Train support vector machine (SVM) classifier for one-class and binary classification

`fitcsvm`

trains or cross-validates a support vector
machine (SVM) model for one-class and two-class (binary) classification on a
low-dimensional or moderate-dimensional predictor data set. `fitcsvm`

supports mapping the predictor data using kernel functions, and supports sequential
minimal optimization (SMO), iterative single data algorithm (ISDA), or
*L*1 soft-margin minimization via quadratic programming for
objective-function minimization.

To train a linear SVM model for binary classification on a high-dimensional data set,
that is, a data set that includes many predictor variables, use `fitclinear`

instead.

For multiclass learning with combined binary SVM models, use error-correcting output
codes (ECOC). For more details, see `fitcecoc`

.

To train an SVM regression model, see `fitrsvm`

for low-dimensional and moderate-dimensional predictor data
sets, or `fitrlinear`

for high-dimensional data
sets.

returns a support vector machine
(SVM) classifier
`Mdl`

= fitcsvm(`Tbl`

,`ResponseVarName`

)`Mdl`

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-class or two-class classification.

specifies options using one or more name-value pair arguments in addition to the
input arguments in previous syntaxes. For example, you can specify the type of
cross-validation, the cost for misclassification, and the type of score
transformation function.`Mdl`

= fitcsvm(___,`Name,Value`

)

`fitcsvm`

trains SVM classifiers for one-class or two-class learning applications. To train SVM classifiers using data with more than two classes, use`fitcecoc`

.`fitcsvm`

supports low-dimensional and moderate-dimensional data sets. For high-dimensional data sets, use`fitclinear`

instead.

Unless your data set is large, always try to standardize the predictors (see

`Standardize`

). Standardization makes predictors insensitive to the scales on which they are measured.It is a good practice to cross-validate using the

`KFold`

name-value pair argument. The cross-validation results determine how well the SVM classifier generalizes.For one-class learning:

The default setting for the name-value pair argument

`Alpha`

can lead to long training times. To speed up training, set`Alpha`

to a vector mostly composed of`0`

s.Set the name-value pair argument

`Nu`

to a value closer to`0`

to yield fewer support vectors and, therefore, a smoother but crude decision boundary.

Sparsity in support vectors is a desirable property of an SVM classifier. To decrease the number of support vectors, set

`BoxConstraint`

to a large value. This action increases the training time.For optimal training time, set

`CacheSize`

as high as the memory limit your computer allows.If you expect many fewer support vectors than observations in the training set, then you can significantly speed up convergence by shrinking the active set using the name-value pair argument

`'ShrinkagePeriod'`

. It is a good practice to specify`'ShrinkagePeriod',1000`

.Duplicate observations that are far from the decision boundary do not affect convergence. However, just a few duplicate observations that occur near the decision boundary can slow down convergence considerably. To speed up convergence, specify

`'RemoveDuplicates',true`

if:Your data set contains many duplicate observations.

You suspect that a few duplicate observations fall near the decision boundary.

To maintain the original data set during training,

`fitcsvm`

must temporarily store separate data sets: the original and one without the duplicate observations. Therefore, if you specify`true`

for data sets containing few duplicates, then`fitcsvm`

consumes close to double the memory of the original data.After training a model, you can generate C/C++ code that predicts labels for new data. Generating C/C++ code requires MATLAB Coder™. For details, see Introduction to Code Generation.

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 equal`BoxConstraint`

.`fitcsvm`

removes observations that have zero weight or prior probability.For two-class learning, if you specify the cost matrix $$\mathcal{C}$$ (see

`Cost`

), then the software updates the class prior probabilities*p*(see`Prior`

) to*p*by incorporating the penalties described in $$\mathcal{C}$$._{c}Specifically,

`fitcsvm`

completes these steps:Compute $${p}_{c}^{\ast}=p\prime \mathcal{C}.$$

Normalize

*p*_{c}^{*}so that the updated prior probabilities sum to 1.$${p}_{c}=\frac{1}{{\displaystyle \sum _{j=1}^{K}{p}_{c,j}^{\ast}}}{p}_{c}^{\ast}.$$

*K*is the number of classes.Reset the cost matrix to the default

$$\mathcal{C}=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right].$$

Remove observations from the training data corresponding to classes with zero prior probability.

For two-class learning,

`fitcsvm`

normalizes all observation weights (see`Weights`

) to sum to 1. The function 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*is the normalized weight for observation_{j}*j*;*p*_{c,k}is the updated prior probability of class*k*(see previous bullet).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$${C}_{j}=n{C}_{0}{w}_{j}^{\ast}.$$

*n*is the training sample size,*C*_{0}is the initial box constraint (see the`'BoxConstraint'`

name-value pair argument), and $${w}_{j}^{\ast}$$ is the total weight of observation*j*(see previous bullet).If you set

`'Standardize',true`

and the`'Cost'`

,`'Prior'`

, or`'Weights'`

name-value pair argument, then`fitcsvm`

standardizes the predictors using their corresponding weighted means and weighted standard deviations. That is,`fitcsvm`

standardizes predictor*j*(*x*) using_{j}$${x}_{j}^{\ast}=\frac{{x}_{j}-{\mu}_{j}^{\ast}}{{\sigma}_{j}^{\ast}}.$$

$${\mu}_{j}^{\ast}=\frac{1}{{\displaystyle \sum _{k}{w}_{k}^{\ast}}}{\displaystyle \sum _{k}{w}_{k}^{\ast}{x}_{jk}}.$$

*x*is observation_{jk}*k*(row) of predictor*j*(column).$${\left({\sigma}_{j}^{\ast}\right)}^{2}=\frac{{v}_{1}}{{v}_{1}^{2}-{v}_{2}}{\displaystyle \sum _{k}{w}_{k}^{\ast}{\left({x}_{jk}-{\mu}_{j}^{\ast}\right)}^{2}}.$$

$${v}_{1}={\displaystyle \sum _{j}{w}_{j}^{\ast}}.$$

$${v}_{2}={\displaystyle \sum _{j}{\left({w}_{j}^{\ast}\right)}^{2}}.$$

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 100

`p`

% 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 100`p`

% 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},...,*α*, such that_{n}$$\sum _{j=1}^{n}{\alpha}_{j}}=n\nu .$$

[1] 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.

[2] Fan, R.-E., P.-H. Chen, and C.-J. Lin. “Working set
selection using second order information for training support vector machines.”
*Journal of Machine Learning Research*, Vol. 6, 2005, pp.
1889–1918.

[3] Hastie, T., R. Tibshirani, and J. Friedman. *The
Elements of Statistical Learning*, Second Edition. NY: Springer,
2008.

[4] Kecman V., T. -M. Huang, and M. Vogt. “Iterative Single
Data Algorithm for Training Kernel Machines from Huge Data Sets: Theory and
Performance.” *Support Vector Machines: Theory and
Applications*. Edited by Lipo Wang, 255–274. Berlin: Springer-Verlag,
2005.

[5] 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.

[6] 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.

`ClassificationPartitionedModel`

| `ClassificationSVM`

| `CompactClassificationSVM`

| `fitSVMPosterior`

| `fitcecoc`

| `fitclinear`

| `predict`

| `quadprog`

| `rng`