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Gaussian process emulator : Currin, C., Mitchell, T., Morris, M., and Ylvisaker, D. (1991) "Bayesian Prediction of Deterministic Functions, with Applications to the Design and Analysis of Computer Experiments," Journal of the American Statistical Association, 86, 953–963. Kimeldorf, G. S. and Wahba, G. (1970) "A corres...
Gradient boosting : Gradient boosting is a machine learning technique based on boosting in a functional space, where the target is pseudo-residuals instead of residuals as in traditional boosting. It gives a prediction model in the form of an ensemble of weak prediction models, i.e., models that make very few assumptio...
Gradient boosting : The idea of gradient boosting originated in the observation by Leo Breiman that boosting can be interpreted as an optimization algorithm on a suitable cost function. Explicit regression gradient boosting algorithms were subsequently developed, by Jerome H. Friedman, (in 1999 and later in 2001) simul...
Gradient boosting : (This section follows the exposition by Cheng Li.) Like other boosting methods, gradient boosting combines weak "learners" into a single strong learner iteratively. It is easiest to explain in the least-squares regression setting, where the goal is to teach a model F to predict values of the form y...
Gradient boosting : Many supervised learning problems involve an output variable y and a vector of input variables x, related to each other with some probabilistic distribution. The goal is to find some function F ^ ( x ) (x) that best approximates the output variable from the values of input variables. This is formali...
Gradient boosting : Gradient boosting is typically used with decision trees (especially CARTs) of a fixed size as base learners. For this special case, Friedman proposes a modification to gradient boosting method which improves the quality of fit of each base learner. Generic gradient boosting at the m-th step would fi...
Gradient boosting : Fitting the training set too closely can lead to degradation of the model's generalization ability, that is, its performance on unseen examples. Several so-called regularization techniques reduce this overfitting effect by constraining the fitting procedure. One natural regularization parameter is t...
Gradient boosting : Gradient boosting can be used in the field of learning to rank. The commercial web search engines Yahoo and Yandex use variants of gradient boosting in their machine-learned ranking engines. Gradient boosting is also utilized in High Energy Physics in data analysis. At the Large Hadron Collider (LHC...
Gradient boosting : The method goes by a variety of names. Friedman introduced his regression technique as a "Gradient Boosting Machine" (GBM). Mason, Baxter et al. described the generalized abstract class of algorithms as "functional gradient boosting". Friedman et al. describe an advancement of gradient boosted model...
Gradient boosting : Gradient boosting can be used for feature importance ranking, which is usually based on aggregating importance function of the base learners. For example, if a gradient boosted trees algorithm is developed using entropy-based decision trees, the ensemble algorithm ranks the importance of features ba...
Gradient boosting : While boosting can increase the accuracy of a base learner, such as a decision tree or linear regression, it sacrifices intelligibility and interpretability. For example, following the path that a decision tree takes to make its decision is trivial and self-explained, but following the paths of hund...
Gradient boosting : AdaBoost Random forest Catboost LightGBM XGBoost Decision tree learning
Gradient boosting : Boehmke, Bradley; Greenwell, Brandon (2019). "Gradient Boosting". Hands-On Machine Learning with R. Chapman & Hall. pp. 221–245. ISBN 978-1-138-49568-5.
Gradient boosting : How to explain gradient boosting Gradient Boosted Regression Trees LightGBM
LPBoost : Linear Programming Boosting (LPBoost) is a supervised classifier from the boosting family of classifiers. LPBoost maximizes a margin between training samples of different classes, and thus also belongs to the class of margin classifier algorithms. Consider a classification function f : X → , \to \, which cla...
LPBoost : As in all boosting classifiers, the final classification function is of the form f ( x ) = ∑ j = 1 J α j h j ( x ) , )=\sum _^\alpha _h_(), where α j are non-negative weightings for weak classifiers h j : X → :\to \ . Each individual weak classifier h j may be just a little bit better than random, but the ...
LPBoost : More generally, let H = =\ be the possibly infinite set of weak classifiers, also termed hypotheses. One way to write down the problem LPBoost solves is as a linear program with infinitely many variables. The primal linear program of LPBoost, optimizing over the non-negative weight vector α , the non-negati...
