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Resubstitution regression loss



    L = resubLoss(Mdl) returns the regression loss by resubstitution (L), or the in-sample regression loss, for the trained regression model Mdl using the training data stored in Mdl.X and the corresponding responses stored in Mdl.Y.

    The interpretation of L depends on the loss function ('LossFun') and weighting scheme (Mdl.W). In general, better models yield smaller loss values. The default 'LossFun' value is 'mse' (mean squared error).


    L = resubLoss(Mdl,Name,Value) specifies additional options using one or more name-value arguments. For example, 'IncludeInteractions',false specifies to exclude interaction terms from a generalized additive model Mdl.


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    Train a generalized additive model (GAM), then calculate the resubstitution loss using the mean squared error (MSE).

    Load the patients data set.

    load patients

    Create a table that contains the predictor variables (Age, Diastolic, Smoker, Weight, Gender, SelfAssessedHealthStatus) and the response variable (Systolic).

    tbl = table(Age,Diastolic,Smoker,Weight,Gender,SelfAssessedHealthStatus,Systolic);

    Train a univariate GAM that contains the linear terms for the predictors in tbl.

    Mdl = fitrgam(tbl,"Systolic")
    Mdl = 
               PredictorNames: {1x6 cell}
                 ResponseName: 'Systolic'
        CategoricalPredictors: [3 5 6]
            ResponseTransform: 'none'
                    Intercept: 122.7800
              NumObservations: 100
      Properties, Methods

    Mdl is a RegressionGAM model object.

    Calculate the resubstitution loss using the mean squared error (MSE).

    L = resubLoss(Mdl)
    L = 4.1957

    Train a generalized additive model (GAM) that contains both linear and interaction terms for predictors, and estimate the regression loss (mean squared error, MSE) with and without interaction terms for the training data and test data. Specify whether to include interaction terms when estimating the regression loss.

    Load the carbig data set, which contains measurements of cars made in the 1970s and early 1980s.

    load carbig

    Specify Acceleration, Displacement, Horsepower, and Weight as the predictor variables (X) and MPG as the response variable (Y).

    X = [Acceleration,Displacement,Horsepower,Weight];
    Y = MPG;

    Partition the data set into two sets: one containing training data, and the other containing new, unobserved test data. Reserve 10 observations for the new test data set.

    rng('default') % For reproducibility
    n = size(X,1);
    newInds = randsample(n,10);
    inds = ~ismember(1:n,newInds);
    XNew = X(newInds,:);
    YNew = Y(newInds);

    Train a generalized additive model that contains all the available linear and interaction terms in X.

    Mdl = fitrgam(X(inds,:),Y(inds),'Interactions','all');

    Mdl is a RegressionGAM model object.

    Compute the resubstitution MSEs (that is, the in-sample MSEs) both with and without interaction terms in Mdl. To exclude interaction terms, specify 'IncludeInteractions',false.

    resubl = resubLoss(Mdl)
    resubl = 0.0292
    resubl_nointeraction = resubLoss(Mdl,'IncludeInteractions',false)
    resubl_nointeraction = 4.7330

    Compute the regression MSEs both with and without interaction terms for the test data set. Use a memory-efficient model object for the computation.

    CMdl = compact(Mdl);

    CMdl is a CompactRegressionGAM model object.

    l = loss(CMdl,XNew,YNew)
    l = 12.8604
    l_nointeraction = loss(CMdl,XNew,YNew,'IncludeInteractions',false)
    l_nointeraction = 15.6741

    Including interaction terms achieves a smaller error for the training data set and test data set.

    Input Arguments

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    Regression machine learning model, specified as a full regression model object, as given in the following table of supported models.

    ModelRegression Model Object
    Generalized additive modelRegressionGAM
    Neural network modelRegressionNeuralNetwork

    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: resubLoss(Mdl,'IncludeInteractions',false) excludes interaction terms from a generalized additive model Mdl.

    Flag to include interaction terms of the model, specified as true or false. This argument is valid only for a generalized additive model (GAM). That is, you can specify this argument only when Mdl is RegressionGAM.

    The default value is true if Mdl contains interaction terms. The value must be false if the model does not contain interaction terms.

    Example: 'IncludeInteractions',false

    Data Types: logical

    Loss function, specified as 'mse' or a function handle.

    • 'mse' — Weighted mean squared error.

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

      lossval = lossfun(Y,YFit,W)

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

      • You specify the function name (lossfun).

      • Y is a length n numeric vector of observed responses, where n is the number of observations in Tbl or X.

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

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

    Example: 'LossFun',@lossfun

    Data Types: char | string | function_handle

    More About

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    Weighted Mean Squared Error

    The weighted mean squared error measures the predictive inaccuracy of regression models. When you compare the same type of loss among many models, a lower error indicates a better predictive model.

    The weighted mean squared error is calculated as follows:



    • n is the number of rows of data.

    • xj is the jth row of data.

    • yj is the true response to xj.

    • f(xj) is the response prediction of the model Mdl to xj.

    • w is the vector of observation weights.


    resubLoss computes the regression loss according to the corresponding loss function of the object (Mdl). For a model-specific description, see the loss function reference pages in the following table.

    ModelRegression Model Object (Mdl)loss Object Function
    Generalized additive modelRegressionGAMloss
    Neural network modelRegressionNeuralNetworkloss

    See Also

    Introduced in R2021a