Superclasses: CompactClassificationSVM
Support vector machine for binary classification
ClassificationSVM
is a support vector machine classifier for
one or twoclass learning. To train a ClassificationSVM
classifier,
use fitcsvm
.
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
.
returns
an SVM classifier (SVMModel
= fitcsvm(TBL
,ResponseVarName
)SVMModel
) trained using the
sample data contained in the table TBL
. ResponseVarName
is
the name of the variable in TBL
that contains
the class labels for one or twoclass classification. For details,
see fitcsvm
.
returns
an SVM classifer trained using the sample data contained in a table
(SVMModel
= fitcsvm(TBL
,formula
)TBL
). formula
is a formula
string that identifies the response and predictor variables in TBL
that
are used for training. For details, see fitcsvm
.
returns
an SVM classifer trained using the predictor variables in table SVMModel
= fitcsvm(TBL
,Y
)TBL
and
class labels in vector Y
. For details, see fitcsvm
.
returns
an SVM classifier trained using the predictors in the matrix SVMModel
=
fitcsvm(X
,Y
)X
and
class labels in the vector Y
for one or twoclass
classification. For details, see fitcsvm
.
returns
a trained SVM classifier with additional options specified by one
or more SVMModel
= fitcsvm(___,Name,Value
)Name,Value
pair arguments, using any of
the previous syntaxes. For example, you can specify the type of cross
validation, the cost for misclassification, or the type of score transformation
function. For namevalue 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.

Numeric vector of trained classifier coefficients from the dual
problem (i.e., the estimated Lagrange multipliers).  

Numeric vector of linear predictor coefficients. If your predictor data contains categorical variables, then
the software uses full dummy encoding for these variables. The software
creates one dummy variable for each level of each categorical variable. If $$f\left(x\right)=\left(x/s\right)\prime \beta +b.$$
If  

Scalar corresponding to the trained classifier bias term.  

Numeric vector of box constraints.
 

Structure array containing:
 

Indices of categorical predictors, stored as a numeric vector.  

List of elements in  

Structure array containing convergence information.
 

Square matrix, where During training, the software updates the prior probabilities by incorporating the penalties described in the cost matrix. Therefore,
This property is readonly. For more details, see Algorithms.  

Expanded predictor names, stored as a cell array of strings. If the model uses encoding for categorical variables, then  

Numeric vector of training data gradient values.  

Logical vector indicating whether a corresponding row in the
predictor data matrix is a support vector.  

Structure array containing the kernel name and parameter values. To display the values of The software accepts  

Object containing parameter values, e.g., the namevalue pair
argument values, used to train the SVM classifier. Access fields of  

Numeric vector of predictor means. If you specify If your predictor data contains categorical variables, then
the software uses full dummy encoding for these variables. The software
creates one dummy variable for each level of each categorical variable. If  

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.,  

Positive scalar representing the ν parameter for oneclass learning.  

Numeric scalar representing the number
of observations in the training data. If the input arguments  

Scalar indicating the expected proportion of outliers in the training data.  

Cell array of strings containing the predictor names, in the
order that they appear in  

Numeric vector of prior probabilities for each class. The order
of the elements of For twoclass learning, if you specify a cost matrix, then the software updates the prior probabilities by incorporating the penalties described in the cost matrix. This property is readonly. For more details, see Algorithms.  

String describing the response variable  

String representing a builtin transformation function, or a function handle for transforming predicted classification scores. To change the score transformation function to, e.g.,
 

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.,  

Numeric vector of predictor standard deviations. If you specify If your predictor data contains categorical variables, then
the software uses full dummy encoding for these variables. The software
creates one dummy variable for each level of each categorical variable. If  

String indicating the solving routine that the software used to train the SVM classifier. To set the solver to, e.g.,  

Matrix containing rows of If you specify  

Numeric vector of support vector class labels.
 

Numeric vector of observation weights that the software used to train the SVM classifier. The length of
 

Numeric matrix of unstandardized predictor values that the software used to train the SVM classifier. Each row of The software excludes predictor data rows removed due to  