LPBoost : Input: Training set X = _,\dots ,_\ , x i ∈ X _\in Training labels Y = ,\dots ,y_\ , y i ∈ \in \ Convergence threshold θ ≥ 0 Output: Classification function f : X → \to \ Initialization Weights, uniform λ n ← 1 ℓ , n = 1 , … , ℓ \leftarrow ,\quad n=1,\dots ,\ell Edge γ ← 0 Hypothesis count J ← 1 Iter...
LPBoost : LPBoost is an ensemble learning method and thus does not dictate the choice of base learners, the space of hypotheses H . Demiriz et al. showed that under mild assumptions, any base learner can be used. If the base learners are particularly simple, they are often referred to as decision stumps. The number of...
LPBoost : Linear Programming Boosting via Column Generation, A. Demiriz and K.P. Bennett and J. Shawe-Taylor. Published 2002 in Kluwer Machine Learning 46, pages 225–254.
Random forest : Random forests or random decision forests is an ensemble learning method for classification, regression and other tasks that works by creating a multitude of decision trees during training. For classification tasks, the output of the random forest is the class selected by most trees. For regression task...
Random forest : The general method of random decision forests was first proposed by Salzberg and Heath in 1993, with a method that used a randomized decision tree algorithm to create multiple trees and then combine them using majority voting. This idea was developed further by Ho in 1995. Ho established that forests of...
Random forest : As part of their construction, random forest predictors naturally lead to a dissimilarity measure among observations. One can analogously define dissimilarity between unlabeled data, by training a forest to distinguish original "observed" data from suitably generated synthetic data drawn from a referenc...
Random forest : Instead of decision trees, linear models have been proposed and evaluated as base estimators in random forests, in particular multinomial logistic regression and naive Bayes classifiers. In cases that the relationship between the predictors and the target variable is linear, the base learners may have a...
Random forest : In machine learning, kernel random forests (KeRF) establish the connection between random forests and kernel methods. By slightly modifying their definition, random forests can be rewritten as kernel methods, which are more interpretable and easier to analyze.
Random forest : While random forests often achieve higher accuracy than a single decision tree, they sacrifice the intrinsic interpretability of decision trees. Decision trees are among a fairly small family of machine learning models that are easily interpretable along with linear models, rule-based models, and attent...
Random forest : Boosting – Method in machine learning Decision tree learning – Machine learning algorithm Ensemble learning – Statistics and machine learning technique Gradient boosting – Machine learning technique Non-parametric statistics – Type of statistical analysisPages displaying short descriptions of redirect t...
Random forest : Random Forests classifier description (Leo Breiman's site) Liaw, Andy & Wiener, Matthew "Classification and Regression by randomForest" R News (2002) Vol. 2/3 p. 18 (Discussion of the use of the random forest package for R)
Random subspace method : In machine learning the random subspace method, also called attribute bagging or feature bagging, is an ensemble learning method that attempts to reduce the correlation between estimators in an ensemble by training them on random samples of features instead of the entire feature set.
Random subspace method : In ensemble learning one tries to combine the models produced by several learners into an ensemble that performs better than the original learners. One way of combining learners is bootstrap aggregating or bagging, which shows each learner a randomly sampled subset of the training points so tha...
Random subspace method : An ensemble of models employing the random subspace method can be constructed using the following algorithm: Let the number of training points be N and the number of features in the training data be D. Let L be the number of individual models in the ensemble. For each individual model l, choose...
Dimensionality reduction : Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally close to its intrinsic dimension. Working in h...
Dimensionality reduction : The process of feature selection aims to find a suitable subset of the input variables (features, or attributes) for the task at hand. The three strategies are: the filter strategy (e.g., information gain), the wrapper strategy (e.g., accuracy-guided search), and the embedded strategy (featur...
Dimensionality reduction : Feature projection (also called feature extraction) transforms the data from the high-dimensional space to a space of fewer dimensions. The data transformation may be linear, as in principal component analysis (PCA), but many nonlinear dimensionality reduction techniques also exist. For multi...
