ClassificationSVM - Support vector machine (SVM) for one-class and binary classification - MATLAB (original) (raw)

Support vector machine (SVM) for one-class and binary classification

Description

ClassificationSVM is a support vector machine (SVM) classifier for one-class and two-class learning. Trained ClassificationSVM classifiers store training data, parameter values, prior probabilities, support vectors, and algorithmic implementation information. Use these classifiers to perform tasks such as fitting a score-to-posterior-probability transformation function (see fitPosterior) and predicting labels for new data (see predict).

Creation

Create a ClassificationSVM object by using fitcsvm.

Properties

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SVM Properties

This property is read-only.

Data Types: single | double

This property is read-only.

Data Types: single | double

This property is read-only.

Bias term, specified as a scalar.

Data Types: single | double

This property is read-only.

Box constraints, specified as a numeric vector of _n_-by-1 box constraints. n is the number of observations in the training data (see the NumObservations property).

If you remove duplicates by using the RemoveDuplicates name-value pair argument of fitcsvm, then for a given set of duplicate observations, MATLAB sums the box constraints and then attributes the sum to one observation. MATLAB attributes the box constraints of 0 to all other observations in the set.

Data Types: single | double

This property is read-only.

Caching information, specified as a structure array. The caching information contains the fields described in this table.

Display the fields of CacheInfo by using dot notation. For example, Mdl.CacheInfo.Size displays the value of the cache size.

Data Types: struct

This property is read-only.

Support vector indicator, specified as an _n_-by-1 logical vector that flags whether a corresponding observation in the predictor data matrix is a Support Vector. n is the number of observations in the training data (see NumObservations).

If you remove duplicates by using the RemoveDuplicates name-value pair argument of fitcsvm, then for a given set of duplicate observations that are support vectors, IsSupportVector flags only one observation as a support vector.

Data Types: logical

This property is read-only.

Data Types: struct

This property is read-only.

Data Types: single | double

This property is read-only.

Proportion of outliers in the training data, specified as a numeric scalar.

Data Types: double

This property is read-only.

Optimization routine used to train the SVM classifier, specified as 'ISDA', 'L1QP', or 'SMO'. For more details, see 'Solver'.

This property is read-only.

Data Types: single | double

This property is read-only.

Data Types: single | double

Other Classification Properties

This property is read-only.

Categorical predictor indices, specified as a vector of positive integers. CategoricalPredictors contains index values indicating that the corresponding predictors are categorical. The index values are between 1 and p, where p is the number of predictors used to train the model. If none of the predictors are categorical, then this property is empty ([]).

Data Types: double

This property is read-only.

Unique class labels used in training, specified as a categorical or character array, logical or numeric vector, or cell array of character vectors.ClassNames has the same data type as the class labelsY. (The software treats string arrays as cell arrays of character vectors.) ClassNames also determines the class order.

Data Types: single | double | logical | char | cell | categorical

This property is read-only.

Data Types: double

This property is read-only.

Data Types: cell

This property is read-only.

Training data gradient values, specified as a numeric vector. The length of Gradient is equal to the number of observations (NumObservations).

Data Types: single | double

This property is read-only.

Parameters used to train the ClassificationSVM model, specified as an object. ModelParameters contains parameter values such as the name-value pair argument values used to train the SVM classifier. ModelParameters does not contain estimated parameters.

Access the properties of ModelParameters by using dot notation. For example, access the initial values for estimatingAlpha by usingMdl.ModelParameters.Alpha.

This property is read-only.

Data Types: single | double

This property is read-only.

Number of observations in the training data stored in X and Y, specified as a numeric scalar.

Data Types: double

This property is read-only.

Predictor variable names, specified as a cell array of character vectors. The order of the elements in PredictorNames corresponds to the order in which the predictor names appear in the training data.

Data Types: cell

This property is read-only.

Data Types: single | double

This property is read-only.

Response variable name, specified as a character vector.

Data Types: char

This property is read-only.

Rows of the original training data stored in the model, specified as a logical vector. This property is empty if all rows are stored inX and Y.

Data Types: logical

Data Types: char | function_handle

This property is read-only.

Data Types: single | double

This property is read-only.

Observation weights used to train the SVM classifier, specified as an _n_-by-1 numeric vector. n is the number of observations (see NumObservations).

fitcsvm normalizes the observation weights specified in the 'Weights' name-value pair argument so that the elements of W within a particular class sum up to the prior probability of that class.

