loss

Find classification error for support vector machine (SVM) classifier

Syntax

L = loss(SVMModel,Tbl,ResponseVarName)

L = loss(SVMModel,Tbl,Y)

L = loss(SVMModel,X,Y)

L = loss(___,Name,Value)

Description

L = loss(SVMModel,Tbl,ResponseVarName) returns the classification error (see Classification Loss), a scalar representing how well the trained support vector machine (SVM) classifier (SVMModel) classifies the predictor data in table Tbl compared to the true class labels in Tbl.ResponseVarName.

The classification loss (L) is a generalization or resubstitution quality measure. Its interpretation depends on the loss function and weighting scheme, but, in general, better classifiers yield smaller classification loss values.

L = loss(SVMModel,Tbl,Y) returns the classification error for the predictor data in table Tbl and the true class labels in Y.

L = loss(SVMModel,X,Y) returns the classification error based on the predictor data in matrix X compared to the true class labels in Y.

L = loss(___,Name,Value) 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 loss function and the classification weights.

Note

If the predictor data in X or Tbl contains any missing values and LossFun is not set to "classifcost", "classiferror", or "mincost", the loss function can return NaN. For more details, see loss can return NaN for predictor data with missing values.

Examples

Determine Test Sample Classification Error of SVM Classifiers

Load the ionosphere data set.

load ionosphere
rng(1); % For reproducibility

Train an SVM classifier. Specify a 15% holdout sample for testing, standardize the data, and specify that 'g' is the positive class.

CVSVMModel = fitcsvm(X,Y,'Holdout',0.15,'ClassNames',{'b','g'},...
    'Standardize',true);
CompactSVMModel = CVSVMModel.Trained{1}; % Extract the trained, compact classifier
testInds = test(CVSVMModel.Partition);   % Extract the test indices
XTest = X(testInds,:);
YTest = Y(testInds,:);

CVSVMModel is a ClassificationPartitionedModel classifier. It contains the property Trained, which is a 1-by-1 cell array holding a CompactClassificationSVM classifier that the software trained using the training set.

Determine how well the algorithm generalizes by estimating the test sample classification error.

L = loss(CompactSVMModel,XTest,YTest)

L = 
0.0787

The SVM classifier misclassifies approximately 8% of the test sample.

Determine Test Sample Hinge Loss of SVM Classifiers

Load the ionosphere data set.

load ionosphere
rng(1); % For reproducibility

Train an SVM classifier. Specify a 15% holdout sample for testing, standardize the data, and specify that 'g' is the positive class.

CVSVMModel = fitcsvm(X,Y,'Holdout',0.15,'ClassNames',{'b','g'},...
    'Standardize',true);
CompactSVMModel = CVSVMModel.Trained{1}; % Extract the trained, compact classifier
testInds = test(CVSVMModel.Partition);   % Extract the test indices
XTest = X(testInds,:);
YTest = Y(testInds,:);

CVSVMModel is a ClassificationPartitionedModel classifier. It contains the property Trained, which is a 1-by-1 cell array holding a CompactClassificationSVM classifier that the software trained using the training set.

Determine how well the algorithm generalizes by estimating the test sample hinge loss.

L = loss(CompactSVMModel,XTest,YTest,'LossFun','hinge')

L = 
0.2998

The hinge loss is approximately 0.3. Classifiers with hinge losses close to 0 are preferred.

Input Arguments

SVMModel — SVM classification model

ClassificationSVM model object | CompactClassificationSVM model object

SVM classification model, specified as a ClassificationSVM model object or CompactClassificationSVM model object returned by fitcsvm or compact, respectively.

Tbl — Sample data

table

Sample data used to train the model, specified as a table. Each row of Tbl corresponds to one observation, and each column corresponds to one predictor variable. Optionally, Tbl can contain additional columns for the response variable and observation weights. Tbl must contain all of the predictors used to train SVMModel. Multicolumn variables and cell arrays other than cell arrays of character vectors are not allowed.

If Tbl contains the response variable used to train SVMModel, then you do not need to specify ResponseVarName or Y.

If you trained SVMModel using sample data contained in a table, then the input data for loss must also be in a table.

