# incrementalClassificationLinear

Binary classification linear model for incremental learning

## Description

incrementalClassificationLinear creates an incrementalClassificationLinear model object, which represents a binary classification linear model for incremental learning. Supported learners include support vector machine (SVM) and logistic regression.

Unlike other Statistics and Machine Learning Toolbox™ model objects, incrementalClassificationLinear can be called directly. Also, you can specify learning options, such as performance metrics configurations, parameter values, and the objective solver, before fitting the model to data. After you create an incrementalClassificationLinear object, it is prepared for incremental learning.

incrementalClassificationLinear is best suited for incremental learning. For a traditional approach to training an SVM or linear model for binary classification (such as creating a model by fitting it to data, performing cross-validation, tuning hyperparameters, and so on), see fitcsvm or fitclinear. For multiclass incremental learning, see incrementalClassificationECOC and incrementalClassificationNaiveBayes.

## Creation

You can create an incrementalClassificationLinear model object in several ways:

• Call the function directly — Configure incremental learning options, or specify initial values for linear model parameters and hyperparameters, by calling incrementalClassificationLinear directly. This approach is best when you do not have data yet or you want to start incremental learning immediately.

• Convert a traditionally trained model — To initialize a binary classification linear model for incremental learning using the model coefficients and hyperparameters of a trained model object, you can convert the traditionally trained model to an incrementalClassificationLinear model object by passing it to the incrementalLearner function. This table contains links to the appropriate reference pages.

• Call an incremental learning functionfit, updateMetrics, and updateMetricsAndFit accept a configured incrementalClassificationLinear model object and data as input, and return an incrementalClassificationLinear model object updated with information learned from the input model and data.

### Syntax

Mdl = incrementalClassificationLinear()
Mdl = incrementalClassificationLinear(Name,Value)

### Description

example

Mdl = incrementalClassificationLinear() returns a default incremental learning model object for binary linear classification, Mdl. Properties of a default model contain placeholders for unknown model parameters. You must train a default model before you can track its performance or generate predictions from it.

example

Mdl = incrementalClassificationLinear(Name,Value) sets properties and additional options using name-value arguments. Enclose each name in quotes. For example, incrementalClassificationLinear('Beta',[0.1 0.3],'Bias',1,'MetricsWarmupPeriod',100) sets the vector of linear model coefficients β to [0.1 0.3], the bias β0 to 1, and the metrics warm-up period to 100.

### Input Arguments

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Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: 'Standardize',true standardizes the predictor data using the predictor means and standard deviations estimated during the estimation period.

Model performance metrics to track during incremental learning, specified as a built-in loss function name, string vector of names, function handle (@metricName), structure array of function handles, or cell vector of names, function handles, or structure arrays.

When Mdl is warm (see IsWarm), updateMetrics and updateMetricsAndFit track performance metrics in the Metrics property of Mdl.

The following table lists the built-in loss function names. You can specify more than one by using a string vector.

NameDescription
"binodeviance"Binomial deviance
"classiferror"Classification error
"exponential"Exponential
"hinge"Hinge
"logit"Logistic
"quadratic"Quadratic

For more details on the built-in loss functions, see loss.

Example: 'Metrics',["classiferror" "hinge"]

To specify a custom function that returns a performance metric, use function handle notation. The function must have this form:

metric = customMetric(C,S)

• The output argument metric is an n-by-1 numeric vector, where each element is the loss of the corresponding observation in the data processed by the incremental learning functions during a learning cycle.

• You specify the function name (customMetric).

• C is an n-by-2 logical matrix with rows indicating the class to which the corresponding observation belongs. The column order corresponds to the class order in the ClassNames property. Create C by setting C(p,q) = 1, if observation p is in class q, for each observation in the specified data. Set the other element in row p to 0.

• S is an n-by-2 numeric matrix of predicted classification scores. S is similar to the score output of predict, where rows correspond to observations in the data and the column order corresponds to the class order in the ClassNames property. S(p,q) is the classification score of observation p being classified in class q.

To specify multiple custom metrics and assign a custom name to each, use a structure array. To specify a combination of built-in and custom metrics, use a cell vector.

Example: 'Metrics',struct('Metric1',@customMetric1,'Metric2',@customMetric2)

Example: 'Metrics',{@customMetric1 @customMetric2 'logit' struct('Metric3',@customMetric3)}

updateMetrics and updateMetricsAndFit store specified metrics in a table in the Metrics property. The data type of Metrics determines the row names of the table.

