Documentation |
Superclasses: CompactClassificationSVM
Support vector machine for binary classification
ClassificationSVM is a support vector machine classifier for one- or two-class learning. Use fitcsvm and the training data to train a ClassificationSVM classifier.
Trained ClassificationSVM classifiers store the training data, parameter values, prior probabilities, support vectors, and algorithmic implementation information. You can use these classifiers to:
Estimate resubstitution predictions. For details, see resubPredict.
Predict labels or posterior probabilities for new data. For details, see predict.
SVMModel = fitcsvm(X,Y) returns a trained SVM classifier (SVMModel) based on the input variables (also known as predictors, features, or attributes) X and output variables (also known as responses or class labels) Y. For details, see fitcsvm.
SVMModel = fitcsvm(X,Y,Name,Value) returns a trained SVM classifier with additional options specified by one or more Name,Value pair arguments. For name-value pair argument details, see fitcsvm.
If you set one of the following five options, then SVMModel is a ClassificationPartitionedModel model: 'CrossVal', 'CVPartition', 'Holdout', 'KFold', or 'Leaveout'. Otherwise, SVMModel is a ClassificationSVM classifier.
Alpha |
Numeric vector of trained classifier coefficients from the dual problem (i.e., the estimated Lagrange multipliers). Alpha has length equal to the number of support vectors in the trained classifier (i.e., sum(SVMModel.IsSupportVector)). | ||||||||||||||||||||||
Beta |
Numeric vector of trained classifier coefficients from the primal linear problem. Beta has length equal to the number of predictors (i.e., size(SVMModel.X,2)). Beta = [] for one-class learning or two-class learning using a nonlinear kernel function. | ||||||||||||||||||||||
Bias |
Scalar corresponding to the trained classifier bias term. | ||||||||||||||||||||||
BoxConstraints |
Numeric vector of box constraints. BoxConstraints has length equal to the number of observations (i.e., size(SVMModel.X,1)). | ||||||||||||||||||||||
CacheInfo |
Structure array containing:
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CategoricalPredictors |
List of categorical predictors, which is always empty ([]) for SVM and discriminant analysis classifiers. | ||||||||||||||||||||||
ClassNames |
List of elements in Y with duplicates removed. ClassNames has the same data type as the data in the argument Y, and therefore can be a categorical or character array, logical or numeric vector, or cell array of strings. | ||||||||||||||||||||||
ConvergenceInfo |
Structure array containing convergence information.
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Cost |
Square matrix, where Cost(i,j) is the cost of classifying a point into class j if its true class is i. During training, the software updates the prior probabilities by incorporating the penalties described in the cost matrix. Therefore,
For more details, see Algorithms. | ||||||||||||||||||||||
Gradient |
Numeric vector of training data gradient values. Gradient has length equal to the number of observations (i.e., size(SVMModel.X,1)). | ||||||||||||||||||||||
IsSupportVector |
Logical vector indicating whether a corresponding row in the predictor data matrix is a support vector. IsSupportVector has length equal to the number of observations (i.e., size(SVMModel.X,1)). | ||||||||||||||||||||||
KernelParameters |
Structure array containing the kernel name and parameter values. To display the values of KernelParameters, use dot notation, e.g., SVMModel.KernelParameters.Scale displays the scale parameter value. The software accepts KernelParameters as inputs, and does not modify them. Alter KernelParameters by setting the appropriate name-value pair arguments when you train the SVM classifier using fitcsvm. | ||||||||||||||||||||||
ModelParameters |
Object containing parameter values, e.g., the name-value pair argument values, used to train the SVM classifier. ModelParameters does not contain estimated parameters. Access fields of ModelParameters using dot notation. For example, access the initial values for estimating Alpha using SVMModel.ModelParameters.Alpha. | ||||||||||||||||||||||
Mu |
Numeric vector of predictor means. If you specify 'Standardize',1 or 'Standardize',true when you train an SVM classifier using fitcsvm, then Mu has length equal to the number of predictors (i.e., size(SVMModel.X,2)). Otherwise, Mu is an empty vector ([]). | ||||||||||||||||||||||
NumIterations |
Positive integer indicating the number of iterations required by the optimization routine to attain convergence. To set a limit on the number of iterations to, e.g., k, specify the name-value pair argument 'IterationLimit',k of fitcsvm. | ||||||||||||||||||||||
Nu |
Positive scalar representing the ν parameter for one-class learning. | ||||||||||||||||||||||
NumObservations |
Numeric scalar representing the number of observations in the training data. If the input arguments X or Y contain missing values, then NumObservations is less than the length of Y. | ||||||||||||||||||||||
OutlierFraction |
Scalar indicating the expected proportion of outliers in the training data. | ||||||||||||||||||||||
PredictorNames |
Cell array of strings containing the predictor names, in the order that they appear in X. | ||||||||||||||||||||||
Prior |
Numeric vector of prior probabilities for each class. The order of the elements of Prior corresponds to the elements of SVMModel.ClassNames. For two-class learning, if you specify a cost matrix, then the software updates the prior probabilities by incorporating the penalties described in the cost matrix. For more details, see Algorithms. | ||||||||||||||||||||||
ResponseName |
String describing the response variable Y. | ||||||||||||||||||||||
ScoreTransform |
String representing a built-in transformation function, or a function handle for transforming predicted classification scores. To change the score transformation function to, e.g., function, use dot notation.
