# loss

Loss of kernel incremental learning model on batch of data

## Description

loss returns the regression or classification loss of a configured incremental learning model for kernel regression (incrementalRegressionKernel object) or binary kernel classification (incrementalClassificationKernel 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 arguments. For example, you can specify the classification loss function and the observation weights.

## 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.

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 kernel model for binary classification. Specify a metrics window size of 1000 observations. Configure the model for loss by fitting it to the first 10 observations.

p = size(X,2);
Mdl = incrementalClassificationKernel(MetricsWindowSize=1000);
initobs = 10;
Mdl = fit(Mdl,X(1:initobs,:),Y(1:initobs));

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

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 observations.

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

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

% Incremental learning
for j = 1:nchunk
ibegin = min(n,numObsPerChunk*(j-1) + 1 + initobs);
iend   = min(n,numObsPerChunk*j + initobs);
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 incrementalClassificationKernel 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 observations, and the fit function fits the model to those observations. loss is agnostic of the metrics warm-up period, so it measures the classification error for all iterations.

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

plot(ce.Variables)
xlim([0 nchunk])
ylabel("Classification Error")
xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"--")
legend(ce.Properties.VariableNames)
xlabel("Iteration")

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.

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

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 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 fitting it to the first 10 observations.

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

Mdl = incrementalRegressionKernel(MetricsWarmupPeriod=1000,MetricsWindowSize=500, ...
Metrics=maemetric);
initobs = 10;
Mdl = fit(Mdl,Xtrain(1:initobs,:),ytrain(1:initobs));

Mdl is an incrementalRegressionKernel 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 - initobs)/numObsPerChunk);
mae = array2table(zeros(nchunk,3),VariableNames=["Cumulative","Window","Chunk"]);

% Incremental fitting
for j = 1:nchunk
ibegin = min(n,numObsPerChunk*(j-1) + 1 + initobs);
iend   = min(n,numObsPerChunk*j + initobs);
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

Mdl is an incrementalRegressionKernel 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 observations, and the fit function fits the model to those observations.

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

plot(mae.Variables)
ylabel("Mean Absolute Deviation")
xlabel("Iteration")
xlim([0 nchunk])
xline(Mdl.EstimationPeriod/numObsPerChunk,"-.")
xline((Mdl.EstimationPeriod + Mdl.MetricsWarmupPeriod)/numObsPerChunk,"--")
legend(mae.Properties.VariableNames)

The plot suggests the following:

• updateMetrics computes the 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 (10 iterations).

• 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 incrementalClassificationKernel or incrementalRegressionKernel 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 predict labels for a batch of observations.

• If Mdl is a converted, traditionally trained model, you can predict labels without any modifications.

• Otherwise, you must fit Mdl to data using fit or updateMetricsAndFit.

Batch of predictor data, specified as a floating-point matrix of n observations and Mdl.NumPredictors predictor variables.

The length of the observation labels Y and the number of observations in X must be equal; Y(j) is the label of observation j (row) in X.

Note

loss supports only floating-point input predictor data. If your input data includes categorical data, you must prepare an encoded version of the categorical data. Use dummyvar to convert each categorical variable to a numeric matrix of dummy variables. Then, concatenate all dummy variable matrices and any other numeric predictors. For more details, see Dummy Variables.

Data Types: single | double

Batch of responses (labels), specified as a categorical, character, or string array, a logical or floating-point vector, or a 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) in X.

For classification problems:

• loss supports binary classification only.

• 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 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: LossFun="epsiloninsensitive",Weights=W returns the epsilon insensitive loss and specifies the observation weights as the vector W.

Loss function, specified as a built-in loss function name or function handle.

• Classification problems: The following table lists the available loss functions when Mdl is an incrementalClassificationKernel 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

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 incrementalRegressionKernel model. Specify one using its corresponding character vector or string scalar.

NameDescriptionLearner 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

Batch of observation weights, specified as 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.

Example: Weights=W specifies the observation weights as the vector W.

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.

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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.

• 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 argument.

Loss FunctionValue of LossFunEquation
Binomial deviance"binodeviance"$L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}.$
Exponential loss"exponential"$L=\sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right).$
Misclassification rate in decimal"classiferror"

$L=\sum _{j=1}^{n}{w}_{j}I\left\{{\stackrel{^}{y}}_{j}\ne {y}_{j}\right\},$

where ${\stackrel{^}{y}}_{j}$ is the class label corresponding to the class with the maximal score, and I{·} is the indicator function.

Hinge loss"hinge"$L=\sum _{j=1}^{n}{w}_{j}\mathrm{max}\left\{0,1-{m}_{j}\right\}.$
Logit loss"logit"$L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left(1+\mathrm{exp}\left(-{m}_{j}\right)\right).$
Quadratic loss"quadratic"$L=\sum _{j=1}^{n}{w}_{j}{\left(1-{m}_{j}\right)}^{2}.$

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

### 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) is the predicted value of observation j of the predictor data X.

• 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 argument.

Loss FunctionValue of LossFunEquation
Epsilon insensitive loss"epsiloninsensitive"$L=\mathrm{max}\left[0,|y-f\left(x\right)|-\epsilon \right].$
Mean squared error"mse"$L={\left[y-f\left(x\right)\right]}^{2}.$

## 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.

## Version History

Introduced in R2022a