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edge

Class: CompactClassificationSVM

Classification edge for support vector machine classifiers

Syntax

  • e = edge(SVMModel,TBL,ResponseVarName)
  • e = edge(SVMModel,TBL,Y)
  • e = edge(___,Name,Value)

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 class labels in TBL.ResponseVarName.

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

e = edge(___,Name,Value) computes the classification edge with additional options specified by one or more Name,Value pair arguments, using any of the previous syntaxes. For example, you can specify observation weights.

example

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

example

e = edge(___,Name,Value) computes the classification edge with additional options specified by one or more Name,Value pair arguments, using any of the previous syntaxes. For example, you can specify observation 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

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

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. For example, if the response variable is stored as TBL.Response, then specify it as 'Response'. Otherwise, the software treats all columns of TBL, including TBL.Response, as predictors.

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.

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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Observation weights, specified as the comma-separated pair consisting of 'Weights' and 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. 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 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.

Data Types: single | double

Output Arguments

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Classification edge, returned as a scalar. e represents the (weighted) mean of the classification margins.

Definitions

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, e.g., to perform feature selection, is to choose the classifier that yields the highest edge.

Classification Margin

The classification margins for binary classification are, 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(x).

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 literature commonly defines the margin as m = yf(x).

If the margins are on the same scale, then they serve as a classification confidence measure, i.e., among multiple classifiers, those that yield larger 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 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 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 =

    5.0765

The estimated test sample margin average is approximately 5.

Load the ionosphere data set.

load ionosphere
rng(1); % For reproducibility

Suppose that the observations were measured sequentially, and that the last 150 observations were better quality due to a technology upgrade. One way to incorporate this advancement is to weigh the better quality observations more than the other observations.

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. It is good practice to specify the class order and standardize the data.

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 =

    4.8341

The test sample weighted average margin is approximately 5.

The classifier edge measures the average of the classifier margins. One way to perform feature selection is to compare test sample edges from multiple models. Based solely on this criterion, 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)
partEdge = edge(PCSVMModel,XTest(:,end-20:end),YTest)
fullEdge =

    2.8319


partEdge =

    1.5541

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

Algorithms

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

mj=2yjf(xj),

where yj ∊ {-1,1}, and f(xj) is the predicted score of observation j for the positive class. However, the literature commonly uses mj = yjf(xj) 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.

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