If you set 'Standardize',true in fitcsvm when training SVMModel, then the software standardizes the columns of the predictor data using the corresponding means in SVMModel.Mu and the standard deviations in SVMModel.Sigma.

Data Types: table

ResponseVarName — Response variable name

name of variable in Tbl

Response variable name, specified as the name of a variable in Tbl. If Tbl contains the response variable used to train SVMModel, then you do not need to specify ResponseVarName.

You must specify ResponseVarName as a character vector or string scalar. For example, if the response variable Y is stored as Tbl.Y, then specify ResponseVarName as 'Y'. Otherwise, the software treats all columns of Tbl, including Y, as predictors when training the model.

The response variable must be a categorical, character, or string array, logical or numeric vector, or cell array of character vectors. If the response variable is a character array, then each element must correspond to one row of the array.

Data Types: char | string

X — Predictor data

numeric matrix

Predictor data, specified as a numeric matrix.

Each row of X corresponds to one observation (also known as an instance or example), and each column corresponds to one variable (also known as a feature). The variables in the columns of X must be the same as the variables that trained the SVMModel classifier.

The length of Y and the number of rows in X must be equal.

If you set 'Standardize',true in fitcsvm to train SVMModel, then the software standardizes the columns of X using the corresponding means in SVMModel.Mu and the standard deviations in SVMModel.Sigma.

Data Types: double | single

Y — Class labels

categorical array | character array | string array | logical vector | numeric vector | cell array of character vectors

Class labels, specified as a categorical, character, or string array, logical or numeric vector, or cell array of character vectors. Y must be the same as the data type of SVMModel.ClassNames. (The software treats string arrays as cell arrays of character vectors.)

The length of Y must equal the number of rows in Tbl or the number of rows in X.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: loss(SVMModel,Tbl,Y,'Weights',W) weighs the observations in each row of Tbl using the corresponding weight in each row of the variable W in Tbl.

LossFun — Loss function

'classiferror' (default) | 'binodeviance' | 'classifcost' | 'exponential' | 'hinge' | 'logit' | 'mincost' | 'quadratic' | function handle

Loss function, specified as a built-in loss function name or a function handle.

• This table lists the available loss functions. Specify one using its corresponding character vector or string scalar.

  Value	Description
  "binodeviance"	Binomial deviance
  "classifcost"	Observed misclassification cost
  "classiferror"	Misclassified rate in decimal
  "exponential"	Exponential loss
  "hinge"	Hinge loss
  "logit"	Logistic loss
  "mincost"	Minimal expected misclassification cost (for classification scores that are posterior probabilities)
  "quadratic"	Quadratic loss

  'mincost' is appropriate for classification scores that are posterior probabilities. You can specify to use posterior probabilities as classification scores for SVM models by setting 'FitPosterior',true when you cross-validate the model using fitcsvm.

• Specify your own function by using function handle notation.

  Suppose that n is the number of observations in X, and K is the number of distinct classes (numel(SVMModel.ClassNames)) used to create the input model (SVMModel). Your function must have this signature

  lossvalue = lossfun(C,S,W,Cost)

  where:

  • The output argument lossvalue is a scalar.

  • You choose the function name (lossfun).

  • C is an n-by-K logical matrix with rows indicating the class to which the corresponding observation belongs. The column order corresponds to the class order in SVMModel.ClassNames.

    Construct C by setting C(p,q) = 1 if observation p is in class q, for each row. Set all other elements of row p to 0.

  • S is an n-by-K numeric matrix of classification scores, similar to the output of predict. The column order corresponds to the class order in SVMModel.ClassNames.

  • W is an n-by-1 numeric vector of observation weights. If you pass W, the software normalizes the weights to sum to 1.

  • Cost is a K-by-K numeric matrix of misclassification costs. For example, Cost = ones(K) – eye(K) specifies a cost of 0 for correct classification and 1 for misclassification.

  Specify your function using 'LossFun',@lossfun.

For more details on loss functions, see Classification Loss.

Example: 'LossFun','binodeviance'

Data Types: char | string | function_handle

Weights — Observation weights

ones(size(X,1),1) (default) | numeric vector | name of variable in Tbl

Observation weights, specified as a numeric vector or the name of a variable in Tbl. The software weighs the observations in each row of X or Tbl with the corresponding weight in Weights.