'Metrics' Value Data TypeDescription of Metrics Property Row NameExample
String or character vectorName of corresponding built-in metricRow name for "classiferror" is "ClassificationError"
Structure arrayField nameRow name for struct('Metric1',@customMetric1) is "Metric1"
Function handle to function stored in a program fileName of functionRow name for @customMetric is "customMetric"
Anonymous functionCustomMetric_j, where j is metric j in MetricsRow name for @(C,S)customMetric(C,S)... is CustomMetric_1

For more details on performance metrics options, see Performance Metrics.

Data Types: char | string | struct | cell | function_handle

Flag to standardize the predictor data, specified as a value in this table.

ValueDescription
'auto'incrementalClassificationLinear determines whether the predictor variables need to be standardized. See Standardize Data.
trueThe software standardizes the predictor data. For more details, see Standardize Data.
falseThe software does not standardize the predictor data.

Example: 'Standardize',true

Data Types: logical | char | string

Flag for shuffling the observations at each iteration, specified as a value in this table.

ValueDescription
trueThe software shuffles the observations in an incoming chunk of data before the fit function fits the model. This action reduces bias induced by the sampling scheme.
falseThe software processes the data in the order received.

This option is valid only when Solver is 'scale-invariant'. When Solver is 'sgd' or 'asgd', the software always shuffles the observations in an incoming chunk of data before processing the data.

Example: 'Shuffle',false

Data Types: logical

## Properties

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You can set most properties by using name-value argument syntax only when you call incrementalClassificationLinear directly. You can set some properties when you call incrementalLearner to convert a traditionally trained model. You cannot set the properties FittedLoss, NumTrainingObservations, Mu, Sigma, SolverOptions, and IsWarm.

### Classification Model Parameters

Linear model coefficients β, specified as a NumPredictors-by-1 numeric vector.

Incremental fitting functions estimate Beta during training. The default initial Beta value depends on how you create the model:

• If you convert a traditionally trained model to create Mdl, the initial value is specified by the corresponding property of the traditionally trained model.

• Otherwise, the initial value is zeros(NumPredictors,1).

Data Types: single | double

Model intercept β0, or bias term, specified as a numeric scalar.

Incremental fitting functions estimate Bias during training. The default initial Bias value depends on how you create the model:

• If you convert a traditionally trained model to create Mdl, the initial value is specified by the corresponding property of the traditionally trained model.

• Otherwise, the initial value is 0.

Data Types: single | double

Unique class labels used in training the model, specified as a categorical, character, or string array, logical or numeric vector, or cell array of character vectors. ClassNames and the response data must have the same data type. (The software treats string arrays as cell arrays of character vectors.)

The default ClassNames value depends on how you create the model:

• If you convert a traditionally trained model to create Mdl, ClassNames is specified by the corresponding property of the traditionally trained model.

• Otherwise, incremental fitting functions infer ClassNames during training.

Data Types: single | double | logical | char | string | cell | categorical

Loss function used to fit the linear model, specified as 'hinge' or 'logit'.

ValueAlgorithmLoss FunctionLearner Value
'hinge'Support vector machineHinge: $\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,1-yf\left(x\right)\right]$'svm'
'logit'Logistic regressionDeviance (logistic): $\ell \left[y,f\left(x\right)\right]=\mathrm{log}\left\{1+\mathrm{exp}\left[-yf\left(x\right)\right]\right\}$'logistic'

Linear classification model type, specified as 'svm' or 'logistic'. incrementalClassificationLinear stores the Learner value as a character vector.

In the following table, $f\left(x\right)=x\beta +b.$

• β is a vector of p coefficients.

• x is an observation from p predictor variables.

• b is the scalar bias.

ValueAlgorithmLoss FunctionFittedLoss Value
'svm'Support vector machineHinge: $\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,1-yf\left(x\right)\right]$'hinge'
'logistic'Logistic regressionDeviance (logistic): $\ell \left[y,f\left(x\right)\right]=\mathrm{log}\left\{1+\mathrm{exp}\left[-yf\left(x\right)\right]\right\}$'logit'

The default Learner value depends on how you create the model:

Data Types: char | string

Number of predictor variables, specified as a nonnegative numeric scalar.

