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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 correspondence between Bayesian estimation on stochastic processes and smoothing by splines," The Annals of Mathematical Statistics, 41, 495–502. O'Hagan, A. (1978) "Curve fitting and optimal design for predictions," Journal of the Royal Statistical Society B, 40, 1–42. O'Hagan, A. (2006) "Bayesian analysis of computer code outputs: A tutorial," Reliability Engineering & System Safety, 91, 1290–1300. Sacks, J., Welch, W. J., Mitchell, T. J., and Wynn, H. P. (1989) "Design and Analysis of Computer Experiments," Statistical Science, 4, 409–423.
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 assumptions about the data, which are typically simple decision trees. When a decision tree is the weak learner, the resulting algorithm is called gradient-boosted trees; it usually outperforms random forest. As with other boosting methods, a gradient-boosted trees model is built in stages, but it generalizes the other methods by allowing optimization of an arbitrary differentiable loss function.
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) simultaneously with the more general functional gradient boosting perspective of Llew Mason, Jonathan Baxter, Peter Bartlett and Marcus Frean. The latter two papers introduced the view of boosting algorithms as iterative functional gradient descent algorithms. That is, algorithms that optimize a cost function over function space by iteratively choosing a function (weak hypothesis) that points in the negative gradient direction. This functional gradient view of boosting has led to the development of boosting algorithms in many areas of machine learning and statistics beyond regression and classification.
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 ^ = F ( x ) =F(x) by minimizing the mean squared error 1 n ∑ i ( y ^ i − y i ) 2 \sum _(_-y_)^ , where i indexes over some training set of size n of actual values of the output variable y : y ^ i = _= the predicted value F ( x i ) ) y i = = the observed value n = the number of samples in y If the algorithm has M stages, at each stage m ( 1 ≤ m ≤ M ), suppose some imperfect model F m (for low m , this model may simply predict y ^ i _ to be y ¯ , the mean of y ). In order to improve F m , our algorithm should add some new estimator, h m ( x ) (x) . Thus, F m + 1 ( x i ) = F m ( x i ) + h m ( x i ) = y i (x_)=F_(x_)+h_(x_)=y_ or, equivalently, h m ( x i ) = y i − F m ( x i ) (x_)=y_-F_(x_) . Therefore, gradient boosting will fit h m to the residual y i − F m ( x i ) -F_(x_) . As in other boosting variants, each F m + 1 attempts to correct the errors of its predecessor F m . A generalization of this idea to loss functions other than squared error, and to classification and ranking problems, follows from the observation that residuals h m ( x i ) (x_) for a given model are proportional to the negative gradients of the mean squared error (MSE) loss function (with respect to F ( x i ) ) ): L M S E = 1 n ∑ i = 1 n ( y i − F ( x i ) ) 2 =\sum _^\left(y_-F(x_)\right)^ − ∂ L M S E ∂ F ( x i ) = 2 n ( y i − F ( x i ) ) = 2 n h m ( x i ) )=(y_-F(x_))=h_(x_) . So, gradient boosting could be generalized to a gradient descent algorithm by plugging in a different loss and its gradient.
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 formalized by introducing some loss function L ( y , F ( x ) ) and minimizing it in expectation: F ^ = arg ⁡ min F E x , y [ L ( y , F ( x ) ) ] =\,\mathbb _[L(y,F(x))] . The gradient boosting method assumes a real-valued y. It seeks an approximation F ^ ( x ) (x) in the form of a weighted sum of M functions h m ( x ) (x) from some class H , called base (or weak) learners: F ^ ( x ) = ∑ m = 1 M γ m h m ( x ) + const (x)=\sum _^\gamma _h_(x)+ . We are usually given a training set ,y_),\dots ,(x_,y_)\ of known values of x and corresponding values of y. In accordance with the empirical risk minimization principle, the method tries to find an approximation F ^ ( x ) (x) that minimizes the average value of the loss function on the training set, i.e., minimizes the empirical risk. It does so by starting with a model, consisting of a constant function F 0 ( x ) (x) , and incrementally expands it in a greedy fashion: F 0 ( x ) = arg ⁡ min h m ∈ H ∑ i = 1 n L ( y i , h m ( x i ) ) (x)=\in ^,h_(x_)) , F m ( x ) = F m − 1 ( x ) + ( a r g m i n h m ∈ H [ ∑ i = 1 n L ( y i , F m − 1 ( x i ) + h m ( x i ) ) ] ) ( x ) (x)=F_(x)+\left(\in \left[^,F_(x_)+h_(x_))\right]\right)(x) , for m ≥ 1 , where h m ∈ H \in is a base learner function. Unfortunately, choosing the best function h m at each step for an arbitrary loss function L is a computationally infeasible optimization problem in general. Therefore, we restrict our approach to a simplified version of the problem. The idea is to apply a steepest descent step to this minimization problem (functional gradient descent). The basic idea is to find a local minimum of the loss function by iterating on F m − 1 ( x ) (x) . In fact, the local maximum-descent direction of the loss function is the negative gradient. Hence, moving a small amount γ such that the linear approximation remains valid: F m ( x ) = F m − 1 ( x ) − γ ∑ i = 1 n ∇ F m − 1 L ( y i , F m − 1 ( x i ) ) (x)=F_(x)-\gamma \sum _^L(y_,F_(x_)) where γ > 0 . For small γ , this implies that L ( y i , F m ( x i ) ) ≤ L ( y i , F m − 1 ( x i ) ) ,F_(x_))\leq L(y_,F_(x_)) . Furthermore, we can optimize γ by finding the γ value for which the loss function has a minimum: γ m = arg ⁡ min γ ∑ i = 1 n L ( y i , F m ( x i ) ) = arg ⁡ min γ ∑ i = 1 n L ( y i , F m − 1 ( x i ) − γ ∇ F m − 1 L ( y i , F m − 1 ( x i ) ) ) . =^,F_(x_))=^,F_(x_)-\gamma \nabla _L(y_,F_(x_))\right). If we considered the continuous case, i.e., where H is the set of arbitrary differentiable functions on R , we would update the model in accordance with the following equations F m ( x ) = F m − 1 ( x ) − γ m ∑ i = 1 n ∇ F m − 1 L ( y i , F m − 1 ( x i ) ) (x)=F_(x)-\gamma _\sum _^L(y_,F_(x_)) where γ m is the step length, defined as γ m = arg ⁡ min γ ∑ i = 1 n L ( y i , F m − 1 ( x i ) − γ ∇ F m − 1 L ( y i , F m − 1 ( x i ) ) ) . =^,F_(x_)-\gamma \nabla _L(y_,F_(x_))\right). In the discrete case however, i.e. when the set H is finite, we choose the candidate function h closest to the gradient of L for which the coefficient γ may then be calculated with the aid of line search on the above equations. Note that this approach is a heuristic and therefore doesn't yield an exact solution to the given problem, but rather an approximation. In pseudocode, the generic gradient boosting method is:
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 fit a decision tree h m ( x ) (x) to pseudo-residuals. Let J m be the number of its leaves. The tree partitions the input space into J m disjoint regions R 1 m , … , R J m m ,\ldots ,R_m and predicts a constant value in each region. Using the indicator notation, the output of h m ( x ) (x) for input x can be written as the sum: h m ( x ) = ∑ j = 1 J m b j m 1 R j m ( x ) , (x)=\sum _^b_\mathbf _(x), where b j m is the value predicted in the region R j m . Then the coefficients b j m are multiplied by some value γ m , chosen using line search so as to minimize the loss function, and the model is updated as follows: F m ( x ) = F m − 1 ( x ) + γ m h m ( x ) , γ m = a r g m i n γ ∑ i = 1 n L ( y i , F m − 1 ( x i ) + γ h m ( x i ) ) . (x)=F_(x)+\gamma _h_(x),\quad \gamma _= \sum _^L(y_,F_(x_)+\gamma h_(x_)). Friedman proposes to modify this algorithm so that it chooses a separate optimal value γ j m for each of the tree's regions, instead of a single γ m for the whole tree. He calls the modified algorithm "TreeBoost". The coefficients b j m from the tree-fitting procedure can be then simply discarded and the model update rule becomes: F m ( x ) = F m − 1 ( x ) + ∑ j = 1 J m γ j m 1 R j m ( x ) , γ j m = a r g m i n γ ∑ x i ∈ R j m L ( y i , F m − 1 ( x i ) + γ ) . (x)=F_(x)+\sum _^\gamma _\mathbf _(x),\quad \gamma _= \sum _\in R_L(y_,F_(x_)+\gamma ). When the loss L ( ⋅ , ⋅ ) is mean-squared error (MSE) the coefficients γ j m coincide with the coefficients of the tree-fitting procedure b j m .
