# fit

Train naive Bayes classification model for incremental learning

## Description

The `fit`

function fits a configured naive Bayes classification model for incremental learning (`incrementalClassificationNaiveBayes`

object) to streaming data. To additionally track performance metrics using the data as it arrives, use `updateMetricsAndFit`

instead.

To fit or cross-validate a naive Bayes classification model to an entire batch of data at once, see `fitcnb`

.

returns a naive Bayes classification model for incremental learning `Mdl`

= fit(`Mdl`

,`X`

,`Y`

)`Mdl`

, which represents the input naive Bayes classification model for
incremental learning `Mdl`

trained using the predictor and response
data, `X`

and `Y`

respectively. Specifically,
`fit`

updates the conditional posterior distribution of the
predictor variables given the data.

## Examples

### Incrementally Train Model with Little Prior Information

Fit an incremental naive Bayes learner when you know only the expected maximum number of classes in the data.

Create an incremental naive Bayes model. Specify that the maximum number of expected classes is 5.

`Mdl = incrementalClassificationNaiveBayes('MaxNumClasses',5)`

Mdl = incrementalClassificationNaiveBayes IsWarm: 0 Metrics: [1x2 table] ClassNames: [1x0 double] ScoreTransform: 'none' DistributionNames: 'normal' DistributionParameters: {} Properties, Methods

`Mdl`

is an `incrementalClassificationNaiveBayes`

model. All its properties are read-only. `Mdl`

can process at most 5 unique classes. By default, the prior class distribution `Mdl.Prior`

is empirical, which means the software updates the prior distribution as it encounters labels.

`Mdl`

must be fit to data before you can use it to perform any other operations.

Load the human activity data set. Randomly shuffle the data.

load humanactivity n = numel(actid); rng(1) % For reproducibility idx = randsample(n,n); X = feat(idx,:); Y = actid(idx);

For details on the data set, enter `Description`

at the command line.

Fit the incremental model to the training data, in chunks of 50 observations at a time, by using the `fit`

function. At each iteration:

Simulate a data stream by processing 50 observations.

Overwrite the previous incremental model with a new one fitted to the incoming observations.

Store the mean of the first predictor in the first class ${\mu}_{11}$ and the prior probability that the subject is moving (

`Y`

> 2) to see how these parameters evolve during incremental learning.

% Preallocation numObsPerChunk = 50; nchunk = floor(n/numObsPerChunk); mu11 = zeros(nchunk,1); priormoved = zeros(nchunk,1); % Incremental fitting for j = 1:nchunk ibegin = min(n,numObsPerChunk*(j-1) + 1); iend = min(n,numObsPerChunk*j); idx = ibegin:iend; Mdl = fit(Mdl,X(idx,:),Y(idx)); mu11(j) = Mdl.DistributionParameters{1,1}(1); priormoved(j) = sum(Mdl.Prior(Mdl.ClassNames > 2)); end

`Mdl`

is an `incrementalClassificationNaiveBayes`

model object trained on all the data in the stream.

To see how the parameters evolve during incremental learning, plot them on separate tiles.

t = tiledlayout(2,1); nexttile plot(mu11) ylabel('\mu_{11}') xlabel('Iteration') axis tight nexttile plot(priormoved) ylabel('\pi(Subject Is Moving)') xlabel(t,'Iteration') axis tight

`fit`

updates the posterior mean of the predictor distribution as it processes each chunk. Because the prior class distribution is empirical, $$\pi $$(subject is moving) changes as `fit`

processes each chunk.

### Specify All Class Names Before Fitting

Fit an incremental naive Bayes learner when you know all the class names in the data.

Consider training a device to predict whether a subject is sitting, standing, walking, running, or dancing based on biometric data measured on the subject. The class names map 1 through 5 to an activity. Also, suppose that the researchers plan to expose the device to each class uniformly.