The default NumPredictors value depends on how you create the model:

• If you convert a traditionally trained model to create Mdl, NumPredictors is specified by the corresponding property of the traditionally trained model.

• If you create Mdl by calling incrementalClassificationLinear directly, you can specify NumPredictors by using name-value argument syntax. If you do not specify the value, then the default value is 0, and incremental fitting functions infer NumPredictors from the predictor data during training.

Data Types: double

Number of observations fit to the incremental model Mdl, specified as a nonnegative numeric scalar. NumTrainingObservations increases when you pass Mdl and training data to fit or updateMetricsAndFit.

Note

If you convert a traditionally trained model to create Mdl, incrementalClassificationLinear does not add the number of observations fit to the traditionally trained model to NumTrainingObservations.

Data Types: double

Prior class probabilities, specified as 'empirical', 'uniform', or a numeric vector. incrementalClassificationLinear stores the Prior value as a numeric vector.

ValueDescription
'empirical'Incremental learning functions infer prior class probabilities from the observed class relative frequencies in the response data during incremental training (after the estimation period EstimationPeriod).
'uniform'For each class, the prior probability is 1/2.
numeric vectorCustom, normalized prior probabilities. The order of the elements of Prior corresponds to the elements of the ClassNames property.

The default Prior value depends on how you create the model:

• If you convert a traditionally trained model to create Mdl, Prior is specified by the corresponding property of the traditionally trained model.

• Otherwise, the default value is 'empirical'.

Data Types: single | double | char | string

Score transformation function describing how incremental learning functions transform raw response values, specified as a character vector, string scalar, or function handle. incrementalClassificationLinear stores the ScoreTransform value as a character vector or function handle.

This table describes the available built-in functions for score transformation.

ValueDescription
"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 + ex)
"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 + ex) – 1

For a MATLAB® function or a function that you define, enter its function handle; for example, 'ScoreTransform',@function, where:

• function accepts an n-by-2 matrix (the original scores) and returns a matrix of the same size (the transformed scores). The column order corresponds to the class order in the ClassNames property.

• n is the number of observations, and row j of the matrix contains the class scores of observation j.

The default ScoreTransform value depends on how you create the model:

• If you convert a traditionally trained model to create Mdl, ScoreTransform is specified by the corresponding property of the traditionally trained model. For example, if the ScoreTransform property of the traditionally trained model is a score-to-posterior-probability transformation function, as computed by fitPosterior or fitSVMPosterior, Mdl.ScoreTransform contains an anonymous function.

• Otherwise, the default value is 'none' (when Learner is 'svm') or 'logit' (when Learner is 'logistic').

Data Types: char | string | function_handle

### Training Parameters

Number of observations processed by the incremental model to estimate hyperparameters before training or tracking performance metrics, specified as a nonnegative integer.

Note

• If Mdl is prepared for incremental learning (all hyperparameters required for training are specified), incrementalClassificationLinear forces EstimationPeriod to 0.

• If Mdl is not prepared for incremental learning, incrementalClassificationLinear sets EstimationPeriod to 1000.

For more details, see Estimation Period.

Data Types: single | double

Linear model intercept inclusion flag, specified as true or false.

ValueDescription
trueincrementalClassificationLinear includes the bias term β0 in the linear model, which incremental fitting functions fit to data.
falseincrementalClassificationLinear sets β0 = 0.

If Bias ≠ 0, FitBias must be true. In other words, incrementalClassificationLinear does not support an equality constraint on β0.

The default FitBias value depends on how you create the model:

• If you convert a traditionally trained linear classification model (ClassificationLinear) to create Mdl, FitBias is specified by the FitBias value of the ModelParameters property of the traditionally trained model.

• Otherwise, the default value is true.

Data Types: logical

Predictor means, specified as a numeric vector.

If Mu is an empty array [] and you specify 'Standardize',true, incremental fitting functions set Mu to the predictor variable means estimated during the estimation period specified by EstimationPeriod.

You cannot specify Mu directly.

Data Types: single | double

Predictor standard deviations, specified as a numeric vector.

If Sigma is an empty array [] and you specify 'Standardize',true, incremental fitting functions set Sigma to the predictor variable standard deviations estimated during the estimation period specified by EstimationPeriod.

You cannot specify Sigma directly.

Data Types: single | double

Objective function minimization technique, specified as 'scale-invariant', 'sgd', or 'asgd'. incrementalClassificationLinear stores the Solver value as a character vector.