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 the number of gradient boosting iterations M (i.e. the number of base models). Increasing M reduces the error on training set, but increases risk of overfitting. An optimal value of M is often selected by monitoring prediction error on a separate validation data set. Another regularization parameter for tree boosting is tree depth. The higher this value the more likely the model will overfit the training data.
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), variants of gradient boosting Deep Neural Networks (DNN) were successful in reproducing the results of non-machine learning methods of analysis on datasets used to discover the Higgs boson. Gradient boosting decision tree was also applied in earth and geological studies – for example quality evaluation of sandstone reservoir.
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 models as Multiple Additive Regression Trees (MART); Elith et al. describe that approach as "Boosted Regression Trees" (BRT). A popular open-source implementation for R calls it a "Generalized Boosting Model", however packages expanding this work use BRT. Yet another name is TreeNet, after an early commercial implementation from Salford System's Dan Steinberg, one of researchers who pioneered the use of tree-based methods.
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 based on entropy as well with the caveat that it is averaged out over all base learners.
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 hundreds or thousands of trees is much harder. To achieve both performance and interpretability, some model compression techniques allow transforming an XGBoost into a single "born-again" decision tree that approximates the same decision function. Furthermore, its implementation may be more difficult due to the higher computational demand.
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 classifies samples from a space X into one of two classes, labelled 1 and -1, respectively. LPBoost is an algorithm for learning such a classification function, given a set of training examples with known class labels. LPBoost is a machine learning technique especially suited for joint classification and feature selection in structured domains.
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 resulting linear combination of many weak classifiers can perform very well. LPBoost constructs f by starting with an empty set of weak classifiers. Iteratively, a single weak classifier to add to the set of considered weak classifiers is selected, added and all the weights α for the current set of weak classifiers are adjusted. This is repeated until no weak classifiers to add remain. The property that all classifier weights are adjusted in each iteration is known as totally-corrective property. Early boosting methods, such as AdaBoost do not have this property and converge slower.
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-negative vector ξ of slack variables and the margin ρ is the following. min α , ξ , ρ − ρ + D ∑ n = 1 ℓ ξ n sb.t. ∑ ω ∈ Ω y n α ω h ( x n ; ω ) + ξ n ≥ ρ , n = 1 , … , ℓ , ∑ ω ∈ Ω α ω = 1 , ξ n ≥ 0 , n = 1 , … , ℓ , α ω ≥ 0 , ω ∈ Ω , ρ ∈ R . ,,\rho &-\rho +D\sum _^\xi _\\&\sum _y_\alpha _h(_;\omega )+\xi _\geq \rho ,\qquad n=1,\dots ,\ell ,\\&\sum _\alpha _=1,\\&\xi _\geq 0,\qquad n=1,\dots ,\ell ,\\&\alpha _\geq 0,\qquad \omega \in \Omega ,\\&\rho \in .\end Note the effects of slack variables ξ ≥ 0 \geq 0 : their one-norm is penalized in the objective function by a constant factor D , which—if small enough—always leads to a primal feasible linear program. Here we adopted the notation of a parameter space Ω , such that for a choice ω ∈ Ω the weak classifier h ( ⋅ ; ω ) : X → \to \ is uniquely defined. When the above linear program was first written down in early publications about boosting methods it was disregarded as intractable due to the large number of variables α . Only later it was discovered that such linear programs can indeed be solved efficiently using the classic technique of column generation.
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 Iterate h ^ ← argmax ω ∈ Ω ∑ n = 1 ℓ y n h ( x n ; ω ) λ n \leftarrow \sum _^y_h(_;\omega )\lambda _ if ∑ n = 1 ℓ y n h ^ ( x n ) λ n + γ ≤ θ ^y_(_)\lambda _+\gamma \leq \theta then break h J ← h ^ \leftarrow J ← J + 1 ( λ , γ ) ← ,\gamma )\leftarrow solution of the LPBoost dual α ← \leftarrow Lagrangian multipliers of solution to LPBoost dual problem f ( x ) := sign ( ∑ j = 1 J α j h j ( x ) ) ):=\left(\sum _^\alpha _h_()\right) Note that if the convergence threshold is set to θ = 0 the solution obtained is the global optimal solution of the above linear program. In practice, θ is set to a small positive value in order obtain a good solution quickly.
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 base learners commonly used with Boosting in the literature is large. For example, if X ⊆ R n \subseteq ^ , a base learner could be a linear soft margin support vector machine. Or even more simple, a simple stump of the form h ( x ; ω ∈ , p ∈ , t ∈ R ) := ;\omega \in \,p\in \,t\in ):=\left\\omega &_\leq t\\-\omega &\end\right.. The above decision stumps looks only along a single dimension p of the input space and simply thresholds the respective column of the sample using a constant threshold t . Then, it can decide in either direction, depending on ω for a positive or negative class. Given weights for the training samples, constructing the optimal decision stump of the above form simply involves searching along all sample columns and determining p , t and ω in order to optimize the gain function.
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 tasks, the output is the average of the predictions of the trees. Random forests correct for decision trees' habit of overfitting to their training set.: 587–588 The first algorithm for random decision forests was created in 1995 by Tin Kam Ho using the random subspace method, which, in Ho's formulation, is a way to implement the "stochastic discrimination" approach to classification proposed by Eugene Kleinberg. An extension of the algorithm was developed by Leo Breiman and Adele Cutler, who registered "Random Forests" as a trademark in 2006 (as of 2019, owned by Minitab, Inc.). The extension combines Breiman's "bagging" idea and random selection of features, introduced first by Ho and later independently by Amit and Geman in order to construct a collection of decision trees with controlled variance.
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 trees splitting with oblique hyperplanes can gain accuracy as they grow without suffering from overtraining, as long as the forests are randomly restricted to be sensitive to only selected feature dimensions. A subsequent work along the same lines concluded that other splitting methods behave similarly, as long as they are randomly forced to be insensitive to some feature dimensions. This observation that a more complex classifier (a larger forest) gets more accurate nearly monotonically is in sharp contrast to the common belief that the complexity of a classifier can only grow to a certain level of accuracy before being hurt by overfitting. The explanation of the forest method's resistance to overtraining can be found in Kleinberg's theory of stochastic discrimination. The early development of Breiman's notion of random forests was influenced by the work of Amit and Geman who introduced the idea of searching over a random subset of the available decisions when splitting a node, in the context of growing a single tree. The idea of random subspace selection from Ho was also influential in the design of random forests. This method grows a forest of trees, and introduces variation among the trees by projecting the training data into a randomly chosen subspace before fitting each tree or each node. Finally, the idea of randomized node optimization, where the decision at each node is selected by a randomized procedure, rather than a deterministic optimization was first introduced by Thomas G. Dietterich. The proper introduction of random forests was made in a paper by Leo Breiman. This paper describes a method of building a forest of uncorrelated trees using a CART like procedure, combined with randomized node optimization and bagging. In addition, this paper combines several ingredients, some previously known and some novel, which form the basis of the modern practice of random forests, in particular: Using out-of-bag error as an estimate of the generalization error. Measuring variable importance through permutation. The report also offers the first theoretical result for random forests in the form of a bound on the generalization error which depends on the strength of the trees in the forest and their correlation.
