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loss

Class: CompactClassificationSVM

Classification error for support vector machine classifiers

Syntax

  • L = loss(SVMModel,TBL,ResponseVarName)
  • L = loss(SVMModel,TBL,Y)

Description

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

loss normalizes the class probabilities in TBL.ResponseVarName to the prior class probabilities fitcsvm used for training, stored in the Prior property of SVMModel.

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

loss normalizes the class probabilities in Y to the prior class probabilities fitcsvm used for training, stored in the Prior property of SVMModel.

example

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

example

L = loss(___,Name,Value) returns the classification error with additional options specified by one or more Name,Value pair arguments, using any of the previous syntaxes. For example, you can specify the loss function or classification weights.

Input Arguments

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

Sample data, 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. Multi-column 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 this method must also be in a table.

Data Types: table

Response variable name, specified as the name of a variable in TBL.

You must specify ResponseVarName as a character vector. For example, if the response variable Y is stored as TBL.Y, then specify it 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 or character 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.

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 making up 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 of 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 standard deviations in SVMModel.Sigma.

Data Types: double | single

Class labels, specified as a categorical or character array, logical or numeric vector, or cell array of character vectors. Y must be the same as the data type of SVMModel.ClassNames.

The length of Y must equal the number of rows of TBL or X must be equal.

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

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Loss function, specified as the comma-separated pair consisting of 'LossFun' and a built-in, loss-function name or function handle.

  • The following lists available loss functions. Specify one using its corresponding character vector.

    ValueDescription
    'binodeviance'Binomial deviance
    'classiferror'Classification error
    'exponential'Exponential
    'hinge'Hinge
    'logit'Logistic
    'mincost'Minimal expected misclassification cost (for classification scores that are posterior probabilities)
    'quadratic'Quadratic

    '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 using function handle notation.

    Suppose that n be the number of observations in X and K be the number of distinct classes (numel(SVMModel.ClassNames), SVMModel is the input model). 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 which class 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. The column order corresponds to the class order in SVMModel.ClassNames. S is a matrix of classification scores, similar to the output of predict.

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

Data Types: char | function_handle

Observation weights, specified as the comma-separated pair consisting of 'Weights' and 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 vector, then the size of Weights must be equal to the number of rows of X or TBL.

If you specify Weights as the name of a variable in TBL, you must do so as a character vector. For example, if the weights are stored as TBL.W, then specify it 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.

Data Types: single | double

Output Arguments

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Classification loss, returned as a scalar. 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 loss values.

Definitions

Classification Loss

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

Suppose that:

  • L is the weighted average classification loss.

  • n is the sample size.

  • For binary classification:

    • yj is the observed class label. The software codes it as –1 or 1 indicating the negative or positive class, respectively.

    • f(Xj) is the raw classification score for observation (row) j of the predictor data X.

    • mj = yjf(Xj) is the classification score for classifying observation j into the class corresponding to yj. Positive values of mj indicate correct classification and do not contribute much to the average loss. Negative values of mj indicate incorrect classification and contribute to the average loss.

  • For algorithms that support multiclass classification (that is, K ≥ 3):

    • yj* is a vector of K – 1 zeros, and a 1 in the position corresponding to the true, observed class yj. For example, if the true class of the second observation is the third class and K = 4, then y*2 = [0 0 1 0]′. The order of the classes corresponds to the order in the ClassNames property of the input model.

    • f(Xj) is the length K vector of class scores for observation j of the predictor data X. The order of the scores corresponds to the order of the classes in the ClassNames property of the input model.

    • mj = yj*f(Xj). Therefore, mj is the scalar classification score that the model predicts for the true, observed class.

  • The weight for observation j is wj. The software normalizes the observation weights so that they sum to the corresponding prior class probability. The software also normalizes the prior probabilities so they sum to 1. Therefore,

    j=1nwj=1.

The supported loss functions are:

  • Binomial deviance, specified using 'LossFun','binodeviance'. Its equation is

    L=j=1nwjlog{1+exp[2mj]}.

  • Exponential loss, specified using 'LossFun','exponential'. Its equation is

    L=j=1nwjexp(mj).

  • Classification error, specified using 'LossFun','classiferror'. It is the weighted fraction of misclassified observations, with equation

    L=j=1nwjI{y^jyj}.

    y^j is the class label corresponding to the class with the maximal posterior probability. I{x} is the indicator function.

  • Hinge loss, specified using 'LossFun','hinge'. Its equation is

    L=j=1nwjmax{0,1mj}.

  • Logit loss, specified using 'LossFun','logit'. Its equation is

    L=j=1nwjlog(1+exp(mj)).

  • Minimal cost, specified using 'LossFun','mincost'. The software computes the weighted minimal cost using this procedure for observations j = 1,...,n:

    1. Estimate the 1-by-K vector of expected classification costs for observation j

      γj=f(Xj)C.

      f(Xj) is the column vector of class posterior probabilities for binary and multiclass classification. C is the cost matrix the input model stores in the property Cost.

    2. For observation j, predict the class label corresponding to the minimum, expected classification cost:

      y^j=minj=1,...,K(γj).

    3. Using C, identify the cost incurred (cj) for making the prediction.

    The weighted, average, minimum cost loss is

    L=j=1nwjcj.

  • Quadratic loss, specified using 'LossFun','quadratic'. Its equation is

    L=j=1nwj(1mj)2.

This figure compares some of the loss functions for one observation over m (some functions are normalized to pass through [0,1]).

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

f(x)=j=1nαjyjG(xj,x)+b,

where (α1,...,αn,b) are the estimated SVM parameters, G(xj,x) is the dot product in the predictor space between x and the support vectors, and the sum includes the training set observations. The score for predicting x into the negative class is –f(x).

If G(xj,x) = xjx (the linear kernel), then the score function reduces to

f(x)=(x/s)β+b.

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

Examples

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Load the ionosphere data set.

load ionosphere
rng(1); % For reproducibility

Train an SVM classifier. Specify a 15% holdout sample for testing. It is good practice to specify the class order and standardize the data.

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 radar returns.

Load the ionosphere data set.

load ionosphere
rng(1); % For reproducibility

Train an SVM classifier. Specify a 15% holdout sample for testing. It is good practice to specify the class order and standardize the data.

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 desirable.

References

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

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