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loss

Loss of linear model for incremental learning on batch of data

Description

loss returns the regression or classification loss of a configured incremental learning model for linear regression (incrementalRegressionLinear object) or linear binary classification (incrementalClassificationLinear object).

To measure model performance on a data stream and store the results in the output model, call updateMetrics or updateMetricsAndFit.

example

L = loss(Mdl,X,Y) returns the loss for the incremental learning model Mdl using the batch of predictor data X and corresponding responses Y.

example

L = loss(Mdl,X,Y,Name,Value) uses additional options specified by one or more name-value pair arguments. For example, you can specify that the columns of the predictor data matrix correspond to observations, or specify the classification loss function .

Examples

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The performance of an incremental model on streaming data is measured in three ways:

  1. Cumulative metrics measure the performance since the start of incremental learning.

  2. Window metrics measure the performance on a specified window of observations. The metrics are updated every time the model processes the specified window.

  3. The loss function measures the performance on a specified batch of data only.

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;

Create an incremental linear SVM model for binary classification. Configure it for loss by specifying the class names, prior class distribution (uniform), and arbitrary coefficient and bias values. Specify a metrics window size of 1000 observations.

p = size(X,2);
Beta = randn(p,1);
Bias = randn(1);
Mdl = incrementalClassificationLinear('Beta',Beta,'Bias',Bias,...
    'ClassNames',unique(Y),'Prior','uniform','MetricsWindowSize',1000);

Mdl is an incrementalClassificationLinear model. All its properties are read-only. Instead of specifying arbitrary values, you can take either of these actions to configure the model:

  • Train an SVM model using fitcsvm or fitclinear on a subset of the data (if available), and then convert the model to an incremental learner by using incrementalLearner.

  • Incrementally fit Mdl to data by using fit.

Simulate a data stream, and perform the following actions on each incoming chunk of 50 observations:

  1. Call updateMetrics to measure the cumulative performance and the performance within a window of observations. Overwrite the previous incremental model with a new one to track performance metrics.

  2. Call loss to measure the model performance on the incoming chunk.

  3. Call fit to fit the model to the incoming chunk. Overwrite the previous incremental model with a new one fitted to the incoming observation.

  4. Store all performance metrics to see how they evolve during incremental learning.

% Preallocation
numObsPerChunk = 50;
nchunk = floor(n/numObsPerChunk);
ce = array2table(zeros(nchunk,3),'VariableNames',["Cumulative" "Window" "Loss"]);

% Incremental learning
for j = 1:nchunk
    ibegin = min(n,numObsPerChunk*(j-1) + 1);
    iend   = min(n,numObsPerChunk*j);
    idx = ibegin:iend;    
    Mdl = updateMetrics(Mdl,X(idx,:),Y(idx));
    ce{j,["Cumulative" "Window"]} = Mdl.Metrics{"ClassificationError",:};
    ce{j,"Loss"} = loss(Mdl,X(idx,:),Y(idx));
    Mdl = fit(Mdl,X(idx,:),Y(idx));
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, updateMetrics checks the performance of the model on the incoming observation, then and the fit function fits the model to that observation. loss is agnostic of the metrics warm-up period, so it measures the classification error for all iterations.

To see how the performance metrics evolved during training, plot them.

figure;
plot(ce.Variables);
xlim([0 nchunk]);
ylim([0 0.05])
ylabel('Classification Error')
xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,'r-.');
legend(ce.Properties.VariableNames)
xlabel('Iteration')

Figure contains an axes. The axes contains 4 objects of type line, constantline. These objects represent Cumulative, Window, Loss.

During the metrics warm-up period (the area to the left of the red line), the yellow line represents the classification error on each incoming chunk of data. After the metrics warm-up period, Mdl tracks the cumulative and window metrics. The cumulative and batch losses converge as the fit function fits the incremental model to the incoming data.

Fit an incremental learning model for regression to streaming data, and compute the mean absolute deviation (MAD) on the incoming data batches.

Load the robot arm data set. Obtain the sample size n and the number of predictor variables p.

load robotarm
n = numel(ytrain);
p = size(Xtrain,2);

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

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

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

  • Specify a metrics window size of 500 observations.

  • Track the mean absolute deviation (MAD) to measure the performance of the model. 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.

  • Configure the model to predict responses by specifying that all regression coefficients and the bias are 0.

maefcn = @(z,zfit,w)(abs(z - zfit));
maemetric = struct("MeanAbsoluteError",maefcn);

Mdl = incrementalRegressionLinear('MetricsWarmupPeriod',1000,'MetricsWindowSize',500,...
    'Metrics',maemetric,'Beta',zeros(p,1),'Bias',0,'EstimationPeriod',0)
Mdl = 
  incrementalRegressionLinear

               IsWarm: 0
              Metrics: [2x2 table]
    ResponseTransform: 'none'
                 Beta: [32x1 double]
                 Bias: 0
              Learner: 'svm'


  Properties, Methods

Mdl is an incrementalRegressionLinear model object configured for incremental learning.