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ShrinkagePeriod |
Nonnegative integer indicating the shrinkage period, i.e., number of iterations between reductions of the active set. To set the shrinkage period to, e.g., sp, specify the name-value pair argument 'ShrinkagePeriod',sp of fitcsvm. | ||||||||||||||||||||||
Sigma |
Numeric vector of predictor standard deviations. If you specify 'Standardize',1 or 'Standardize',true when you train the SVM classifier, then Sigma has length equal to the number of predictors (i.e., size(SVMModel.X,2)). Otherwise, Sigma is an empty vector ([]). | ||||||||||||||||||||||
Solver |
String indicating the solving routine that the software used to train the SVM classifier. To set the solver to, e.g., solver, specify the name-value pair argument 'Solver',solver of fitcsvm. | ||||||||||||||||||||||
SupportVectors |
Matrix containing rows of X that the software considers the support vectors. If you specify 'Standardize',1 or 'Standardize',true, then SupportVectors are the standardized rows of X. | ||||||||||||||||||||||
SupportVectorLabels |
Numeric vector of support vector class labels. SupportVectorLabels has length equal to the number of support vectors (i.e., sum(SVMModel.IsSupportVector)). +1 indicates that the corresponding support vector is in the positive class (SVMModel.ClassNames{2}). -1 indicates that the corresponding support vector is in the negative class (SVMModel.ClassNames{1}). | ||||||||||||||||||||||
W |
Numeric vector of observation weights that the software used to train the SVM classifier. The length of W is SVMModel.NumObservations. fitcsvm normalizes Weights so that the elements of W within a particular class sum up to the prior probability of that class. | ||||||||||||||||||||||
X |
Numeric matrix of unstandardized predictor values that the software used to train the SVM classifier. Each row of X corresponds to one observation, and each column corresponds to one variable. The software excludes predictor data rows removed due to NaNs from X. | ||||||||||||||||||||||
Y |
Categorical or character array, logical or numeric vector, or cell array of strings representing the observed class labels that the software used to train the SVM classifier. Y is the same data type as the input argument Y of fitcsvm. Each row of Y represents the observed classification of the corresponding row of X. The software excludes elements removed due to NaNs from Y. |
compact | Compact support vector machine classifier |
crossval | Cross-validated support vector machine classifier |
fitPosterior | Fit posterior probabilities |
resubEdge | Classification edge for support vector machine classifiers by resubstitution |
resubLoss | Classification loss for support vector machine classifiers by resubstitution |
resubMargin | Classification margins for support vector machine classifiers by resubstitution |
resubPredict | Predict support vector machine classifier resubstitution responses |
resume | Resume training support vector machine classifier |
edge | Classification edge for support vector machine classifiers |
fitPosterior | Fit posterior probabilities |
loss | Classification error for support vector machine classifiers |
margin | Classification margins for support vector machine classifiers |
predict | Predict labels for support vector machine classifiers |
A parameter that controls the maximum penalty imposed on margin-violating observations, and aids in preventing overfitting (regularization).
If you increase the box constraint, then the SVM classifier assigns fewer support vectors. However, increasing the box constraint can lead to longer training times.
The Gram matrix of a set of n vectors {x_{1},..,x_{n}; x_{j} ∊ R^{p}} is an n-by-n matrix with element (j,k) defined as G(x_{j},x_{k}) = <ϕ(x_{j}),ϕ(x_{k})>, an inner product of the transformed predictors using the kernel function ϕ.
