used rather than transverse profile angle and transverse pressure angle. In helical teeth, the profiles may be considered in different planes, and in specifications it is essential to use terms that indicate the direction of the plane in which the profile angle or the pressure angle lies, such as transverse profile angle, normal pressure angle, axial profile angle.
Types.
Standard.
In tools for cutting, grinding, and gaging gear teeth, the profile angle is the angle between a cutting edge or a cutting surface, and some principal direction such as that of a shank, an axis, or a plane of rotation.
Standard profile angles are established in connection with standard proportions of gear teeth and standard gear cutting tools. Involute gears operate together correctly after a change of center distance, and gears designed for a different center distance can be generated correctly by standard tools. A change of center distance is accomplished by changes in operating values for pitch diameter, circular pitch, diametral pitch, pressure angle, and tooth thicknesses or backlash. The same involute gear may be used under conditions that change its operating pitch diameter and pressure angle. Unless there is a good reason for doing otherwise, it is practical to consider that the pitch and the profile angle of a single gear correspond to the pitch and the profile angle of the hob or cutter used to generate its teeth.
Transverse.
The transverse pressure angle and transverse profile angle are the pressure angle and the profile angle in a transverse plane.
Normal.
Normal pressure angle and normal profile angle are the pressure and profile angles in a normal plane of a helical or a spiral tooth. In a spiral bevel gear, unless otherwise specified, profile angle means normal profile angle at the mean cone distance.
Axial.
Axial pressure angle and axial profile angle are the pressure angle and the profile angle in an axial plane of a helical gear or a worm, or of a spiral bevel gear.


Monitoring and surveillance agents

Monitoring and surveillance agents (also known as predictive agents) are a type of intelligent agent software that observes and reports on computer equipment. Monitoring and surveillance agents are often used to monitor complex computer networks to predict when a crash or some other defect may occur. Another type of monitoring and surveillance agent works on computer networks keeping track of the configuration of each computer connected to the network. It tracks and updates the central configuration database when anything on any computer changes, such as the number or type of disk drives. An important task in managing networks lies in prioritizing traffic and shaping bandwidth.


FOG Project

The FOG Project is a software project that implements FOG (Free and Open-source Ghost), a software tool that can deploy disk images of Microsoft Windows and Linux using the Preboot Execution Environment. It makes use of TFTP, the Apache webserver and iPXE. It is written in PHP.
The configuration tool developed by the FOG Project makes it possible to do remote system administration of the computers in a network. FOG depends on Partclone to copy the disk image.


Panasonic Electric Works

 is a Japanese company specializing in the production of industrial devices. It can trace its beginnings to a firm that was founded in 1918 by Konosuke Matsushita. Matsushita began making the flashlight components for bicycles, then progressed to making lighting fixtures.
During World War II, the company manufactured everything from airplane propellers to light sockets. At the conclusion of World War II the U.S.A. forced the company to split into two separate companies, Matsushita Electric Works, Ltd. (MEW), and Matsushita Electric Industrial Co., Ltd. (MEI) (which became Panasonic).
MEW conducts business in automation controls, electronic materials, lighting products, information equipment, and wiring products, building products and home appliances. In 2004 MEW began a pursuit of collaborate business ties with its brother company, MEI.
In 2005, the company was renamed from Aromat to Panasonic Electric Works.
On July 29, 2010, Panasonic reached an agreement to acquire the remaining shares of Panasonic Electric Works and Sanyo shares for $9.4 billion.
In 2007, Panasonic acquired Indian company Anchor Electricals Pvt. Ltd.


