margin

Classification margins for Gaussian kernel classification model

Syntax

m = margin(Mdl,X,Y)

m = margin(Mdl,Tbl,ResponseVarName)

m = margin(Mdl,Tbl,Y)

Description

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

m = margin(Mdl,Tbl,ResponseVarName) returns the classification margins for the trained kernel classifier Mdl using the predictor data in table Tbl and the class labels in Tbl.ResponseVarName.

m = margin(Mdl,Tbl,Y) returns the classification margins for the classifier Mdl using the predictor data in table Tbl and the class labels in vector Y.

Examples

Estimate Test-Set Margins

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

load ionosphere

Partition the data set into training and test sets. Specify a 30% holdout sample for the test set.

rng('default') % For reproducibility
Partition = cvpartition(Y,'Holdout',0.30);
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 margins and test-set margins.

mTrain = margin(Mdl,X(trainingInds,:),Y(trainingInds));
mTest = margin(Mdl,X(testInds,:),Y(testInds));

Plot both sets of margins using box plots.

boxplot([mTrain; mTest],[zeros(size(mTrain,1),1); ones(size(mTest,1),1)], ...
    'Labels',{'Training set','Test set'});
title('Training-Set and Test-Set Margins')

The margin distribution of the training set is situated higher than the margin distribution of the test set.

Feature Selection Using Test-Set Margins

Perform feature selection by comparing test-set margins from multiple models. Based solely on this criterion, the classifier with the larger margins is the better 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').

load ionosphere

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 10% of the predictor variables.

p = size(X,2); % Number of predictors
idxPart = randsample(p,ceil(0.1*p));

Train two binary kernel classification models: one that uses all of the predictors, and one that uses the random 10%.

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

Mdl and PMdl are ClassificationKernel models.

Estimate the test-set margins for each classifier.

fullMargins = margin(Mdl,XTest,YTest);
partMargins = margin(PMdl,XTest(:,idxPart),YTest);

Plot the distribution of the margin sets using box plots.

boxplot([fullMargins partMargins], ...
    'Labels',{'All Predictors','10% of the Predictors'});
title('Test-Set Margins')

The margin distribution of PMdl is situated higher than the margin distribution of Mdl. Therefore, the PMdl model is the better classifier.

Input Arguments

Mdl — Binary kernel classification model

ClassificationKernel model object

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

X — Predictor data

n-by-p numeric matrix

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

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.

• 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 must be equal to the number of observations in X or Tbl.

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

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 the predictors used to train Mdl. Multicolumn variables and cell arrays other than cell arrays of character vectors are not allowed.

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

If you train Mdl using sample data contained in a table, then the input data for margin must also be in a 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 Mdl, then you do not need to specify ResponseVarName.

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

The response variable must be a categorical, character, or string array, a logical or numeric vector, or a 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

Output Arguments

m — Classification margins

numeric column vector

Classification margins, returned as an n-by-1 numeric column vector, where n is the number of observations in X.

More About

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

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

• $$T(·)$$ 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).

Extended Capabilities

Tall Arrays

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

Usage notes and limitations:

• margin does not support tall table data.

For more information, see Tall Arrays.