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 reference distribution. A random forest dissimilarity is attractive because it handles mixed variable types very well, is invariant to monotonic transformations of the input variables, and is robust to outlying observations. Random forest dissimilarity easily deals with a large number of semi-continuous variables due to its intrinsic variable selection; for example, the "Addcl 1" random forest dissimilarity weighs the contribution of each variable according to how dependent it is on other variables. Random forest dissimilarity has been used in a variety of applications, e.g. to find clusters of patients based on tissue marker data.
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 an equally high accuracy as the ensemble learner.
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 attention-based models. This interpretability is one of the main advantages of decision trees. It allows developers to confirm that the model has learned realistic information from the data and allows end-users to have trust and confidence in the decisions made by the model. For example, following the path that a decision tree takes to make its decision is quite trivial, but following the paths of tens or hundreds of trees is much harder. To achieve both performance and interpretability, some model compression techniques allow transforming a random forest into a minimal "born-again" decision tree that faithfully reproduces the same decision function. Another limitation of random forests is that if features are linearly correlated with the target, random forest may not enhance the accuracy of the base learner. Likewise in problems with multiple categorical variables.
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 targets Randomized algorithm – Algorithm that employs a degree of randomness as part of its logic or procedure
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 that the learners will produce different models that can be sensibly averaged. In bagging, one samples training points with replacement from the full training set. The random subspace method is similar to bagging except that the features ("attributes", "predictors", "independent variables") are randomly sampled, with replacement, for each learner. Informally, this causes individual learners to not over-focus on features that appear highly predictive/descriptive in the training set, but fail to be as predictive for points outside that set. For this reason, random subspaces are an attractive choice for high-dimensional problems where the number of features is much larger than the number of training points, such as learning from fMRI data or gene expression data. The random subspace method has been used for decision trees; when combined with "ordinary" bagging of decision trees, the resulting models are called random forests. It has also been applied to linear classifiers, support vector machines, nearest neighbours and other types of classifiers. This method is also applicable to one-class classifiers. The random subspace method has also been applied to the portfolio selection problem showing its superiority to the conventional resampled portfolio essentially based on Bagging. To tackle high-dimensional sparse problems, a framework named Random Subspace Ensemble (RaSE) was developed. RaSE combines weak learners trained in random subspaces with a two-layer structure and iterative process. RaSE has been shown to enjoy appealing theoretical properties and practical performance.
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 nl (nl < N) to be the number of input points for l. It is common to have only one value of nl for all the individual models. For each individual model l, create a training set by choosing dl features from D with replacement and train the model. Now, to apply the ensemble model to an unseen point, combine the outputs of the L individual models by majority voting or by combining the posterior probabilities.
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 high-dimensional spaces can be undesirable for many reasons; raw data are often sparse as a consequence of the curse of dimensionality, and analyzing the data is usually computationally intractable. Dimensionality reduction is common in fields that deal with large numbers of observations and/or large numbers of variables, such as signal processing, speech recognition, neuroinformatics, and bioinformatics. Methods are commonly divided into linear and nonlinear approaches. Linear approaches can be further divided into feature selection and feature extraction. Dimensionality reduction can be used for noise reduction, data visualization, cluster analysis, or as an intermediate step to facilitate other analyses.
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 (features are added or removed while building the model based on prediction errors). Data analysis such as regression or classification can be done in the reduced space more accurately than in the original space.
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 multidimensional data, tensor representation can be used in dimensionality reduction through multilinear subspace learning.
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), linear discriminant analysis (LDA), canonical correlation analysis (CCA), or non-negative matrix factorization (NMF) techniques to pre-process the data, followed by clustering via k-NN on feature vectors in a reduced-dimension space. In machine learning, this process is also called low-dimensional embedding. For high-dimensional datasets (e.g., when performing similarity search on live video streams, DNA data, or high-dimensional time series), running a fast approximate k-NN search using locality-sensitive hashing, random projection, "sketches", or other high-dimensional similarity search techniques from the VLDB conference toolbox may be the only feasible option.
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 numerical chemical information in the form of molecular descriptors and correlates these to the response activity/property/toxicity using statistical techniques. While QSAR is essentially a similarity-based approach, the occurrence of activity/property cliffs may greatly reduce the predictive accuracy of the developed models. The novel Arithmetic Residuals in K-groups Analysis (ARKA) approach is a supervised dimensionality reduction technique that can easily identify activity cliffs in a data set. Activity cliffs are similar in their structures but differ considerably in their activity. The basic idea of the ARKA descriptors is to group the conventional QSAR descriptors based on a predefined criterion and then assign weightage to each descriptor in each group. ARKA descriptors have also been used to develop classification-based and regression-based QSAR models with acceptable quality statistics. == References ==
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 (up to approximately 1 billion) number of unknowns. The charge-based BEM solves an integral equation of the potential theory written in terms of the induced surface charge density. This formulation is naturally combined with fast multipole method (FMM) acceleration, and the entire method is known as charge-based BEM-FMM. The combination of BEM and FMM is a common technique in different areas of computational electromagnetics and, in the context of bioelectromagnetism, it provides improvements over the finite element method.
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-compartment conducting media, the surface charge density formulation first appeared in discretized form (for faceted interfaces) in the 1964 paper by Gelernter and Swihart. A subsequent continuous form, including time-dependent and dielectric effects, appeared in the 1967 paper by Barnard, Duck, and Lynn. The charge-based BEM has also been formulated for conducting, dielectric, and magnetic media, and used in different applications. In 2009, Greengard et al. successfully applied the charge-based BEM with fast multipole acceleration to molecular electrostatics of dielectrics. A similar approach to realistic modeling of the human brain with multiple conducting compartments was first described by Makarov et al. in 2018. Along with this, the BEM-based multilevel fast multipole method has been widely used in radar and antenna studies at microwave frequencies as well as in acoustics.
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 classified as one of the following: A conservative field E i ^ derived from an impressed density of EEG or MEG current sources in a homogeneous infinite medium with the conductivity σ at the source location; An instantaneous solenoidal field E i ^ of an induction coil obtained from Faraday's law of induction in a homogeneous infinite medium (air), when transcranial magnetic stimulation (TMS) problems are concerned; A surface field E i ^ derived from an impressed surface current density J i = σ E i ^=\sigma \mathbf ^ of current electrodes injecting electric current at a boundary of a compartment with conductivity σ when transcranial direct-current stimulation (tDCS) or deep brain stimulation (DBS) are concerned; A conservative field E i ^ of charges deposited on voltage electrodes for tDCS or DBS. This specific problem requires a coupled treatment since these charges will depend on the environment; In application to multiscale modeling, a field E i ^ obtained from any other macroscopic numerical solution in a small (mesoscale or microscale) spatial domain within the brain. For example, a constant field can be used. When the impressed field is "turned on", free charges located within a conducting volume D immediately begin to redistribute and accumulate at the boundaries (interfaces) of regions of different conductivity in D. A surface charge density ρ ( r ) ) appears on the conductivity interfaces. This charge density induces a secondary conservative electric field E s ^ following Coulomb's law. One example is a human under a direct current powerline with the known field E i ^ directed down. The superior surface of the human's conducting body will be charged negatively while its inferior portion is charged positively. These surface charges create a secondary electric field that effectively cancels or blocks the primary field everywhere in the body so that no current will flow within the body under DC steady state conditions. Another example is a human head with electrodes attached. At any conductivity interface with a normal vector n pointing from an "inside" (-) compartment of conductivity σ − to an "outside" (+) compartment of conductivity σ + , Kirchhoff's current law requires continuity of the normal component of the electric current density. This leads to the interfacial boundary condition in the form for every facet at a triangulated interface. As long as σ ± are different from each other, the two normal components of the electric field, E ± ⋅ n ^\cdot \mathbf , must also be different. Such a jump across the interface is only possible when a sheet of surface charge exists at that interface. Thus, if an electric current or voltage is applied, the surface charge density follows. The goal of the numerical analysis is to find the unknown surface charge distribution and thus the total electric field E = E i + E s =\mathbf ^+\mathbf ^ (and the total electric potential if required) anywhere in space.
