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incrementalRegressionKernel

Kernel regression model for incremental learning

    Description

    The incrementalRegressionKernel function creates an incrementalRegressionKernel model object, which represents a binary Gaussian kernel regression model for incremental learning. The kernel model maps data in a low-dimensional space into a high-dimensional space, then fits a linear model in the high-dimensional space. Supported linear models include support vector machine (SVM) and least-squares regression.

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

    incrementalRegressionKernel is best suited for incremental learning. For a traditional approach to training a kernel regression model (such as creating a model by fitting it to data, performing cross-validation, tuning hyperparameters, and so on), see fitrkernel.

    Creation

    You can create an incrementalRegressionKernel model object in several ways:

    • Call the function directly — Configure incremental learning options, or specify learner-specific options, by calling incrementalRegressionKernel 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 model for incremental learning using the model parameters and hyperparameters of a trained model object (RegressionKernel), you can convert the traditionally trained model to an incrementalRegressionKernel model object by passing it to the incrementalLearner function.

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

    Description

    example

    Mdl = incrementalRegressionKernel() returns a default incremental learning model object for binary Gaussian kernel regression, 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 = incrementalRegressionKernel(Name=Value) sets properties and additional options using name-value arguments. For example, incrementalRegressionKernel(Solver="sgd",LearnRateSchedule="constant") specifies to use the stochastic gradient descent (SGD) solver with a constant learning rate.

    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.

    Example: Metrics="mse",MetricsWarmupPeriod=100 sets the model performance metric to the weighted mean squared error and the metrics warm-up period to 100.

    Regression Options

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    Random number stream for reproducibility of data transformation, specified as a random stream object. For details, see Random Feature Expansion.

    Use RandomStream to reproduce the random basis functions used by incrementalRegressionKernel to transform the predictor data to a high-dimensional space. For details, see Managing the Global Stream Using RandStream and Creating and Controlling a Random Number Stream.

    Example: RandomStream=RandStream("mlfg6331_64")

    SGD and ASGD (Average SGD) Solver Options

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

    The number of observations in 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.

    Example: BatchSize=5

    Data Types: single | double

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

    Example: Lambda=0.01

    Data Types: single | double

    Initial learning rate, specified as "auto" or a positive 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.^2,2))) at the end of EstimationPeriod.

    Example: LearnRate=0.001

    Data Types: single | double | char | string

    Learning rate schedule, specified as a value in this table, where LearnRate specifies the initial learning rate ɣ0.

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

    The learning rate at learning cycle t is

    γt=γ0(1+λγ0t)c.

    • λ is the value of Lambda.

    • If Solver is "sgd", c = 1.

    • If Solver is "asgd":

      • c = 2/3 if Learner is "leastsquares".

      • c = 3/4 if Learner is "svm" [4].

    Example: LearnRateSchedule="constant"

    Data Types: char | string

    Adaptive Scale-Invariant Solver Options

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    Flag for shuffling the observations at each iteration, specified as logical 1 (true) or 0 (false).

    ValueDescription
    logical 1 (true)The 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.
    logical 0 (false)The software processes the data in the order received.

    Example: Shuffle=false

    Data Types: logical

    Performance Metrics Options

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    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 and which learners, specified in Learner, support them. You can specify more than one loss function by using a string vector.

    NameDescriptionLearner Supporting Metric
    "epsiloninsensitive"Epsilon insensitive loss"svm"
    "mse"Weighted mean squared error"svm" and "leastsquares"

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

    Example: Metrics=["epsiloninsensitive","mse"]

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

    metric = customMetric(Y,YFit)

    • 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).

    • Y is a length n numeric vector of observed responses, where n is the sample size.

    • YFit is a length n numeric vector of corresponding predicted responses.

    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,"mse",struct(Metric3=@customMetric3)}

    updateMetrics and updateMetricsAndFit store specified metrics in a table in the property Metrics. 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 "epsiloninsensitive" is "EpsilonInsensitiveLoss"
    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 @(Y,YFit)customMetric(Y,YFit)... is CustomMetric_1

    By default:

    • Metrics is "epsiloninsensitive" if Learner is "svm".

    • Metrics is "mse" if Learner is "leastsquares".

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

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

    Properties

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

    Regression Model Parameters

    This property is read-only.

    Half of the width of the epsilon insensitive band, specified as "auto" or a nonnegative scalar. incrementalRegressionKernel stores the Epsilon value as a numeric scalar.

    If you specify "auto" when you call incrementalRegressionKernel, incremental fitting functions estimate Epsilon during the estimation period, specified by EstimationPeriod, using this procedure:

    • If iqr(Y) ≠ 0, Epsilon is iqr(Y)/13.49, where Y is the estimation period response data.

