# ClassificationNaiveBayes class

Superclasses: CompactClassificationNaiveBayes

Naive Bayes classification

## Description

ClassificationNaiveBayes is a naive Bayes classifier for multiclass learning. Use fitcnb and the training data to train a ClassificationNaiveBayes classifier.

Trained ClassificationNaiveBayes classifiers store the training data, parameter values, data distribution, and prior probabilities. You can use these classifiers to:

• Estimate resubstitution predictions. For details, see resubPredict.

• Predict labels or posterior probabilities for new data. For details, see predict.

## Construction

Create a ClassificationNaiveBayes object by using fitcnb.

## Properties

expand all

Categorical predictor indices, specified as a vector of positive integers. CategoricalPredictors contains index values corresponding to the columns of the predictor data that contain categorical predictors. If none of the predictors are categorical, then this property is empty ([]).

Data Types: single | double

Multivariate multinomial levels, specified as a cell vector of numeric vectors. CategoricalLevels has length equal to the number of predictors (size(X,2)).

The cells of CategoricalLevels correspond to predictors that you specified as 'mvmn' (i.e., having a multivariate multinomial distribution) during training. Cells that do not correspond to a multivariate multinomial distribution are empty ([]).

If predictor j is multivariate multinomial, then CategoricalLevels{j} is a list of all distinct values of predictor j in the sample (NaNs removed from unique(X(:,j))).

Data Types: cell

Distinct class names, specified as a categorical or character array, logical or numeric vector, or cell vector of character vectors.

ClassNames is the same data type as Y, and has K elements or rows for character arrays. (The software treats string arrays as cell arrays of character vectors.)

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

Misclassification cost, specified as a K-by-K square matrix.

The value of Cost(i,j) is the cost of classifying a point into class j if its true class is i. The order of the rows and columns of Cost correspond to the order of the classes in ClassNames.

The value of Cost does not influence training. You can reset Cost after training Mdl using dot notation, e.g., Mdl.Cost = [0 0.5; 1 0];.

Data Types: double | single

Predictor distributions fitcnb uses to model the predictors, specified as a character vector or cell array of character vectors.

This table summarizes the available distributions.

ValueDescription
'kernel'Kernel smoothing density estimate.
'mn'Multinomial bag-of-tokens model. Indicates that all predictors have this distribution.
'mvmn'Multivariate multinomial distribution.
'normal'Normal (Gaussian) distribution.

If Distribution is a 1-by-P cell array of character vectors, then the software models feature j using the distribution in element j of the cell array.

Data Types: char | cell

Distribution parameter estimates, specified as a cell array. DistributionParameters is a K-by-D cell array, where cell (k,d) contains the distribution parameter estimates for instances of predictor d in class k. The order of the rows corresponds to the order of the classes in the property ClassNames, and the order of the predictors corresponds to the order of the columns of X.

If class k has no observations for predictor j, then Distribution{k,j} is empty ([]).

The elements of DistributionParameters depends on the distributions of the predictors. This table describes the values in DistributionParameters{k,j}.

Distribution of Predictor jValue
kernelA KernelDistribution model. Display properties using cell indexing and dot notation. For example, to display the estimated bandwidth of the kernel density for predictor 2 in the third class, use Mdl.DistributionParameters{3,2}.BandWidth.
mnA scalar representing the probability that token j appears in class k. For details, see Algorithms.
mvmnA numeric vector containing the probabilities for each possible level of predictor j in class k. The software orders the probabilities by the sorted order of all unique levels of predictor j (stored in the property CategoricalLevels). For more details, see Algorithms.
normalA 2-by-1 numeric vector. The first element is the sample mean and the second element is the sample standard deviation.

Data Types: cell

Expanded predictor names, stored as a cell array of character vectors.

If the model uses encoding for categorical variables, then ExpandedPredictorNames includes the names that describe the expanded variables. Otherwise, ExpandedPredictorNames is the same as PredictorNames.

Data Types: cell

Description of the cross-validation optimization of hyperparameters, specified as a BayesianOptimization object or a table of hyperparameters and associated values. This property is nonempty if the 'OptimizeHyperparameters' name-value pair argument is nonempty when you create the model. The value of HyperparameterOptimizationResults depends on the setting of the Optimizer field in the HyperparameterOptimizationOptions structure when you create the model, as described in this table.

Value of Optimizer FieldValue of HyperparameterOptimizationResults
'bayesopt' (default)Object of class BayesianOptimization
'gridsearch' or 'randomsearch'Table of hyperparameters used, observed objective function values (cross-validation loss), and rank of observations from lowest (best) to highest (worst)

Kernel smoother types, specified as a character vector or cell array of character vectors. Kernel has length equal to the number of predictors (size(X,2)). Kernel{j} corresponds to predictor j, and contains a character vector describing the type of kernel smoother. This table describes the supported kernel smoother types. Let I{u} denote the indicator function.