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 uniform surface charge density ρ m . If a volumetric tetrahedral mesh were present, the charged facets would belong to tetrahedra with different conductivity values. We first compute the electric field E m + _^ at the point r m + δ n m _+\delta \mathbf _ , for δ → 0 + i.e., just outside facet 𝑚 at its center. This field contains three contributions: The continuous impressed electric field E i ^ itself; An electric field of the m-th charged facet itself. Very close to the facet, it can be approximated as the electric field of an infinite sheet of uniform surface charge ρ m . By Gauss's Law, it is given by + ρ m / 2 ε 0 ⋅ n m /2\varepsilon _\cdot \mathbf _ where ε 0 is a background electrical permittivity; An electric field generated by all other facets t n , which we approximate as point charges of charge A n ρ n \rho _ at each center r n _ . A similar treatment holds for the electric field E m − _^ just inside facet 𝑚, but the electric field of the flat sheet of charge changes its sign. Using Coulomb's law to calculate the contribution of facets different from t m , we find From this equation, we see that the normal component of the electric field indeed undergoes a jump through the charged interface. This is equivalent to a jump relation of the potential theory. As a second step, the two expressions for E m ± _^ are substituted into the interfacial boundary condition σ − E m − ⋅ n m = σ + E m + ⋅ n m \mathbf _^\cdot \mathbf _=\sigma ^\mathbf _^\cdot \mathbf _ , applied to every facet 𝑚. This operation leads to a system of linear equations for unknown charge densities ρ m which solves the problem: where K m = σ − − σ + σ − + σ + =-\sigma ^+\sigma ^ is the electric conductivity contrast at the m-th facet. The normalization constant ε 0 will cancel out after the solution is substituted in the expression for E s ^ and becomes redundant.
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 ignored. Next, the sum is computed and the initial guess is refined, etc. This solution employs the simple Jacobi iterative method. The more rigorous generalized minimum residual method (GMRES) yields a much faster convergence of the BEM-FMM. In either case, the major work is in computing the underbraced sum in the system of equations above for every m at every iteration; this operation corresponds to a repetitive matrix-vector multiplication. However, one can recognize this sum as an electric field (times 1 2 π ϵ 0 ) of M charges to be computed at M observation points. Such a computation is exactly the task of the fast multipole method, which performs fast matrix-by-vector multiplication in O ( M log ⁡ M ) ) or even O ( M ) operations instead of O ( M 2 ) ) . The FMM3D library realized in both Python and MATLAB can be used for this purpose. It is therefore unnecessary to form or store the dense system matrix typical for the standard BEM.
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 electric current continuity. It is a Fredholm integral equation of the second kind Its derivation does not involve Green's identities (integrations by parts) and is applicable to non-nested geometries. When the Galerkin method is applied and the same zeroth-order basis functions (with a constant charge density for each facet) are still used on triangulated interfaces, we obtain exactly the same discretization as before if we replace the double integrals over surfaces S m and S n of triangles t m and t n , respectively, by ∫ S m ∫ S n r − r ′ \| r − r ′ \| 3 d s ( r ′ ) d s ( r ) ≈ A m A n r m − r n \| r m − r n \| 3 , \int _ -\mathbf -\mathbf \|^ds(\mathbf )ds(\mathbf )\approx _-\mathbf _ _-\mathbf _\|^, where A n is the surface area of the triangle t n . This approximation is only valid when \| r m − r n \| _-\mathbf _\| is much larger than a typical facet size i.e., in the "far field". Otherwise, semi-analytical formulae and Gaussian quadratures for triangles should be used. Typically, 4 to 32 such neighbor integrals per facet should be precomputed, stored, and then used at every iteration. This is an important correction to the plain fast multipole method in the "near field" which should also be used in the simple discrete formulation derived above. Such a correction makes it possible to obtain an unconstrained numerical (but not anatomical) resolution in the brain.
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 mater, arachnoid mater, and pia mater). This is particularly important for accurate transcranial direct-current stimulation and electroconvulsive therapy dosage predictions. The BEM-FMM allows for straightforward adaptive mesh refinement including multiple extracerebral brain compartments. Another application is modeling electric field perturbations within densely packed neuronal/axonal arbor. Such perturbations change the biophysical activating function. A charge-based BEM formulation is being developed for promising bi-domain biophysical modeling of axonal processes. In its present form, the charge-based BEM-FMM is applicable to multi-compartment piecewise homogeneous media only; it cannot handle macroscopically anisotropic tissues. Additionally, the maximum number of facets (degrees of freedom) is limited to approximately 10 9 for typical academic computer hardware resources used as of 2023.
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 to principal component analysis, it provides a means of displaying or summarising a set of data in two-dimensional graphical form. Its aim is to display in a biplot any structure hidden in the multivariate setting of the data table. As such it is a technique from the field of multivariate ordination. Since the variant of CA described here can be applied either with a focus on the rows or on the columns it should in fact be called simple (symmetric) correspondence analysis. It is traditionally applied to the contingency table of a pair of nominal variables where each cell contains either a count or a zero value. If more than two categorical variables are to be summarized, a variant called multiple correspondence analysis should be chosen instead. CA may also be applied to binary data given the presence/absence coding represents simplified count data i.e. a 1 describes a positive count and 0 stands for a count of zero. Depending on the scores used CA preserves the chi-square distance between either the rows or the columns of the table. Because CA is a descriptive technique, it can be applied to tables regardless of a significant chi-squared test. Although the χ 2 statistic used in inferential statistics and the chi-square distance are computationally related they should not be confused since the latter works as a multivariate statistical distance measure in CA while the χ 2 statistic is in fact a scalar not a metric.
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 matrix C of size m × n where m is the number of rows and n is the number of columns. In the following mathematical description of the method capital letters in italics refer to a matrix while letters in italics refer to vectors. Understanding the following computations requires knowledge of matrix algebra.