If you specify Weights as a numeric vector, then the size of Weights must be equal to the number of rows in X or Tbl.

If you specify Weights as the name of a variable in Tbl, you must do so as a character vector or string scalar. For example, if the weights are stored as Tbl.W, then specify Weights as 'W'. Otherwise, the software treats all columns of Tbl, including Tbl.W, as predictors.

If you do not specify your own loss function, then the software normalizes Weights to sum up to the value of the prior probability in the respective class.

Example: 'Weights','W'

Data Types: single | double | char | string

More About

Classification Loss

Classification loss functions measure the predictive inaccuracy of classification models. When you compare the same type of loss among many models, a lower loss indicates a better predictive model.

Consider the following scenario.

• L is the weighted average classification loss.

• n is the sample size.

• y_j is the observed class label. The software codes it as –1 or 1, indicating the negative or positive class (or the first or second class in the ClassNames property), respectively.

• f(X_j) is the positive-class classification score for observation (row) j of the predictor data X.

• m_j = y_jf(X_j) is the classification score for classifying observation j into the class corresponding to y_j. Positive values of m_j indicate correct classification and do not contribute much to the average loss. Negative values of m_j indicate incorrect classification and contribute significantly to the average loss.

• The weight for observation j is w_j. The software normalizes the observation weights so that they sum to the corresponding prior class probability stored in the Prior property. Therefore,

  $${\displaystyle \sum _{j=1}^n{w}_j}=1.$$

Given this scenario, the following table describes the supported loss functions that you can specify by using the LossFun name-value argument.

Loss Function	Value of LossFun	Equation
Binomial deviance	"binodeviance"	$$L={\displaystyle \sum _{j=1}^n{w}_j\log \left\{1+\exp \left[-2m_j\right]\right\}}.$$
Observed misclassification cost	"classifcost"	$$L={\displaystyle \sum _{j=1}^n{w}_j}{c}_{y_j{\widehat{y}}_j},$$ where $${\widehat{y}}_j$$ is the class label corresponding to the class with the maximal score, and $$c_{y_j{\widehat{y}}_j}$$ is the user-specified cost of classifying an observation into class $${\widehat{y}}_j$$ when its true class is yj.
Misclassified rate in decimal	"classiferror"	$$L={\displaystyle \sum _{j=1}^n{w}_j}I\left\{{\widehat{y}}_j\ne y_j\right\},$$ where I{·} is the indicator function.
Cross-entropy loss	"crossentropy"	"crossentropy" is appropriate only for neural network models. The weighted cross-entropy loss is $$L=-{\displaystyle \sum _{j=1}^n\frac{{\tilde{w}}_j\log (m_j)}{Kn}},$$ where the weights $${\tilde{w}}_j$$ are normalized to sum to n instead of 1.
Exponential loss	"exponential"	$$L={\displaystyle \sum _{j=1}^n{w}_j\exp \left(-m_j\right)}.$$
Hinge loss	"hinge"	$$L={\displaystyle \sum }_{j=1}^n{w}_j\max \left\{0,1-m_j\right\}.$$
Logistic loss	"logit"	$$L={\displaystyle \sum _{j=1}^n{w}_j\log \left(1+\exp \left(-m_j\right)\right)}.$$
Minimal expected misclassification cost	"mincost"	"mincost" is appropriate only if classification scores are posterior probabilities. The software computes the weighted minimal expected classification cost using this procedure for observations j = 1,...,n. Estimate the expected misclassification cost of classifying the observation Xj into the class k: $${\gamma }_{jk}={\left(f{\left(X_j\right)}^{\prime }C\right)}_k.$$ f(Xj) is the column vector of class posterior probabilities for the observation Xj. C is the cost matrix stored in the Cost property of the model. For observation j, predict the class label corresponding to the minimal expected misclassification cost: $${\widehat{y}}_j=\underset{k=1,\mathrm{...},K}{\text{argmin}}{\gamma }_{jk}.$$ Using C, identify the cost incurred (cj) for making the prediction. The weighted average of the minimal expected misclassification cost loss is $$L={\displaystyle \sum _{j=1}^n{w}_j{c}_j}.$$
Quadratic loss	"quadratic"	$$L={\displaystyle \sum _{j=1}^n{w}_j{\left(1-m_j\right)}^2}.$$

If you use the default cost matrix (whose element value is 0 for correct classification and 1 for incorrect classification), then the loss values for "classifcost", "classiferror", and "mincost" are identical. For a model with a nondefault cost matrix, the "classifcost" loss is equivalent to the "mincost" loss most of the time. These losses can be different if prediction into the class with maximal posterior probability is different from prediction into the class with minimal expected cost. Note that "mincost" is appropriate only if classification scores are posterior probabilities.

