ClassificationNaiveBayes

Naive Bayes classification for multiclass classification

Description

ClassificationNaiveBayes is a Naive Bayes classifier for multiclass learning. Trained ClassificationNaiveBayes classifiers store the training data, parameter values, data distribution, and prior probabilities. Use these classifiers to perform tasks such as estimating resubstitution predictions (see resubPredict) and predicting labels or posterior probabilities for new data (see predict).

Creation

Create a ClassificationNaiveBayes object by using fitcnb.

Properties

Predictor Properties

PredictorNames — Predictor names

cell array of character vectors

This property is read-only.

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

ExpandedPredictorNames — Expanded predictor names

cell array of character vectors

This property is read-only.

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

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

CategoricalPredictors — Categorical predictor indices

vector of positive integers | []

This property is read-only.

Categorical predictor indices, specified as a vector of positive integers. CategoricalPredictors contains index values indicating that the corresponding predictors are categorical. The index values are between 1 and p, where p is the number of predictors used to train the model. If none of the predictors are categorical, then this property is empty ([]).

Data Types: single | double

CategoricalLevels — Multivariate multinomial levels

cell array

This property is read-only.

Multivariate multinomial levels, specified as a cell array. The length of CategoricalLevels is equal to the number of predictors (size(X,2)).

The cells of CategoricalLevels correspond to predictors that you specify as 'mvmn' during training, that is, they have a multivariate multinomial distribution. 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 are removed from unique(X(:,j)).

X — Unstandardized predictors

numeric matrix

This property is read-only.

Unstandardized predictors used to train the naive Bayes classifier, specified as a numeric matrix. Each row of X corresponds to one observation, and each column corresponds to one variable. The software excludes observations containing at least one missing value, and removes corresponding elements from Y.

Predictor Distribution Properties

DistributionNames — Predictor distributions

'normal' (default) | 'kernel' | 'mn' | 'mvmn' | cell array of character vectors

This property is read-only.

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

Value	Description
'kernel'	Kernel smoothing density estimate
'mn'	Multinomial distribution. If you specify mn, then all features are components of a multinomial distribution. Therefore, you cannot include 'mn' as an element of a string array or a cell array of character vectors. For details, see Estimated Probability for Multinomial Distribution.
'mvmn'	Multivariate multinomial distribution. For details, see Estimated Probability for Multivariate Multinomial Distribution.
'normal'	Normal (Gaussian) distribution

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

Example: 'mn'

Example: {'kernel','normal','kernel'}

Data Types: char | string | cell

DistributionParameters — Distribution parameter estimates

cell array

This property is read-only.

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 the Distribution{k,j} is empty ([]).

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

Distribution of Predictor j	Value of Cell Array for Predictor j and Class k
kernel	A 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.
mn	A scalar representing the probability that token j appears in class k. For details, see Estimated Probability for Multinomial Distribution.
mvmn	A 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 Estimated Probability for Multivariate Multinomial Distribution.
normal	A 2-by-1 numeric vector. The first element is the sample mean and the second element is the sample standard deviation. For more details, see Normal Distribution Estimators

Kernel — Kernel smoother type

'normal' (default) | 'box' | cell array | ...

This property is read-only.

Kernel smoother type, specified as the name of a kernel or a cell array of kernel names. The length of Kernel is 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. If a cell is empty ([]), then fitcnb did not fit a kernel distribution to the corresponding predictor.

This table describes the supported kernel smoother types. I{u} denotes the indicator function.

Value	Kernel	Formula
'box'	Box (uniform)	$$f(x)=0.5I\left\{\left|x\right|\le 1\right\}$$
'epanechnikov'	Epanechnikov	$$f(x)=0.75\left(1-x^2\right)I\left\{\left|x\right|\le 1\right\}$$
'normal'	Gaussian	$$f(x)=\frac{1}{\sqrt{2\pi }}\exp \left(-0.5x^2\right)$$
'triangle'	Triangular	$$f(x)=\left(1-\left|x\right|\right)I\left\{\left|x\right|\le 1\right\}$$

Example: 'box'

Example: {'epanechnikov','normal'}

Data Types: char | string | cell

Mu — Predictor means

numeric vector | []

Since R2023b

This property is read-only.

Predictor means, specified as a numeric vector. If you specify Standardize as 1 or true when you train the naive Bayes classifier using fitcnb, then the length of the Mu vector is equal to the number of predictors. The vector contains 0 values for predictors with nonkernel distributions, such as categorical predictors (see DistributionNames).

If you set Standardize to 0 or false when you train the naive Bayes classifier using fitcnb, then the Mu value is an empty vector ([]).

Data Types: double

Sigma — Predictor standard deviations

numeric vector | []

Since R2023b

This property is read-only.