Rule based DFM analysis for deep drawing. Deep drawing is a widely used cold sheet metal forming process to draw the sheet metal in forming dye of desirable cross-section using mechanical force of the punch. DFM refers to design for manufacturability. DFA refers to design for assembly. DFMA stands for design for manufacture and assembly. It is a practice for designing the engineering components keeping manufacturing and assembly aspects in mind. DFMA tries to tackle the problems that may come during the manufacturing and assembly at the design stage itself. Changes in the parts design to remove these problems while keeping the functionality of the parts intact. This is done to reduce the cost of iterations thus making the manufacturing of components more efficient and economical.
In the deep drawing process, a blank of sheet metal (usually circular) is placed on the die. The die is fixed to the base. The metal blank is held in position on the die using blank holder. Mechanical force is applied on the part of the metal blank above the die cavity through a punch. As the punch force increases the metal flows from the flange region in to the die cavity.
Here is the Rule based DFM analysis for Deep drawing process. These rules can be incorporated at the design stage to improve the efficiency of the process:
Material of Sheet Metal.
As the deep drawing is a cold forming operation, the germane properties of the sheet metal are formability, ductility and yield strength. The material should have good formability and ductility so that it can be drawn into the desired shape without any cracks. The yield strength of the material should be low facilitating initiation of the flow of metal without tearing near the punch radius.
Clearance between Punch and Die.
Clearance between the punch and die guides the flow of the metal into the die. Clearance should be more than the metal thickness to avoid concentration of metal at the top of the die cavity. Clearance should not be as large so that the flow of metal into the die region becomes unrestricted leading to the wrinkling of wall.
Die corner radius.
Radius of curvature at the die where the metal enters from the flange region into the die region is an important geometrical parameter. If the die corner radius is small than wrinkling near the flange region becomes more prominent. Too small die corner radius results in cracks due to sharp change in the direction of metal flow. Generally it should be 5-10 times the sheet thickness.
Punch corner radius.
As the metal draws into the die the thickness of the sheet decreases near in the lower region of the punch. Maximum reduction happens near punch corner because the metal flow decreases significantly here. Too sharp corner results in cracks near the punch base. Corner radius of punch should be 4-10 times the sheet thickness.
Blank holding force.
The friction in the flange region is mainly affected by blank holding force. Blank holding force is required for checking the amount of the metal flow in to the die. The low value of blank holding force results in wrinkling in the flange region and too high value of holding force results in increase in the drawing force due to the increase in the friction between the flange region. The blank holding force should be just enough to restrict the flow of the metal.
Drawing Ratio.
Measurement of the amount of drawing performed on a sheet metal blank is quantified using drawing ratio. The higher the drawing ratio, the more extreme the amount of deep drawing. Due to the geometry, forces, metal flow and material properties of the work, there is a limit to the amount of deep drawing that can be performed on a sheet metal blank in a single operation. The drawing ratio is roughly calculated as,
DR = Db/Dp.
Db is the diameter of the blank and Dp is the diameter of the punch. For shapes that are noncircular the maximum diameter is sometimes used, or occasionally drawing ratio is calculated using surface areas. The limit to the drawing ratio for an operation is usually 2 or under


DFM Guidelines for Hot Metal Extrusion Process

Extrusion is a metal forming process to form parts with constant cross-section along its length. This process uses a metal billet or ingot which is inserted in a chamber. One side of this contains a die to produce the desired cross section and the other side a hydraulic ram is present to push the metal billet or ingot. Metal flows around the profile of the die and after solidification takes the desired shape.
Extrusion process can be done with the material hot or cold, but most of the metals are heated before the process, if high surface finish and tight tolerances are required then the material is not heated.
DFM stands for design for manufacturing, so as the name suggest the design is manufacturing friendly, in simple terms design that can be manufactured easily and cheaply. DFM guidelines define a set of rules for a person designing a product to ease the manufacturing process, reduce cost and time. For example, if a hole is to be drilled, if the designer specifies a standard hole size then it reduces the cost because the drill bits of unusual sizes are not readily available they have to be custom made.
Material based guidelines.
Nowadays, quite a wide variety of metals are currently extruded commercially, the most common are (in order of decreasing extrudability): Aluminium, Magnesium and their alloys, Copper and Copper alloys, low-carbon and medium-carbon steels, low-alloy steels and stainless steels. So obviously as the extrudability decrease the cost of production increases