Perform incremental learning. At each iteration:

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

  • Call updateMetrics to compute cumulative and window metrics on the incoming chunk of data. Overwrite the previous incremental model with a new one fitted to overwrite the previous metrics.

  • Call loss to compute the MAD on the incoming chunk of data. Whereas the cumulative and window metrics require that custom losses return the loss for each observation, loss requires the loss on the entire chunk. Compute the mean of the absolute deviation.

  • Call fit to fit the incremental model to the incoming chunk of data.

  • Store the cumulative, window, and chunk metrics to see how they evolve during incremental learning.

% Preallocation
numObsPerChunk = 50;
nchunk = floor(n/numObsPerChunk);
mae = array2table(zeros(nchunk,3),'VariableNames',["Cumulative" "Window" "Chunk"]);

% Incremental fitting
for j = 1:nchunk
    ibegin = min(n,numObsPerChunk*(j-1) + 1);
    iend   = min(n,numObsPerChunk*j);
    idx = ibegin:iend;    
    Mdl = updateMetrics(Mdl,Xtrain(idx,:),ytrain(idx));
    mae{j,1:2} = Mdl.Metrics{"MeanAbsoluteError",:};
    mae{j,3} = loss(Mdl,Xtrain(idx,:),ytrain(idx),'LossFun',@(x,y,w)mean(maefcn(x,y,w)));
    Mdl = fit(Mdl,Xtrain(idx,:),ytrain(idx));
end

IncrementalMdl is an incrementalRegressionLinear model object trained on all the data in the stream. During incremental learning and after the model is warmed up, updateMetrics checks the performance of the model on the incoming observation, and the fit function fits the model to that observation.

Plot the performance metrics to see how they evolved during incremental learning.

figure;
h = plot(mae.Variables);
ylabel('Mean Absolute Deviation')
xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,'r-.');
xlabel('Iteration')
legend(h,mae.Properties.VariableNames)

Figure contains an axes. The axes contains 4 objects of type line, constantline. These objects represent Cumulative, Window, Chunk.

The plot suggests the following:

  • updateMetrics computes performance metrics after the metrics warm-up period only.

  • updateMetrics computes the cumulative metrics during each iteration.

  • updateMetrics computes the window metrics after processing 500 observations

  • Because Mdl was configured to predict observations from the beginning of incremental learning, loss can compute the MAD on each incoming chunk of data.

Input Arguments

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Incremental learning model, specified as an incrementalClassificationLinear or incrementalRegressionLinear 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.

You must configure Mdl to compute its loss on a batch of observations.

  • If Mdl is a converted, traditionally trained model, you can compute its loss without any modifications.

  • Otherwise, Mdl must satisfy the following criteria, which you can specify directly or by fitting Mdl to data using fit or updateMetricsAndFit.

    • If Mdl is an incrementalRegressionLinear model, its model coefficients Mdl.Beta and bias Mdl.Bias must be nonempty arrays.

    • If Mdl is an incrementalClassificationLinear model, its model coefficients Mdl.Beta and bias Mdl.Bias must be nonempty arrays, the class names Mdl.ClassNames must contain two classes, and the prior class distribution Mdl.Prior must contain known values.

    • Regardless of object type, if you configure the model so that functions standardize predictor data, the predictor means Mdl.Mu and standard deviations Mdl.Sigma must be nonempty arrays.

Batch of predictor data with which to compute the loss, specified as a floating-point matrix of n observations and Mdl.NumPredictors predictor variables. The value of the 'ObservationsIn' name-value pair argument determines the orientation of the variables and observations.

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 or column) in X.

Note

loss supports only floating-point input predictor data. If the input model Mdl represents a converted, traditionally trained model fit to categorical data, use dummyvar to convert each categorical variable to a numeric matrix of dummy variables, and concatenate all dummy variable matrices and any other numeric predictors. For more details, see Dummy Variables.

Data Types: single | double

Batch of labels with which to compute the loss, specified as a categorical, character, or string array, logical or floating-point vector, or cell array of character vectors for classification problems; or a floating-point vector for regression problems.

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 or column) in X.

For classification problems:

  • loss supports binary classification only.

  • When the ClassNames property of the input model Mdl is nonempty, the following conditions apply:

    • If Y contains a label that is not a member of Mdl.ClassNames, loss issues an error.

    • The data type of Y and Mdl.ClassNames must be the same.

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

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'ObservationsIn','columns','Weights',W specifies that the columns of the predictor matrix correspond to observations, and the vector W contains observation weights to apply.

Loss function, specified as the comma-separated pair consisting of 'LossFun' and a built-in loss function name or function handle.

  • Classification problems: The following table lists the available loss functions when Mdl is an incrementalClassificationLinear model. Specify one using its corresponding character vector or string scalar.