Data Types: single | double

This property is read-only.

Unstandardized predictors used to train the SVM classifier, specified as a numeric matrix or table.

Each row of X corresponds to one observation, and each column corresponds to one variable.

Data Types: single | double

This property is read-only.

Class labels used to train the SVM classifier, specified as a categorical or character array, logical or numeric vector, or cell array of character vectors. Y is the same data type as the input argument Y of fitcsvm. (The software treats string arrays as cell arrays of character vectors.)

Each row of Y represents the observed classification of the corresponding row of X.

Data Types: single | double | logical | char | cell | categorical

Convergence Control Properties

This property is read-only.

Convergence information, specified as a structure array.

Data Types: struct

This property is read-only.

Number of iterations required by the optimization routine to attain convergence, specified as a positive integer.

To set the limit on the number of iterations to 1000, for example, specify 'IterationLimit',1000 when you train the SVM classifier using fitcsvm.

Data Types: double

This property is read-only.

Number of iterations between reductions of the active set, specified as a nonnegative integer.

To set the shrinkage period to 1000, for example, specify 'ShrinkagePeriod',1000 when you train the SVM classifier using fitcsvm.

Data Types: single | double

Hyperparameter Optimization Properties

This property is read-only.

Description of the cross-validation optimization of hyperparameters, specified as a BayesianOptimization object or a table of hyperparameters and associated values. This property is nonempty when the 'OptimizeHyperparameters' name-value pair argument of fitcsvm is nonempty at creation. The value of HyperparameterOptimizationResults depends on the setting of the Optimizer field in the HyperparameterOptimizationOptions structure of fitcsvm at creation, as described in this table.

Object Functions

Examples

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Load Fisher's iris data set. Remove the sepal lengths and widths and all observed setosa irises.

load fisheriris inds = ~strcmp(species,'setosa'); X = meas(inds,3:4); y = species(inds);

Train an SVM classifier using the processed data set.

SVMModel = ClassificationSVM ResponseName: 'Y' CategoricalPredictors: [] ClassNames: {'versicolor' 'virginica'} ScoreTransform: 'none' NumObservations: 100 Alpha: [24×1 double] Bias: -14.4149 KernelParameters: [1×1 struct] BoxConstraints: [100×1 double] ConvergenceInfo: [1×1 struct] IsSupportVector: [100×1 logical] Solver: 'SMO'

Properties, Methods

SVMModel is a trained ClassificationSVM classifier. Display the properties of SVMModel. For example, to determine the class order, use dot notation.

classOrder = SVMModel.ClassNames

classOrder = 2×1 cell {'versicolor'} {'virginica' }

The first class ('versicolor') is the negative class, and the second ('virginica') is the positive class. You can change the class order during training by using the 'ClassNames' name-value pair argument.

Plot a scatter diagram of the data and circle the support vectors.

sv = SVMModel.SupportVectors; figure gscatter(X(:,1),X(:,2),y) hold on plot(sv(:,1),sv(:,2),'ko','MarkerSize',10) legend('versicolor','virginica','Support Vector') hold off

The support vectors are observations that occur on or beyond their estimated class boundaries.

You can adjust the boundaries (and, therefore, the number of support vectors) by setting a box constraint during training using the 'BoxConstraint' name-value pair argument.

Load the ionosphere data set.

Train and cross-validate an SVM classifier. Standardize the predictor data and specify the order of the classes.

rng(1); % For reproducibility CVSVMModel = fitcsvm(X,Y,'Standardize',true,... 'ClassNames',{'b','g'},'CrossVal','on')

CVSVMModel = ClassificationPartitionedModel CrossValidatedModel: 'SVM' PredictorNames: {'x1' 'x2' 'x3' 'x4' 'x5' 'x6' 'x7' 'x8' 'x9' 'x10' 'x11' 'x12' 'x13' 'x14' 'x15' 'x16' 'x17' 'x18' 'x19' 'x20' 'x21' 'x22' 'x23' 'x24' 'x25' 'x26' 'x27' 'x28' 'x29' 'x30' 'x31' 'x32' 'x33' 'x34'} ResponseName: 'Y' NumObservations: 351 KFold: 10 Partition: [1×1 cvpartition] ClassNames: {'b' 'g'} ScoreTransform: 'none'

Properties, Methods

CVSVMModel is a ClassificationPartitionedModel cross-validated SVM classifier. By default, the software implements 10-fold cross-validation.