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 complicated scatter plot. In fact it consists of two scatter plots printed one upon the other, one set of points for the rows and one for the columns. But being a biplot a clear interpretation rule relates the two coordinate matrices used. Usually the first two dimensions of the CA solution are plotted because they encompass the maximum of information about the data table that can be displayed in 2D although other combinations of dimensions may be investigated by a biplot. A biplot is in fact a low dimensional mapping of a part of the information contained in the original table. As a rule of thumb that set (rows or columns) which should be analysed with respect to its composition as measured by the other set is displayed in principal coordinates while the other set is displayed in standard coordinates. E.g. a table displaying voting districts in rows and political parties in columns with the cells containing the counted votes may be displayed with the districts (rows) in principal coordinates when the focus is on ordering districts according to similar voting. Traditionally, originating from the French tradition in CA, early CA biplots mapped both entities in the same coordinate version, usually principal coordinates, but this kind of display is misleading insofar as: "Although this is called a biplot, it does not have any useful inner product relationship between the row and column scores" as Brian Ripley, maintainer of R package MASS points out correctly. Today that kind of display should be avoided since laymen usually are not aware of the lacking relation between the two point sets. A scaling 1 biplot (rows in principal coordinates, columns in standard coordinates) is interpreted as follows: The distances between row points approximate their chi-square distance. Points close to each other represent rows with very similar values in the original data table. I.e they may exhibit rather similar frequencies in case of count data or closely related binary values in case of presence/absence data. (Column) points in standard coordinates represent the vertices of the vector space i.e. the outer corner of something that in multidimensional space has the shape of an irregular polyhedron. Project row points onto the line connecting the origin and the standard coordinate of a column; if the projected position along that connection line is close to the position of the standard coordinate, that row point is strongly associated with this column i.e. in case of count data the row has a high frequency of that category and in case of presence/absence data the row is likely to exhibit a 1 in that column. Row points whose projection would require to elongate the connection line beyond the origin have a lower than average value in that column.
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 analysis to many categorical variables is called multiple correspondence analysis. An adaptation of correspondence analysis to the problem of discrimination based upon qualitative variables (i.e., the equivalent of discriminant analysis for qualitative data) is called discriminant correspondence analysis or barycentric discriminant analysis. In the social sciences, correspondence analysis, and particularly its extension multiple correspondence analysis, was made known outside France through French sociologist Pierre Bourdieu's application of it.
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: ade4::dudi.coa(), ca::ca() , ExPosition::epCA(), FactoMineR::CA(), MASS::corresp(), vegan::cca(). The easiest approach for beginners is ca::ca() as there is an extensive text book accompanying that package. The Freeware PAST (PAleontological STatistics) offers (simple symmetric) correspondence analysis via the menu "Multivariate/Ordination/Correspondence (CA)".
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 moments of streams (these calculations require counting of the number of occurrences for the distinct elements of the stream). The sketch is nearly identical to the Feature hashing algorithm by John Moody, but differs in its use of hash functions with low dependence, which makes it more practical. In order to still have a high probability of success, the median trick is used to aggregate multiple count sketches, rather than the mean. These properties allow use for explicit kernel methods, bilinear pooling in neural networks and is a cornerstone in many numerical linear algebra algorithms.
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 element q can be approximated, although extremely poorly, by the expected value E [ C ⋅ s ( q ) ] [C\cdot s(q)] ; a straightforward way to improve the variance of the previous estimate is to use an array of different hash functions s i , each connected to its own counter C i . For each element q, the E [ C i ⋅ s i ( q ) ] = n ( q ) [C_\cdot s_(q)]=n(q) still holds, so averaging across the i range will tighten the approximation; the previous construct still has a major deficiency: if a lower-frequency-but-still-important output element a exhibits a hash collision with a high-frequency element, n ( a ) estimate can be significantly affected. Avoiding this requires reducing the frequency of collision counter updates between any two distinct elements. This is achieved by replacing each C i in the previous construct with an array of m counters (making the counter set into a two-dimensional matrix C i , j ), with index j of a particular counter to be incremented/decremented selected via another set of hash functions h i that map element q into the range . Since E [ C i , h i ( q ) ⋅ s i ( q ) ] = n ( q ) [C_(q)\cdot s_(q)]=n(q) , averaging across all values of i will work.
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 chosen be pairwise independent. 2. For each item q i in the stream, add s j ( q i ) (q_) to the h j ( q i ) (q_) th bucket of the j th hash. At the end of this process, one has w d sums ( C i j ) ) where C i , j = ∑ h i ( k ) = j s i ( k ) . =\sum _(k)=js_(k). To estimate the count of q s one computes the following value: The values s i ( q ) ⋅ C i , h i ( q ) (q)\cdot C_(q) are unbiased estimates of how many times q has appeared in the stream. The estimate r q has variance O ( m i n ) \^/w^,m_^/w\) , where m 1 is the length of the stream and m 2 2 ^ is ∑ q ( ∑ i [ q i = q ] ) 2 (\sum _[q_=q])^ . Furthermore, r q is guaranteed to never be more than 2 m 2 / w / off from the true value, with probability 1 − e − O ( t ) .
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 this equals C ( x ⊗ x T ) ) – a count sketch C of the outer product of vectors, where ⊗ denotes Kronecker product. The fast Fourier transform can be used to do fast convolution of count sketches. By using the face-splitting product such structures can be computed much faster than normal matrices.
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. "Multispectral Periocular Classification WithMultimodal Compact Multi-Linear Pooling" [1]. IEEE Access, Vol. 5. 2017. Ahle, Thomas; Knudsen, Jakob (2019-09-03). "Almost Optimal Tensor Sketch". ResearchGate. Retrieved 2020-07-11.
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 inherent in most other multivariate analyses when applied to gradient data.
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 sometimes erroneously referred to as DECORANA; however, DCA is the underlying algorithm, while DECORANA is a tool implementing it.
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 late successional ones (see example below). When such data are analysed by a standard ordination such as a correspondence analysis: the ordination scores of the samples will exhibit the 'edge effect', i.e. the variance of the scores at the beginning and the end of a regular succession of species will be considerably smaller than that in the middle, when presented as a graph the points will be seen to follow a horseshoe shaped curve rather than a straight line ('arch effect'), even though the process under analysis is a steady and continuous change that human intuition would prefer to see as a linear trend. Outside ecology, the same artifacts occur when gradient data are analysed (e.g. soil properties along a transect running between 2 different geologies, or behavioural data over the lifespan of an individual) because the curved projection is an accurate representation of the shape of the data in multivariate space. Ter Braak and Prentice (1987, p. 121) cite a simulation study analysing two-dimensional species packing models resulting in a better performance of DCA compared to CA.
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 in which the 1st ordination axis distorts into the 2nd axis. It then divides the first axis into segments (default = 26), and rescales each segment to have mean value of zero on the 2nd axis - this effectively squashes the curve flat. It also rescales the axis so that the ends are no longer compressed relative to the middle, so that 1 DCA unit approximates to the same rate of turnover all the way through the data: the rule of thumb is that 4 DCA units mean that there has been a total turnover in the community. Ter Braak and Prentice (1987, p. 122) warn against the non-linear rescaling of the axes due to robustness issues and recommend using detrending-by-polynomials only.
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 a null model by Monte-Carlo permutation analysis.
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 at the edges each sample contains five species. The plot of the first two axes of the correspondence analysis result on the right hand side clearly shows the disadvantages of this procedure: the edge effect, i.e. the points are clustered at the edges of the first axis, and the arch effect.
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 Technique. Vegetatio 42, 47–58. Oksanen J and Minchin PR (1997). Instability of ordination results under changes in input data order: explanation and remedies. Journal of vegetation science 8, 447–454 Shaw PJA (2003). Multivariate Statistics for the Environmental Sciences. London: Hodder Arnold Ter Braak, C.J.F. and Prentice, I.C. (1988). A Theory of Gradient Analysis. Advances in Ecological Research 18, 271–371. ISBN 0-12-013918-9. Reprinted in: Ter Braak, C.J.F. (1987). Unimodal models to relate species to environment. Wageningen: PhD thesis Agricultural Mathematics Group, 101–146.
