Documentation

# edge

Classification edge for Gaussian kernel classification model

## Syntax

e = edge(Mdl,X,Y)
e = edge(Mdl,X,Y,'Weights',weights)

## Description

example

e = edge(Mdl,X,Y) returns the classification edge for the binary Gaussian kernel classification model Mdl using the predictor data in X and the corresponding class labels in Y.

e = edge(Mdl,X,Y,'Weights',weights) returns the weighted classification edges for the observation weights supplied in weights.

## Examples

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Load the ionosphere data set. This data set has 34 predictors and 351 binary responses for radar returns, either bad ('b') or good ('g').

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

rng('default') % For reproducibility
Partition = cvpartition(Y,'Holdout',0.15);
trainingInds = training(Partition); % Indices for the training set
testInds = test(Partition); % Indices for the test set

Train a binary kernel classification model using the training set.

Mdl = fitckernel(X(trainingInds,:),Y(trainingInds));

Estimate the training-set edge and the test-set edge.

eTrain = edge(Mdl,X(trainingInds,:),Y(trainingInds))
eTrain = 2.1703
eTest = edge(Mdl,X(testInds,:),Y(testInds))
eTest = 1.5643

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

Load the ionosphere data set. This data set has 34 predictors and 351 binary responses for radar returns, either bad ('b') or good ('g').

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

rng('default') % For reproducibility
Partition = cvpartition(Y,'Holdout',0.15);
trainingInds = training(Partition); % Indices for the training set
XTrain = X(trainingInds,:);
YTrain = Y(trainingInds);
testInds = test(Partition); % Indices for the test set
XTest = X(testInds,:);
YTest = Y(testInds);

Randomly choose half of the predictor variables.

p = size(X,2); % Number of predictors
idxPart = randsample(p,ceil(0.5*p));

Train two binary kernel classification models: one that uses all of the predictors, and one that uses half of the predictors.

Mdl = fitckernel(XTrain,YTrain);
PMdl = fitckernel(XTrain(:,idxPart),YTrain);

Mdl and PMdl are ClassificationKernel models.

Estimate the test-set edge for each classifier.

fullEdge = edge(Mdl,XTest,YTest)
fullEdge = 1.6335
partEdge = edge(PMdl,XTest(:,idxPart),YTest)
partEdge = 2.0205

Based on the test-set edges, the classifier that uses half of the predictors is the better model.

## Input Arguments

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Binary kernel classification model, specified as a ClassificationKernel model object. You can create a ClassificationKernel model object using fitckernel.

Predictor data, specified as an n-by-p numeric matrix, where n is the number of observations and p is the number of predictors used to train Mdl.

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

Data Types: single | double

Class labels, specified as a categorical, character, or string array, logical or numeric vector, or cell array of character vectors.

• The data type of Y must be the same as the data type of Mdl.ClassNames. (The software treats string arrays as cell arrays of character vectors.)

• The distinct classes in Y must be a subset of Mdl.ClassNames.

• If Y is a character array, then each element must correspond to one row of the array.

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

Data Types: categorical | char | string | logical | single | double | cell

Observation weights, specified as a positive numeric vector of length n, where n is the number of observations in X. If you supply weights, edge computes the weighted classification edge.

The default value is ones(n,1).

edge normalizes weights to sum up to the value of the prior probability in the respective class.

Data Types: double | single

## Output Arguments

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Classification edge, returned as a numeric scalar.

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### Classification Edge

The classification edge is the weighted mean of the classification margins.

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

For kernel classification models, the raw classification score for classifying the observation x, a row vector, into the positive class is defined by

$f\left(x\right)=T\left(x\right)\beta +b.$

• $T\left(·\right)$ is a transformation of an observation for feature expansion.

• β is the estimated column vector of coefficients.

• b is the estimated scalar bias.

The raw classification score for classifying x into the negative class is f(x). The software classifies observations into the class that yields a positive score.

If the kernel classification model consists of logistic regression learners, then the software applies the 'logit' score transformation to the raw classification scores (see ScoreTransform).