For nonlinear SVM, the algorithm forms a Gram matrix using the predictor matrix columns. The dual formalization replaces the inner product of the predictors with corresponding elements of the resulting Gram matrix (called the "kernel trick"). Subsequently, nonlinear SVM operates in the transformed predictor space to find a separating hyperplane.
KKT complementarity conditions are optimization constraints required for optimal nonlinear programming solutions.
In SVM, the KKT complementarity conditions are
$$\{\begin{array}{l}{\alpha}_{j}\left[{y}_{j}\left(w\prime \varphi \left({x}_{j}\right)+b\right)-1+{\xi}_{j}\right]=0\\ {\xi}_{j}\left(C-{\alpha}_{j}\right)=0\end{array}$$
for all j = 1,...,n, where w_{j} is a weight, ϕ is a kernel function (see Gram matrix), and ξ_{j} is a slack variable. If the classes are perfectly separable, then ξ_{j} = 0 for all j = 1,...,n.
One-class learning, or unsupervised SVM, aims at separating data from the origin in the high-dimensional, predictor space (not the original predictor space), and is an algorithm used for outlier detection.
The algorithm resembles that of SVM for binary classification. The objective is to minimize dual expression
$$0.5{\displaystyle \sum _{jk}{\alpha}_{j}}{\alpha}_{k}G({x}_{j},{x}_{k})$$
with respect to $${\alpha}_{1},\mathrm{...},{\alpha}_{n}$$, subject to
$$\sum {\alpha}_{j}}=n\nu $$
and $$0\le {\alpha}_{j}\le 1$$ for all j = 1,...,n. G(x_{j},x_{k},) is element (j,k) of the Gram matrix.
A small value of ν leads to fewer support vectors, and, therefore, a smooth, crude decision boundary. A large value of ν leads to more support vectors, and therefore, a curvy, flexible decision boundary. The optimal value of ν should be large enough to capture the data complexity and small enough to avoid overtraining. Also, 0 < ν ≤ 1.
For more details, see [3].
Support vectors are observations corresponding to strictly positive estimates of α_{1},...,α_{n}.
SVM classifiers that yield fewer support vectors for a given training set are more desirable.
The SVM binary classification algorithm searches for an optimal hyperplane that separates the data into two classes. For separable classes, the optimal hyperplane maximizes a margin (space that does not contain any observations) surrounding itself, which creates boundaries for the positive and negative classes. For inseparable classes, the objective is the same, but the algorithm imposes a penalty on the length of the margin for every observation that is on the wrong side of its class boundary.
The linear SVM score function is
$$f(x)=x\prime \beta +{\beta}_{0},$$
where:
x is an observation (corresponding to a row of X).
The vector β contains the coefficients that define an orthogonal vector to the hyperplane (corresponding to SVMModel.Beta). For separable data, the optimal margin length is $$2/\Vert \beta \Vert .$$
β_{0} is the bias term (corresponding to SVMModel.Bias).
The root of f(x) for particular coefficients defines a hyperplane. For a particular hyperplane, f(z) is the distance from point z to the hyperplane.
An SVM classifier searches for the maximum margin length, while keeping observations in the positive (y = 1) and negative (y = –1) classes separate. Therefore:
For separable classes, the objective is to minimize $$\Vert \beta \Vert $$ with respect to the β and β_{0} subject to y_{j}f(x_{j}) ≥ 1, for all j = 1,..,n. This is the primal formalization for separable classes.
For inseparable classes, SVM uses slack variables (ξ_{j}) to penalize the objective function for observations that cross the margin boundary for their class. ξ_{j} = 0 for observations that do not cross the margin boundary for their class, otherwise ξ_{j} ≥ 0.
The objective is to minimize$$0.5{\Vert \beta \Vert}^{2}+C{\displaystyle \sum {\xi}_{j}}$$ with respect to the β, β_{0}, and ξ_{j} subject to $${y}_{j}f({x}_{j})\ge 1-{\xi}_{j}$$ and $${\xi}_{j}\ge 0$$ for all j = 1,..,n, and for a positive scalar box constraint C. This is the primal formalization for inseparable classes.