Predictor standard deviations, specified as a numeric vector. If you specify Standardize as 1 or true when you train the naive Bayes classifier using fitcnb, then the length of the Sigma vector is equal to the number of predictors. The vector contains 1 values for predictors with nonkernel distributions, such as categorical predictors (see DistributionNames).

If you set Standardize to 0 or false when you train the naive Bayes classifier using fitcnb, then the Sigma value is an empty vector ([]).

Data Types: double

Support — Kernel smoother density support

cell array

This property is read-only.

Kernel smoother density support, specified as a cell array. The length of Support is equal to the number of predictors (size(X,2)). The cells represent the regions to which fitcnb applies the kernel density. If a cell is empty ([]), then fitcnb did not fit a kernel distribution to the corresponding predictor.

This table describes the supported options.

Value	Description
1-by-2 numeric row vector	The density support applies to the specified bounds, for example [L,U], where L and U are the finite lower and upper bounds, respectively.
'positive'	The density support applies to all positive real values.
'unbounded'	The density support applies to all real values.

Width — Kernel smoother window width

numeric matrix

This property is read-only.

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 fitcnb did not fit predictor j using a kernel density.

Response Properties

ClassNames — Unique class names

categorical array | character array | logical vector | numeric vector | cell array of character vectors

This property is read-only.

Unique class names used in the training model, specified as a categorical or character array, logical or numeric vector, or cell array of character vectors.

ClassNames has 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 | string | logical | double | cell

ResponseName — Response variable name

character vector

This property is read-only.

Response variable name, specified as a character vector.

Data Types: char | string

Y — Class labels

categorical array | character array | logical vector | numeric vector | cell array of character vectors

This property is read-only.

Class labels used to train the naive Bayes classifier, specified as a categorical or character array, logical or numeric vector, or cell array of character vectors. Each row of Y represents the observed classification of the corresponding row of X.

Y has the same data type as the data in Y used for training the model. (The software treats string arrays as cell arrays of character vectors.)

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

Training Properties

ModelParameters — Parameter values used to train model

object

This property is read-only.

Parameter values used to train the ClassificationNaiveBayes model, specified as an object. ModelParameters contains parameter values such as the name-value pair argument values used to train the naive Bayes classifier.

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

NumObservations — Number of training observations

numeric scalar

This property is read-only.

Number of training observations in the training data stored in X and Y, specified as a numeric scalar.

Prior — Prior probabilities

numeric vector

Prior probabilities, specified as a numeric vector. The order of the elements in Prior corresponds to the elements of Mdl.ClassNames.

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

The value of Prior does not affect the best-fitting model. Therefore, you can reset Prior after training Mdl using dot notation.

Example: Mdl.Prior = [0.2 0.8]

Data Types: double | single

W — Observation weights

vector of nonnegative values

This property is read-only.

Observation weights, specified as a vector of nonnegative values with the same number of rows as Y. Each entry in W specifies the relative importance of the corresponding observation in Y. fitcnb normalizes the value you set for the 'Weights' name-value pair argument, so that the weights within a particular class sum to the prior probability for that class.

Classifier Properties

Cost — Misclassification cost

square matrix

Misclassification cost, specified as a numeric square matrix, where Cost(i,j) is the cost of classifying a point into class j if its true class is i. The rows correspond to the true class and the columns correspond to the predicted class. The order of the rows and columns of Cost corresponds to the order of the classes in ClassNames.

The misclassification cost matrix must have zeros on the diagonal.

The value of Cost does not influence training. You can reset Cost after training Mdl using dot notation.

Example: Mdl.Cost = [0 0.5 ; 1 0]

Data Types: double | single

HyperparameterOptimizationResults — Cross-validation optimization of hyperparameters

BayesianOptimization object | table

This property is read-only.

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.

Value of Optimizer Option	Value 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)

ScoreTransform — Classification score transformation

'none' (default) | 'doublelogit' | 'invlogit' | 'ismax' | 'logit' | function handle | ...

Classification score transformation, specified as a character vector or function handle. This table summarizes the available character vectors.

Value	Description
"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 + e–x)
"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 + e–x) – 1

For a MATLAB^® function or a function you define, use its function handle for the score transformation. The function handle must accept a matrix (the original scores) and return a matrix of the same size (the transformed scores).

Example: Mdl.ScoreTransform = 'logit'

Data Types: char | string | function handle

Object Functions

compact	Reduce size of machine learning model
compareHoldout	Compare accuracies of two classification models using new data
crossval	Cross-validate machine learning model
edge	Classification edge for naive Bayes classifier
incrementalLearner	Convert naive Bayes classification model to incremental learner
lime	Local interpretable model-agnostic explanations (LIME)
logp	Log unconditional probability density for naive Bayes classifier
loss	Classification loss for naive Bayes classifier
margin	Classification margins for naive Bayes classifier
partialDependence	Compute partial dependence
plotPartialDependence	Create partial dependence plot (PDP) and individual conditional expectation (ICE) plots
predict	Classify observations using naive Bayes classifier
resubEdge	Resubstitution classification edge
resubLoss	Resubstitution classification loss
resubMargin	Resubstitution classification margin
resubPredict	Classify training data using trained classifier
shapley	Shapley values
testckfold	Compare accuracies of two classification models by repeated cross-validation

Examples

Train Naive Bayes Classifier

Create a naive Bayes classifier for Fisher's iris data set. Then, specify prior probabilities after training the classifier.