    • If iqr(Y) = 0 or before you fit Mdl to data, Epsilon is 0.1.

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

    • If you convert a traditionally trained model whose Learner property is 'svm', Epsilon is specified by the corresponding property of the traditionally trained model.

    • Otherwise, the default value is "auto".

    If Learner is "leastsquares", you cannot set Epsilon and its value is NaN.

    Data Types: single | double

    This property is read-only.

    Loss function used to fit the linear model, specified as 'epsiloninsensitive' or 'mse'.

    ValueAlgorithmLoss FunctionLearner Value
    'epsiloninsensitive'Support vector machine regressionEpsilon insensitive: [y,f(x)]=max[0,|yf(x)|ε]'svm'
    'mse'Linear regression through ordinary least squaresMean squared error (MSE): [y,f(x)]=12[yf(x)]2'leastsquares'

    This property is read-only.

    Kernel scale parameter, specified as "auto" or a positive scalar. incrementalRegressionKernel stores the KernelScale value as a numeric scalar. The software obtains a random basis for feature expansion by using the kernel scale parameter. For details, see Random Feature Expansion.

    If you specify "auto" when creating the model object, the software selects an appropriate kernel scale parameter using a heuristic procedure. This procedure uses subsampling, so estimates can vary from one call to another. Therefore, to reproduce results, set a random number seed by using rng before training.

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

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

    • Otherwise, the default value is 1.

    Data Types: char | string | single | double

    This property is read-only.

    Linear regression model type, specified as "svm" or "leastsquares". incrementalRegressionKernel stores the Learner value as a character vector.

    In the following table, f(x)=T(x)β+b.

    • x is an observation (row vector) from p predictor variables.

    • T(·) is a transformation of an observation (row vector) for feature expansion. T(x) maps x in p to a high-dimensional space (m).

    • β is a vector of coefficients.

    • b is the scalar bias.

    ValueAlgorithmLoss FunctionFittedLoss Value
    "svm"Support vector machine regressionEpsilon insensitive: [y,f(x)]=max[0,|yf(x)|ε]'epsiloninsensitive'
    "leastsquares"Linear regression through ordinary least squaresMean squared error (MSE): [y,f(x)]=12[yf(x)]2'mse'

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

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

    • Otherwise, the default value is "svm".

    This property is read-only.

    Number of dimensions of the expanded space, specified as "auto" or a positive integer. incrementalRegressionKernel stores the NumExpansionDimensions value as a numeric scalar.

    For "auto", the software selects the number of dimensions using 2.^ceil(min(log2(p)+5,15)), where p is the number of predictors. For details, see Random Feature Expansion.

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

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

    • Otherwise, the default value is "auto".

    Data Types: char | string | single | double

    This property is read-only.

    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 incrementalRegressionKernel 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

    This property is read-only.

    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, incrementalRegressionKernel does not add the number of observations fit to the traditionally trained model to NumTrainingObservations.

    Data Types: double

    This property is read-only.

    Response transformation function, specified as "none" or a function handle. incrementalRegressionKernel stores the ResponseTransform value as a character vector or function handle.

    ResponseTransform describes how incremental learning functions transform raw response values.

    For a MATLAB® function or a function that you define, enter its function handle; for example, ResponseTransform=@function, where function accepts an n-by-1 vector (the original responses) and returns a vector of the same length (the transformed responses).

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

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

    • Otherwise, the default value is "none".

    Data Types: char | string | function_handle

    Training Parameters

    This property is read-only.

    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), incrementalRegressionKernel forces EstimationPeriod to 0.

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

    For more details, see Estimation Period.

    Data Types: single | double

    This property is read-only.

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

    ValueDescriptionNotes
    "scale-invariant"

    Adaptive scale-invariant solver for incremental learning [1]

    • 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) [2][3]

    • To train effectively with SGD, specify adequate values for hyperparameters using options listed in SGD and ASGD (Average SGD) Solver Options.

    • 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, specify adequate values for hyperparameters using options listed in SGD and ASGD (Average SGD) Solver Options.

    • 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 convert a traditionally trained model to create Mdl, the Solver name-value argument of the incrementalLearner function sets this property. The default value of the argument is "scale-invariant".

    • Otherwise, the default value is "scale-invariant".

    Data Types: char | string

    This property is read-only.

    Objective solver configurations, specified as a structure array. The fields of SolverOptions depend on Solver.

    You can specify the field values using the corresponding name-value arguments when you create the model object by calling incrementalRegressionKernel directly, or when you convert a traditionally trained model using the incrementalLearner function.

