# updateMetricsAndFit

Update performance metrics in naive Bayes incremental learning classification model given new data and train model

## Description

Given streaming data, `updateMetricsAndFit`

first evaluates the performance of a configured naive Bayes classification model for incremental learning (`incrementalClassificationNaiveBayes`

object) by calling `updateMetrics`

on incoming data. Then `updateMetricsAndFit`

fits the model to that data by calling `fit`

. In other words, `updateMetricsAndFit`

performs *prequential evaluation* because it treats each incoming chunk of data as a test set, and tracks performance metrics measured cumulatively and over a specified window [1].

`updateMetricsAndFit`

provides a simple way to update model performance metrics and train the model on each chunk of data. Alternatively, you can perform the operations separately by calling `updateMetrics`

and then `fit`

, which allows for more flexibility (for example, you can decide whether you need to train the model based on its performance on a chunk of data).

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

= updateMetricsAndFit(`Mdl`

,`X`

,`Y`

)`Mdl`

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

with the following modifications:

`updateMetricsAndFit`

measures the model performance on the incoming predictor and response data,`X`

and`Y`

respectively. When the input model is*warm*(`Mdl.IsWarm`

is`true`

),`updateMetricsAndFit`

overwrites previously computed metrics, stored in the`Metrics`

property, with the new values. Otherwise,`updateMetricsAndFit`

stores`NaN`

values in`Metrics`

instead.`updateMetricsAndFit`

fits the modified model to the incoming data by updating the conditional posterior mean and standard deviation of each predictor variable, given the class, and stores the new estimates, among other configurations, in the output model`Mdl`

.

## Examples

### Update Performance Metrics and Train Model on Data Stream

Create a naive Bayes classification model for incremental learning by calling `incrementalClassificationNaiveBayes`

and specifying a maximum of 5 expected classes in the data.

`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 object. All its properties are read-only.

`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.

Implement incremental learning by performing the following actions at each iteration:

Simulate a data stream by processing a chunk of 50 observations.

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

Store the conditional mean of the first predictor in the first class ${\mu}_{11}$, the cumulative metrics, and the window metrics to see how they evolve during incremental learning.

% Preallocation numObsPerChunk = 50; nchunk = floor(n/numObsPerChunk); mc = array2table(zeros(nchunk,2),'VariableNames',["Cumulative" "Window"]); mu11 = 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 = updateMetricsAndFit(Mdl,X(idx,:),Y(idx)); mc{j,:} = Mdl.Metrics{"MinimalCost",:}; mu11(j + 1) = Mdl.DistributionParameters{1,1}(1); end

Now, `Mdl`

is an `incrementalClassificationNaiveBayes`

model object trained on all the data in the stream. During incremental learning and after the model is warmed up, `updateMetricsAndFit`

checks the performance of the model on the incoming observations, and then fits the model to those observations.

To see how the performance metrics and ${\mu}_{11}$ evolve during training, plot them on separate tiles.

t = tiledlayout(2,1); nexttile plot(mu11) ylabel('\mu_{11}') xlim([0 nchunk]) nexttile h = plot(mc.Variables); xlim([0 nchunk]) ylabel('Minimal Cost') xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,'r-.') legend(h,mc.Properties.VariableNames) xlabel(t,'Iteration')

The plot indicates that `updateMetricsAndFit`

performs the following actions:

Fit ${\mu}_{11}$ during all incremental learning iterations.

Compute the performance metrics after the metrics warm-up period only.

Compute the cumulative metrics during each iteration.

Compute the window metrics after processing 200 observations (4 iterations).

### 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 fit it to the data in one call. 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 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. Specify tracking the misclassification error rate during incremental learning.

IncrementalMdl = incrementalLearner(TTMdl,'Metrics',"classiferror")

IncrementalMdl = incrementalClassificationNaiveBayes IsWarm: 1 Metrics: [2x2 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`

.

**Track Performance Metrics and Fit Model**

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

function. At each iteration:

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

Call

`updateMetricsAndFit`

to update the cumulative and window performance metrics of the model given the incoming chunk of observations, and then fit the model to the data. Overwrite the previous incremental model with a new one. Specify the observation weights.Store the misclassification error rate.

% 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 = updateMetricsAndFit(IncrementalMdl,Xil(idx,:),Yil(idx),... 'Weights',Wil(idx)); mc{j,:} = IncrementalMdl.Metrics{"ClassificationError",:}; end

Now, `IncrementalMdl`

is an `incrementalClassificationNaiveBayes`

model object trained on all the data in the stream.

Create a trace plot of the misclassification error rate.

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

The cumulative loss initially jumps, but stabilizes around 0.05, whereas the window loss jumps throughout the training.