ValueDescriptionNotes
'scale-invariant'

• This algorithm is parameter free and can adapt to differences in predictor scales. Try this algorithm before using SGD or ASGD.

• To shuffle an incoming chunk of data before the fit function fits the model, set Shuffle to true.

'sgd'Stochastic gradient descent (SGD) [3][2]

• To train effectively with SGD, standardize the data and specify adequate values for hyperparameters using options listed in SGD and ASGD Solver Parameters.

• The fit function always shuffles an incoming chunk of data before fitting the model.

'asgd'Average stochastic gradient descent (ASGD) [4]

• To train effectively with ASGD, standardize the data and specify adequate values for hyperparameters using options listed in SGD and ASGD Solver Parameters.

• The fit function always shuffles an incoming chunk of data before fitting the model.

The default Solver value depends on how you create the model:

• If you create Mdl by calling incrementalClassificationLinear directly, the default value is 'scale-invariant'.

• If you convert a traditionally trained linear classification model (ClassificationLinear) to create Mdl, and the traditionally trained model's Regularization property is 'ridge (L2)' and ModelParameters.Solver is 'sgd' or 'asgd', Solver is specified by the Solver value of the ModelParameters property of the traditionally trained model.

• Otherwise, the Solver name-value argument of the incrementalLearner function sets this property. The default value of the argument is 'scale-invariant'.

Data Types: char | string

Objective solver configurations, specified as a structure array. The fields of SolverOptions are properties specific to the specified solver Solver.

Data Types: struct

### SGD and ASGD Solver Parameters

Mini-batch size, specified as a positive integer. At each learning cycle during training, incrementalClassificationLinear uses BatchSize observations to compute the subgradient.

The number of observations for the last mini-batch (last learning cycle in each function call of fit or updateMetricsAndFit) can be smaller than BatchSize. For example, if you supply 25 observations to fit or updateMetricsAndFit, the function uses 10 observations for the first two learning cycles and 5 observations for the last learning cycle.

The default BatchSize value depends on how you create the model:

• If you create Mdl by calling incrementalClassificationLinear directly, the default value is 10.

• If you convert a traditionally trained linear classification model (ClassificationLinear) to create Mdl, and the traditionally trained model's Regularization property is 'ridge (L2)' and ModelParameters.Solver is 'sgd' or 'asgd', BatchSize is specified by the BatchSize value of the ModelParameters property of the traditionally trained model.

• Otherwise, the BatchSize name-value argument of the incrementalLearner function sets this property. The default value of the argument is 10.

Data Types: single | double

Ridge (L2) regularization term strength, specified as a nonnegative scalar.

The default Lambda value depends on how you create the model:

• If you create Mdl by calling incrementalClassificationLinear directly, the default value is 1e-5.

• If you convert a traditionally trained linear classification model (ClassificationLinear) to create Mdl, and the traditionally trained model's Regularization property is 'ridge (L2)' and ModelParameters.Solver is 'sgd' or 'asgd', Lambda is specified by the corresponding property of the traditionally trained model.

• Otherwise, the Lambda name-value argument of the incrementalLearner function sets this property. The default value of the argument is 1e-5.

Data Types: double | single

Initial learning rate, specified as 'auto' or a positive scalar. incrementalClassificationLinear stores the LearnRate value as a numeric scalar.

The learning rate controls the optimization step size by scaling the objective subgradient. LearnRate specifies an initial value for the learning rate, and LearnRateSchedule determines the learning rate for subsequent learning cycles.

When you specify 'auto':

• The initial learning rate is 0.7.

• If EstimationPeriod > 0, fit and updateMetricsAndFit change the rate to 1/sqrt(1+max(sum(X.^2,obsDim))) at the end of EstimationPeriod. The obsDim value is 1 if the observations compose the columns of the predictor data; otherwise, the value is 2.

The default LearnRate value depends on how you create the model:

• If you create Mdl by calling incrementalClassificationLinear directly, the default value is 'auto'.

• If you convert a traditionally trained linear classification model (ClassificationLinear) to create Mdl, and the traditionally trained model's Regularization property is 'ridge (L2)' and ModelParameters.Solver is 'sgd' or 'asgd', LearnRate is specified by the LearnRate value of the ModelParameters property of the traditionally trained model.

• Otherwise, the LearnRate name-value argument of the incrementalLearner function sets this property. The default value of the argument is 'auto'.