This figure compares the loss functions (except "classifcost", "crossentropy", and "mincost") over the score m for one observation. Some functions are normalized to pass through the point (0,1).

Classification Score

The SVM classification score for classifying observation x is the signed distance from x to the decision boundary ranging from -∞ to +∞. A positive score for a class indicates that x is predicted to be in that class. A negative score indicates otherwise.

The positive class classification score $$f(x)$$ is the trained SVM classification function. $$f(x)$$ is also the numerical predicted response for x, or the score for predicting x into the positive class.

$$f(x)={\displaystyle \sum _{j=1}^n{\alpha }_j}{y}_{j}G(x_j,x)+b,$$

where $$({\alpha }_1,\mathrm{...},{\alpha }_n,b)$$ are the estimated SVM parameters, $$G(x_j,x)$$ is the dot product in the predictor space between x and the support vectors, and the sum includes the training set observations. The negative class classification score for x, or the score for predicting x into the negative class, is –f(x).

If G(x_j,x) = x_j′x (the linear kernel), then the score function reduces to

$$f\left(x\right)=\left(x/s\right)\prime \beta +b.$$

s is the kernel scale and β is the vector of fitted linear coefficients.

For more details, see Understanding Support Vector Machines.

References

[1] Hastie, T., R. Tibshirani, and J. Friedman. The Elements of Statistical Learning, second edition. Springer, New York, 2008.

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

The loss function fully supports tall arrays. For more information, see Tall Arrays.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

• The loss function does not support one-class classification models.

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

Version History

Introduced in R2014a

R2022a: loss returns a different value for a model with a nondefault cost matrix

If you specify a nondefault cost matrix when you train the input model object, the loss function returns a different value compared to previous releases.

The loss function uses the prior probabilities stored in the Prior property to normalize the observation weights of the input data. Also, the function uses the cost matrix stored in the Cost property if you specify the LossFun name-value argument as "classifcost" or "mincost". The way the function uses the Prior and Cost property values has not changed. However, the property values stored in the input model object have changed for a model with a nondefault cost matrix, so the function might return a different value.

For details about the property value change, see Cost property stores the user-specified cost matrix.

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.

R2022a: loss can return NaN for predictor data with missing values

The loss function no longer omits an observation with a NaN score when computing the weighted average classification loss. Therefore, loss can now return NaN when the predictor data X or the predictor variables in Tbl contain any missing values, and the name-value argument LossFun is not specified as "classifcost", "classiferror", or "mincost". In most cases, if the test set observations do not contain missing predictors, the loss function does not return NaN.

This change improves the automatic selection of a classification model when you use fitcauto. Before this change, the software might select a model (expected to best classify new data) with few non-NaN predictors.

If loss in your code returns NaN, you can update your code to avoid this result by doing one of the following:

• Remove or replace the missing values by using rmmissing or fillmissing, respectively.

• Specify the name-value argument LossFun as "classifcost", "classiferror", or "mincost".

The following table shows the classification models for which the loss object function might return NaN. For more details, see the Compatibility Considerations for each loss function.

Model Type	Full or Compact Model Object	loss Object Function
Discriminant analysis classification model	ClassificationDiscriminant, CompactClassificationDiscriminant	loss
Ensemble of learners for classification	ClassificationEnsemble, CompactClassificationEnsemble	loss
Gaussian kernel classification model	ClassificationKernel	loss
k-nearest neighbor classification model	ClassificationKNN	loss
Linear classification model	ClassificationLinear	loss
Neural network classification model	ClassificationNeuralNetwork, CompactClassificationNeuralNetwork	loss
Support vector machine (SVM) classification model	ClassificationSVM, CompactClassificationSVM	loss