Dimensionality reduction : For high-dimensional datasets, dimension reduction is usually performed prior to applying a k-nearest neighbors (k-NN) algorithm in order to mitigate the curse of dimensionality. Feature extraction and dimension reduction can be combined in one step, using principal component analysis (PCA), ...
Dimensionality reduction : A dimensionality reduction technique that is sometimes used in neuroscience is maximally informative dimensions, which finds a lower-dimensional representation of a dataset such that as much information as possible about the original data is preserved.
Dimensionality reduction : JMLR Special Issue on Variable and Feature Selection ELastic MAPs Archived 2011-07-20 at the Wayback Machine Locally Linear Embedding Visual Comparison of various dimensionality reduction methods A Global Geometric Framework for Nonlinear Dimensionality Reduction
ARKA descriptors in QSAR : One of the most commonly used in silico approaches for assessing new molecules' activity/property/toxicity is the Quantitative Structure-Activity/Property/Toxicity Relationship (QSAR/QSPR/QSTR), which generates predictive models for efficiently predicting query compounds . QSAR/QSPR/QSTR uses...
Charge based boundary element fast multipole method : The charge-based formulation of the boundary element method (BEM) is a dimensionality reduction numerical technique that is used to model quasistatic electromagnetic phenomena in highly complex conducting media (targeting, e.g., the human brain) with a very large (u...
Charge based boundary element fast multipole method : Along with more common electric potential-based BEM, the quasistatic charge-based BEM, derived in terms of the single-layer (charge) density, for a single-compartment medium has been known in the potential theory since the beginning of the 20th century. For multi-co...
Charge based boundary element fast multipole method : The charge-based BEM is based on the concept of an impressed (or primary) electric field E i ^ and a secondary electric field E s ^ . The impressed field is usually known a priori or is trivial to find. For the human brain, the impressed electric field can be clas...
Charge based boundary element fast multipole method : Below, a derivation is given based on Gauss's law and Coulomb's law. All conductivity interfaces, denoted by S, are discretized into planar triangular facets t m with centers r m _ . Assume that an m-th facet with the normal vector n m _ and area A m carries a u...
Charge based boundary element fast multipole method : For modern characterizations of brain topologies with ever-increasing levels of complexity, the above system of equations for ρ m is very large; it is therefore solved iteratively. An initial guess for ρ m is the last term on its right-hand side while the sum is i...
Charge based boundary element fast multipole method : The system of equations formulated above is derived with the collocation method and is less accurate. The corresponding integral equation is obtained from the local jump relations of the potential theory and the local interfacial boundary condition of normal electri...
Charge based boundary element fast multipole method : Applications of the charge-based BEM-FMM include modeling brain stimulation with near real-time accurate TMS computations as well as neurophysiological recordings. They also include modeling challenging mesoscale head topologies such as thin brain membranes (dura ma...
Charge based boundary element fast multipole method : Computational electromagnetics Boundary element method Fast multipole method Computational neuroscience Transcranial magnetic stimulation Transcranial direct-current stimulation Electroencephalography Magnetoencephalography
Charge based boundary element fast multipole method : A survey on integral equations for bioelectric modeling, preprint. Flatiron Institute - Simons Foundation FMM3D GitHub Project Site. == References ==
Correspondence analysis : Correspondence analysis (CA) is a multivariate statistical technique proposed by Herman Otto Hartley (Hirschfeld) and later developed by Jean-Paul Benzécri. It is conceptually similar to principal component analysis, but applies to categorical rather than continuous data. In a similar manner t...
Correspondence analysis : Like principal components analysis, correspondence analysis creates orthogonal components (or axes) and, for each item in a table i.e. for each row, a set of scores (sometimes called factor scores, see Factor analysis). Correspondence analysis is performed on the data table, conceived as matri...
Correspondence analysis : The visualization of a CA result always starts with displaying the scree plot of the principal inertia values to evaluate the success of summarizing spread by the first few singular vectors. The actual ordination is presented in a graph which could - at first look - be confused with a complica...
Correspondence analysis : Several variants of CA are available, including detrended correspondence analysis (DCA) and canonical correspondence analysis (CCA). The latter (CCA) is used when there is information about possible causes for the similarities between the investigated entities. The extension of correspondence ...