Data Types: single | double | char | string

Learning rate schedule, specified as 'decaying' or 'constant', where LearnRate specifies the initial learning rate ɣ0. incrementalClassificationLinear stores the LearnRateSchedule value as a character vector.

ValueDescription
'constant'The learning rate is ɣ0 for all learning cycles.
'decaying'

The learning rate at learning cycle t is

${\gamma }_{t}=\frac{{\gamma }_{0}}{{\left(1+\lambda {\gamma }_{0}t\right)}^{c}}.$

• λ is the value of Lambda.

• If Solver is 'sgd', then c = 1.

• If Solver is 'asgd', then c is 0.75 [4].

The default LearnRateSchedule value depends on how you create the model:

• If you convert a traditionally trained model to create Mdl, the LearnRateSchedule name-value argument of the incrementalLearner function sets this property. The default value of the argument is 'decaying'.

• Otherwise, the default value is 'decaying'.

Data Types: char | string

### Performance Metrics Parameters

Flag indicating whether the incremental model tracks performance metrics, specified as logical 0 (false) or 1 (true).

The incremental model Mdl is warm (IsWarm becomes true) after incremental fitting functions fit (EstimationPeriod + MetricsWarmupPeriod) observations to the incremental model.

ValueDescription
true or 1The incremental model Mdl is warm. Consequently, updateMetrics and updateMetricsAndFit track performance metrics in the Metrics property of Mdl.
false or 0updateMetrics and updateMetricsAndFit do not track performance metrics.

Data Types: logical

Model performance metrics updated during incremental learning by updateMetrics and updateMetricsAndFit, specified as a table with two columns and m rows, where m is the number of metrics specified by the Metrics name-value argument.

The columns of Metrics are labeled Cumulative and Window.

• Cumulative: Element j is the model performance, as measured by metric j, from the time the model became warm (IsWarm is 1).

• Window: Element j is the model performance, as measured by metric j, evaluated over all observations within the window specified by the MetricsWindowSize property. The software updates Window after it processes MetricsWindowSize observations.

Rows are labeled by the specified metrics. For details, see the Metrics name-value argument of incrementalLearner or incrementalClassificationLinear.

Data Types: table

Number of observations the incremental model must be fit to before it tracks performance metrics in its Metrics property, specified as a nonnegative integer.

The default MetricsWarmupPeriod value depends on how you create the model:

• If you convert a traditionally trained model to create Mdl, the MetricsWarmupPeriod name-value argument of the incrementalLearner function sets this property. The default value of the argument is 0.

• Otherwise, the default value is 1000.

For more details, see Performance Metrics.

Data Types: single | double

Number of observations to use to compute window performance metrics, specified as a positive integer.

The default MetricsWindowSize value depends on how you create the model:

• If you convert a traditionally trained model to create Mdl, the MetricsWindowSize name-value argument of the incrementalLearner function sets this property. The default value of the argument is 200.

• Otherwise, the default value is 200.

For more details on performance metrics options, see Performance Metrics.

Data Types: single | double

## Object Functions

 fit Train linear model for incremental learning updateMetricsAndFit Update performance metrics in linear incremental learning model given new data and train model updateMetrics Update performance metrics in linear incremental learning model given new data loss Loss of linear incremental learning model on batch of data predict Predict responses for new observations from linear incremental learning model perObservationLoss Per observation classification error of model for incremental learning reset Reset incremental classification model

## Examples

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Create a default incremental linear SVM model for binary classification.

Mdl = incrementalClassificationLinear()
Mdl = incrementalClassificationLinear IsWarm: 0 Metrics: [1x2 table] ClassNames: [1x0 double] ScoreTransform: 'none' Beta: [0x1 double] Bias: 0 Learner: 'svm' Properties, Methods 

Mdl is an incrementalClassificationLinear 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.

Responses can be one of five classes: Sitting, Standing, Walking, Running, or Dancing. Dichotomize the response by identifying whether the subject is moving (actid > 2).

Y = Y > 2;

Fit the incremental model to the training data by using the updateMetricsAndFit 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 ${\beta }_{1}$, the cumulative metrics, and the window metrics to see how they evolve during incremental learning.