    Data Types: struct

    Performance Metrics Parameters

    This property is read-only.

    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

    This property is read-only.

    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 incrementalRegressionKernel.

    Data Types: table

    This property is read-only.

    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

    This property is read-only.

    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

    fitTrain kernel model for incremental learning
    updateMetricsUpdate performance metrics in kernel incremental learning model given new data
    updateMetricsAndFitUpdate performance metrics in kernel incremental learning model given new data and train model
    lossLoss of kernel incremental learning model on batch of data
    predictPredict responses for new observations from kernel incremental learning model
    perObservationLossPer observation regression error of model for incremental learning
    resetReset incremental regression model

    Examples

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    Create an incremental kernel model without any prior information. Track the model performance on streaming data, and fit the model to the data.

    Create a default incremental kernel SVM model for regression.

    Mdl = incrementalRegressionKernel()
    Mdl = 
      incrementalRegressionKernel
    
                        IsWarm: 0
                       Metrics: [1x2 table]
             ResponseTransform: 'none'
        NumExpansionDimensions: 0
                   KernelScale: 1
    
    
      Properties, Methods
    
    
    Mdl.EstimationPeriod
    ans = 1000
    

    Mdl is an incrementalRegressionKernel model object. All its properties are read-only.

    Mdl must be fit to data before you can use it to perform any other operations. The software sets the estimation period to 1000 because half the width of the epsilon insensitive band Epsilon is unknown. You can set Epsilon to a positive floating-point scalar by using the Epsilon name-value argument. This action results in a default estimation period of 0.

    Load the robot arm data set.

    load robotarm

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

    Fit the incremental model to the training data by using the updateMetricsAndFit function. To simulate a data stream, fit the model in 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 cumulative metrics, window metrics, and number of training observations to see how they evolve during incremental learning.

    % Preallocation
    n = numel(ytrain);
    numObsPerChunk = 50;
    nchunk = floor(n/numObsPerChunk);
    ei = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]); 
    numtrainobs = zeros(nchunk+1,1);
    
    % Incremental fitting
    for j = 1:nchunk
        ibegin = min(n,numObsPerChunk*(j-1) + 1);
        iend   = min(n,numObsPerChunk*j);
        idx = ibegin:iend;    
        Mdl = updateMetricsAndFit(Mdl,Xtrain(idx,:),ytrain(idx));
        ei{j,:} = Mdl.Metrics{"EpsilonInsensitiveLoss",:};
        numtrainobs(j+1) = Mdl.NumTrainingObservations;
    end

    Mdl is an incrementalRegressionKernel model object trained on all the data in the stream. While updateMetricsAndFit processes the first 1000 observations, it stores the response values to estimate Epsilon; the function does not fit the model until after this estimation period. 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.

    Plot a trace plot of the number of training observations and the performance metrics on separate tiles.

    t = tiledlayout(2,1);
    nexttile
    plot(numtrainobs)
    xlim([0 nchunk])
    ylabel("Number of Training Observations")
    xline(Mdl.EstimationPeriod/numObsPerChunk,"-.")
    xline((Mdl.EstimationPeriod + Mdl.MetricsWarmupPeriod)/numObsPerChunk,"--")
    nexttile
    plot(ei.Variables)
    xlim([0 nchunk])
    ylabel("Epsilon Insensitive Loss")
    xline(Mdl.EstimationPeriod/numObsPerChunk,"-.")
    xline((Mdl.EstimationPeriod + Mdl.MetricsWarmupPeriod)/numObsPerChunk,"--")
    legend(ei.Properties.VariableNames,Location="best")
    xlabel(t,"Iteration")

    Figure contains 2 axes objects. Axes object 1 contains 3 objects of type line, constantline. Axes object 2 contains 4 objects of type line, constantline. These objects represent Cumulative, Window.

    The plot suggests that updateMetricsAndFit does the following:

    • After the estimation period (first 20 iterations), fit the model 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 regression learner by specifying a metrics warm-up period and a metrics window size. Train the model by using SGD, and adjust the SGD batch size, learning rate, and regularization parameter.

    Load the robot arm data set.

    load robotarm
    n = numel(ytrain);

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

    Create an incremental kernel model for regression. Configure the model as follows:

    • Specify the SGD solver.

    • Assume that these settings work well for the problem: a ridge regularization parameter value of 0.001, SGD batch size of 20, learning rate of 0.002, and half the width of the epsilon insensitive band for SVM of 0.05.

    • Specify a metrics warm-up period of 1000 observations.