Rule-based DFM analysis for electric discharge machining

Electrical discharge machining (or EDM) is one of the most accurate manufacturing processes available for creating complex or simple shapes and geometries within parts and assemblies. A machining method typically used for hard metals, EDM makes it possible to work with metals for which traditional machining techniques are ineffective.
Design for manufacturability (also sometimes known as design for manufacturing or DFM) is the general engineering art of designing products in such a way that they are easy to manufacture. The concept exists in almost all engineering disciplines, but the implementation differs widely depending on the manufacturing technology. DFM describes the process of designing or engineering a product in order to facilitate the manufacturing process in order to reduce its manufacturing costs. DFM will allow potential problems to be fixed in the design phase which is the least expensive place to address them. Other factors may affect the manufacturability such as the type of raw material, the form of the raw material, dimensional tolerances, and secondary processing such as finishing.
Depending on various types of manufacturing processes there are set guidelines for DFM practices. These DFM guidelines help to precisely define various tolerances, rules and common manufacturing checks related to DFM. Rule based guidelines which can be referred to while designing parts are mentioned below. The parts are designed considering manufacturability with electrical discharge machining in mind.
Mechanical design considerations.
Minimum internal corner radius.
The minimum internal corner radius of the feature will dictate the maximum wire diameter that can be used. The wire diameter needs to be less than double the minimum internal corner radius for successful machining. However, the amount of final overcut and a small amount of maneuvering need to be taken into account for the corner to be generated. For small diameter wires, the following are recommended:
Surface finishing.
Surface finishing comprises the small local deviations of a surface from the perfectly flat ideal. It is one of the important factors that controls friction and transfer layer formation during sliding.
Many wire EDM machines have adopted the pulse generating circuit using low power for ignition and high power for machining. However, it is not suitable for finishing process since the energy generated by the high voltage sub-circuit is too high to obtain a desired fine surface. Relaxing the surface finish allows the manufacturer to produce the part with fewer passes, at a higher current level and a higher metal-removal rate, enabling lower production time and cost.
Material removal.
The removal of material in EDM is associated with the erosive effects produced when discrete and spatial discharge occurs between the tool and work-piece electrodes. Short duration sparks generated between these two electrodes. The generator releases electrical energy, which is responsible for melting a small quantity of material from both the electrodes. The part should be designed and prepared such that the amount of stock removed by EDM is relatively small. Traditional machining techniques, such as milling can be used to remove bulk of stock with the finishing operations performed by EDM.
Simultaneous machining.
EDM enhanced with CNC systems is a highly competitive model for making forging dies, casting tooling, plastic injection molds and tooling for powder metals. It enables the user to machine simultaneously multiple highly precise parts from a single clamping. Designs should be considered such that several parts can be stacked and machined simultaneously or a single part can have several EDM operations performed simultaneously.
Enlarging holes.
When existing holes are to be enlarged or reshaped by EDM, through holes are preferred to blind holes as they permit easier flow of dielectric fluid past the area being machined.
Sharp corners.
When cutting sharp corners, the wire dwells longer by the inside radius causing a slight overcut. On the outside radius, it speeds, leaving a slight undercut. Hence, sharp corners should be avoided while designing part.
Galvanic corrosion.
Galvanic corrosion is an electrochemical process in which one metal corrodes preferentially to another when both metals are in electrical contact, in the presence of an electrolyte. In EDM, there will be some degree of material exchange between the wire or the probe and the base material. Electrodes and base material should be chosen to prevent galvanic corrosion as far as possible.


PLC technician

PLC technicians design, program, repair, and maintain programmable logic controller (PLC) systems used within manufacturing and service industries ranging from industrial packaging to commercial car washes and traffic lights.
Scope of work.
PLC technicians are knowledgeable in overall plant systems and the interactions of processes. They install and service a variety of systems including safety and security, energy delivery (hydraulic, pneumatic and electrical), communication, and process control systems. They also install and service measuring and indicating instruments to monitor process control variables associated with PLCs, and monitor the operation of PLC equipment. PLC technicians work with final control devices such as valves, actuators and positioners to manipulate the process medium. They install and terminate electrical, pneumatic, and fluid connections. They also work on network and signal transmission systems such as fibre optic and wireless.
Along with the calibration, repair, adjustment, and replacement of components, PLC technicians inspect and test the operation of instruments and systems to diagnose faults and verify repairs. They establish and optimize process control strategies, and configure related systems such as Distributed Control Systems (DCSs), Supervisory Control & Data Acquisition (SCADA), and Human Machine Interfaces (HMIs). PLC technicians maintain backups, documentation, and software revisions as part of maintaining these computer-based control systems. Scheduled maintenance and the commissioning of systems are also important aspects of the work. PLC technicians consult technical documentation, drawings, schematics, and manuals. They may assist engineering in plant design, modification and hazard analysis, and work with plant operators to optimize plant controls.
PLC technicians use hand, power, and electronic tools, test equipment, and material handling equipment. They work on a variety of systems including primary control elements, transmitters, analyzers, sensors, detectors, signal conditioners, recorders, controllers, and final control elements (actuators, valve positioners, etc.). These instruments measure and control variables such as pressure, flow, temperature, level, motion, force, and chemical composition. PLC systems designed and maintained by PLC technicians range from high-speed robotic assembly to conveyors, to batch mixers, to DCS and SCADA systems. PLC systems are often found within industrial and manufacturing plants, such as food processing facilities. Alternate job titles include PLC engineer, Automation Technician, Field Technician, or Controls Technician.
Education, training and skills.
PLC technician educational courses and programs integrate PLC programming with mechanics, electronics and process controls, They also commonly include coursework in hydraulics, pneumatics, robotics, DCS, SCADA, electrical circuits, electrical machinery and human-machine interfaces. Typical courses include math, communications, circuits, digital devices, and electrical controls. Other courses include robotics, automation, electrical motor controls, programmable logic controllers, and computer-aided design. When performing their duties, PLC technicians must comply with federal, jurisdictional, industrial, and site-specific standards, codes, and regulations. They must ensure that all processes operate and are maintained within these set standards, codes, and regulations. Keeping up-to-date with advances in technology in the industry is important. Key attributes for PLC Technicians are critical thinking skills, manual dexterity, mechanical aptitude, attention to detail, strong problem-solving skills, communication skills, and mathematical and scientific aptitude.
Employers generally prefer applicants who have completed a PLC technician certificate or related associate degree. These programs can be completed at Colleges and Universities in either an in-class or online format. Some Colleges, such as George Brown College, offer an online PLC Technician program that uses simulation software, PLCLogix, to complete PLC lab projects and assignments. Certification by accredited schools and third-party organizations can enhance employment opportunities and keep PLC technicians current and up-to-date. In addition to Colleges and Universities, other organizations and companies also offer credential programs in PLCs, including equipment manufacturers such as Rockwell and professional associations, such as the Electronics Technicians Association, Robotics Industries Association and the Manufacturing Skill Standards Council.
Career opportunities.
PLC technicians install and repair industrial electronic equipment (including input/output networks, data highways, variable speed drives, and process control equipment) and write PLC programs for a wide variety of automated control systems, ranging from simple on–off controls to robotics. PLC technicians also find employment in the industrial engineering field where they are actively involved in the design and implementation of PLC control systems.
Career opportunities for PLC Technicians include a wide range of manufacturing and service industries such as automotive, pharmaceutical, power distribution, food processing, mining, and transportation. Other career prospects include areas such as machine assembly, troubleshooting, and testing, systems integration, application support, maintenance, component testing and assembly, automation programming, robot maintenance and programming, technical sales and services.
PLC Technicians work mainly indoors, on the plant floor and sometimes in cramped conditions. They may be required to stand for prolonged periods of time and be exposed to high noise, fumes and heat levels. Because this is such an important job, they must pay close attention to safety and may be called out in emergencies. Constant learning may be required to keep up with new technology.
Primarily work in this area is full-time and can be in shifts. Employers who hire PLC Technicians include:


