margin

Classification margins for naive Bayes classifier

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 Margin (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 for Mdl using the predictor data in table tbl and the class labels in vector Y.

m = margin(Mdl,X,Y) returns the classification margins for Mdl using the predictor data in matrix X and the class labels in Y.

m is returned as a numeric vector with the same length as Y. The software estimates each entry of m using the trained naive Bayes classifier Mdl, the corresponding row of X, and the true class label Y.

Examples

Estimate Test Sample Classification Margins of Naive Bayes Classifier

Estimate the test sample classification margins of a naive Bayes classifier. An observation margin is the observed true class score minus the maximum false class score among all scores in the respective class.

Load the fisheriris data set. Create X as a numeric matrix that contains four petal measurements for 150 irises. Create Y as a cell array of character vectors that contains the corresponding iris species.

load fisheriris
X = meas;
Y = species;
rng('default')  % for reproducibility

Randomly partition observations into a training set and a test set with stratification, using the class information in Y. Specify a 30% holdout sample for testing.

cv = cvpartition(Y,'HoldOut',0.30);

Extract the training and test indices.

trainInds = training(cv);
testInds = test(cv);

Specify the training and test data sets.

XTrain = X(trainInds,:);
YTrain = Y(trainInds);
XTest = X(testInds,:);
YTest = Y(testInds);

Train a naive Bayes classifier using the predictors XTrain and class labels YTrain. A recommended practice is to specify the class names. fitcnb assumes that each predictor is conditionally and normally distributed.

Mdl = fitcnb(XTrain,YTrain,'ClassNames',{'setosa','versicolor','virginica'})

Mdl = 
  ClassificationNaiveBayes
              ResponseName: 'Y'
     CategoricalPredictors: []
                ClassNames: {'setosa'  'versicolor'  'virginica'}
            ScoreTransform: 'none'
           NumObservations: 105
         DistributionNames: {'normal'  'normal'  'normal'  'normal'}
    DistributionParameters: {3x4 cell}


  Properties, Methods

Mdl is a trained ClassificationNaiveBayes classifier.

Estimate the test sample classification margins.

m = margin(Mdl,XTest,YTest);
median(m)

ans = 1.0000

Display the histogram of the test sample classification margins.

histogram(m,length(unique(m)),'Normalization','probability')
xlabel('Test Sample Margins')
ylabel('Probability')
title('Probability Distribution of the Test Sample Margins')

Classifiers that yield relatively large margins are preferred.

Select Naive Bayes Classifier Features by Examining Test Sample Margins

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

Load the fisheriris data set. Specify the predictors X and class labels Y.

load fisheriris
X = meas;
Y = species;
rng('default')  % for reproducibility

Randomly partition observations into a training set and a test set with stratification, using the class information in Y. Specify a 30% holdout sample for testing. Partition defines the data set partition.

cv = cvpartition(Y,'Holdout',0.30);

Extract the training and test indices.

trainInds = training(cv);
testInds = test(cv);

Specify the training and test data sets.

XTrain = X(trainInds,:);
YTrain = Y(trainInds);
XTest = X(testInds,:);
YTest = Y(testInds);

Define these two data sets:

• fullX contains all predictors.

• partX contains the last two predictors.

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

Train a naive Bayes classifier for each predictor set.

fullMdl = fitcnb(fullX,YTrain);
partMdl = fitcnb(partX,YTrain);

fullMdl and partMdl are trained ClassificationNaiveBayes classifiers.

Estimate the test sample margins for each classifier.

fullM = margin(fullMdl,XTest,YTest);
median(fullM)

ans = 1.0000

partM = margin(partMdl,XTest(:,3:4),YTest);
median(partM)

ans = 1.0000

Display the distribution of the margins for each model using boxplots.

boxplot([fullM partM],'Labels',{'All Predictors','Two Predictors'})
ylim([0.98 1.01]) % Modify the y-axis limits to see the boxes
title('Boxplots of Test Sample Margins')

The margins for fullMdl (all predictors model) and partMdl (two predictors model) have a similar distribution with the same median. partMdl is less complex but has outliers.

Input Arguments

Mdl — Naive Bayes classification model

ClassificationNaiveBayes model object | CompactClassificationNaiveBayes model object

Naive Bayes classification model, specified as a ClassificationNaiveBayes model object or CompactClassificationNaiveBayes model object returned by fitcnb 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. 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. Optionally, tbl can contain additional columns for the response variable and observation weights.

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 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.

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

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 in the columns of X must be the same as the variables that trained the Mdl classifier.

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 | 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 have the same data type as Mdl.ClassNames. (The software treats string arrays as cell arrays of character vectors.)

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

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

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.

When choosing among multiple classifiers to perform a task such as feature section, choose the classifier that yields the highest edge.

Classification Margin

The classification margin for each observation is the difference between the score for the true class and the maximal score for the false classes. Margins provide a classification confidence measure; among multiple classifiers, those that yield larger margins (on the same scale) 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 assumed 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.