Alternatively, you can cross-validate a trained ClassificationSVM classifier by passing it to crossval.

Inspect one of the trained folds using dot notation.

ans = CompactClassificationSVM ResponseName: 'Y' CategoricalPredictors: [] ClassNames: {'b' 'g'} ScoreTransform: 'none' Alpha: [78×1 double] Bias: -0.2210 KernelParameters: [1×1 struct] Mu: [0.8888 0 0.6320 0.0406 0.5931 0.1205 0.5361 0.1286 0.5083 0.1879 0.4779 0.1567 0.3924 0.0875 0.3360 0.0789 0.3839 9.6066e-05 0.3562 -0.0308 0.3398 -0.0073 0.3590 -0.0628 0.4064 -0.0664 0.5535 -0.0749 0.3835 … ] (1×34 double) Sigma: [0.3149 0 0.5033 0.4441 0.5255 0.4663 0.4987 0.5205 0.5040 0.4780 0.5649 0.4896 0.6293 0.4924 0.6606 0.4535 0.6133 0.4878 0.6250 0.5140 0.6075 0.5150 0.6068 0.5222 0.5729 0.5103 0.5061 0.5478 0.5712 0.5032 … ] (1×34 double) SupportVectors: [78×34 double] SupportVectorLabels: [78×1 double]

Properties, Methods

Each fold is a CompactClassificationSVM classifier trained on 90% of the data.

Estimate the generalization error.

genError = kfoldLoss(CVSVMModel)

On average, the generalization error is approximately 12%.

More About

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A box constraint is a parameter that controls the maximum penalty imposed on margin-violating observations, which helps to prevent 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,..,xn; xj ∊ Rp} is an _n_-by-n matrix with element (j,k) defined as G(xj,xk) = <_ϕ_(_xj_),_ϕ_(_xk_)>, an inner product of the transformed predictors using the kernel function ϕ.

For nonlinear SVM, the algorithm forms a Gram matrix using the rows of the predictor data_X_. The dual formalization replaces the inner product of the observations in X with corresponding elements of the resulting Gram matrix (called the “kernel trick”). Consequently, 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

for all j = 1,...,n, where f(xj)=ϕ(xj)′β+b, ϕ is a kernel function (see Gram matrix), and ξj is a slack variable. If the classes are perfectly separable, then ξj = 0 for all j = 1,...,n.

One-class learning, or unsupervised SVM, aims to separate 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 the dual expression

with respect to α1,...,αn, subject to

and 0≤αj≤1 for all j = 1,...,n. The value of_G_(xj,xk) is in 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 [4].

Support vectors are observations corresponding to strictly positive estimates of _α_1,...,αn.

SVM classifiers that yield fewer support vectors for a given training set are preferred.

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

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 Mdl.Beta). For separable data, the optimal margin length is 2/‖β‖.
• b is the bias term (corresponding to Mdl.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.

The algorithm searches for the maximum margin length, while keeping observations in the positive (y = 1) and negative (y = –1) classes separate.

• For separable classes, the objective is to minimize ‖β‖ with respect to the β and_b_ subject to_yj_ f(xj) ≥ 1, for all j = 1,..,n. This is the_primal_ formalization for separable classes.
• For inseparable classes, the algorithm uses slack variables (ξj) to penalize the objective function for observations that cross the margin boundary for their class.ξj = 0 for observations that do not cross the margin boundary for their class, otherwise_ξj_ ≥ 0.
  The objective is to minimize 0.5‖β‖2+C∑ξj with respect to the β,b, and ξj subject to yjf(xj)≥1−ξj and ξj≥0 for all j = 1,..,n, and for a positive scalar box constraint C. This is the primal formalization for inseparable classes.

The algorithm uses the Lagrange multipliers method to optimize the objective, which introduces_n_ coefficients_α_1,...,αn (corresponding to Mdl.Alpha). The dual formalizations for linear SVM are as follows:

• For separable classes, minimize
  with respect to_α_1,...,αn, subject to ∑αjyj=0, αj ≥ 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≤αj≤C for all j = 1,..,n.