MIDACO

MIDACO (Mixed Integer Distributed Ant Colony Optimization) is a software package for numerical optimization based on evolutionary computing.
MIDACO was created in collaboration of
European Space Agency and EADS Astrium to solve constrained mixed-integer non-linear (MINLP) space applications.
MIDACO holds several record solutions on interplanetary spaceflight trajectory design problems made publicly available by European Space Agency. MIDACO is included in software packages like TOMLAB, Astos, and SigmaXL.


Nonnegative matrix

In mathematics, a nonnegative matrix, written
is a matrix in which all the elements are equal to or greater than zero, that is,
A positive matrix is a matrix in which all the elements are strictly greater than zero. The set of positive matrices is a subset of all non-negative matrices. While such matrices are commonly found, the term is only occasionally used due to the possible confusion with positive-definite matrices, which are different. A matrix which is both non-negative and is positive semidefinite is called a doubly non-negative matrix.
A rectangular non-negative matrix can be approximated by a decomposition with two other non-negative matrices via non-negative matrix factorization.
Eigenvalues and eigenvectors of square positive matrices are described by the Perron–Frobenius theorem.
Inversion.
The inverse of any non-singular M-matrix is a non-negative matrix. If the non-singular M-matrix is also symmetric then it is called a Stieltjes matrix.
The inverse of a non-negative matrix is usually not non-negative. The exception is the non-negative monomial matrices: a non-negative matrix has non-negative inverse if and only if it is a (non-negative) monomial matrix. Note that thus the inverse of a positive matrix is not positive or even non-negative, as positive matrices are not monomial, for dimension .
Specializations.
There are a number of groups of matrices that form specializations of non-negative matrices, e.g. stochastic matrix; doubly stochastic matrix; symmetric non-negative matrix.


