edge

Find classification edge for support vector machine (SVM) classifier

Syntax

e = edge(SVMModel,Tbl,ResponseVarName)

e = edge(SVMModel,Tbl,Y)

e = edge(SVMModel,X,Y)

e = edge(___,'Weights',weights)

Description

e = edge(SVMModel,Tbl,ResponseVarName) returns the classification edge (e) for the support vector machine (SVM) classifier SVMModel using the predictor data in table Tbl and the class labels in Tbl.ResponseVarName.

The classification edge (e) is a scalar value that represents the weighted mean of the classification margins.

e = edge(SVMModel,Tbl,Y) returns the classification edge (e) for the SVM classifier SVMModel using the predictor data in table Tbl and the class labels in Y.

e = edge(SVMModel,X,Y) returns the classification edge for SVMModel using the predictor data in matrix X and the class labels in Y.

e = edge(___,'Weights',weights) computes the classification edge for the observation weights supplied in weights using any of the input arguments in the previous syntaxes.

Note

If the predictor data X or the predictor variables in Tbl contain any missing values, the edge function can return NaN. For more details, see edge can return NaN for predictor data with missing values.

Examples

Estimate Test Sample Edge 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 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.

Estimate the test sample edge.

e = edge(CompactSVMModel,XTest,YTest)

e = 
5.0766

The margin average of the test sample is approximately 5.

Estimate Test Sample Weighted Margin Mean of SVM Classifiers

Suppose that the observations in a data set are measured sequentially, and that the last 150 observations have better quality due to a technology upgrade. Incorporate this advancement by weighing the better quality observations more than the other observations.

Load the ionosphere data set.

load ionosphere
rng(1); % For reproducibility

Define a weight vector that weighs the better quality observations two times the other observations.

n = size(X,1);
weights = [ones(n-150,1);2*ones(150,1)];

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

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

CVSVMModel is a trained 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.

Estimate the test sample weighted edge using the weighting scheme.

e = edge(CompactSVMModel,XTest,YTest,'Weights',wTest)

e = 
4.8340

The weighted average margin of the test sample is approximately 5.

Select SVM Classifier Features by Comparing Test Sample Edges

Perform feature selection by comparing test sample edges from multiple models. Based solely on this comparison, the classifier with the highest edge is the best classifier.

Load the ionosphere data set.

load ionosphere
rng(1); % For reproducibility

Partition the data set into training and test sets. Specify a 15% holdout sample for testing.

Partition = cvpartition(Y,'Holdout',0.15);
testInds = test(Partition); % Indices for the test set
XTest = X(testInds,:);
YTest = Y(testInds,:);

Partition defines the data set partition.

Define these two data sets:

• fullX contains all predictors (except the removed column of 0s).

• partX contains the last 20 predictors.

fullX = X;
partX = X(:,end-20:end);

Train SVM classifiers for each predictor set. Specify the partition definition.

FullCVSVMModel = fitcsvm(fullX,Y,'CVPartition',Partition);
PartCVSVMModel = fitcsvm(partX,Y,'CVPartition',Partition);
FCSVMModel = FullCVSVMModel.Trained{1};
PCSVMModel = PartCVSVMModel.Trained{1};

FullCVSVMModel and PartCVSVMModel are ClassificationPartitionedModel classifiers. They contain the property Trained, which is a 1-by-1 cell array holding a CompactClassificationSVM classifier that the software trained using the training set.

Estimate the test sample edge for each classifier.

fullEdge = edge(FCSVMModel,XTest,YTest)

fullEdge = 
2.8319

partEdge = edge(PCSVMModel,XTest(:,end-20:end),YTest)

partEdge = 
1.5542

The edge for the classifier trained on the complete data set is greater, suggesting that the classifier trained with all the predictors is better.

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 edge 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

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

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.

If you specify ResponseVarName, then you must do so as a character vector or string scalar. For example, if the response variable is stored as Tbl.Response, then specify ResponseVarName as 'Response'. Otherwise, the software treats all columns of Tbl, including Tbl.Response, as predictors.

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

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.

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.

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 supply weights, edge computes the weighted classification edge. The software weights the observations in each row of X or Tbl with the corresponding weight in weights.

Example: 'Weights','W'

Data Types: single | double | char | string

More About

Classification Edge

The edge is the weighted mean of the classification margins.

The weights are the prior class probabilities. If you supply weights, then the software normalizes them to sum to the prior probabilities in the respective classes. The software uses the renormalized weights to compute the weighted mean.

One way to choose among multiple classifiers, for example, to perform feature selection, is to choose the classifier that yields the highest edge.

Classification Margin

The classification margin for binary classification is, for each observation, the difference between the classification score for the true class and the classification score for the false class.

The software defines the classification margin for binary classification as

$$m=2yf\left(x\right).$$

x is an observation. If the true label of x is the positive class, then y is 1, and –1 otherwise. f(x) is the positive-class classification score for the observation x. The classification margin is commonly defined as m = yf(x).

If the margins are on the same scale, then they serve as a classification confidence measure. Among multiple classifiers, those that yield greater margins are better.

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.

Algorithms

For binary classification, the software defines the margin for observation j, m_j, as

$$m_j=2y_{j}f(x_j),$$

where y_j ∊ {-1,1}, and f(x_j) is the predicted score of observation j for the positive class. However, m_j = y_jf(x_j) is commonly used to define the margin.

References

[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.

Extended Capabilities

Tall Arrays

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

The edge 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 edge 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: edge 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 edge function returns a different value compared to previous releases.

The edge function uses the prior probabilities stored in the Prior property to normalize the observation weights of the input data. The way the function uses the Prior property value has not changed. However, the property value stored in the input model object has changed for a model with a nondefault cost matrix, so the function can 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: edge can return NaN for predictor data with missing values

The edge function no longer omits an observation with a NaN score when computing the weighted mean of the classification margins. Therefore, edge can now return NaN when the predictor data X or the predictor variables in Tbl contain any missing values. In most cases, if the test set observations do not contain missing predictors, the edge 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 edge in your code returns NaN, you can update your code to avoid this result. Remove or replace the missing values by using rmmissing or fillmissing, respectively.

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

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