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 software including the function decorana: Detrended Correspondence Analysis and Basic Reciprocal Averaging from Hill and Gauch (1980)
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 clustering (zero elasticity) to the estimators located closely to linear PCA manifolds (for high bending and low stretching modules). With some intermediate values of the elasticity coefficients, this system effectively approximates non-linear principal manifolds. This approach is based on a mechanical analogy between principal manifolds, that are passing through "the middle" of the data distribution, and elastic membranes and plates. The method was developed by A.N. Gorban, A.Y. Zinovyev and A.A. Pitenko in 1996–1998.
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 divided into classes K j = =\_s\ . The approximation energy D is the distortion D = 1 2 ∑ j = 1 k ∑ s ∈ K j ‖ s − w j ‖ 2 \sum _^\sum _\\|s-_\\|^ , which is the energy of the springs with unit elasticity which connect each data point with its host node. It is possible to apply weighting factors to the terms of this sum, for example to reflect the standard deviation of the probability density function of any subset of data points \ . On the set of nodes an additional structure is defined. Some pairs of nodes, ( w i , w j ) _,_) , are connected by elastic edges. Call this set of pairs E . Some triplets of nodes, ( w i , w j , w k ) _,_,_) , form bending ribs. Call this set of triplets G . The stretching energy is U E = 1 2 λ ∑ ( w i , w j ) ∈ E ‖ w i − w j ‖ 2 =\lambda \sum __,_)\in E\\|_-_\\|^ , The bending energy is U G = 1 2 μ ∑ ( w i , w j , w k ) ∈ G ‖ w i − 2 w j + w k ‖ 2 =\mu \sum __,_,_)\in G\\|_-2_+_\\|^ , where λ and μ are the stretching and bending moduli respectively. The stretching energy is sometimes referred to as the membrane, while the bending energy is referred to as the thin plate term. For example, on the 2D rectangular grid the elastic edges are just vertical and horizontal edges (pairs of closest vertices) and the bending ribs are the vertical or horizontal triplets of consecutive (closest) vertices. The total energy of the elastic map is thus U = D + U E + U G . +U_. The position of the nodes _\ is determined by the mechanical equilibrium of the elastic map, i.e. its location is such that it minimizes the total energy U .
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 _\ ; If no change, terminate. This expectation-maximization algorithm guarantees a local minimum of U . For improving the approximation various additional methods are proposed. For example, the softening strategy is used. This strategy starts with a rigid grids (small length, small bending and large elasticity modules λ and μ coefficients) and finishes with soft grids (small λ and μ ). The training goes in several epochs, each epoch with its own grid rigidness. Another adaptive strategy is growing net: one starts from a small number of nodes and gradually adds new nodes. Each epoch goes with its own number of nodes.
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 visualisation of data of various nature. The method is applied in quantitative biology for reconstructing the curved surface of a tree leaf from a stack of light microscopy images. This reconstruction is used for quantifying the geodesic distances between trichomes and their patterning, which is a marker of the capability of a plant to resist to pathogenes. Recently, the method is adapted as a support tool in the decision process underlying the selection, optimization, and management of financial portfolios. The method of elastic maps has been systematically tested and compared with several machine learning methods on the applied problem of identification of the flow regime of a gas-liquid flow in a pipe. There are various regimes: Single phase water or air flow, Bubbly flow, Bubbly-slug flow, Slug flow, Slug-churn flow, Churn flow, Churn-annular flow, and Annular flow. The simplest and most common method used to identify the flow regime is visual observation. This approach is, however, subjective and unsuitable for relatively high gas and liquid flow rates. Therefore, the machine learning methods are proposed by many authors. The methods are applied to differential pressure data collected during a calibration process. The method of elastic maps provided a 2D map, where the area of each regime is represented. The comparison with some other machine learning methods is presented in Table 1 for various pipe diameters and pressure. Here, ANN stands for the backpropagation artificial neural networks, SVM stands for the support vector machine, SOM for the self-organizing maps. The hybrid technology was developed for engineering applications. In this technology, elastic maps are used in combination with Principal Component Analysis (PCA), Independent Component Analysis (ICA) and backpropagation ANN. The textbook provides a systematic comparison of elastic maps and self-organizing maps (SOMs) in applications to economic and financial decision-making. == References ==
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 avoid the curse of dimensionality, improve the compatibility of the data with a certain learning model class, to encode inherent symmetries present in the input space. The central premise when using feature selection is that data sometimes contains features that are redundant or irrelevant, and can thus be removed without incurring much loss of information. Redundancy and irrelevance are two distinct notions, since one relevant feature may be redundant in the presence of another relevant feature with which it is strongly correlated. Feature extraction creates new features from functions of the original features, whereas feature selection finds a subset of the features. Feature selection techniques are often used in domains where there are many features and comparatively few samples (data points).
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 error rate. This is an exhaustive search of the space, and is computationally intractable for all but the smallest of feature sets. The choice of evaluation metric heavily influences the algorithm, and it is these evaluation metrics which distinguish between the three main categories of feature selection algorithms: wrappers, filters and embedded methods. Wrapper methods use a predictive model to score feature subsets. Each new subset is used to train a model, which is tested on a hold-out set. Counting the number of mistakes made on that hold-out set (the error rate of the model) gives the score for that subset. As wrapper methods train a new model for each subset, they are very computationally intensive, but usually provide the best performing feature set for that particular type of model or typical problem. Filter methods use a proxy measure instead of the error rate to score a feature subset. This measure is chosen to be fast to compute, while still capturing the usefulness of the feature set. Common measures include the mutual information, the pointwise mutual information, Pearson product-moment correlation coefficient, Relief-based algorithms, and inter/intra class distance or the scores of significance tests for each class/feature combinations. Filters are usually less computationally intensive than wrappers, but they produce a feature set which is not tuned to a specific type of predictive model. This lack of tuning means a feature set from a filter is more general than the set from a wrapper, usually giving lower prediction performance than a wrapper. However the feature set doesn't contain the assumptions of a prediction model, and so is more useful for exposing the relationships between the features. Many filters provide a feature ranking rather than an explicit best feature subset, and the cut off point in the ranking is chosen via cross-validation. Filter methods have also been used as a preprocessing step for wrapper methods, allowing a wrapper to be used on larger problems. One other popular approach is the Recursive Feature Elimination algorithm, commonly used with Support Vector Machines to repeatedly construct a model and remove features with low weights. Embedded methods are a catch-all group of techniques which perform feature selection as part of the model construction process. The exemplar of this approach is the LASSO method for constructing a linear model, which penalizes the regression coefficients with an L1 penalty, shrinking many of them to zero. Any features which have non-zero regression coefficients are 'selected' by the LASSO algorithm. Improvements to the LASSO include Bolasso which bootstraps samples; Elastic net regularization, which combines the L1 penalty of LASSO with the L2 penalty of ridge regression; and FeaLect which scores all the features based on combinatorial analysis of regression coefficients. AEFS further extends LASSO to nonlinear scenario with autoencoders. These approaches tend to be between filters and wrappers in terms of computational complexity. In traditional regression analysis, the most popular form of feature selection is stepwise regression, which is a wrapper technique. It is a greedy algorithm that adds the best feature (or deletes the worst feature) at each round. The main control issue is deciding when to stop the algorithm. In machine learning, this is typically done by cross-validation. In statistics, some criteria are optimized. This leads to the inherent problem of nesting. More robust methods have been explored, such as branch and bound and piecewise linear network.