Support vector machine

In machine learning, support vector machines (SVMs, also support vector networks) are supervised learning models with associated learning algorithms that analyze data for classification and regression analysis. Developed at AT&T Bell Laboratories by Vladimir Vapnik with colleagues (Boser et al., 1992, Guyon et al., 1993, Cortes and Vapnik, 1995, Vapnik et al., 1997) SVMs are one of the most robust prediction methods, being based on statistical learning frameworks or VC theory proposed by Vapnik (1982, 1995) and Chervonenkis (1974). Given a set of training examples, each marked as belonging to one of two categories, an SVM training algorithm builds a model that assigns new examples to one category or the other, making it a non-probabilistic binary linear classifier (although methods such as Platt scaling exist to use SVM in a probabilistic classification setting). SVM maps training examples to points in space so as to maximise the width of the gap between the two categories. New examples are then mapped into that same space and predicted to belong to a category based on which side of the gap they fall.
In addition to performing linear classification, SVMs can efficiently perform a non-linear classification using what is called the , implicitly mapping their inputs into high-dimensional feature spaces.
The support vector clustering algorithm, created by Hava Siegelmann and Vladimir Vapnik, applies the statistics of support vectors, developed in the support vector machines algorithm, to categorize unlabeled data. These data sets require unsupervised learning approaches, which attempt to find natural clustering of the data to groups and, then, to map new data according to these clusters.
Motivation.
Classifying data is a common task in machine learning.
Suppose some given data points each belong to one of two classes, and the goal is to decide which class a "new" data point will be in. In the case of support vector machines, a data point is viewed as a formula_1-dimensional vector (a list of formula_1 numbers), and we want to know whether we can separate such points with a formula_3-dimensional hyperplane. This is called a linear classifier. There are many hyperplanes that might classify the data. One reasonable choice as the best hyperplane is the one that represents the largest separation, or margin, between the two classes. So we choose the hyperplane so that the distance from it to the nearest data point on each side is maximized. If such a hyperplane exists, it is known as the "maximum-margin hyperplane" and the linear classifier it defines is known as a "maximum-margin classifier"; or equivalently, the "perceptron of optimal stability".
More formally, a support vector machine constructs a hyperplane or set of hyperplanes in a high or infinite-dimensional space, which can be used for classification, regression, or other tasks like outliers detection. Intuitively, a good separation is achieved by the hyperplane that has the largest distance to the nearest training-data point of any class (so-called functional margin), since in general the larger the margin, the lower the generalization error of the classifier. A lower generalization error means that the implementer is less likely to experience overfitting.
Whereas the original problem may be stated in a finite-dimensional space, it often happens that the sets to discriminate are not linearly separable in that space. For this reason, it was proposed that the original finite-dimensional space be mapped into a much higher-dimensional space, presumably making the separation easier in that space. To keep the computational load reasonable, the mappings used by SVM schemes are designed to ensure that dot products of pairs of input data vectors may be computed easily in terms of the variables in the original space, by defining them in terms of a kernel function formula_4 selected to suit the problem. The hyperplanes in the higher-dimensional space are defined as the set of points whose dot product with a vector in that space is constant, where such a set of vectors is an orthogonal (and thus minimal) set of vectors that defines a hyperplane. The vectors defining the hyperplanes can be chosen to be linear combinations with parameters formula_5 of images of feature vectors formula_6 that occur in the data base. With this choice of a hyperplane, the points formula_7 in the feature space that are mapped into the hyperplane are defined by the relation formula_8 Note that if formula_4 becomes small as formula_10 grows further away from formula_7, each term in the sum measures the degree of closeness of the test point formula_7 to the corresponding data base point formula_6. In this way, the sum of kernels above can be used to measure the relative nearness of each test point to the data points originating in one or the other of the sets to be discriminated. Note the fact that the set of points formula_7 mapped into any hyperplane can be quite convoluted as a result, allowing much more complex discrimination between sets that are not convex at all in the original space.
Applications.
SVMs can be used to solve various real-world problems:
History.
The original SVM algorithm was invented by Vladimir N. Vapnik and Alexey Ya. Chervonenkis in 1964. In 1992, Bernhard Boser, Isabelle Guyon and Vladimir Vapnik suggested a way to create nonlinear classifiers by applying the kernel trick to maximum-margin hyperplanes. The "soft margin" incarnation, as is commonly used in software packages, was proposed by Corinna Cortes and Vapnik in 1993 and published in 1995.
Linear SVM.
We are given a training dataset of formula_15 points of the form
formula_16
where the formula_17 are either 1 or −1, each indicating the class to which the point formula_18 belongs. Each formula_19 is a formula_1-dimensional real vector. We want to find the "maximum-margin hyperplane" that divides the group of points formula_19 for which formula_22 from the group of points for which formula_23, which is defined so that the distance between the hyperplane and the nearest point formula_19 from either group is maximized.
Any hyperplane can be written as the set of points formula_25 satisfying
formula_26
where formula_27 is the (not necessarily normalized) normal vector to the hyperplane. This is much like Hesse normal form, except that formula_27 is not necessarily a unit vector. The parameter formula_29 determines the offset of the hyperplane from the origin along the normal vector formula_27.