% Preallocation numObsPerChunk = 50; nchunk = floor(n/numObsPerChunk); ce = array2table(zeros(nchunk,2),'VariableNames',["Cumulative" "Window"]); beta1 = zeros(nchunk,1); % Incremental learning 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)); ce{j,:} = Mdl.Metrics{"ClassificationError",:}; beta1(j + 1) = Mdl.Beta(1); end

IncrementalMdl is an incrementalClassificationLinear 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 ${\beta }_{1}$ evolve during training, plot them on separate tiles.

t = tiledlayout(2,1); nexttile plot(beta1) ylabel('\beta_1') xlim([0 nchunk]) nexttile h = plot(ce.Variables); xlim([0 nchunk]) ylabel('Classification Error') xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,'g-.') legend(h,ce.Properties.VariableNames) xlabel(t,'Iteration')

The plot suggests that updateMetricsAndFit does the following:

• Fit ${\beta }_{1}$ 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).

Prepare an incremental binary SVM learner by specifying a metrics warm-up period, during which the updateMetricsAndFit function only fits the model. Specify a metrics window size of 500 observations. Train the model by using SGD, and adjust the SGD batch size, learning rate, and regularization parameter.

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

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

For details on the data set, enter Description at the command line.

Responses can be one of five classes: Sitting, Standing, Walking, Running, or Dancing. Dichotomize the response by identifying whether the subject is moving (actid > 2).

Y = Y > 2;

Create an incremental linear model for binary classification. Configure the model as follows:

• Specify that the incremental fitting functions process the raw (unstandardized) predictor data.

• Specify the SGD solver.

• Assume that a ridge regularization parameter value of 0.001, SGD batch size of 20, and learning rate of 0.002 work well for the problem.

• Specify a metrics warm-up period of 5000 observations.

• Specify a metrics window size of 500 observations.

• Track the classification and hinge error metrics to measure the performance of the model.

Mdl = incrementalClassificationLinear('Standardize',false, ... 'Solver','sgd','Lambda',0.001,'BatchSize',20,'LearnRate',0.002, ... 'MetricsWarmupPeriod',5000,'MetricsWindowSize',500, ... 'Metrics',{'classiferror' 'hinge'})
Mdl = incrementalClassificationLinear IsWarm: 0 Metrics: [2x2 table] ClassNames: [1x0 double] ScoreTransform: 'none' Beta: [0x1 double] Bias: 0 Learner: 'svm' Properties, Methods 

Mdl is an incrementalClassificationLinear model object configured for incremental learning.

Fit the incremental model to the rest of the data by using the updateMetricsAndFit function. At each iteration:

• Simulate a data stream by processing a chunk of 50 observations. Note that the chunk size is different from SGD batch size.

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

• Store the estimated coefficient ${\beta }_{10}$, the cumulative metrics, and the window metrics to see how they evolve during incremental learning.

% Preallocation numObsPerChunk = 50; nchunk = floor(n/numObsPerChunk); ce = array2table(zeros(nchunk,2),'VariableNames',["Cumulative" "Window"]); hinge = array2table(zeros(nchunk,2),'VariableNames',["Cumulative" "Window"]); beta10 = 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)); ce{j,:} = Mdl.Metrics{"ClassificationError",:}; hinge{j,:} = Mdl.Metrics{"HingeLoss",:}; beta10(j + 1) = Mdl.Beta(10); end

Mdl is an incrementalClassificationLinear 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 ${\beta }_{10}$ evolve during training, plot them on separate tiles.

tiledlayout(2,2) nexttile plot(beta10) ylabel('\beta_{10}') xlim([0 nchunk]); xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,'g-.') xlabel('Iteration') nexttile h = plot(ce.Variables); xlim([0 nchunk]); ylabel('Classification Error') xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,'g-.') legend(h,ce.Properties.VariableNames) xlabel('Iteration') nexttile h = plot(hinge.Variables); xlim([0 nchunk]); ylabel('Hinge Loss') xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,'g-.') legend(h,hinge.Properties.VariableNames) xlabel('Iteration')

The plot suggests that updateMetricsAndFit does the following:

• Fit ${\beta }_{10}$ 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 500 observations (10 iterations).

Train a linear model for binary classification by using fitclinear, convert it to an incremental learner, track its performance, and fit it to streaming data. Carry over training options from traditional to incremental learning.

Load the human activity data set. Randomly shuffle the data. Orient the observations of the predictor data in columns.

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.