Create an incremental naive Bayes model for multiclass learning. Specify the class names and the uniform prior class distribution.

classnames = 1:5; Mdl = incrementalClassificationNaiveBayes('ClassNames',classnames,'Prior','uniform')

Mdl = incrementalClassificationNaiveBayes IsWarm: 0 Metrics: [1x2 table] ClassNames: [1 2 3 4 5] ScoreTransform: 'none' DistributionNames: 'normal' DistributionParameters: {5x0 cell} Properties, Methods

`Mdl`

is an `incrementalClassificationNaiveBayes`

model object. All its properties are read-only. During training, observed labels must be in `Mdl.ClassNames`

.

`Mdl`

must be fit to data before you can use it to perform any other operations.

Load the human activity data set. Randomly shuffle the data.

load humanactivity n = numel(actid); rng(1); % For reproducibility idx = randsample(n,n); X = feat(idx,:); Y = actid(idx);

For details on the data set, enter `Description`

at the command line.

Fit the incremental model to the training data by using the `fit`

function. Simulate a data stream by processing chunks of 50 observations at a time. At each iteration:

Process 50 observations.

Overwrite the previous incremental model with a new one fitted to the incoming observations.

Store the mean of the first predictor in the first class ${\mu}_{11}$ and the prior probability that the subject is moving (

`Y`

> 2) to see how these parameters evolve during incremental learning.

% Preallocation numObsPerChunk = 50; nchunk = floor(n/numObsPerChunk); mu11 = zeros(nchunk,1); priormoved = zeros(nchunk,1); % Incremental fitting for j = 1:nchunk ibegin = min(n,numObsPerChunk*(j-1) + 1); iend = min(n,numObsPerChunk*j); idx = ibegin:iend; Mdl = fit(Mdl,X(idx,:),Y(idx)); mu11(j) = Mdl.DistributionParameters{1,1}(1); priormoved(j) = sum(Mdl.Prior(Mdl.ClassNames > 2)); end

`Mdl`

is an `incrementalClassificationNaiveBayes`

model object trained on all the data in the stream.

To see how the parameters evolve during incremental learning, plot them on separate tiles.

t = tiledlayout(2,1); nexttile plot(mu11) ylabel('\mu_{11}') xlabel('Iteration') axis tight nexttile plot(priormoved) ylabel('\pi(Subject Is Moving)') xlabel(t,'Iteration') axis tight

`fit`

updates the posterior mean of the predictor distribution as it processes each chunk. Because the prior class distribution is specified as uniform, $$\pi $$(subject is moving) = 0.6 and does not change as `fit`

processes each chunk.

### Specify Observation Weights

Train a naive Bayes classification model by using `fitcnb`

, convert it to an incremental learner, track its performance on streaming data, and then fit the model to the data. Specify observation weights.

**Load and Preprocess Data**

Load the human activity data set. Randomly shuffle the data.

load humanactivity rng(1); % For reproducibility n = numel(actid); idx = randsample(n,n); X = feat(idx,:); Y = actid(idx);

For details on the data set, enter `Description`

at the command line.

Suppose that the data from a stationary subject (`Y`

<= 2) has double the quality of the data from a moving subject. Create a weight variable that assigns a weight of 2 to observations from a stationary subject and 1 to a moving subject.

W = ones(n,1) + (Y <=2);

**Train Naive Bayes Classification Model**

Fit a naive Bayes classification model to a random sample of half the data.

```
idxtt = randsample([true false],n,true);
TTMdl = fitcnb(X(idxtt,:),Y(idxtt),'Weights',W(idxtt))
```

TTMdl = ClassificationNaiveBayes ResponseName: 'Y' CategoricalPredictors: [] ClassNames: [1 2 3 4 5] ScoreTransform: 'none' NumObservations: 12053 DistributionNames: {1x60 cell} DistributionParameters: {5x60 cell} Properties, Methods

`TTMdl`

is a `ClassificationNaiveBayes`

model object representing a traditionally trained naive Bayes classification model.

**Convert Trained Model**

Convert the traditionally trained model to a naive Bayes classification model for incremental learning.