Hard-margin.
If the training data is linearly separable, we can select two parallel hyperplanes that separate the two classes of data, so that the distance between them is as large as possible. The region bounded by these two hyperplanes is called the "margin", and the maximum-margin hyperplane is the hyperplane that lies halfway between them. With a normalized or standardized dataset, these hyperplanes can be described by the equations
and
Geometrically, the distance between these two hyperplanes is formula_33, so to maximize the distance between the planes we want to minimize formula_34. The distance is computed using the distance from a point to a plane equation. We also have to prevent data points from falling into the margin, we add the following constraint: for each formula_35 either
formula_36
or
formula_37
These constraints state that each data point must lie on the correct side of the margin.
This can be rewritten as
We can put this together to get the optimization problem:
formula_38
The formula_27 and formula_40 that solve this problem determine our classifier, formula_41 where formula_42 is the sign function.
An important consequence of this geometric description is that the max-margin hyperplane is completely determined by those formula_19 that lie nearest to it. These formula_19 are called "support vectors".
Soft-margin.
To extend SVM to cases in which the data are not linearly separable, the "hinge loss" function is helpful
formula_45
Note that formula_17 is the "i"-th target (i.e., in this case, 1 or −1), and formula_47 is the "i"-th output.
This function is zero if the constraint in is satisfied, in other words, if formula_19 lies on the correct side of the margin. For data on the wrong side of the margin, the function's value is proportional to the distance from the margin.
The goal of the optimization then is to minimize
formula_49
where the parameter formula_50 determines the trade-off between increasing the margin size and ensuring that the formula_19 lie on the correct side of the margin. By deconstructing the hinge loss, this optimization problem can be massaged into the following:
formula_52
Thus, for large values of formula_53, it will behave similar to the hard-margin SVM, if the input data are linearly classifiable, but will still learn if a classification rule is viable or not. (formula_54 is inversely related to formula_53, e.g. in "LIBSVM".)
Nonlinear Kernels.
The original maximum-margin hyperplane algorithm proposed by Vapnik in 1963 constructed a linear classifier. However, in 1992, Bernhard Boser, Isabelle Guyon and Vladimir Vapnik suggested a way to create nonlinear classifiers by applying the kernel trick (originally proposed by Aizerman et al.) to maximum-margin hyperplanes. The resulting algorithm is formally similar, except that every dot product is replaced by a nonlinear kernel function. This allows the algorithm to fit the maximum-margin hyperplane in a transformed feature space. The transformation may be nonlinear and the transformed space high-dimensional; although the classifier is a hyperplane in the transformed feature space, it may be nonlinear in the original input space.
It is noteworthy that working in a higher-dimensional feature space increases the generalization error of support vector machines, although given enough samples the algorithm still performs well.
Some common kernels include:
The kernel is related to the transform formula_65 by the equation formula_66. The value is also in the transformed space, with formula_67. Dot products with for classification can again be computed by the kernel trick, i.e. formula_68.
Computing the SVM classifier.
Computing the (soft-margin) SVM classifier amounts to minimizing an expression of the form
We focus on the soft-margin classifier since, as noted above, choosing a sufficiently small value for formula_54 yields the hard-margin classifier for linearly classifiable input data. The classical approach, which involves reducing to a quadratic programming problem, is detailed below. Then, more recent approaches such as sub-gradient descent and coordinate descent will be discussed.
Primal.
Minimizing can be rewritten as a constrained optimization problem with a differentiable objective function in the following way.
For each formula_70 we introduce a variable formula_71. Note that formula_72 is the smallest nonnegative number satisfying formula_73
Thus we can rewrite the optimization problem as follows
formula_74
This is called the "primal" problem.
Dual.
By solving for the Lagrangian dual of the above problem, one obtains the simplified problem
formula_75
This is called the "dual" problem. Since the dual maximization problem is a quadratic function of the formula_76 subject to linear constraints, it is efficiently solvable by quadratic programming algorithms.
Here, the variables formula_76 are defined such that
formula_78
Moreover, formula_79 exactly when formula_80 lies on the correct side of the margin, and formula_81 when formula_80 lies on the margin's boundary. It follows that formula_27 can be written as a linear combination of the support vectors.
The offset, formula_84, can be recovered by finding an formula_80 on the margin's boundary and solving
formula_86
Kernel trick.
Suppose now that we would like to learn a nonlinear classification rule which corresponds to a linear classification rule for the transformed data points formula_89 Moreover, we are given a kernel function formula_90 which satisfies formula_91.
We know the classification vector formula_27 in the transformed space satisfies
formula_93
where, the formula_94 are obtained by solving the optimization problem
formula_95
The coefficients formula_76 can be solved for using quadratic programming, as before. Again, we can find some index formula_97 such that formula_81, so that formula_99 lies on the boundary of the margin in the transformed space, and then solve
formula_100
Finally,
formula_101
Modern methods.
Recent algorithms for finding the SVM classifier include sub-gradient descent and coordinate descent. Both techniques have proven to offer significant advantages over the traditional approach when dealing with large, sparse datasets—sub-gradient methods are especially efficient when there are many training examples, and coordinate descent when the dimension of the feature space is high.