SVM uses the Lagrange multipliers method to optimize the objective. This introduces n coefficients α_{1},...,α_{n} (corresponding to SVMModel.Alpha). The dual formalizations for linear SVM are:
For separable classes, minimize
$$0.5{\displaystyle \sum _{j=1}^{n}{\displaystyle \sum}_{k=1}^{n}}{\alpha}_{j}{\alpha}_{k}{y}_{j}{y}_{k}{x}_{j}\prime {x}_{k}-{\displaystyle \sum}_{j=1}^{n}{\alpha}_{j}$$
with respect to α_{1},...,α_{n}, subject to $$\sum {\alpha}_{j}}{y}_{j}=0$$, α_{j} ≥ 0 for all j = 1,...,n, and Karush-Kuhn-Tucker (KKT) complementarity conditions.
For inseparable classes, the objective is the same as for separable classes, except for the additional condition $$0\le {\alpha}_{j}\le C$$ for all j = 1,..,n.
The resulting score function is
$$f(x)={\displaystyle \sum _{j=1}^{n}{\widehat{\alpha}}_{j}}{y}_{j}x\prime {x}_{j}+\widehat{b}.$$
The score function is free of the estimate of β as a result of the primal formalization.
In some cases, there is a nonlinear boundary separating the classes. Nonlinear SVM works in a transformed predictor space to find an optimal, separating hyperplane.
The dual formalization for nonlinear SVM is
$$0.5{\displaystyle \sum _{j=1}^{n}{\displaystyle \sum _{k=1}^{n}{\alpha}_{j}}}{\alpha}_{k}{y}_{j}{y}_{k}G({x}_{j},{x}_{k})-{\displaystyle \sum _{j=1}^{n}{\alpha}_{j}}$$
with respect to α_{1},...,α_{n}, subject to $$\sum {\alpha}_{j}}{y}_{j}=0$$, $$0\le {\alpha}_{j}\le C$$ for all j = 1,..,n, and the KKT complementarity conditions.G(x_{k},x_{j}) are elements of the Gram matrix. The resulting score function is
$$f(x)={\displaystyle \sum _{j=1}^{n}{\alpha}_{j}}{y}_{j}G(x,{x}_{j})+b.$$
For more details, see Understanding Support Vector Machines, [1], and [2].
Value. To learn how value classes affect copy operations, see Copying Objects in the MATLAB documentation.
All solvers implement L1 soft-margin minimization.
fitcsvm and svmtrain use, among other algorithms, SMO for optimization. The software implements SMO differently between the two functions, but numerical studies show that there is sensible agreement in the results.
For one-class learning, the software estimates the Lagrange multipliers, α_{1},...,α_{n}, such that
$$\sum _{j=1}^{n}{\alpha}_{j}}=n\nu .$$
For two-class learning, if you specify a cost matrix C, then the software updates the class prior probabilities (p) to p_{c} by incorporating the penalties described in C. The formula for the updated prior probability vector is
$${p}_{c}=\frac{p\prime C}{\sum p\prime C}.$$
Subsequently, the software resets the cost matrix to the default:
$$C=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right].$$
If you set 'Standardize',true when you train the SVM classifier using fitcsvm, then the software trains the classifier using the standardized predictor matrix, but stores the unstandardized data in the classifier property X. However, if you standardize the data, then the data size in memory doubles until optimization ends.
If you set 'Standardize',true and any of 'Cost', 'Prior', or 'Weights', then the software standardizes the predictors using their corresponding weighted means and weighted standard deviations.
Let p be the proportion of outliers you expect in the training data. If you use 'OutlierFraction',p when you train the SVM classifier using fitcsvm, then:
For one-class learning, the software trains the bias term such that 100p% of the observations in the training data have negative scores.
The software implements robust learning for two-class learning. In other words, the software attempts to remove 100p% of the observations when the optimization algorithm converges. The removed observations correspond to gradients that are large in magnitude.
[1] Hastie, T., R. Tibshirani, and J. Friedman. The Elements of Statistical Learning, Second Edition. NY: Springer, 2008.
[2] Scholkopf, B., J. C. Platt, J. C. Shawe-Taylor, A. J. Smola, and R. C. Williamson. "Estimating the Support of a High-Dimensional Distribution." Neural Comput., Vol. 13, Number 7, 2001, pp. 1443–1471.
[3] Christianini, N., and J. C. Shawe-Taylor. An Introduction to Support Vector Machines and Other Kernel-Based Learning Methods. Cambridge, UK: Cambridge University Press, 2000.
[4] Scholkopf, B. and A. Smola. Learning with Kernels: Support Vector Machines, Regularization, Optimization and Beyond, Adaptive Computation and Machine Learning Cambridge, MA: The MIT Press, 2002.