    • Specify a metrics window size of 500 observations.

    • Track the epsilon insensitive loss, MSE, and mean absolute error (MAE) to measure the performance of the model. The software supports epsilon insensitive loss and MSE. Create an anonymous function that measures the absolute error of each new observation. Create a structure array containing the name MeanAbsoluteError and its corresponding function.

    maefcn = @(z,zfit)abs(z - zfit);
    maemetric = struct("MeanAbsoluteError",maefcn);
    
    Mdl = incrementalRegressionKernel(Solver="sgd", ...
        Lambda=0.001,BatchSize=20,LearnRate=0.002,Epsilon=0.05, ...
        MetricsWarmupPeriod=1000,MetricsWindowSize=500, ...
        Metrics={"epsiloninsensitive","mse",maemetric})
    Mdl = 
      incrementalRegressionKernel
    
                        IsWarm: 0
                       Metrics: [3x2 table]
             ResponseTransform: 'none'
        NumExpansionDimensions: 0
                   KernelScale: 1
    
    
      Properties, Methods
    
    

    Mdl is an incrementalRegressionKernel model object configured for incremental learning without an estimation period.

    Fit the incremental model to 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 the SGD batch size.

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

    • Store the cumulative metrics, window metrics, and number of training observations to see how they evolve during incremental learning.

    % Preallocation
    numObsPerChunk = 50;
    nchunk = floor(n/numObsPerChunk);
    ei = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]);
    mse = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]);
    mae = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]);  
    numtrainobs = zeros(nchunk,1);
    
    % Incremental fitting
    rng("default") % For reproducibility
    for j = 1:nchunk
        ibegin = min(n,numObsPerChunk*(j-1) + 1);
        iend   = min(n,numObsPerChunk*j);
        idx = ibegin:iend;    
        Mdl = updateMetricsAndFit(Mdl,Xtrain(idx,:),ytrain(idx));
        ei{j,:} = Mdl.Metrics{"EpsilonInsensitiveLoss",:};
        mse{j,:} = Mdl.Metrics{"MeanSquaredError",:};
        mae{j,:} = Mdl.Metrics{"MeanAbsoluteError",:};
        numtrainobs(j) = Mdl.NumTrainingObservations;
    end

    Mdl is an incrementalRegressionKernel 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.

    Plot a trace plot of the number of training observations and the performance metrics on separate tiles.

    t = tiledlayout(4,1);
    nexttile
    plot(numtrainobs)
    xlim([0 nchunk])
    ylabel(["Number of","Training Observations"])
    xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"--")
    nexttile
    plot(ei.Variables)
    xlim([0 nchunk])
    ylabel(["Epsilon Insensitive","Loss"])
    xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"--")
    legend(ei.Properties.VariableNames)
    nexttile
    plot(mse.Variables)
    xlim([0 nchunk])
    ylabel("MSE")
    xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"--")
    legend(mse.Properties.VariableNames)
    nexttile
    plot(mae.Variables)
    xlim([0 nchunk])
    ylabel("MAE")
    xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"--")
    legend(mae.Properties.VariableNames)
    xlabel(t,"Iteration")

    Figure contains 4 axes objects. Axes object 1 contains 2 objects of type line, constantline. Axes object 2 contains 3 objects of type line, constantline. These objects represent Cumulative, Window. Axes object 3 contains 3 objects of type line, constantline. These objects represent Cumulative, Window. Axes object 4 contains 3 objects of type line, constantline. These objects represent Cumulative, Window.

    The plot suggests that updateMetricsAndFit does the following:

    • Fit the model 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 kernel regression model by using fitrkernel, 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 and Preprocess Data

    Load the 2015 NYC housing data set, and shuffle the data. For more details on the data, see NYC Open Data.

    load NYCHousing2015
    rng(1) % For reproducibility
    n = size(NYCHousing2015,1);
    idxshuff = randsample(n,n);
    NYCHousing2015 = NYCHousing2015(idxshuff,:);

    Suppose that the data collected from Manhattan (BOROUGH = 1) was collected using a new method that doubles its quality. Create a weight variable that attributes 2 to observations collected from Manhattan, and 1 to all other observations.

    NYCHousing2015.W = ones(n,1) + (NYCHousing2015.BOROUGH == 1);

    Extract the response variable SALEPRICE from the table. For numerical stability, scale SALEPRICE by 1e6.

    Y = NYCHousing2015.SALEPRICE/1e6;
    NYCHousing2015.SALEPRICE = [];

    To reduce computational cost for this example, remove the NEIGHBORHOOD column, which contains a categorical variable with 254 categories.