IncrementalMdl = incrementalLearner(TTMdl)

IncrementalMdl = incrementalClassificationNaiveBayes IsWarm: 1 Metrics: [1x2 table] ClassNames: [1 2 3 4 5] ScoreTransform: 'none' DistributionNames: {1x60 cell} DistributionParameters: {5x60 cell} Properties, Methods

`IncrementalMdl`

is an `incrementalClassificationNaiveBayes`

model. Because class names are specified in `IncrementalMdl.ClassNames`

, labels encountered during incremental learning must be in `IncrementalMdl.ClassNames`

.

**Separately Track Performance Metrics and Fit Model**

Perform incremental learning on the rest of the data by using the `updateMetrics`

and `fit`

functions. At each iteration:

Simulate a data stream by processing 50 observations at a time.

Call

`updateMetrics`

to update the cumulative and window minimal cost of the model given the incoming chunk of observations. Overwrite the previous incremental model to update the losses in the`Metrics`

property. Note that the function does not fit the model to the chunk of data—the chunk is "new" data for the model. Specify the observation weights.Store the minimal cost.

Call

`fit`

to fit the model to the incoming chunk of observations. Overwrite the previous incremental model to update the model parameters. Specify the observation weights.

% Preallocation idxil = ~idxtt; nil = sum(idxil); numObsPerChunk = 50; nchunk = floor(nil/numObsPerChunk); mc = array2table(zeros(nchunk,2),'VariableNames',["Cumulative" "Window"]); Xil = X(idxil,:); Yil = Y(idxil); Wil = W(idxil); % Incremental fitting for j = 1:nchunk ibegin = min(nil,numObsPerChunk*(j-1) + 1); iend = min(nil,numObsPerChunk*j); idx = ibegin:iend; IncrementalMdl = updateMetrics(IncrementalMdl,Xil(idx,:),Yil(idx), ... 'Weights',Wil(idx)); mc{j,:} = IncrementalMdl.Metrics{"MinimalCost",:}; IncrementalMdl = fit(IncrementalMdl,Xil(idx,:),Yil(idx),'Weights',Wil(idx)); end

`IncrementalMdl`

is an `incrementalClassificationNaiveBayes`

model object trained on all the data in the stream.

Alternatively, you can use `updateMetricsAndFit`

to update performance metrics of the model given a new chunk of data, and then fit the model to the data.

Plot a trace plot of the performance metrics.

h = plot(mc.Variables); xlim([0 nchunk]) ylabel('Minimal Cost') legend(h,mc.Properties.VariableNames) xlabel('Iteration')

The cumulative loss gradually stabilizes, whereas the window loss jumps throughout the training.

### Perform Conditional Training

Incrementally train a naive Bayes classification model only when its performance degrades.

Load the human activity data set. Randomly shuffle the data.

load humanactivity n = numel(actid); rng(1) % For reproducibility idx = randsample(n,n); X = feat(idx,:); Y = actid(idx);

For details on the data set, enter `Description`

at the command line.

Configure a naive Bayes classification model for incremental learning so that the maximum number of expected classes is 5, the tracked performance metric includes the misclassification error rate, and the metrics window size is 1000. Fit the configured model to the first 1000 observations.

Mdl = incrementalClassificationNaiveBayes('MaxNumClasses',5,'MetricsWindowSize',1000, ... 'Metrics','classiferror'); initobs = 1000; Mdl = fit(Mdl,X(1:initobs,:),Y(1:initobs));

`Mdl`

is an `incrementalClassificationNaiveBayes`

model object.

Perform incremental learning, with conditional fitting, by following this procedure for each iteration:

Simulate a data stream by processing a chunk of 100 observations at a time.

Update the model performance on the incoming chunk of data.

Fit the model to the chunk of data only when the misclassification error rate is greater than 0.05.

When tracking performance and fitting, overwrite the previous incremental model.

Store the misclassification error rate and the mean of the first predictor in the second class ${\mu}_{21}$ to see how they evolve during training.

Track when

`fit`

trains the model.