The resulting score function is

b^ is the estimate of the bias and α^j is the _j_th estimate of the vector α^, j = 1,...,n. Written this way, the score function is free of the estimate of β as a result of the primal formalization.

The SVM algorithm classifies a new observation z using sign(f^(z)).

In some cases, a nonlinear boundary separates the classes. Nonlinear SVM works in a transformed predictor space to find an optimal, separating hyperplane.

The dual formalization for nonlinear SVM is

with respect to_α_1,...,αn, subject to ∑αjyj=0, 0≤αj≤C for all j = 1,..,n, and the KKT complementarity conditions.G(xk,xj) are elements of the Gram matrix. The resulting score function is

For more details, see Understanding Support Vector Machines, [3], and [1].

Algorithms

• For the mathematical formulation of the SVM binary classification algorithm, see Support Vector Machines for Binary Classification and Understanding Support Vector Machines.
• NaN, <undefined>, empty character vector (''), empty string (""), and<missing> values indicate missing values.fitcsvm removes entire rows of data corresponding to a missing response. When computing total weights (see the next bullets),fitcsvm ignores any weight corresponding to an observation with at least one missing predictor. This action can lead to unbalanced prior probabilities in balanced-class problems. Consequently, observation box constraints might not equalBoxConstraint.
• If you specify the Cost,Prior, and Weights name-value arguments, the output model object stores the specified values in the Cost,Prior, and W properties, respectively. TheCost property stores the user-specified cost matrix (C) without modification. The Prior andW properties store the prior probabilities and observation weights, respectively, after normalization. For model training, the software updates the prior probabilities and observation weights to incorporate the penalties described in the cost matrix. For details, see Misclassification Cost Matrix, Prior Probabilities, and Observation Weights.
  Note that the Cost and Prior name-value arguments are used for two-class learning. For one-class learning, theCost and Prior properties store0 and 1, respectively.
• For two-class learning, fitcsvm assigns a box constraint to each observation in the training data. The formula for the box constraint of observation_j_ is
  where C_0 is the initial box constraint (see the BoxConstraint name-value argument), and_wj* is the observation weight adjusted by Cost and Prior for observation j. For details about the observation weights, seeAdjust Prior Probabilities and Observation Weights for Misclassification Cost Matrix.
• If you specify Standardize astrue and set the Cost,Prior, or Weights name-value argument, then fitcsvm standardizes the predictors using their corresponding weighted means and weighted standard deviations. That is, fitcsvm standardizes predictor j (xj) using
  where xjk is observation_k_ (row) of predictor j (column), and
• Assume that p is the proportion of outliers that you expect in the training data, and that you set 'OutlierFraction',p.
  • 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.
• If your predictor data contains categorical variables, then the software generally uses full dummy encoding for these variables. The software creates one dummy variable for each level of each categorical variable.
  • The PredictorNames property stores one element for each of the original predictor variable names. For example, assume that there are three predictors, one of which is a categorical variable with three levels. Then PredictorNames is a 1-by-3 cell array of character vectors containing the original names of the predictor variables.
  • The ExpandedPredictorNames property stores one element for each of the predictor variables, including the dummy variables. For example, assume that there are three predictors, one of which is a categorical variable with three levels. Then ExpandedPredictorNames is a 1-by-5 cell array of character vectors containing the names of the predictor variables and the new dummy variables.
  • Similarly, the Beta property stores one beta coefficient for each predictor, including the dummy variables.
  • The SupportVectors property stores the predictor values for the support vectors, including the dummy variables. For example, assume that there are m support vectors and three predictors, one of which is a categorical variable with three levels. Then SupportVectors is an _n_-by-5 matrix.
  • The X property stores the training data as originally input and does not include the dummy variables. When the input is a table,X contains only the columns used as predictors.
• For predictors specified in a table, if any of the variables contain ordered (ordinal) categories, the software uses ordinal encoding for these variables.
  • For a variable with k ordered levels, the software creates k – 1 dummy variables. The _j_th dummy variable is –1 for levels up to j, and +1 for levels j + 1 through k.
  • The names of the dummy variables stored in the ExpandedPredictorNames property indicate the first level with the value +1. The software stores k – 1 additional predictor names for the dummy variables, including the names of levels 2, 3, ..., k.
• All solvers implement _L_1 soft-margin minimization.
• For one-class learning, the software estimates the Lagrange multipliers, _α_1,...,αn, such that

References

[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., 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.