Sub-gradient descent.
Sub-gradient descent algorithms for the SVM work directly with the expression
formula_102
Note that formula_103 is a convex function of formula_27 and formula_40. As such, traditional gradient descent (or SGD) methods can be adapted, where instead of taking a step in the direction of the function's gradient, a step is taken in the direction of a vector selected from the function's sub-gradient. This approach has the advantage that, for certain implementations, the number of iterations does not scale with formula_15, the number of data points.
Coordinate descent.
Coordinate descent algorithms for the SVM work from the dual problem
formula_107
For each formula_108, iteratively, the coefficient formula_76 is adjusted in the direction of formula_110. Then, the resulting vector of coefficients formula_111 is projected onto the nearest vector of coefficients that satisfies the given constraints. (Typically Euclidean distances are used.) The process is then repeated until a near-optimal vector of coefficients is obtained. The resulting algorithm is extremely fast in practice, although few performance guarantees have been proven.
Empirical risk minimization.
The soft-margin support vector machine described above is an example of an empirical risk minimization (ERM) algorithm for the "hinge loss". Seen this way, support vector machines belong to a natural class of algorithms for statistical inference, and many of its unique features are due to the behavior of the hinge loss. This perspective can provide further insight into how and why SVMs work, and allow us to better analyze their statistical properties.
Risk minimization.
In supervised learning, one is given a set of training examples formula_112 with labels formula_113, and wishes to predict formula_114 given formula_115. To do so one forms a hypothesis, formula_103, such that formula_117 is a "good" approximation of formula_114. A "good" approximation is usually defined with the help of a "loss function," formula_119, which characterizes how bad formula_120 is as a prediction of formula_10. We would then like to choose a hypothesis that minimizes the "expected risk:"
formula_122
In most cases, we don't know the joint distribution of formula_123 outright. In these cases, a common strategy is to choose the hypothesis that minimizes the "empirical risk:"
formula_124
Under certain assumptions about the sequence of random variables formula_125 (for example, that they are generated by a finite Markov process), if the set of hypotheses being considered is small enough, the minimizer of the empirical risk will closely approximate the minimizer of the expected risk as formula_15 grows large. This approach is called "empirical risk minimization," or ERM.
Regularization and stability.
In order for the minimization problem to have a well-defined solution, we have to place constraints on the set formula_127 of hypotheses being considered. If formula_127 is a normed space (as is the case for SVM), a particularly effective technique is to consider only those hypotheses formula_129 for which formula_130 . This is equivalent to imposing a "regularization penalty" formula_131, and solving the new optimization problem
formula_132
This approach is called "Tikhonov regularization."
More generally, formula_133 can be some measure of the complexity of the hypothesis formula_103, so that simpler hypotheses are preferred.
SVM and the hinge loss.
Recall that the (soft-margin) SVM classifier formula_135 is chosen to minimize the following expression:
formula_136
In light of the above discussion, we see that the SVM technique is equivalent to empirical risk minimization with Tikhonov regularization, where in this case the loss function is the hinge loss
formula_137
From this perspective, SVM is closely related to other fundamental classification algorithms such as regularized least-squares and logistic regression. The difference between the three lies in the choice of loss function: regularized least-squares amounts to empirical risk minimization with the square-loss, formula_138; logistic regression employs the log-loss,
formula_139
Target functions.
The difference between the hinge loss and these other loss functions is best stated in terms of "target functions -" the function that minimizes expected risk for a given pair of random variables formula_140.
In particular, let formula_141 denote formula_10 conditional on the event that formula_143. In the classification setting, we have:
formula_144
The optimal classifier is therefore:
formula_145
For the square-loss, the target function is the conditional expectation function, formula_146; For the logistic loss, it's the logit function, formula_147. While both of these target functions yield the correct classifier, as formula_148, they give us more information than we need. In fact, they give us enough information to completely describe the distribution of formula_149.
On the other hand, one can check that the target function for the hinge loss is "exactly" formula_150. Thus, in a sufficiently rich hypothesis space—or equivalently, for an appropriately chosen kernel—the SVM classifier will converge to the simplest function (in terms of formula_151) that correctly classifies the data. This extends the geometric interpretation of SVM—for linear classification, the empirical risk is minimized by any function whose margins lie between the support vectors, and the simplest of these is the max-margin classifier.
Properties.
SVMs belong to a family of generalized linear classifiers and can be interpreted as an extension of the perceptron. They can also be considered a special case of Tikhonov regularization. A special property is that they simultaneously minimize the empirical "classification error" and maximize the "geometric margin"; hence they are also known as maximum margin classifiers.
A comparison of the SVM to other classifiers has been made by Meyer, Leisch and Hornik.
Parameter selection.
The effectiveness of SVM depends on the selection of kernel, the kernel's parameters, and soft margin parameter formula_54.
