updateMetricsAndFit

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

Since R2021a

Syntax

Mdl = updateMetricsAndFit(Mdl,X,Y)

Mdl = updateMetricsAndFit(Mdl,X,Y,'Weights',Weights)

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).

Mdl = updateMetricsAndFit(Mdl,X,Y) returns a naive Bayes classification model for incremental learning Mdl, which is the input naive Bayes classification model for incremental learning Mdl with the following modifications:

1. 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.

2. 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.

Mdl = updateMetricsAndFit(Mdl,X,Y,'Weights',Weights) also sets observation weights Weights.

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: {}

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}

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}

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:

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

2. 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.

3. 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(j) is the label of observation 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(j) is the label of observation 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:

  1. Appends any new labels in Y to the tail of Mdl.ClassNames.

  2. 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.

  3. 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_i is the weight for observation i. The software normalizes weights within a class such that they sum to the prior probability for that class.

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

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

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 k in the property DistributionParameters{k,j}. With additive smoothing [2], the estimated probability is

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

where:

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

• n_k is the number of observations in class k.

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

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

Estimated Probability for Multivariate Multinomial Distribution

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

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

2. For each class k, the software counts instances of each categorical level using the list stored in CategoricalLevels{j}.

3. The software stores the probability that predictor j in class k has level L in the property DistributionParameters{k,j}, for all levels in CategoricalLevels{j}. With additive smoothing [2], the estimated probability is

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

   where:

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

   • n_k is the number of observations in class k.

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

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

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

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

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:

    1. Store a buffer of length Mdl.MetricsWindowSize for each specified metric, and store a buffer of observation weights.

    2. 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.

    3. 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.