    NameDescription
    "binodeviance"Binomial deviance
    "classiferror" (default)Misclassification rate in decimal
    "exponential"Exponential loss
    "hinge"Hinge loss
    "logit"Logistic loss
    "quadratic"Quadratic loss

    For more details, see Classification Loss.

    Logistic regression learners return posterior probabilities as classification scores, but SVM learners do not (see predict).

    To specify a custom loss function, use function handle notation. The function must have this form:

    lossval = lossfcn(C,S,W)

    • The output argument lossval is an n-by-1 floating-point vector, where lossval(j) is the classification loss of observation j.

    • You specify the function name (lossfcn).

    • 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.

    • W is an n-by-1 numeric vector of observation weights.

  • Regression problems: The following table lists the available loss functions when Mdl is an incrementalRegressionLinear model. Specify one using its corresponding character vector or string scalar.

    NameDescriptionLearners Supporting Metric
    "epsiloninsensitive"Epsilon insensitive loss'svm'
    "mse" (default)Weighted mean squared error'svm' and 'leastsquares'

    For more details, see Regression Loss.

    To specify a custom loss function, use function handle notation. The function must have this form:

    lossval = lossfcn(Y,YFit,W)

    • The output argument lossval is a floating-point scalar.

    • You specify the function name (lossfcn).

    • Y is a length n numeric vector of observed responses.

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

    • W is an n-by-1 numeric vector of observation weights.

Example: 'LossFun',"mse"

Example: 'LossFun',@lossfcn

Data Types: char | string | function_handle

Predictor data observation dimension, specified as the comma-separated pair consisting of 'ObservationsIn' and 'columns' or 'rows'.

Data Types: char | string

Batch of observation weights, specified as the comma-separated pair consisting of 'Weights' and a floating-point vector of positive values. loss weighs the observations in the input data with the corresponding values in Weights. The size of Weights must equal n, which is the number of observations in the input data.

By default, Weights is ones(n,1).

For more details, see Observation Weights.

Data Types: double | single

Output Arguments

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Classification or regression loss, returned as a numeric scalar. The interpretation of L depends on Weights and LossFun.

More About

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Classification Loss

Classification loss functions measure the predictive inaccuracy of classification models. When you compare the same type of loss among many models, a lower loss indicates a better predictive model.

Consider the following scenario.

  • L is the weighted average classification loss.

  • n is the sample size.

  • For binary classification:

    • yj is the observed class label. The software codes it as –1 or 1, indicating the negative or positive class (or the first or second class in the ClassNames property), respectively.

    • f(Xj) is the positive-class classification score for observation (row) j of the predictor data X.

    • mj = yjf(Xj) is the classification score for classifying observation j into the class corresponding to yj. Positive values of mj indicate correct classification and do not contribute much to the average loss. Negative values of mj indicate incorrect classification and contribute significantly to the average loss.

  • The weight for observation j is wj.

Given this scenario, the following table describes the supported loss functions that you can specify by using the 'LossFun' name-value pair argument.

Loss FunctionValue of LossFunEquation
Binomial deviance"binodeviance"L=j=1nwjlog{1+exp[2mj]}.
Exponential loss"exponential"L=j=1nwjexp(mj).
Misclassification rate in decimal"classiferror"

L=j=1nwjI{y^jyj}.

y^j is the class label corresponding to the class with the maximal score. I{·} is the indicator function.

Hinge loss"hinge"L=j=1nwjmax{0,1mj}.
Logit loss"logit"L=j=1nwjlog(1+exp(mj)).
Quadratic loss"quadratic"L=j=1nwj(1mj)2.

This figure compares the loss functions over the score m for one observation. Some functions are normalized to pass through the point (0,1).

Comparison of classification losses for different loss functions

Regression Loss

Regression loss functions measure the predictive inaccuracy of regression models. When you compare the same type of loss among many models, a lower loss indicates a better predictive model.

Consider the following scenario.

  • L is the weighted average classification loss.

  • n is the sample size.

  • yj is the observed response of observation j.

  • f(Xj) = β0 + xjβ is the predicted value of observation j of the predictor data X, where β0 is the bias and β is the vector of coefficients.

  • The weight for observation j is wj.

Given this scenario, the following table describes the supported loss functions that you can specify by using the 'LossFun' name-value pair argument.

Loss FunctionValue of LossFunEquation
Epsilon insensitive loss"epsiloninsensitive"

L=max[0,|yf(x)|ε].

Mean squared error"mse"

L=[yf(x)]2.

Algorithms

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Observation Weights

For classification problems, if the prior class probability distribution is known (in other words, the prior distribution is not empirical), loss normalizes observation weights to sum to the prior class probabilities in the respective classes. This action implies that observation weights are the respective prior class probabilities by default.

For regression problems or if the prior class probability distribution is empirical, the software normalizes the specified observation weights to sum to 1 each time you call loss.

Extended Capabilities

Introduced in R2020b