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

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`CategoricalPredictors` — Indices of categorical predictors
vector of positive integers

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

`CategoricalLevels` — Multivariate multinomial levels
cell vector of numeric vectors

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

`ClassNames` — Distinct class names
categorical array | character array | logical vector | numeric vector | cell array of character vectors

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.)

`Cost` — Misclassification cost
square matrix

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

`DistributionNames` — Predictor distributions
`'normal'` (default) | `'kernel'` | `'mn'` | `'mvmn'` | cell array of character vectors

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.

Value	Description
`'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

`DistributionParameters` — Distribution parameter estimates
cell array

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 j	Value
`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 Algorithms.
`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 Algorithms.
`normal`	A 2-by-1 numeric vector. The first element is the sample mean and the second element is the sample standard deviation.

Data Types: cell

`ExpandedPredictorNames` — Expanded predictor names
cell array of character vectors

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

`HyperparameterOptimizationResults` — Description of cross-validation optimization of hyperparameters
`BayesianOptimization` object | table

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` Field	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)

`Kernel` — Kernel smoother types
`'normal'` (default) | `'box'` | `'epanechnikov'` | `'triangle'` | cell array of character vectors

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.

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

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

Data Types: char | cell

`ModelParameters` — Parameter values used to train
object

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.

Property	Purpose
`DistributionNames`	Data distribution or distributions. This is the same value as the property `DistributionNames`.
`Kernel`	Kernel smoother type. This is the same as the property `Kernel`.
`Method`	Training method. For naive Bayes, the value is `'NaiveBayes'`.
`Support`	Kernel-smoothing density support. This is the same as the property `Support`.
`Type`	Learning type. For classification, the value is `'classification'`.
`Width`	Kernel 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.

`NumObservations` — Number of training observations
numeric scalar

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

`PredictorNames` — Predictor names
cell array of character vectors

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

`Prior` — Class prior probabilities
numeric vector

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

`ResponseName` — Response name
character vector

Response name, specified as a character vector.

Data Types: char

`ScoreTransform` — Classification score transformation function
`'doublelogit'` | `'invlogit'` | `'ismax'` | `'logit'` | `'none'` | function handle | ...

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';

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 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

`Support` — Kernel smoother density support
cell vector

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.

Value	Description
1-by-2 numeric row vector	For 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.

`W` — Observation weights
numeric vector

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

`Width` — Kernel smoother window width
numeric matrix

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.

`X` — Unstandardized predictor data
numeric matrix

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

`Y` — Observed class labels
categorical array | character array | logical vector | numeric vector | cell array of character vectors

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.

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

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Train a Naive Bayes Classifier

Open Live Script

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

Load Fisher's iris data.

load fisheriris
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.

More About

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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 this observation. The number of categories (bins) in this multinomial model is the number of distinct tokens, that is, the 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. Though 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:

Estimates the densities of the predictors within each class.
Models posterior probabilities according to Bayes rule. That is, for all k = 1,...,K,

$\hat{P} (Y = k | X_{1}, .., X_{P}) = \frac{π (Y = k) \prod_{j = 1}^{P} P (X_{j} | Y = k)}{\sum_{k = 1}^{K} π (Y = k) \prod_{j = 1}^{P} P (X_{j} | Y = k)},$
where:
- Y is the random variable corresponding to the class index of an observation.
- X₁,...,X_P are the random predictors of an observation.
- $π (Y = k)$ is the prior probability that a class index is k.
Classifies an observation by estimating the posterior probability for each class, and then assigns the observation to the class yielding the maximum posterior probability.

If the predictors compose a multinomial distribution, then the posterior probability $\hat{P} (Y = k | X_{1}, .., X_{P}) \propto π (Y = k) P_{m n} (X_{1}, ..., X_{P} | Y = k),$ where $P_{m n} (X_{1}, ..., X_{P} | Y = k)$ is the probability mass function of a multinomial distribution.

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

$P (token j | class k) = \frac{1 + c_{j | k}}{P + c_{k}},$
where:
- $c_{j | k} = n_{k} \frac{\sum_{i : y_{i} \in class k}^{} x_{i j} w_{i}^{}}{\sum_{i : y_{i} \in class k}^{} w_{i}};$ 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 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
  
  $P (predictor j = L | class k) = \frac{1 + m_{j | k} (L)}{m_{j} + m_{k}},$
  where:
  - $m_{j | k} (L) = n_{k} \frac{\sum_{i : y_{i} \in class k}^{} I {x_{i j} = L} w_{i}^{}}{\sum_{i : y_{i} \in class k}^{} w_{i}^{}};$ 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.
  - $I {x_{i j} = L} = 1$ if x_ij = 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.
  - 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, 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.