Responses can be one of five classes: Sitting, Standing, Walking, Running, or Dancing. Dichotomize the response by identifying whether the subject is moving (actid > 2).

Y = Y > 2;

Suppose that the data collected when the subject was idle (Y = false) has double the quality than when the subject was moving. Create a weight variable that attributes 2 to observations collected from an idle subject, and 1 to a moving subject.

W = ones(n,1) + ~Y;

Train Linear Model for Binary Classification

Fit a linear model for binary classification to a random sample of half the data.

idxtt = randsample([true false],n,true); TTMdl = fitclinear(X(:,idxtt),Y(idxtt),'ObservationsIn','columns', ... 'Weights',W(idxtt))
TTMdl = ClassificationLinear ResponseName: 'Y' ClassNames: [0 1] ScoreTransform: 'none' Beta: [60x1 double] Bias: -0.1107 Lambda: 8.2967e-05 Learner: 'svm' Properties, Methods 

TTMdl is a ClassificationLinear model object representing a traditionally trained linear model for binary classification.

Convert Trained Model

Convert the traditionally trained classification model to a binary classification linear model for incremental learning.

IncrementalMdl = incrementalLearner(TTMdl)
IncrementalMdl = incrementalClassificationLinear IsWarm: 1 Metrics: [1x2 table] ClassNames: [0 1] ScoreTransform: 'none' Beta: [60x1 double] Bias: -0.1107 Learner: 'svm' Properties, Methods 

Separately Track Performance Metrics and Fit Model

Perform incremental learning on the rest of the data by using the updateMetrics and fit functions. Simulate a data stream by processing 50 observations at a time. At each iteration:

1. Call updateMetrics to update the cumulative and window classification error 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 that the observations are oriented in columns, and specify the observation weights.

2. Call fit to fit the model to the incoming chunk of observations. Overwrite the previous incremental model to update the model parameters. Specify that the observations are oriented in columns, and specify the observation weights.

3. Store the classification error and first estimated coefficient ${\beta }_{1}$.

% Preallocation idxil = ~idxtt; nil = sum(idxil); numObsPerChunk = 50; nchunk = floor(nil/numObsPerChunk); ce = array2table(zeros(nchunk,2),'VariableNames',["Cumulative" "Window"]); beta1 = [IncrementalMdl.Beta(1); zeros(nchunk,1)]; 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), ... 'ObservationsIn','columns','Weights',Wil(idx)); ce{j,:} = IncrementalMdl.Metrics{"ClassificationError",:}; IncrementalMdl = fit(IncrementalMdl,Xil(:,idx),Yil(idx), ... 'ObservationsIn','columns','Weights',Wil(idx)); beta1(j + 1) = IncrementalMdl.Beta(end); end

IncrementalMdl is an incrementalClassificationLinear 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 and estimated coefficient ${\beta }_{1}$.

t = tiledlayout(2,1); nexttile h = plot(ce.Variables); xlim([0 nchunk]) ylabel('Classification Error') legend(h,ce.Properties.VariableNames) nexttile plot(beta1) ylabel('\beta_1') xlim([0 nchunk]) xlabel(t,'Iteration')

The cumulative loss is stable and decreases gradually, whereas the window loss jumps.

${\beta }_{1}$ changes abruptly at first, then gradually levels off as fit processes more chunks.

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## Tips

• After creating a model, you can generate C/C++ code that performs incremental learning on a data stream. Generating C/C++ code requires MATLAB Coder™. For details, see Introduction to Code Generation.

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## References

[1] Kempka, Michał, Wojciech Kotłowski, and Manfred K. Warmuth. "Adaptive Scale-Invariant Online Algorithms for Learning Linear Models." Preprint, submitted February 10, 2019. https://arxiv.org/abs/1902.07528.

[2] Langford, J., L. Li, and T. Zhang. “Sparse Online Learning Via Truncated Gradient.” J. Mach. Learn. Res., Vol. 10, 2009, pp. 777–801.

[3] Shalev-Shwartz, S., Y. Singer, and N. Srebro. “Pegasos: Primal Estimated Sub-Gradient Solver for SVM.” Proceedings of the 24th International Conference on Machine Learning, ICML ’07, 2007, pp. 807–814.

[4] Xu, Wei. “Towards Optimal One Pass Large Scale Learning with Averaged Stochastic Gradient Descent.” CoRR, abs/1107.2490, 2011.

## Version History

Introduced in R2020b