Correspondence analysis : The data visualization system Orange include the module: orngCA. The statistical programming language R includes several packages, which offer a function for (simple symmetric) correspondence analysis. Using the R notation [package_name::function_name] the packages and respective functions are...
Correspondence analysis : Formal concept analysis == References ==
Count sketch : Count sketch is a type of dimensionality reduction that is particularly efficient in statistics, machine learning and algorithms. It was invented by Moses Charikar, Kevin Chen and Martin Farach-Colton in an effort to speed up the AMS Sketch by Alon, Matias and Szegedy for approximating the frequency mome...
Count sketch : The inventors of this data structure offer the following iterative explanation of its operation: at the simplest level, the output of a single hash function s mapping stream elements q into is feeding a single up/down counter C. After a single pass over the data, the frequency n ( q ) of a stream eleme...
Count sketch : 1. For constants w and t (to be defined later) independently choose d = 2 t + 1 random hash functions h 1 , … , h d ,\dots ,h_ and s 1 , … , s d ,\dots ,s_ such that h i : [ n ] → [ w ] :[n]\to [w] and s i : [ n ] → :[n]\to \ . It is necessary that the hash families from which h i and s i are chose...
Count sketch : The count sketch projection of the outer product of two vectors is equivalent to the convolution of two component count sketches. The count sketch computes a vector convolution C ( 1 ) x ∗ C ( 2 ) x T x\ast C^x^ , where C ( 1 ) and C ( 2 ) are independent count sketch matrices. Pham and Pagh show that ...
Count sketch : Count–min sketch is a version of algorithm with smaller memory requirements (and weaker error guarantees as a tradeoff). Tensor sketch
Count sketch : Charikar, Moses; Chen, Kevin; Farach-Colton, Martin (2004). "Finding frequent items in data streams" (PDF). Theoretical Computer Science. 312 (1). Elsevier BV: 3–15. doi:10.1016/s0304-3975(03)00400-6. ISSN 0304-3975. Faisal M. Algashaam; Kien Nguyen; Mohamed Alkanhal; Vinod Chandran; Wageeh Boles. "Multi...
Detrended correspondence analysis : Detrended correspondence analysis (DCA) is a multivariate statistical technique widely used by ecologists to find the main factors or gradients in large, species-rich but usually sparse data matrices that typify ecological community data. DCA is frequently used to suppress artifacts ...
Detrended correspondence analysis : DCA was created in 1979 by Mark Hill of the United Kingdom's Institute for Terrestrial Ecology (now merged into Centre for Ecology and Hydrology) and implemented in FORTRAN code package called DECORANA (Detrended Correspondence Analysis), a correspondence analysis method. DCA is some...
Detrended correspondence analysis : According to Hill and Gauch, DCA suppresses two artifacts inherent in most other multivariate analyses when applied to gradient data. An example is a time-series of plant species colonising a new habitat; early successional species are replaced by mid-successional species, then by la...
Detrended correspondence analysis : DCA is an iterative algorithm that has shown itself to be a highly reliable and useful tool for data exploration and summary in community ecology (Shaw 2003). It starts by running a standard ordination (CA or reciprocal averaging) on the data, to produce the initial horse-shoe curve ...
Detrended correspondence analysis : No significance tests are available with DCA, although there is a constrained (canonical) version called DCCA in which the axes are forced by Multiple linear regression to correlate optimally with a linear combination of other (usually environmental) variables; this allows testing of...
Detrended correspondence analysis : The example shows an ideal data set: The species data is in rows, samples in columns. For each sample along the gradient, a new species is introduced but another species is no longer present. The result is a sparse matrix. Ones indicate the presence of a species in a sample. Except a...
Detrended correspondence analysis : An open source implementation of DCA, based on the original FORTRAN code, is available in the vegan R-package.
Detrended correspondence analysis : Eigenanalysis Ordination (statistics) Seriation (archaeology) – including additional examples for the arch effect Principal Component Analysis
Detrended correspondence analysis : Hill, M.O. (1979). DECORANA — A FORTRAN program for Detrended Correspondence Analysis and Reciprocal Averaging. Section of Ecology and Systematics, Cornell University, Ithaca, New York, 52pp. Hill, M.O. and Gauch, H.G. (1980). Detrended Correspondence Analysis: An Improved Ordination...