Load the fisheriris data set. Create X as a numeric matrix that contains four 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;

Train a naive Bayes classifier using the predictors X and class labels Y. fitcnb assumes each predictor is independent and fits each predictor using a normal distribution by default.

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}

Mdl is a trained ClassificationNaiveBayes classifier. Some of the Mdl properties appear in the Command Window.

Display the properties of Mdl using dot notation. For example, display the class names and prior probabilities.

Mdl.ClassNames

ans = 3x1 cell
    {'setosa'    }
    {'versicolor'}
    {'virginica' }

Mdl.Prior

ans = 1×3

    0.3333    0.3333    0.3333

The order of the class prior probabilities in Mdl.Prior 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. Alternatively, you can set the prior probabilities when calling fitcnb by using the 'Prior' name-value pair argument.

Set the prior probabilities after training the classifier by using dot notation. 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 now use this trained classifier to perform additional tasks. For example, you can label new measurements using predict or cross-validate the classifier using crossval.

Train and Cross-Validate Naive Bayes Classifier

Train and cross-validate a naive Bayes classifier. fitcnb implements 10-fold cross-validation by default. Then, estimate the cross-validated classification error.

Load the ionosphere data set. Remove the first two predictors for stability.

load ionosphere
X = X(:,3:end);
rng('default')  % for reproducibility

Train and cross-validate a naive Bayes classifier using the predictors X and class labels Y. A recommended practice is to specify the class names. fitcnb assumes that each predictor is conditionally and normally distributed.

CVMdl = fitcnb(X,Y,'ClassNames',{'b','g'},'CrossVal','on')

CVMdl = 
  ClassificationPartitionedModel
    CrossValidatedModel: 'NaiveBayes'
         PredictorNames: {'x1'  'x2'  'x3'  'x4'  'x5'  'x6'  'x7'  'x8'  'x9'  'x10'  'x11'  'x12'  'x13'  'x14'  'x15'  'x16'  'x17'  'x18'  'x19'  'x20'  'x21'  'x22'  'x23'  'x24'  'x25'  'x26'  'x27'  'x28'  'x29'  'x30'  'x31'  'x32'}
           ResponseName: 'Y'
        NumObservations: 351
                  KFold: 10
              Partition: [1x1 cvpartition]
             ClassNames: {'b'  'g'}
         ScoreTransform: 'none'

CVMdl is a ClassificationPartitionedModel cross-validated, naive Bayes classifier. Alternatively, you can cross-validate a trained ClassificationNaiveBayes model by passing it to crossval.

Display the first training fold of CVMdl using dot notation.

CVMdl.Trained{1}

ans = 
  CompactClassificationNaiveBayes
              ResponseName: 'Y'
     CategoricalPredictors: []
                ClassNames: {'b'  'g'}
            ScoreTransform: 'none'
         DistributionNames: {1x32 cell}
    DistributionParameters: {2x32 cell}

Each fold is a CompactClassificationNaiveBayes model trained on 90% of the data.

Full and compact naive Bayes models are not used for predicting on new data. Instead, use them to estimate the generalization error by passing CVMdl to kfoldLoss.

genError = kfoldLoss(CVMdl)

genError = 
0.1852

On average, the generalization error is approximately 19%.

You can specify a different conditional distribution for the predictors, or tune the conditional distribution parameters to reduce the generalization error.

More About

Bag-of-Tokens Model

In the bag-of-tokens model, the value of predictor j is the nonnegative number of occurrences of token j in the observation. The number of categories (bins) in the multinomial model is the number of distinct tokens (number of predictors).

Naive Bayes

Naive Bayes is a classification algorithm that applies density estimation to the data.

The algorithm leverages Bayes theorem, and (naively) assumes that the predictors are conditionally independent, given the class. Although the assumption is usually violated in practice, naive Bayes classifiers tend to yield posterior distributions that are robust to biased class density estimates, particularly where the posterior is 0.5 (the decision boundary) [1].