Categorical or character array, logical or numeric vector, or
cell array of strings representing the observed class labels used
to train the SVM classifier. Each row of The software excludes elements removed due to 
compact  Compact support vector machine classifier 
crossval  Crossvalidated 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 
compareHoldout  Compare accuracies of two classification models using new data 
discardSupportVectors  Discard support vectors for linear support vector machine models 
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 marginviolating 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 nbyn 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
$$\{\begin{array}{l}{\alpha}_{j}\left[{y}_{j}f\left({x}_{j}\right)1+{\xi}_{j}\right]=0\\ {\xi}_{j}\left(C{\alpha}_{j}\right)=0\end{array}$$
for all j = 1,...,n, where $$f\left({x}_{j}\right)=\varphi \left({x}_{j}\right)\prime \beta +b,$$ 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.
Oneclass learning, or unsupervised SVM, aims at separating data from the origin in the highdimensional, 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 [5].
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
$$f(x)=x\prime \beta +b,$$
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 $$2/\Vert \beta \Vert .$$
b 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.
The algorithm 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 b subject to y_{j}f(x_{j}) ≥ 1, for all j = 1,..,n. This is the primal formalization for separable classes.
For inseparable classes, the algorithm 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, b, 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.
The algorithm 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 KarushKuhnTucker (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
$$\widehat{f}(x)={\displaystyle \sum _{j=1}^{n}{\widehat{\alpha}}_{j}}{y}_{j}x\prime {x}_{j}+\widehat{b}.$$
$$\widehat{b}$$ is the estimate of the bias and $${\widehat{\alpha}}_{j}$$ is the jth estimate of the vector $$\widehat{\alpha}$$, 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 $$\text{sign}\left(\widehat{f}\left(z\right)\right).$$
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
$$\widehat{f}(x)={\displaystyle \sum _{j=1}^{n}{\widehat{\alpha}}_{j}}{y}_{j}G(x,{x}_{j})+\widehat{b}.$$
For more details, see Understanding Support Vector Machines, [1], and [3].
Value. To learn how value classes affect copy operations, see Copying Objects in the MATLAB documentation.
NaN
, <undefined>
,
and empty strings (''
) indicate missing values. fitcsvm
removes
entire rows of data corresponding to a missing response. When computing
total weights (see the bullets below), fitcsvm
ignores
any missing predictor observation. This can lead to unbalanced prior
probabilities in balancedclass problems. Consequently, observation
box constraints might not equal BoxConstraint
.
fitcsvm
removes observations that
have zero weight or prior probability.
For twoclass learning, if you specify the cost matrix $$\mathcal{C}$$ (see Cost
),
then the software updates the class prior probabilities p (see Prior
)
to p_{c} by incorporating the
penalties described in $$\mathcal{C}$$.
Specifically, fitcsvm
:
Computes $${p}_{c}^{\ast}=p\prime \mathcal{C}.$$
Normalizes p_{c}^{*} so that the updated prior probabilities sum 1:
$${p}_{c}=\frac{1}{{\displaystyle \sum _{j=1}^{K}{p}_{c,j}^{\ast}}}{p}_{c}^{\ast}.$$
K is the number of classes.
Resets the cost matrix to the default:
$$\mathcal{C}=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right].$$
Removes observations from the training data corresponding to classes with zero prior probability.
For twoclass learning, fitcsvm
normalizes
all observation weights (see Weights
) to sum
to 1. Then, renormalizes the normalized weights to sum up to the updated,
prior probability of the class to which the observation belongs. That
is, the total weight for observation j in class k is
$${w}_{j}^{\ast}=\frac{{w}_{j}}{{\displaystyle \sum _{\forall j\in \text{Class}k}{w}_{j}}}{p}_{c,k}.$$
w_{j} is the normalized weight for observation j; p_{c,k} is the updated prior probability of class k (see previous bullet).
For twoclass learning, fitcsvm
assigns
a box constraint to each observation in the training data. The formula
for the box constraint of observation j is
$${C}_{j}=n{C}_{0}{w}_{j}^{\ast}.$$
n is
the training sample size, C_{0} is
the initial box constraint (see BoxConstraint
),
and $${w}_{j}^{\ast}$$ is
the total weight of observation j (see previous
bullet).
If you set 'Standardize',true
and
any of 'Cost'
, 'Prior'
, or 'Weights'
,
then fitcsvm
standardizes the predictors using
their corresponding weighted means and weighted standard deviations.
That is, fitcsvm
standardizes predictor j (x_{j})
using
$${x}_{j}^{\ast}=\frac{{x}_{j}{\mu}_{j}^{\ast}}{{\sigma}_{j}^{\ast}}.$$
where
$${\mu}_{j}^{\ast}=\frac{1}{{\displaystyle \sum _{k}{w}_{k}^{\ast}}}{\displaystyle \sum _{k}{w}_{k}^{\ast}{x}_{jk}},$$
x_{jk} is observation k (row) of predictor j (column).
$${\left({\sigma}_{j}^{\ast}\right)}^{2}=\frac{{v}_{1}}{{v}_{1}^{2}{v}_{2}}{\displaystyle \sum _{k}{w}_{k}^{\ast}{\left({x}_{jk}{\mu}_{j}^{\ast}\right)}^{2}},$$
$${v}_{1}={\displaystyle \sum _{j}{w}_{j}^{\ast}}.$$
$${v}_{2}={\displaystyle \sum _{j}{\left({w}_{j}^{\ast}\right)}^{2}}.$$
Let op
be the proportion of outliers
you expect in the training data. If you use 'OutlierFraction',op
when
you train the SVM classifier using fitcsvm
, then:
For oneclass learning, the software trains the bias
term such that 100op
% of the observations in the
training data have negative scores.
The software implements robust learning for
twoclass learning. In other words, the software attempts to remove
100op
% 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 1by3 cell array of strings 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 1by5 cell array of strings 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 nby5
matrix.
The X
property stores the training
data as originally input. It 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 having k ordered levels, the software creates k – 1 dummy variables. The jth 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 L1 softmargin 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 oneclass learning, the software estimates the Lagrange multipliers, α_{1},...,α_{n}, such that
$$\sum _{j=1}^{n}{\alpha}_{j}}=n\nu .$$
[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. ShaweTaylor, A. J. Smola, and R. C. Williamson. "Estimating the Support of a HighDimensional Distribution." Neural Comput., Vol. 13, Number 7, 2001, pp. 1443–1471.
[3] Christianini, N., and J. C. ShaweTaylor. An Introduction to Support Vector Machines and Other KernelBased 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.