Detrended correspondence analysis : PAST (PAlaeontological STatistics) — free software including DCA with modifications according to Oksanen and Minchin (1997) WINBASP — free software including DCA with detrending-by-polynomials according to Ter Braak and Prentice (1988) vegan: Community Ecology Package for R — free so...
Elastic map : Elastic maps provide a tool for nonlinear dimensionality reduction. By their construction, they are a system of elastic springs embedded in the data space. This system approximates a low-dimensional manifold. The elastic coefficients of this system allow the switch from completely unstructured k-means clu...
Elastic map : Let S be a data set in a finite-dimensional Euclidean space. Elastic map is represented by a set of nodes w j _ in the same space. Each datapoint s ∈ S has a host node, namely the closest node w j _ (if there are several closest nodes then one takes the node with the smallest number). The data set S is...
Elastic map : For a given splitting of dataset S in classes K j , minimization of the quadratic functional U is a linear problem with the sparse matrix of coefficients. Therefore, similar to principal component analysis or k-means, a splitting method is used: For given _\ find \ ; For given \ minimize U and find...
Elastic map : Most important applications of the method and free software are in bioinformatics for exploratory data analysis and visualisation of multidimensional data, for data visualisation in economics, social and political sciences, as an auxiliary tool for data mapping in geographic informational systems and for ...
Feature selection : In machine learning, feature selection is the process of selecting a subset of relevant features (variables, predictors) for use in model construction. Feature selection techniques are used for several reasons: simplification of models to make them easier to interpret, shorter training times, to avo...
Feature selection : A feature selection algorithm can be seen as the combination of a search technique for proposing new feature subsets, along with an evaluation measure which scores the different feature subsets. The simplest algorithm is to test each possible subset of features finding the one which minimizes the er...
Feature selection : Subset selection evaluates a subset of features as a group for suitability. Subset selection algorithms can be broken up into wrappers, filters, and embedded methods. Wrappers use a search algorithm to search through the space of possible features and evaluate each subset by running a model on the s...
Feature selection : The choice of optimality criteria is difficult as there are multiple objectives in a feature selection task. Many common criteria incorporate a measure of accuracy, penalised by the number of features selected. Examples include Akaike information criterion (AIC) and Mallows's Cp, which have a penalt...
Feature selection : Filter feature selection is a specific case of a more general paradigm called structure learning. Feature selection finds the relevant feature set for a specific target variable whereas structure learning finds the relationships between all the variables, usually by expressing these relationships as...
Feature selection : There are different Feature Selection mechanisms around that utilize mutual information for scoring the different features. They usually use all the same algorithm: Calculate the mutual information as score for between all features ( f i ∈ F \in F ) and the target class (c) Select the feature with t...
Feature selection : For high-dimensional and small sample data (e.g., dimensionality > 105 and the number of samples < 103), the Hilbert-Schmidt Independence Criterion Lasso (HSIC Lasso) is useful. HSIC Lasso optimization problem is given as H S I C L a s s o : min x 1 2 ∑ k , l = 1 n x k x l HSIC ( f k , f l ) − ∑ k =...
Feature selection : The correlation feature selection (CFS) measure evaluates subsets of features on the basis of the following hypothesis: "Good feature subsets contain features highly correlated with the classification, yet uncorrelated to each other". The following equation gives the merit of a feature subset S cons...
Feature selection : The features from a decision tree or a tree ensemble are shown to be redundant. A recent method called regularized tree can be used for feature subset selection. Regularized trees penalize using a variable similar to the variables selected at previous tree nodes for splitting the current node. Regul...
Feature selection : A metaheuristic is a general description of an algorithm dedicated to solve difficult (typically NP-hard problem) optimization problems for which there is no classical solving methods. Generally, a metaheuristic is a stochastic algorithm tending to reach a global optimum. There are many metaheuristi...
Feature selection : Some learning algorithms perform feature selection as part of their overall operation. These include: ⁠ l 1 ⁠-regularization techniques, such as sparse regression, LASSO, and ⁠ l 1 ⁠-SVM Regularized trees, e.g. regularized random forest implemented in the RRF package Decision tree Memetic algorith...