ValueKernelFormula
'box'Box (uniform)

$f\left(x\right)=0.5I\left\{|x|\le 1\right\}$

'epanechnikov'Epanechnikov

$f\left(x\right)=0.75\left(1-{x}^{2}\right)I\left\{|x|\le 1\right\}$

'normal'Gaussian

$f\left(x\right)=\frac{1}{\sqrt{2\pi }}\mathrm{exp}\left(-0.5{x}^{2}\right)$

'triangle'Triangular

$f\left(x\right)=\left(1-|x|\right)I\left\{|x|\le 1\right\}$

If a cell is empty ([]), then the software did not fit a kernel distribution to the corresponding predictor.

Data Types: char | cell

Parameter values used to train the classifier (such as the name-value pair argument values), specified as an object. This table summarizes the properties of ModelParameters. The properties correspond to the name-value pair argument values set for training the classifier.

PropertyPurpose
DistributionNamesData distribution or distributions. This is the same value as the property DistributionNames.
KernelKernel smoother type. This is the same as the property Kernel.
MethodTraining method. For naive Bayes, the value is 'NaiveBayes'.
SupportKernel-smoothing density support. This is the same as the property Support.
TypeLearning type. For classification, the value is 'classification'.
WidthKernel smoothing window width. This is the same as the property Width.

Access fields of ModelParameters using dot notation. For example, access the kernel support using Mdl.ModelParameters.Support.

Number of training observations, specified as a numeric scalar.

If X or Y contain missing values, then NumObservations might be less than the length of Y.

Data Types: double

Predictor names, specified as a cell array of character vectors. The order of the elements in PredictorNames corresponds to the order in X.

Data Types: cell

Class prior probabilities, specified as a numeric row vector. Prior is a 1-by-K vector, and the order of its elements correspond to the elements of ClassNames.

fitcnb normalizes the prior probabilities you set using the name-value pair parameter 'Prior' so that sum(Prior) = 1.

The value of Prior does not change the best-fitting model. Therefore, you can reset Prior after training Mdl using dot notation, e.g., Mdl.Prior = [0.2 0.8];.

Data Types: double | single

Response name, specified as a character vector.

Data Types: char

Classification score transformation function, specified as a character vector or function handle.

To change the score transformation function to e.g., function, use dot notation.

• For a built-in function, enter this code and replace function with a value in the table.

Mdl.ScoreTransform = 'function';

ValueDescription
'doublelogit'1/(1 + e–2x)
'invlogit'log(x / (1 – x))
'ismax'Sets the score for the class with the largest score to 1, and sets the scores for all other classes to 0
'logit'1/(1 + ex)
'none' or 'identity'x (no transformation)
'sign'–1 for x < 0
0 for x = 0
1 for x > 0
'symmetric'2x – 1
'symmetricismax'Sets the score for the class with the largest score to 1, and sets the scores for all other classes to –1
'symmetriclogit'2/(1 + ex) – 1

• For a MATLAB® function, or a function that you define, enter its function handle.

Mdl.ScoreTransform = @function;

function should accept a matrix (the original scores) and return a matrix of the same size (the transformed scores).

Data Types: char | function_handle

Kernel smoother density support, specified as a cell vector. Support has length equal to the number of predictors (size(X,2)). The cells represent the regions to apply the kernel density.

This table describes the supported options.

ValueDescription
1-by-2 numeric row vectorFor example, [L,U], where L and U are the finite lower and upper bounds, respectively, for the density support.
'positive'The density support is all positive real values.
'unbounded'The density support is all real values.

If a cell is empty ([]), then the software did not fit a kernel distribution to the corresponding predictor.

Observation weights, specified as a numeric vector.

The length of W is NumObservations.

fitcnb normalizes the value you set for the name-value pair parameter 'Weights' so that the weights within a particular class sum to the prior probability for that class.

Data Types: double

Kernel smoother window width, specified as a numeric matrix. Width is a K-by-P matrix, where K is the number of classes in the data, and P is the number of predictors (size(X,2)).

Width(k,j) is the kernel smoother window width for the kernel smoothing density of predictor j within class k. NaNs in column j indicate that the software did not fit predictor j using a kernel density.

Unstandardized predictor data, specified as a numeric matrix. X has NumObservations rows and P columns.

Each row of X corresponds to one observation, and each column corresponds to one variable.

The software excludes rows removed due to missing values from X.