    NYCHousing2015.NEIGHBORHOOD = [];

    Create dummy variable matrices from the other categorical predictors.

    catvars = ["BOROUGH","BUILDINGCLASSCATEGORY"];
    dumvarstbl = varfun(@(x)dummyvar(categorical(x)),NYCHousing2015, ...
        InputVariables=catvars);
    dumvarmat = table2array(dumvarstbl);
    NYCHousing2015(:,catvars) = [];

    Treat all other numeric variables in the table as predictors of sales price. Concatenate the matrix of dummy variables to the rest of the predictor data.

    idxnum = varfun(@isnumeric,NYCHousing2015,OutputFormat="uniform");
    X = [dumvarmat NYCHousing2015{:,idxnum}];

    Train Kernel Regression Model

    Fit a kernel regression model to a random sample of half the data. Specify the observation weights.

    idxtt = randsample([true false],n,true);
    Mdl = fitrkernel(X(idxtt,:),Y(idxtt),Weights=NYCHousing2015.W(idxtt))
    Mdl = 
      RegressionKernel
                  ResponseName: 'Y'
                       Learner: 'svm'
        NumExpansionDimensions: 2048
                   KernelScale: 1
                        Lambda: 2.1977e-05
                 BoxConstraint: 1
                       Epsilon: 0.0547
    
    
      Properties, Methods
    
    

    Mdl is a RegressionKernel model object representing a traditionally trained kernel regression model.

    Convert Trained Model

    Convert the traditionally trained kernel regression model to a model for incremental learning.

    IncrementalMdl = incrementalLearner(Mdl)
    IncrementalMdl = 
      incrementalRegressionKernel
    
                        IsWarm: 1
                       Metrics: [1x2 table]
             ResponseTransform: 'none'
        NumExpansionDimensions: 2048
                   KernelScale: 1
    
    
      Properties, Methods
    
    

    IncrementalMdl is an incrementalRegressionKernel model object configured for incremental learning.

    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 500 observations at a time. At each iteration:

    1. Call updateMetrics to update the cumulative and window epsilon insensitive loss of the model given the incoming chunk of observations. Overwrite the previous incremental model to update 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.

    2. 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.

    3. Store the losses and number of training observations.

    % Preallocation
    idxil = ~idxtt;
    nil = sum(idxil);
    numObsPerChunk = 500;
    nchunk = floor(nil/numObsPerChunk);
    ei = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]);
    numtrainobs = zeros(nchunk,1);
    Xil = X(idxil,:);
    Yil = Y(idxil);
    Wil = NYCHousing2015.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));
        ei{j,:} = IncrementalMdl.Metrics{"EpsilonInsensitiveLoss",:};
        IncrementalMdl = fit(IncrementalMdl,Xil(idx,:),Yil(idx), ...
            Weights=Wil(idx));
        numtrainobs(j) = IncrementalMdl.NumTrainingObservations;
    end

    IncrementalMdl is an incrementalRegressionKernel 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 number of training observations and the performance metrics on separate tiles.

    t = tiledlayout(2,1);
    nexttile
    plot(numtrainobs)
    xlim([0 nchunk])
    ylabel("Number of Training Observations")
    nexttile
    plot(ei.Variables)
    xlim([0 nchunk])
    ylabel("Epsilon Insensitive Loss")
    legend(ei.Properties.VariableNames)
    xlabel(t,"Iteration")

    Figure contains 2 axes objects. Axes object 1 contains an object of type line. Axes object 2 contains 2 objects of type line. These objects represent Cumulative, Window.

    The cumulative loss gradually changes with each iteration (chunk of 500 observations), whereas the window loss jumps. Because the metrics window is 200 by default, updateMetrics measures the performance based on the latest 200 observations in each 500 observation chunk.

    More About

    expand all

    Algorithms

    expand all

    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.

    [5] Rahimi, A., and B. Recht. “Random Features for Large-Scale Kernel Machines.” Advances in Neural Information Processing Systems. Vol. 20, 2008, pp. 1177–1184.

    [6] Le, Q., T. Sarlós, and A. Smola. “Fastfood — Approximating Kernel Expansions in Loglinear Time.” Proceedings of the 30th International Conference on Machine Learning. Vol. 28, No. 3, 2013, pp. 244–252.

    [7] Huang, P. S., H. Avron, T. N. Sainath, V. Sindhwani, and B. Ramabhadran. “Kernel methods match Deep Neural Networks on TIMIT.” 2014 IEEE International Conference on Acoustics, Speech and Signal Processing. 2014, pp. 205–209.

    Version History

    Introduced in R2022a