A common choice is a Gaussian kernel, which has a single parameter "formula_153". The best combination of formula_54 and formula_153 is often selected by a grid search with exponentially growing sequences of formula_54 and "formula_153", for example, formula_158; formula_159. Typically, each combination of parameter choices is checked using cross validation, and the parameters with best cross-validation accuracy are picked. Alternatively, recent work in Bayesian optimization can be used to select formula_54 and "formula_153" , often requiring the evaluation of far fewer parameter combinations than grid search. The final model, which is used for testing and for classifying new data, is then trained on the whole training set using the selected parameters.
Issues.
Potential drawbacks of the SVM include the following aspects:
Extensions.
Support vector clustering (SVC).
SVC is a similar method that also builds on kernel functions but is appropriate for unsupervised learning.
Multiclass SVM.
Multiclass SVM aims to assign labels to instances by using support vector machines, where the labels are drawn from a finite set of several elements.
The dominant approach for doing so is to reduce the single multiclass problem into multiple binary classification problems. Common methods for such reduction include:
Crammer and Singer proposed a multiclass SVM method which casts the multiclass classification problem into a single optimization problem, rather than decomposing it into multiple binary classification problems. See also Lee, Lin and Wahba and Van den Burg and Groenen.
Transductive support vector machines.
Transductive support vector machines extend SVMs in that they could also treat partially labeled data in semi-supervised learning by following the principles of transduction. Here, in addition to the training set formula_162, the learner is also given a set
formula_163
of test examples to be classified. Formally, a transductive support vector machine is defined by the following primal optimization problem:
Minimize (in formula_164)
formula_165
subject to (for any formula_166 and any formula_167)
formula_168
and
formula_169
Transductive support vector machines were introduced by Vladimir N. Vapnik in 1998.
Structured SVM.
SVMs have been generalized to structured SVMs, where the label space is structured and of possibly infinite size.
Regression.
A version of SVM for regression was proposed in 1996 by Vladimir N. Vapnik, Harris Drucker, Christopher J. C. Burges, Linda Kaufman and Alexander J. Smola. This method is called support vector regression (SVR). The model produced by support vector classification (as described above) depends only on a subset of the training data, because the cost function for building the model does not care about training points that lie beyond the margin. Analogously, the model produced by SVR depends only on a subset of the training data, because the cost function for building the model ignores any training data close to the model prediction. Another SVM version known as least-squares support vector machine (LS-SVM) has been proposed by Suykens and Vandewalle.
Training the original SVR means solving
where formula_6 is a training sample with target value formula_17. The inner product plus intercept formula_174 is the prediction for that sample, and formula_175 is a free parameter that serves as a threshold: all predictions have to be within an formula_175 range of the true predictions. Slack variables are usually added into the above to allow for errors and to allow approximation in the case the above problem is infeasible.
Bayesian SVM.
In 2011 it was shown by Polson and Scott that the SVM admits a Bayesian interpretation through the technique of data augmentation. In this approach the SVM is viewed as a graphical model (where the parameters are connected via probability distributions). This extended view allows the application of Bayesian techniques to SVMs, such as flexible feature modeling, automatic hyperparameter tuning, and predictive uncertainty quantification. Recently, a scalable version of the Bayesian SVM was developed by Florian Wenzel, enabling the application of Bayesian SVMs to big data. Florian Wenzel developed two different versions, a variational inference (VI) scheme for the Bayesian kernel support vector machine (SVM) and a stochastic version (SVI) for the linear Bayesian SVM.
Implementation.
The parameters of the maximum-margin hyperplane are derived by solving the optimization. There exist several specialized algorithms for quickly solving the quadratic programming (QP) problem that arises from SVMs, mostly relying on heuristics for breaking the problem down into smaller, more manageable chunks.
Another approach is to use an interior-point method that uses Newton-like iterations to find a solution of the Karush–Kuhn–Tucker conditions of the primal and dual problems.
Instead of solving a sequence of broken-down problems, this approach directly solves the problem altogether. To avoid solving a linear system involving the large kernel matrix, a low-rank approximation to the matrix is often used in the kernel trick.
Another common method is Platt's sequential minimal optimization (SMO) algorithm, which breaks the problem down into 2-dimensional sub-problems that are solved analytically, eliminating the need for a numerical optimization algorithm and matrix storage. This algorithm is conceptually simple, easy to implement, generally faster, and has better scaling properties for difficult SVM problems.
The special case of linear support vector machines can be solved more efficiently by the same kind of algorithms used to optimize its close cousin, logistic regression; this class of algorithms includes sub-gradient descent (e.g., PEGASOS) and coordinate descent (e.g., LIBLINEAR). LIBLINEAR has some attractive training-time properties. Each convergence iteration takes time linear in the time taken to read the train data, and the iterations also have a Q-linear convergence property, making the algorithm extremely fast.
The general kernel SVMs can also be solved more efficiently using sub-gradient descent (e.g. P-packSVM), especially when parallelization is allowed.
Kernel SVMs are available in many machine-learning toolkits, including LIBSVM, MATLAB, SAS, SVMlight, kernlab, scikit-learn, Shogun, Weka, Shark, JKernelMachines, OpenCV and others.
Preprocessing of data (standardization) is highly recommended to enhance accuracy of classification. There are a few methods of standardization, such as min-max, normalization by decimal scaling, Z-score. Subtraction of mean and division by variance of each feature is usually used for SVM.