margin

Classification margins for naive Bayes classifiers

Syntax

m = margin(Mdl,tbl,ResponseVarName)

m = margin(Mdl,tbl,Y)

m = margin(Mdl,X,Y)

Description

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

m = margin(Mdl,X,Y) returns the classification margins (m) for the trained naive Bayes classifier Mdl using the predictor data X and class labels Y.

Input Arguments

Mdl — Naive Bayes classifier

ClassificationNaiveBayes model | CompactClassificationNaiveBayes model

Naive Bayes classifier, specified as a ClassificationNaiveBayes model or CompactClassificationNaiveBayes model returned by fitcnb or compact, respectively.

tbl — Sample data

table

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

If you trained Mdl using sample data contained in a table, then the input data for this method must also be in a table.

Data Types: table

ResponseVarName — Response variable name

name of a variable in tbl

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

You must specify ResponseVarName as a character vector or string scalar. 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, 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

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 making up the columns of X should be the same as the variables that trained Mdl.

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

Data Types: double | single

Y — Class labels

categorical array | character array | string array | logical vector | vector of numeric values | 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 Mdl.ClassNames. (The software treats string arrays as cell arrays of character vectors.)

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

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

Output Arguments

m — Classification margins

numeric vector

Classification margins, returned as a numeric vector.

m has the same length equal to size(X,1). Each entry of m is the classification margin of the corresponding observation (row) of X and element of Y.

Examples

Estimate Test-Sample Classification Margins of Naive Bayes Classifiers

Load Fisher's iris data set.

load fisheriris
X = meas;    % Predictors
Y = species; % Response
rng(1);

Train a naive Bayes classifier. Specify a 30% holdout sample for testing. It is good practice to specify the class order. Assume that each predictor is conditionally normally distributed given its label.

CVMdl = fitcnb(X,Y,'Holdout',0.30,...
    'ClassNames',{'setosa','versicolor','virginica'});
CMdl = CVMdl.Trained{1}; ...
    % Extract the trained, compact classifier
testInds = test(CVMdl.Partition);   % Extract the test indices
XTest = X(testInds,:);
YTest = Y(testInds);

CVMdl is a ClassificationPartitionedModel classifier. It contains the property Trained, which is a 1-by-1 cell array holding a CompactClassificationNaiveBayes classifier that the software trained using the training set.

Estimate the test sample classification margins. Display the distribution of the margins using a boxplot.

m = margin(CMdl,XTest,YTest);

figure;
boxplot(m);
title 'Distribution of the Test-Sample Margins';

An observation margin is the observed true class score minus the maximum false class score among all scores in the respective class. Classifiers that yield relatively large margins are desirable.

Select Naive Bayes Classifier Features by Examining Test Sample Margins

The classifier margins measure, for each observation, the difference between the true class observed score and the maximal false class score for a particular class. One way to perform feature selection is to compare test sample margins from multiple models. Based solely on this criterion, the model with the highest margins is the best model.

Load Fisher's iris data set.

load fisheriris
X = meas;    % Predictors
Y = species; % Response
rng(1);

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

Partition = cvpartition(Y,'Holdout',0.30);
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.

• partX contains the last 2 predictors.

fullX = X;
partX = X(:,3:4);

Train naive Bayes classifiers for each predictor set. Specify the partition definition.

FCVMdl = fitcnb(fullX,Y,'CVPartition',Partition);
PCVMdl = fitcnb(partX,Y,'CVPartition',Partition);
FCMdl = FCVMdl.Trained{1};
PCMdl = PCVMdl.Trained{1};

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

Estimate the test sample margins for each classifier. Display the distributions of the margins for each model using boxplots.

fullM = margin(FCMdl,XTest,YTest);
partM = margin(PCMdl,XTest(:,3:4),YTest);

figure;
boxplot([fullM partM],'Labels',{'All Predictors','Two Predictors'})
h = gca;
h.YLim = [0.98 1.01]; % Modify axis to see boxes.
title 'Boxplots of Test-Sample Margins';

The margins have a similar distribution, but PCMdl is less complex.

More About

Classification Edge

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

If you supply weights, then the software normalizes them to sum to the prior probability of their respective class. The software uses the normalized 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 are, for each observation, the difference between the score for the true class and maximal score for the false classes. Provided that they are on the same scale, margins serve as a classification confidence measure, i.e., among multiple classifiers, those that yield larger margins are better.

Posterior Probability

The posterior probability is the probability that an observation belongs in a particular class, given the data.

For naive Bayes, the posterior probability that a classification is k for a given observation (x₁,...,x_P) is

$$\widehat{P}\left(Y=k|x_1,\mathrm{..},x_P\right)=\frac{P\left(X_1,\mathrm{...},X_P|y=k\right)\pi \left(Y=k\right)}{P\left(X_1,\mathrm{...},X_P\right)},$$

where:

• $$P\left(X_1,\mathrm{...},X_P|y=k\right)$$ is the conditional joint density of the predictors given they are in class k. Mdl.DistributionNames stores the distribution names of the predictors.

• π(Y = k) is the class prior probability distribution. Mdl.Prior stores the prior distribution.

• $$P\left(X_1,\mathrm{..},X_P\right)$$ is the joint density of the predictors. The classes are discrete, so $$P(X_1,\mathrm{...},X_P)={\displaystyle \sum _{k=1}^{K}P}(X_1,\mathrm{...},X_P|y=k)\pi (Y=k).$$

Prior Probability

The prior probability of a class is the believed relative frequency with which observations from that class occur in a population.

Score

The naive Bayes score is the class posterior probability given the observation.

Extended Capabilities

Tall Arrays

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

This function fully supports tall arrays. For more information, see Tall Arrays (MATLAB).