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 subset. Wrappers can be computationally expensive and have a risk of over fitting to the model. Filters are similar to wrappers in the search approach, but instead of evaluating against a model, a simpler filter is evaluated. Embedded techniques are embedded in, and specific to, a model. Many popular search approaches use greedy hill climbing, which iteratively evaluates a candidate subset of features, then modifies the subset and evaluates if the new subset is an improvement over the old. Evaluation of the subsets requires a scoring metric that grades a subset of features. Exhaustive search is generally impractical, so at some implementor (or operator) defined stopping point, the subset of features with the highest score discovered up to that point is selected as the satisfactory feature subset. The stopping criterion varies by algorithm; possible criteria include: a subset score exceeds a threshold, a program's maximum allowed run time has been surpassed, etc. Alternative search-based techniques are based on targeted projection pursuit which finds low-dimensional projections of the data that score highly: the features that have the largest projections in the lower-dimensional space are then selected. Search approaches include: Exhaustive Best first Simulated annealing Genetic algorithm Greedy forward selection Greedy backward elimination Particle swarm optimization Targeted projection pursuit Scatter search Variable neighborhood search Two popular filter metrics for classification problems are correlation and mutual information, although neither are true metrics or 'distance measures' in the mathematical sense, since they fail to obey the triangle inequality and thus do not compute any actual 'distance' – they should rather be regarded as 'scores'. These scores are computed between a candidate feature (or set of features) and the desired output category. There are, however, true metrics that are a simple function of the mutual information; see here. Other available filter metrics include: Class separability Error probability Inter-class distance Probabilistic distance Entropy Consistency-based feature selection Correlation-based feature selection
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 penalty of 2 for each added feature. AIC is based on information theory, and is effectively derived via the maximum entropy principle. Other criteria are Bayesian information criterion (BIC), which uses a penalty of log ⁡ n for each added feature, minimum description length (MDL) which asymptotically uses log ⁡ n , Bonferroni / RIC which use 2 log ⁡ p , maximum dependency feature selection, and a variety of new criteria that are motivated by false discovery rate (FDR), which use something close to 2 log ⁡ p q . A maximum entropy rate criterion may also be used to select the most relevant subset of features.
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 a graph. The most common structure learning algorithms assume the data is generated by a Bayesian Network, and so the structure is a directed graphical model. The optimal solution to the filter feature selection problem is the Markov blanket of the target node, and in a Bayesian Network, there is a unique Markov Blanket for each node.
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 the largest score (e.g. argmax f i ∈ F ( I ( f i , c ) ) \in F (I(f_,c)) ) and add it to the set of selected features (S) Calculate the score which might be derived from the mutual information Select the feature with the largest score and add it to the set of select features (e.g. argmax f i ∈ F ( I d e r i v e d ( f i , c ) ) \in F (I_(f_,c)) ) Repeat 3. and 4. until a certain number of features is selected (e.g. \| S \| = l ) The simplest approach uses the mutual information as the "derived" score. However, there are different approaches, that try to reduce the redundancy between features.
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 = 1 n x k HSIC ( f k , c ) + λ ‖ x ‖ 1 , s.t. x 1 , … , x n ≥ 0 , :\min _ \sum _^x_x_(f_,f_)-\sum _^x_(f_,c)+\lambda \\|\mathbf \\|_,\quad \ x_,\ldots ,x_\geq 0, where HSIC ( f k , c ) = tr ( K ¯ ( k ) L ¯ ) (f_,c)=( ^ ) is a kernel-based independence measure called the (empirical) Hilbert-Schmidt independence criterion (HSIC), tr ( ⋅ ) (\cdot ) denotes the trace, λ is the regularization parameter, K ¯ ( k ) = Γ K ( k ) Γ ^=\mathbf \mathbf ^\mathbf and L ¯ = Γ L Γ =\mathbf \mathbf \mathbf are input and output centered Gram matrices, K i , j ( k ) = K ( u k , i , u k , j ) ^=K(u_,u_) and L i , j = L ( c i , c j ) =L(c_,c_) are Gram matrices, K ( u , u ′ ) and L ( c , c ′ ) are kernel functions, Γ = I m − 1 m 1 m 1 m T =\mathbf _-\mathbf _\mathbf _^ is the centering matrix, I m _ is the m-dimensional identity matrix (m: the number of samples), 1 m _ is the m-dimensional vector with all ones, and ‖ ⋅ ‖ 1 is the ℓ 1 -norm. HSIC always takes a non-negative value, and is zero if and only if two random variables are statistically independent when a universal reproducing kernel such as the Gaussian kernel is used. The HSIC Lasso can be written as H S I C L a s s o : min x 1 2 ‖ L ¯ − ∑ k = 1 n x k K ¯ ( k ) ‖ F 2 + λ ‖ x ‖ 1 , s.t. x 1 , … , x n ≥ 0 , :\min _ \left\\| -\sum _^x_ ^\right\\|_^+\lambda \\|\mathbf \\|_,\quad \ x_,\ldots ,x_\geq 0, where ‖ ⋅ ‖ F is the Frobenius norm. The optimization problem is a Lasso problem, and thus it can be efficiently solved with a state-of-the-art Lasso solver such as the dual augmented Lagrangian method.
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 consisting of k features: M e r i t S k = k r c f ¯ k + k ( k − 1 ) r f f ¯ . _=. Here, r c f ¯ is the average value of all feature-classification correlations, and r f f ¯ is the average value of all feature-feature correlations. The CFS criterion is defined as follows: C F S = max S k [ r c f 1 + r c f 2 + ⋯ + r c f k k + 2 ( r f 1 f 2 + ⋯ + r f i f j + ⋯ + r f k f k − 1 ) ] . =\max _\left[+r_+\cdots +r_f_+\cdots +r_f_+\cdots +r_f_)\right]. The r c f i and r f i f j f_ variables are referred to as correlations, but are not necessarily Pearson's correlation coefficient or Spearman's ρ. Hall's dissertation uses neither of these, but uses three different measures of relatedness, minimum description length (MDL), symmetrical uncertainty, and relief. Let xi be the set membership indicator function for feature fi; then the above can be rewritten as an optimization problem: C F S = max x ∈ n [ ( ∑ i = 1 n a i x i ) 2 ∑ i = 1 n x i + ∑ i ≠ j 2 b i j x i x j ] . =\max _^\left[^a_x_)^^x_+\sum _2b_x_x_\right]. The combinatorial problems above are, in fact, mixed 0–1 linear programming problems that can be solved by using branch-and-bound algorithms.
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. Regularized trees only need build one tree model (or one tree ensemble model) and thus are computationally efficient. Regularized trees naturally handle numerical and categorical features, interactions and nonlinearities. They are invariant to attribute scales (units) and insensitive to outliers, and thus, require little data preprocessing such as normalization. Regularized random forest (RRF) is one type of regularized trees. The guided RRF is an enhanced RRF which is guided by the importance scores from an ordinary random forest.
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 metaheuristics, from a simple local search to a complex global search algorithm.
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 algorithm Random multinomial logit (RMNL) Auto-encoding networks with a bottleneck-layer Submodular feature selection Local learning based feature selection. Compared with traditional methods, it does not involve any heuristic search, can easily handle multi-class problems, and works for both linear and nonlinear problems. It is also supported by a strong theoretical foundation. Numeric experiments showed that the method can achieve a close-to-optimal solution even when data contains >1M irrelevant features. Recommender system based on feature selection. The feature selection methods are introduced into recommender system research.
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 Discovery and Data Mining. Springer. ISBN 0-7923-8198-X. Liu, Huan; Yu, Lei (2005). "Toward Integrating Feature Selection Algorithms for Classification and Clustering". IEEE Transactions on Knowledge and Data Engineering. 17 (4): 491–502. doi:10.1109/TKDE.2005.66. S2CID 1607600.
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 selection program FEAST (Open source Feature Selection algorithms in C and MATLAB)
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 random variables, a gCCA generalizes PCA to more than two sets of random variables. The canonical variables represent those common factors that can be found by a large PCA of all of the transformed random variables after each set underwent its own PCA.