Data Types: double

Observed class labels, specified as a categorical or character array, logical or numeric vector, or cell array of character vectors. Y is the same data type as the input argument Y of fitcnb. (The software treats string arrays as cell arrays of character vectors.)

Each row of Y represents the observed classification of the corresponding row of X.

The software excludes elements removed due to missing values from Y.

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

## Methods

 compact Compact naive Bayes classifier crossval Cross-validated naive Bayes classifier resubEdge Classification edge for naive Bayes classifiers by resubstitution resubLoss Classification loss for naive Bayes classifiers by resubstitution resubMargin Classification margins for naive Bayes classifiers by resubstitution resubPredict Predict resubstitution labels of naive Bayes classifier

### Inherited Methods

 edge Classification edge for naive Bayes classifiers logP Log unconditional probability density for naive Bayes classifier loss Classification error for naive Bayes classifier margin Classification margins for naive Bayes classifiers predict Predict labels using naive Bayes classification model

## Copy Semantics

Value. To learn how value classes affect copy operations, see Copying Objects (MATLAB).

## Examples

collapse all

Construct a naive Bayes classifier for Fisher's iris data. Also, specify prior probabilities after training.

X = meas;
Y = species;

X is a numeric matrix that contains four petal measurements for 150 irises. Y is a cell array of character vectors that contains the corresponding iris species.

Train a naive Bayes classifier.

Mdl = fitcnb(X,Y)
Mdl =
ClassificationNaiveBayes
ResponseName: 'Y'
CategoricalPredictors: []
ClassNames: {'setosa'  'versicolor'  'virginica'}
ScoreTransform: 'none'
NumObservations: 150
DistributionNames: {'normal'  'normal'  'normal'  'normal'}
DistributionParameters: {3x4 cell}

Properties, Methods

Mdl is a trained ClassificationNaiveBayes classifier, and some of its properties display in the Command Window. By default, the software treats each predictor as independent, and fits them using normal distributions.

To access the properties of Mdl, use dot notation.

Mdl.ClassNames
ans = 3x1 cell
{'setosa'    }
{'versicolor'}
{'virginica' }

Mdl.Prior
ans = 1×3

0.3333    0.3333    0.3333

Mdl.Prior contains the class prior probabilities, which are settable using the name-value pair argument 'Prior' in fitcnb. The order of the class prior probabilities corresponds to the order of the classes in Mdl.ClassNames. By default, the prior probabilities are the respective relative frequencies of the classes in the data.

You can also reset the prior probabilities after training. For example, set the prior probabilities to 0.5, 0.2, and 0.3 respectively.

Mdl.Prior = [0.5 0.2 0.3];

You can pass Mdl to e.g., predict to label new measurements, or crossval to cross validate the classifier.

expand all

## Algorithms

• If you specify 'DistributionNames','mn' when training Mdl using fitcnb, then the software fits a multinomial distribution using the bag-of-tokens model. The software stores the probability that token j appears in class k in the property DistributionParameters{k,j}. Using additive smoothing [2], the estimated probability is

where:

• which is the weighted number of occurrences of token j in class k.

• nk is the number of observations in class k.

• ${w}_{i}^{}$ is the weight for observation i. The software normalizes weights within a class such that they sum to the prior probability for that class.

• ${c}_{k}=\sum _{j=1}^{P}{c}_{j|k};$ which is the total weighted number of occurrences of all tokens in class k.

• If you specify 'DistributionNames','mvmn' when training Mdl using fitcnb, then:

1. For each predictor, the software collects a list of the unique levels, stores the sorted list in CategoricalLevels, and considers each level a bin. Each predictor/class combination is a separate, independent multinomial random variable.

2. For predictor j in class k, the software counts instances of each categorical level using the list stored in CategoricalLevels{j}.

3. The software stores the probability that predictor j, in class k, has level L in the property DistributionParameters{k,j}, for all levels in CategoricalLevels{j}. Using additive smoothing [2], the estimated probability is

where:

• which is the weighted number of observations for which predictor j equals L in class k.

• nk is the number of observations in class k.

• $I\left\{{x}_{ij}=L\right\}=1$ if xij = L, 0 otherwise.

• ${w}_{i}^{}$ is the weight for observation i. The software normalizes weights within a class such that they sum to the prior probability for that class.

• mj is the number of distinct levels in predictor j.

• mk is the weighted number of observations in class k.

## References

[1] Hastie, T., R. Tibshirani, and J. Friedman. The Elements of Statistical Learning, Second Edition. NY: Springer, 2008.

[2] Manning, C. D., P. Raghavan, and M. Schütze. Introduction to Information Retrieval, NY: Cambridge University Press, 2008.