[5] 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.

Extended Capabilities

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Usage notes and limitations:

• The predict and update functions support code generation.
• To integrate the prediction of an SVM classification model into Simulink®, you can use the ClassificationSVM Predict block in the Statistics and Machine Learning Toolbox™ library or a MATLAB Function block with the predict function.
• When you train an SVM model by using fitcsvm, the following restrictions apply.
  • The value of the 'ScoreTransform' name-value pair argument cannot be an anonymous function. For generating code that predicts posterior probabilities given new observations, pass a trained SVM model tofitPosterior or fitSVMPosterior. TheScoreTransform property of the returned model contains an anonymous function that represents the score-to-posterior-probability function and is configured for code generation.
  • For fixed-point code generation, the value of the 'ScoreTransform' name-value pair argument cannot be'invlogit'. Also, the value of the 'KernelFunction' name-value pair argument must be'gaussian', 'linear', or'polynomial'.
  • For fixed-point code generation and code generation with a coder configurer, the following additional restrictions apply.
    * Categorical predictors (logical, categorical, char,string, or cell) are not supported. You cannot use theCategoricalPredictors name-value argument. To include categorical predictors in a model, preprocess them by using dummyvar before fitting the model.
    * Class labels with the categorical data type are not supported. Both the class label value in the training data (Tbl orY) and the value of the ClassNames name-value argument cannot be an array with the categorical data type.

For more information, see Introduction to Code Generation.

Usage notes and limitations:

• The following object functions fully support GPU arrays:
  • compact
  • crossval
  • discardSupportVectors
  • fitPosterior
  • gather
  • resubEdge
  • resubLoss
  • resubMargin
  • resubPredict
  • resume
• The following object functions offer limited support for GPU arrays:
  • compareHoldout
  • edge
  • loss
  • margin
  • partialDependence
  • plotPartialDependence
  • predict
• The object functions execute on a GPU if at least one of the following applies:
  • The model was fitted with GPU arrays.
  • The predictor data that you pass to the object function is a GPU array.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2014a

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Starting in R2023b, training observations with missing predictor values are included in the X, Y, and W data properties. The RowsUsed property indicates the training observations stored in the model, rather than those used for training. Observations with missing predictor values continue to be omitted from the model training process.

In previous releases, the software omitted training observations that contained missing predictor values from the data properties of the model.

Starting in R2022a, the Cost property stores the user-specified cost matrix, so that you can compute the observed misclassification cost using the specified cost value. The software stores normalized prior probabilities (Prior) and observation weights (W) that do not reflect the penalties described in the cost matrix. To compute the observed misclassification cost, specify theLossFun name-value argument as "classifcost" when you call the loss or resubLoss function.

Note that model training has not changed and, therefore, the decision boundaries between classes have not changed.

For training, the fitting function updates the specified prior probabilities by incorporating the penalties described in the specified cost matrix, and then normalizes the prior probabilities and observation weights. This behavior has not changed. In previous releases, the software stored the default cost matrix in the Cost property and stored the prior probabilities and observation weights used for training in thePrior and W properties, respectively. Starting in R2022a, the software stores the user-specified cost matrix without modification, and stores normalized prior probabilities and observation weights that do not reflect the cost penalties. For more details, see Misclassification Cost Matrix, Prior Probabilities, and Observation Weights.

Some object functions use the Cost, Prior, and W properties:

• The loss and resubLoss functions use the cost matrix stored in the Cost property if you specify the LossFun name-value argument as"classifcost" or "mincost".
• The loss and edge functions use the prior probabilities stored in the Prior property to normalize the observation weights of the input data.
• The resubLoss and resubEdge functions use the observation weights stored in the W property.

If you specify a nondefault cost matrix when you train a classification model, the object functions return a different value compared to previous releases.

If you want the software to handle the cost matrix, prior probabilities, and observation weights in the same way as in previous releases, adjust the prior probabilities and observation weights for the nondefault cost matrix, as described in Adjust Prior Probabilities and Observation Weights for Misclassification Cost Matrix. Then, when you train a classification model, specify the adjusted prior probabilities and observation weights by using the Prior and Weights name-value arguments, respectively, and use the default cost matrix.