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 method can be used for finding those harmful common factors that create cross-correlation between the blocks. However, no optimal HWB solution exists if the random variables do not contain enough information on all of the new regression parameters.
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, Z.D., Heng, P.A., Xia, D.S. (2005) "A Theorem on the Generalized Canonical Projective Vectors". Pattern Recognition 38 (3) 449 Kettenring, J. R. (1971) "Canonical analysis of several sets of variables". "Biometrika" 58 (3) 433
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 other. ICA was invented by Jeanny Hérault and Christian Jutten in 1985. ICA is a special case of blind source separation. A common example application of ICA is the "cocktail party problem" of listening in on one person's speech in a noisy room.
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 possible to separate these contributing sources from the observed total signal. When the statistical independence assumption is correct, blind ICA separation of a mixed signal gives very good results. It is also used for signals that are not supposed to be generated by mixing for analysis purposes. A simple application of ICA is the "cocktail party problem", where the underlying speech signals are separated from a sample data consisting of people talking simultaneously in a room. Usually the problem is simplified by assuming no time delays or echoes. Note that a filtered and delayed signal is a copy of a dependent component, and thus the statistical independence assumption is not violated. Mixing weights for constructing the M observed signals from the N components can be placed in an M × N matrix. An important thing to consider is that if N sources are present, at least N observations (e.g. microphones if the observed signal is audio) are needed to recover the original signals. When there are an equal number of observations and source signals, the mixing matrix is square ( M = N ). Other cases of underdetermined ( M < N ) and overdetermined ( M > N ) have been investigated. The success of ICA separation of mixed signals relies on two assumptions and three effects of mixing source signals. Two assumptions: The source signals are independent of each other. The values in each source signal have non-Gaussian distributions. Three effects of mixing source signals: Independence: As per assumption 1, the source signals are independent; however, their signal mixtures are not. This is because the signal mixtures share the same source signals. Normality: According to the Central Limit Theorem, the distribution of a sum of independent random variables with finite variance tends towards a Gaussian distribution.Loosely speaking, a sum of two independent random variables usually has a distribution that is closer to Gaussian than any of the two original variables. Here we consider the value of each signal as the random variable. Complexity: The temporal complexity of any signal mixture is greater than that of its simplest constituent source signal. Those principles contribute to the basic establishment of ICA. If the signals extracted from a set of mixtures are independent and have non-Gaussian distributions or have low complexity, then they must be source signals.
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 two broadest definitions of independence for ICA are Minimization of mutual information Maximization of non-Gaussianity The Minimization-of-Mutual information (MMI) family of ICA algorithms uses measures like Kullback-Leibler Divergence and maximum entropy. The non-Gaussianity family of ICA algorithms, motivated by the central limit theorem, uses kurtosis and negentropy. Typical algorithms for ICA use centering (subtract the mean to create a zero mean signal), whitening (usually with the eigenvalue decomposition), and dimensionality reduction as preprocessing steps in order to simplify and reduce the complexity of the problem for the actual iterative algorithm. Whitening and dimension reduction can be achieved with principal component analysis or singular value decomposition. Whitening ensures that all dimensions are treated equally a priori before the algorithm is run. Well-known algorithms for ICA include infomax, FastICA, JADE, and kernel-independent component analysis, among others. In general, ICA cannot identify the actual number of source signals, a uniquely correct ordering of the source signals, nor the proper scaling (including sign) of the source signals. ICA is important to blind signal separation and has many practical applications. It is closely related to (or even a special case of) the search for a factorial code of the data, i.e., a new vector-valued representation of each data vector such that it gets uniquely encoded by the resulting code vector (loss-free coding), but the code components are statistically independent.
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-cluster assignment, network tomography and internet resource management. Let x 1 , x 2 , … , x m ,x_,\ldots ,x_ be the set of binary variables from m monitors and y 1 , y 2 , … , y n ,y_,\ldots ,y_ be the set of binary variables from n sources. Source-monitor connections are represented by the (unknown) mixing matrix G , where g i j = 1 =1 indicates that signal from the i-th source can be observed by the j-th monitor. The system works as follows: at any time, if a source i is active ( y i = 1 =1 ) and it is connected to the monitor j ( g i j = 1 =1 ) then the monitor j will observe some activity ( x j = 1 =1 ). Formally we have: x i = ⋁ j = 1 n ( g i j ∧ y j ) , i = 1 , 2 , … , m , =\bigvee _^(g_\wedge y_),i=1,2,\ldots ,m, where ∧ is Boolean AND and ∨ is Boolean OR. Noise is not explicitly modelled, rather, can be treated as independent sources. The above problem can be heuristically solved by assuming variables are continuous and running FastICA on binary observation data to get the mixing matrix G (real values), then apply round number techniques on G to obtain the binary values. This approach has been shown to produce a highly inaccurate result. Another method is to use dynamic programming: recursively breaking the observation matrix X into its sub-matrices and run the inference algorithm on these sub-matrices. The key observation which leads to this algorithm is the sub-matrix X 0 ^ of X where x i j = 0 , ∀ j =0,\forall j corresponds to the unbiased observation matrix of hidden components that do not have connection to the i -th monitor. Experimental results from show that this approach is accurate under moderate noise levels. The Generalized Binary ICA framework introduces a broader problem formulation which does not necessitate any knowledge on the generative model. In other words, this method attempts to decompose a source into its independent components (as much as possible, and without losing any information) with no prior assumption on the way it was generated. Although this problem appears quite complex, it can be accurately solved with a branch and bound search tree algorithm or tightly upper bounded with a single multiplication of a matrix with a vector.
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 Sejnowski introduced a fast and efficient ICA algorithm based on infomax, a principle introduced by Ralph Linsker in 1987. A link exists between maximum-likelihood estimation and Infomax approaches. A quite comprehensive tutorial on the maximum-likelihood approach to ICA has been published by J-F. Cardoso in 1998. There are many algorithms available in the literature which do ICA. A largely used one, including in industrial applications, is the FastICA algorithm, developed by Hyvärinen and Oja, which uses the negentropy as cost function, already proposed 7 years before by Pierre Comon in this context. Other examples are rather related to blind source separation where a more general approach is used. For example, one can drop the independence assumption and separate mutually correlated signals, thus, statistically "dependent" signals. Sepp Hochreiter and Jürgen Schmidhuber showed how to obtain non-linear ICA or source separation as a by-product of regularization (1999). Their method does not require a priori knowledge about the number of independent sources.
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 primary visual neurons predicting stock market prices mobile phone communications colour based detection of the ripeness of tomatoes removing artifacts, such as eye blinks, from EEG data. predicting decision-making using EEG analysis of changes in gene expression over time in single cell RNA-sequencing experiments. studies of the resting state network of the brain. astronomy and cosmology finance
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 chapter ) Hyvärinen, A.; Oja, E. (2000): "Independent Component Analysis: Algorithms and Application", Neural Networks, 13(4-5):411-430. (Technical but pedagogical introduction). Comon, P.; Jutten C., (2010): Handbook of Blind Source Separation, Independent Component Analysis and Applications. Academic Press, Oxford UK. ISBN 978-0-12-374726-6 Lee, T.-W. (1998): Independent component analysis: Theory and applications, Boston, Mass: Kluwer Academic Publishers, ISBN 0-7923-8261-7 Acharyya, Ranjan (2008): A New Approach for Blind Source Separation of Convolutive Sources - Wavelet Based Separation Using Shrinkage Function ISBN 3-639-07797-0 ISBN 978-3639077971 (this book focuses on unsupervised learning with Blind Source Separation)

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