Naive Bayes classifiers assign observations to the most probable class (in other words, the maximum a posteriori decision rule). Explicitly, the algorithm takes these steps:

1. Estimate the densities of the predictors within each class.

2. Model posterior probabilities according to Bayes rule. That is, for all k = 1,...,K,

   $$\widehat{P}\left(Y=k|X_1,\mathrm{..},X_P\right)=\frac{\pi \left(Y=k\right){\displaystyle \prod _{j=1}^{P}P}\left(X_j|Y=k\right)}{{\displaystyle \sum }_{k=1}^K\pi \left(Y=k\right){\displaystyle \prod _{j=1}^{P}P}\left(X_j|Y=k\right)},$$

   where:

   • Y is the random variable corresponding to the class index of an observation.

   • X₁,...,X_P are the random predictors of an observation.

   • $$\pi \left(Y=k\right)$$ is the prior probability that a class index is k.

3. Classify an observation by estimating the posterior probability for each class, and then assign the observation to the class yielding the maximum posterior probability.

If the predictors compose a multinomial distribution, then the posterior probability$$\widehat{P}\left(Y=k|X_1,\mathrm{..},X_P\right)\propto \pi \left(Y=k\right)P_{mn}\left(X_1,\mathrm{...},X_P|Y=k\right),$$ where $$P_{mn}\left(X_1,\mathrm{...},X_P|Y=k\right)$$ is the probability mass function of a multinomial distribution.

Algorithms

Normal Distribution Estimators

If predictor variable j has a conditional normal distribution (see the DistributionNames property), the software fits the distribution to the data by computing the class-specific weighted mean and the unbiased estimate of the weighted standard deviation. For each class k:

• The weighted mean of predictor j is

  $${\overline{x}}_{j|k}=\frac{{\displaystyle \sum _{\{i:y_i=k\}}{w}_i{x}_{ij}}}{{\displaystyle \sum _{\{i:y_i=k\}}{w}_i}},$$

  where 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.

• The unbiased estimator of the weighted standard deviation of predictor j is

  $$s_{j|k}={\left[\frac{{\displaystyle \sum _{\{i:y_i=k\}}{w}_i{\left(x_{ij}-{\overline{x}}_{j|k}\right)}^2}}{z_{1|k}-\frac{z_{2|k}}{z_{1|k}}}\right]}^{1/2},$$

  where z_1|k is the sum of the weights within class k and z_2|k is the sum of the squared weights within class k.

Estimated Probability for Multinomial Distribution

If all predictor variables compose a conditional multinomial distribution (see the DistributionNames property), the software fits the 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}. With additive smoothing [2], the estimated probability is

$$P(\text{token }j|\text{class }k)=\frac{1+c_{j|k}}{P+c_k},$$

where:

• $$c_{j|k}=n_k\frac{{\displaystyle \sum _{\{i:y_i=k\}}^{}{x}_{ij}}{w}_i^{}}{{\displaystyle \sum _{\{i:y_i=k\}}^{}{w}_i}},$$ which is the weighted number of occurrences of token j in class k.

• n_k is the number of observations in class k.

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

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

Estimated Probability for Multivariate Multinomial Distribution

If predictor variable j has a conditional multivariate multinomial distribution (see the DistributionNames property), the software follows this procedure:

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

2. For each 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}. With additive smoothing [2], the estimated probability is

   $$P\left(\text{predictor }j=L|\text{class }k\right)=\frac{1+m_{j|k}(L)}{m_j+m_k},$$

   where:

   • $$m_{j|k}(L)=n_k\frac{{\displaystyle \sum _{\{i:y_i=k\}}^{}I\{x_{ij}=L\}{w}_i^{}}}{{\displaystyle \sum _{\{i:y_i=k\}}^{}{w}_i^{}}},$$ which is the weighted number of observations for which predictor j equals L in class k.

   • n_k is the number of observations in class k.

   • $$I\left\{x_{ij}=L\right\}=1$$ if x_ij = L, and 0 otherwise.

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

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

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

References

[1] Hastie, Trevor, Robert Tibshirani, and Jerome Friedman. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. 2nd ed. Springer Series in Statistics. New York, NY: Springer, 2009. https://doi.org/10.1007/978-0-387-84858-7.

[2] Manning, Christopher D., Prabhakar Raghavan, and Hinrich Schütze. Introduction to Information Retrieval, NY: Cambridge University Press, 2008.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• The predict function supports code generation.

• When you train a naive Bayes model by using fitcnb, the following restrictions apply.

  • The value of the 'DistributionNames' name-value pair argument cannot contain 'mn'.

  • The value of the 'ScoreTransform' name-value pair argument cannot be an anonymous function.

For more information, see Introduction to Code Generation.

Version History

Introduced in R2014b

R2023b: Naive Bayes models support standardization of kernel-distributed predictors

fitcnb supports the standardization of predictors with kernel distributions. That is, you can specify the Standardize name-value argument as true when the DistributionNames name-value argument includes at least one "kernel" distribution. Naive Bayes models include Mu and Sigma properties that contain the means and standard deviations, respectively, used to standardize the predictors before training. The properties are empty when fitcnb does not perform any standardization.