Feature selection : Cluster analysis Data mining Dimensionality reduction Feature extraction Hyperparameter optimization Model selection Relief (feature selection)
Feature selection : Guyon, Isabelle; Elisseeff, Andre (2003). "An Introduction to Variable and Feature Selection". Journal of Machine Learning Research. 3: 1157–1182. Harrell, F. (2001). Regression Modeling Strategies. Springer. ISBN 0-387-95232-2. Liu, Huan; Motoda, Hiroshi (1998). Feature Selection for Knowledge Disc...
Feature selection : Feature Selection Package, Arizona State University (Matlab Code) NIPS challenge 2003 (see also NIPS) Naive Bayes implementation with feature selection in Visual Basic Archived 2009-02-14 at the Wayback Machine (includes executable and source code) Minimum-redundancy-maximum-relevance (mRMR) feature...
Generalized canonical correlation : In statistics, the generalized canonical correlation analysis (gCCA), is a way of making sense of cross-correlation matrices between the sets of random variables when there are more than two sets. While a conventional CCA generalizes principal component analysis (PCA) to two sets of ...
Generalized canonical correlation : The Helmert-Wolf blocking (HWB) method of estimating linear regression parameters can find an optimal solution only if all cross-correlations between the data blocks are zero. They can always be made to vanish by introducing a new regression parameter for each common factor. The gCCA...
Generalized canonical correlation : Afshin-Pour, B.; Hossein-Zadeh, G.A. Strother, S.C.; Soltanian-Zadeh, H. (2012), "Enhancing reproducibility of fMRI statistical maps using generalized canonical correlation analysis in NPAIRS framework", NeuroImage 60(4): 1970–1981. doi:10.1016/j.neuroimage.2012.01.137 Sun, Q.S., Liu...
Generalized canonical correlation : FactoMineR (free exploratory multivariate data analysis software linked to R)
Independent component analysis : In signal processing, independent component analysis (ICA) is a computational method for separating a multivariate signal into additive subcomponents. This is done by assuming that at most one subcomponent is Gaussian and that the subcomponents are statistically independent from each ot...
Independent component analysis : Independent component analysis attempts to decompose a multivariate signal into independent non-Gaussian signals. As an example, sound is usually a signal that is composed of the numerical addition, at each time t, of signals from several sources. The question then is whether it is poss...
Independent component analysis : ICA finds the independent components (also called factors, latent variables or sources) by maximizing the statistical independence of the estimated components. We may choose one of many ways to define a proxy for independence, and this choice governs the form of the ICA algorithm. The t...
Independent component analysis : Linear independent component analysis can be divided into noiseless and noisy cases, where noiseless ICA is a special case of noisy ICA. Nonlinear ICA should be considered as a separate case.
Independent component analysis : A special variant of ICA is binary ICA in which both signal sources and monitors are in binary form and observations from monitors are disjunctive mixtures of binary independent sources. The problem was shown to have applications in many domains including medical diagnosis, multi-cluste...
Independent component analysis : The early general framework for independent component analysis was introduced by Jeanny Hérault and Bernard Ans from 1984, further developed by Christian Jutten in 1985 and 1986, and refined by Pierre Comon in 1991, and popularized in his paper of 1994. In 1995, Tony Bell and Terry Sejn...
Independent component analysis : ICA can be extended to analyze non-physical signals. For instance, ICA has been applied to discover discussion topics on a bag of news list archives. Some ICA applications are listed below: optical Imaging of neurons neuronal spike sorting face recognition modelling receptive fields of ...
Independent component analysis : ICA can be applied through the following software: SAS PROC ICA R ICA package scikit-learn Python implementation sklearn.decomposition.FastICA mlpack C++ implementation of RADICAL (The Robust Accurate, Direct ICA aLgorithm (RADICAL).) [1]
Independent component analysis : Comon, Pierre (1994): "Independent Component Analysis: a new concept?", Signal Processing, 36(3):287–314 (The original paper describing the concept of ICA) Hyvärinen, A.; Karhunen, J.; Oja, E. (2001): Independent Component Analysis, New York: Wiley, ISBN 978-0-471-40540-5 ( Introductory...

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