## Input Arguments

`Mdl`

— Naive Bayes classification model for incremental learning

`incrementalClassificationNaiveBayes`

model object

Naive Bayes classification model for incremental learning to measure the performance
of and then to fit to 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.

If `Mdl.IsWarm`

is `false`

,
`updateMetricsAndFit`

does not track the performance of the model. For more
details, see Performance Metrics.

`X`

— Chunk of predictor data

floating-point matrix

Chunk of predictor data to measure the model performance with and then fit the model
to, 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, `updateMetricsAndFit`

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`

,
`updateMetricsAndFit`

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 measure the model performance with and then fit the model to, 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`

.`updateMetricsAndFit`

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.
`updateMetricsAndFit`

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, `updateMetricsAndFit`

ignores the
observation. Consequently, `updateMetricsAndFit`

uses fewer than *n*
observations to compute the model performance and 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`

, `incrementalClassificationNaiveBayes`

.

If the model is not warm, `updateMetricsAndFit`

does not compute
performance metrics. As a result, the `Metrics`

property of
`Mdl`

remains completely composed of `NaN`

values.
If the model is warm, `updateMetricsAndFit`

computes the cumulative and
window performance metrics on the new data `X`

and
`Y`

, and overwrites the corresponding elements of
`Mdl.Metrics`

. For more details, see Performance Metrics.

In addition to updating distribution model parameters, `updateMetricsAndFit`

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`

,`updateMetricsAndFit`

: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`

),`updateMetricsAndFit`

sets misclassification costs of processed classes to`1`

and unprocessed classes to`NaN`

. For example, if`updateMetricsAndFit`

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).

## 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 [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*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 [2], 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*.

### Performance Metrics

`updateMetricsAndFit`

tracks model performance metrics, specified by the row labels of the table in`Mdl.Metrics`

, from new data only when the incremental model is*warm*(`IsWarm`

property is`true`

).If you create an incremental model by using

`incrementalLearner`

and`MetricsWarmupPeriod`

is 0 (default for`incrementalLearner`

), the model is warm at creation.Otherwise, an incremental model becomes warm after an incremental fitting function, such as

`updateMetricsAndFit`

, performs both of these actions:Fit the incremental model to

`Mdl.MetricsWarmupPeriod`

observations, which is the*metrics warm-up period*.Fit the incremental model to all expected classes (see the

`MaxNumClasses`

and`ClassNames`

arguments of`incrementalClassificationNaiveBayes`

).

`Mdl.Metrics`

stores two forms of each performance metric as variables (columns) of a table,`Cumulative`

and`Window`

, with individual metrics in rows. When the incremental model is warm,`updateMetricsAndFit`

updates the metrics at the following frequencies:`Cumulative`

— The function computes cumulative metrics since the start of model performance tracking. The function updates metrics every time you call the function and bases the calculation on the entire supplied data set.`Window`

— The function computes metrics based on all observations within a window determined by the`Mdl.MetricsWindowSize`

property.`Mdl.MetricsWindowSize`

also determines the frequency at which the software updates`Window`

metrics. For example, if`Mdl.MetricsWindowSize`

is 20, the function computes metrics based on the last 20 observations in the supplied data (`X((end – 20 + 1):end,:)`

and`Y((end – 20 + 1):end)`

).Incremental functions that track performance metrics within a window use the following process:

Store a buffer of length

`Mdl.MetricsWindowSize`

for each specified metric, and store a buffer of observation weights.Populate elements of the metrics buffer with the model performance based on batches of incoming observations, and store corresponding observation weights in the weights buffer.

When the buffer is full, overwrite

`Mdl.Metrics.Window`

with the weighted average performance in the metrics window. If the buffer is overfills when the function processes a batch of observations, the latest incoming`Mdl.MetricsWindowSize`

observations enter the buffer, and the earliest observations are removed from the buffer. For example, suppose`Mdl.MetricsWindowSize`

is 20, the metrics buffer has 10 values from a previously processed batch, and 15 values are incoming. To compose the length 20 window, the function uses the measurements from the 15 incoming observations and the latest 5 measurements from the previous batch.

The software omits an observation with a

`NaN`

score when computing the`Cumulative`

and`Window`

performance metric values.

### Observation Weights

For each conditional predictor distribution, `updateMetricsAndFit`

computes the weighted average and standard deviation.

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

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 `updateMetricsAndFit`

.

## References

[1] Bifet, Albert, Ricard Gavaldá, Geoffrey Holmes, and Bernhard Pfahringer. *Machine Learning for Data Streams with Practical Example in MOA*. Cambridge, MA: The MIT Press, 2007.

[2] 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.

## See Also

### Objects

### Functions

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