Naive Bayes classification for multiclass classification
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).
Create a ClassificationNaiveBayes object by using fitcnb.
PredictorNames — Predictor namesThis 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 namesThis 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 integersThis property is read-only.
Categorical predictor indices, specified as a vector of
positive integers. CategoricalPredictors contains index values
corresponding to the columns of 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 levelsThis 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 predictorsThis 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.
DistributionNames — Predictor distributions'normal' (default) | 'kernel' | 'mn' | 'mvmn' | cell array of character vectorsThis 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 estimatesThis 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{
is empty (k,j}[]).
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. |
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) |
|
'epanechnikov' | Epanechnikov |
|
'normal' | Gaussian |
|
'triangle' | Triangular |
|
Example: 'box'
Example: {'epanechnikov','normal'}
Data Types: char | string | cell
Support — Kernel smoother density supportThis 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 widthThis 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(
is the kernel smoother window width for the kernel smoothing density of
predictor k,j)j within class
k. NaNs in column
j indicate that fitcnb did not fit
predictor j using a kernel density.
ClassNames — Unique class namesThis 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 nameThis property is read-only.
Response variable name, specified as a character vector.
Data Types: char | string
Y — Class labelsThis 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
ModelParameters — Parameter values used to train modelThis property is read-only.
Parameter values used to train the ClassificationNaiveBayes
model, specified as a structure array. ModelParameters contains
parameter values such as the name-value pair argument values used to train the naive
Bayes classifier.
Access the fields of ModelParameters by using dot notation. For
example, access the kernel support using
Mdl.ModelParameters.Support.
NumObservations — Number of training observationsThis property is read-only.
Number of training observations in the training data stored in
X and Y, specified as a numeric
scalar.
Prior — Prior probabilitiesPrior 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 weightsThis 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.
Cost — Misclassification costMisclassification 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 hyperparametersBayesianOptimization object | tableThis 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 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) |
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
compact | Reduce size of naive Bayes classifier |
crossval | Cross-validate naive Bayes classifier |
edge | Classification edge for naive Bayes classifier |
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 for naive Bayes classifier |
resubLoss | Resubstitution classification loss for naive Bayes classifier |
resubMargin | Resubstitution classification margins for naive Bayes classifier |
resubPredict | Classify observations using 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 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;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}
Properties, Methods
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 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: {1x32 cell}
ResponseName: 'Y'
NumObservations: 351
KFold: 10
Partition: [1x1 cvpartition]
ClassNames: {'b' 'g'}
ScoreTransform: 'none'
Properties, Methods
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}
Properties, Methods
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.
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 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:
Estimate the densities of the predictors within each class.
Model posterior probabilities according to Bayes rule. That is, for all k = 1,...,K,
where:
Y is the random variable corresponding to the class index of an observation.
X1,...,XP are the random predictors of an observation.
is the prior probability that a class index is k.
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 where is the probability mass function of a multinomial distribution.
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{.
With additive smoothing [2], the estimated
probability isk,j}
where:
which is the weighted number of occurrences of token j in class k.
nk is the number of observations in class k.
is the weight for observation i. The software normalizes weights within a class so that they sum to the prior probability for that class.
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 the software takes these steps:
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
combination of predictor and class is a separate, independent multinomial random
variable.
For predictor j in class k, the
software counts instances of each categorical level using the list stored in
CategoricalLevels{.j}
The software stores the probability that predictor
j in class k has level
L in the property
DistributionParameters{,
for all levels in
k,j}CategoricalLevels{. With
additive smoothing [2], the
estimated probability isj}
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.
if xij = L, and 0 otherwise.
is the weight for observation i. The software normalizes weights within a class so 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.
[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.
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.
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