% Preallocation numObsPerChunk = 100; nchunk = floor((n - initobs)/numObsPerChunk); mu21 = zeros(nchunk,1); ce = array2table(nan(nchunk,2),'VariableNames',["Cumulative" "Window"]); trained = false(nchunk,1); % Incremental fitting for j = 1:nchunk ibegin = min(n,numObsPerChunk*(j-1) + 1 + initobs); iend = min(n,numObsPerChunk*j + initobs); idx = ibegin:iend; Mdl = updateMetrics(Mdl,X(idx,:),Y(idx)); ce{j,:} = Mdl.Metrics{"ClassificationError",:}; if ce{j,2} > 0.05 Mdl = fit(Mdl,X(idx,:),Y(idx)); trained(j) = true; end mu21(j) = Mdl.DistributionParameters{2,1}(1); end

`Mdl`

is an `incrementalClassificationNaiveBayes`

model object trained on all the data in the stream.

To see how the model performance and ${\mu}_{21}$ evolve during training, plot them on separate tiles.

t = tiledlayout(2,1); nexttile plot(mu21) hold on plot(find(trained),mu21(trained),'r.') xlim([0 nchunk]) ylabel('\mu_{21}') legend('\mu_{21}','Training occurs','Location','best') hold off nexttile plot(ce.Variables) xlim([0 nchunk]) ylabel('Misclassification Error Rate') legend(ce.Properties.VariableNames,'Location','best') xlabel(t,'Iteration')

The trace plot of ${\mu}_{21}$ shows periods of constant values, during which the loss within the previous observation window is at most 0.05.

## Input Arguments

`Mdl`

— Naive Bayes classification model for incremental learning

`incrementalClassificationNaiveBayes`

model object

Naive Bayes classification model for incremental learning to fit to streaming data,
specified as an `incrementalClassificationNaiveBayes`

model object. You can create
`Mdl`

directly or by converting a supported, traditionally trained
machine learning model using the `incrementalLearner`

function. For more details, see the corresponding
reference page.

`X`

— Chunk of predictor data

floating-point matrix

Chunk of predictor data to which the model is fit, specified as an
*n*-by-`Mdl.NumPredictors`

floating-point
matrix.

The length of the observation labels `Y`

and the number of
observations in `X`

must be equal;
`Y(`

is the label of observation
* j*)

*j*(row) in

`X`

.**Note**

If `Mdl.NumPredictors`

= 0, `fit`

infers the number of predictors from `X`

, and sets the corresponding
property of the output model. Otherwise, if the number of predictor variables in the
streaming data changes from `Mdl.NumPredictors`

,
`fit`

issues an error.

**Data Types: **`single`

| `double`

`Y`

— Chunk of labels

categorical array | character array | string array | logical vector | floating-point vector | cell array of character vectors

Chunk of labels to which the model is fit, specified as a categorical, character, or string array, logical or floating-point vector, or cell array of character vectors.

The length of the observation labels `Y`

and the number of observations in
`X`

must be equal; `Y(`

is the label of observation * j*)

*j*(row) in

`X`

.`fit`

issues an error when one or both of these conditions are met:

`Y`

contains a new label and the maximum number of classes has already been reached (see the`MaxNumClasses`

and`ClassNames`

arguments of`incrementalClassificationNaiveBayes`

).The

`ClassNames`

property of the input model`Mdl`

is nonempty, and the data types of`Y`

and`Mdl.ClassNames`

are different.

**Data Types: **`char`

| `string`

| `cell`

| `categorical`

| `logical`

| `single`

| `double`

`Weights`

— Chunk of observation weights

floating-point vector of positive values

Chunk of observation weights, specified as a floating-point vector of positive values.
`fit`

weighs the observations in `X`

with the corresponding values in `Weights`

. The size of
`Weights`

must equal *n*, the number of
observations in `X`

.

By default, `Weights`

is `ones(`

.* n*,1)

For more details, including normalization schemes, see Observation Weights.

**Data Types: **`double`

| `single`

**Note**

If an observation (predictor or label) or weight contains at
least one missing (`NaN`

) value, `fit`

ignores the
observation. Consequently, `fit`

uses fewer than *n*
observations to create an updated model, where *n* is the number of
observations in `X`

.

## Output Arguments

`Mdl`

— Updated naive Bayes classification model for incremental learning

`incrementalClassificationNaiveBayes`

model object

Updated naive Bayes classification model for incremental learning, returned as an
incremental learning model object of the same data type as the input model
`Mdl`

, an `incrementalClassificationNaiveBayes`

object.