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 class labels input argument value (Y) cannot be a categorical array.
- Code generation does not support categorical predictors (logical, categorical, char, string, or cell). If you supply training data in a table, the predictors must be numeric (double or single). Also, you cannot use the 'CategoricalPredictors' name-value pair argument.
- The value of the 'DistributionNames' name-value pair argument cannot contain 'mn' or 'mvmn'.
- The value of the 'ClassNames' name-value pair argument cannot be a categorical array.
- The value of the 'ScoreTransform' name-value pair argument cannot be an anonymous function.

For more information, see Introduction to Code Generation.

Documentation

ClassificationNaiveBayes class

Description

Construction

Properties

CategoricalPredictors — Indices of categorical predictors vector of positive integers

CategoricalLevels — Multivariate multinomial levels cell vector of numeric vectors

ClassNames — Distinct class names categorical array | character array | logical vector | numeric vector | cell array of character vectors

Cost — Misclassification cost square matrix

DistributionNames — Predictor distributions 'normal' (default) | 'kernel' | 'mn' | 'mvmn' | cell array of character vectors

DistributionParameters — Distribution parameter estimates cell array

ExpandedPredictorNames — Expanded predictor names cell array of character vectors

HyperparameterOptimizationResults — Description of cross-validation optimization of hyperparameters BayesianOptimization object | table

Kernel — Kernel smoother types 'normal' (default) | 'box' | 'epanechnikov' | 'triangle' | cell array of character vectors

ModelParameters — Parameter values used to train object

NumObservations — Number of training observations numeric scalar

PredictorNames — Predictor names cell array of character vectors

Prior — Class prior probabilities numeric vector

ResponseName — Response name character vector

ScoreTransform — Classification score transformation function 'doublelogit' | 'invlogit' | 'ismax' | 'logit' | 'none' | function handle | ...

Support — Kernel smoother density support cell vector

W — Observation weights numeric vector

Width — Kernel smoother window width numeric matrix

X — Unstandardized predictor data numeric matrix

Y — Observed class labels categorical array | character array | logical vector | numeric vector | cell array of character vectors

Methods

Inherited Methods

Copy Semantics

Examples

Train a Naive Bayes Classifier

More About

Bag-of-Tokens Model

Naive Bayes

Algorithms

References

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™.

See Also

Topics

Statistics and Machine Learning Toolbox Documentation

Support

`CategoricalPredictors` — Indices of categorical predictors
vector of positive integers

`CategoricalLevels` — Multivariate multinomial levels
cell vector of numeric vectors

`ClassNames` — Distinct class names
categorical array | character array | logical vector | numeric vector | cell array of character vectors

`Cost` — Misclassification cost
square matrix

`DistributionNames` — Predictor distributions
`'normal'` (default) | `'kernel'` | `'mn'` | `'mvmn'` | cell array of character vectors

`DistributionParameters` — Distribution parameter estimates
cell array

`ExpandedPredictorNames` — Expanded predictor names
cell array of character vectors

`HyperparameterOptimizationResults` — Description of cross-validation optimization of hyperparameters
`BayesianOptimization` object | table

`Kernel` — Kernel smoother types
`'normal'` (default) | `'box'` | `'epanechnikov'` | `'triangle'` | cell array of character vectors

`ModelParameters` — Parameter values used to train
object

`NumObservations` — Number of training observations
numeric scalar

`PredictorNames` — Predictor names
cell array of character vectors

`Prior` — Class prior probabilities
numeric vector

`ResponseName` — Response name
character vector

`ScoreTransform` — Classification score transformation function
`'doublelogit'` | `'invlogit'` | `'ismax'` | `'logit'` | `'none'` | function handle | ...

`Support` — Kernel smoother density support
cell vector

`W` — Observation weights
numeric vector

`Width` — Kernel smoother window width
numeric matrix

`X` — Unstandardized predictor data
numeric matrix

`Y` — Observed class labels
categorical array | character array | logical vector | numeric vector | cell array of character vectors

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.