In addition to updating distribution model parameters, `fit`

performs the following actions when `Y`

contains expected, but unprocessed, classes:

If you do not specify all expected classes by using the

`ClassNames`

name-value argument when you create the input model`Mdl`

using`incrementalClassificationNaiveBayes`

,`fit`

:Appends any new labels in

`Y`

to the tail of`Mdl.ClassNames`

.Expands

`Mdl.Cost`

to a*c*-by-*c*matrix, where*c*is the number of classes in`Mdl.ClassNames`

. The resulting misclassification cost matrix is balanced.Expands

`Mdl.Prior`

to a length*c*vector of an updated empirical class distribution.

If you specify all expected classes when you create the input model

`Mdl`

or convert a traditionally trained naive Bayes model using`incrementalLearner`

, but you do not specify a misclassification cost matrix (`Mdl.Cost`

),`fit`

sets misclassification costs of processed classes to`1`

and unprocessed classes to`NaN`

. For example, if`fit`

processes the first two classes of a possible three classes,`Mdl.Cost`

is`[0 1 NaN; 1 0 NaN; 1 1 0]`

.

## 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).

## Tips

Unlike traditional training, incremental learning might not have a separate test (holdout) set. Therefore, to treat each incoming chunk of data as a test set, pass the incremental model and each incoming chunk to

`updateMetrics`

before training the model on the same data.

## 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 biased (maximum likelihood) 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*is the weight for observation_{i}*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$${s}_{j|k}={\left[\frac{{\displaystyle \sum _{\{i:{y}_{i}=k\}}{w}_{i}{\left({x}_{ij}-{\overline{x}}_{j|k}\right)}^{2}}}{{\displaystyle \sum _{\{i:{y}_{i}=k\}}{w}_{i}}}\right]}^{1/2}.$$

### 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

*in the property*

`k`

`DistributionParameters{``k`

,`j`

}

.
With additive smoothing [1], 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*is the number of observations in class_{k}*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: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 each class

*k*, the software counts instances of each categorical level using the list stored in`CategoricalLevels{`

.}`j`

The software stores the probability that predictor

in class`j`

has level`k`

*L*in the property`DistributionParameters{`

, for all levels in,`k`

}`j`

`CategoricalLevels{`

. With additive smoothing [1], the estimated probability is}`j`

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

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*is the number of observations in class_{k}*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*is the number of distinct levels in predictor_{j}*j*.*m*is the weighted number of observations in class_{k}*k*.

### Observation Weights

For each conditional predictor distribution, `fit`

computes the weighted average and standard deviation.

If the prior class probability distribution is known (in other words, the prior distribution is not empirical), `fit`

normalizes observation weights to sum to the prior class probabilities in the respective classes. This action implies that the default observation weights are the respective prior class probabilities.

If the prior class probability distribution is empirical, the software normalizes the specified observation weights to sum to 1 each time you call `fit`

.

## References

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

## Version History

**Introduced in R2021a**

### R2021b: Naive Bayes incremental fitting functions compute biased (maximum likelihood) standard deviations for conditionally normal predictor variables

Starting in R2021b, naive Bayes incremental fitting functions `fit`

and `updateMetricsAndFit`

compute
biased (maximum likelihood) estimates of the weighted standard deviations for conditionally
normal predictor variables during training. In other words, for each class
*k*, incremental fitting functions normalize the sum of square weighted
deviations of the conditionally normal predictor
*x _{j}* by the sum of the weights in class

*k*. Before R2021b, naive Bayes incremental fitting functions computed the unbiased standard deviation, like

`fitcnb`

. The currently returned weighted standard deviation estimates differ
from those computed before R2021b by a factor of $$1-\frac{{\displaystyle \sum _{\{i:{y}_{i}=k\}}{w}_{i}{}^{2}}}{{\left({\displaystyle \sum _{\{i:{y}_{i}=k\}}{w}_{i}}\right)}^{2}}.$$

The factor approaches 1 as the sample size increases.

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