resubPredict

Predict responses for training data using trained regression model

Syntax

yFit = resubPredict(Mdl)

yFit = resubPredict(Mdl,Name,Value)

[yFit,ySD,yInt] = resubPredict(___)

Description

yFit = resubPredict(Mdl) returns a vector of predicted responses for the trained regression model Mdl using the predictor data stored in Mdl.X.

yFit = resubPredict(Mdl,Name,Value) specifies options using one or more name-value arguments. For example, 'IncludeInteractions',true specifies to include interaction terms in computations for generalized additive models.

[yFit,ySD,yInt] = resubPredict(___) also returns the standard deviations and prediction intervals of the response variable, evaluated at each observation in the predictor data Mdl.X, using any of the input argument combinations in the previous syntaxes. This syntax applies only to generalized additive models for which IsStandardDeviationFit is true, and to Gaussian process regression models for which the PredictMethod is not 'bcd'.

Examples

Resubstitution Predictions

Train a generalized additive model (GAM), then predict responses for the training data.

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 = 
  RegressionGAM
            PredictorNames: {'Age'  'Diastolic'  'Smoker'  'Weight'  'Gender'  'SelfAssessedHealthStatus'}
              ResponseName: 'Systolic'
     CategoricalPredictors: [3 5 6]
         ResponseTransform: 'none'
                 Intercept: 122.7800
    IsStandardDeviationFit: 0
           NumObservations: 100

Mdl is a RegressionGAM model object.

Predict responses for the training set.

yFit = resubPredict(Mdl);

Create a table containing the observed response values and the predicted response values. Display the first eight rows of the table.

t = table(tbl.Systolic,yFit, ...
    'VariableNames',{'Observed Value','Predicted Value'});
head(t)

    Observed Value    Predicted Value
    ______________    _______________

         124              124.75     
         109              109.48     
         125              122.89     
         117              115.87     
         122              121.61     
         121              122.02     
         130              126.39     
         115              115.95     

Compute Prediction Intervals

Train a Gaussian process regression (GPR) model by using the fitrgp function. Then predict responses for the training data and estimate prediction intervals of the responses at each observation in the training data by using the resubPredict function.

Generate a training data set.

rng(1) % For reproducibility
n = 100000;
X = linspace(0,1,n)';
X = [X,X.^2];
y = 1 + X*[1;2] + sin(20*X*[1;-2]) + 0.2*randn(n,1);

Train a GPR model using the squared exponential kernel function. Estimate parameters by using the subset of regressors ('sr') approximation method, and make predictions using the subset of data ('sd') method. Use 50 points in the active set, and specify 'sgma' (sparse greedy matrix approximation) method for active set selection. Because the scales of the first and second predictors are different, standardize the data set.

gprMdl = fitrgp(X,y,'KernelFunction','squaredExponential', ...
    'FitMethod','sr','PredictMethod','sd', ...
    'ActiveSetSize',50,'ActiveSetMethod','sgma','Standardize',true);

fitrgp accepts any combination of fitting, prediction, and active set selection methods. However, if you train a model using the block coordinate descent prediction method ('PredictMethod','bcd'), you cannot use the model to compute the standard deviations of the predicted responses; therefore, you also cannot use the model to compute the prediction intervals. For more details, see Tips.

Use the trained model to predict responses for the training data and to estimate the prediction intervals of the predicted responses.

[ypred,~,yci] = resubPredict(gprMdl);

Plot the true responses, predicted responses, and prediction intervals.

figure
plot(y,'r')
hold on
plot(ypred,'b')
plot(yci(:,1),'k--')
plot(yci(:,2),'k--')
legend('True responses','GPR predictions','95% prediction limits','Location','Best')
xlabel('X')
ylabel('y')
hold off

Compute the mean squared error loss on the training data using the trained GPR model.

L = resubLoss(gprMdl)

L = 0.0523

Compare GAMs by Examining Resubstitution Predictions

Predict responses for a training data set using a generalized additive model (GAM) that contains both linear and interaction terms for predictors. Specify whether to include interaction terms when predicting responses.

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;

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

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

Mdl is a RegressionGAM model object.

Predict the responses using both linear and interaction terms, and then using only linear terms. To exclude interaction terms, specify 'IncludeInteractions',false.

yFit = resubPredict(Mdl);
yFit_nointeraction = resubPredict(Mdl,'IncludeInteractions',false);

Create a table containing the observed response values and the predicted response values. Display the first eight rows of the table.

t = table(Mdl.Y,yFit,yFit_nointeraction, ...
    'VariableNames',{'Observed Response', ...
    'Predicted Response','Predicted Response Without Interactions'});
head(t)

    Observed Response    Predicted Response    Predicted Response Without Interactions
    _________________    __________________    _______________________________________

           18                  18.026                           17.22                 
           15                  15.003                          15.791                 
           18                  17.663                           16.18                 
           16                  16.178                          15.536                 
           17                  17.107                          17.361                 
           15                  14.943                          14.424                 
           14                  14.119                          14.981                 
           14                  13.864                          13.498                 

Input Arguments

Mdl — Regression machine learning model

full regression model object

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

Model	Regression Model Object
Gaussian process regression model	RegressionGP
Generalized additive model (GAM)	RegressionGAM
Neural network model	RegressionNeuralNetwork

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: 'Alpha',0.01,'IncludeInteractions',false specifies the confidence level as 99% and excludes interaction terms from computations for a generalized additive model.

Alpha — Significance level

0.05 (default) | numeric scalar in [0,1]

Significance level for the confidence level of the prediction intervals yInt, specified as a numeric scalar in the range [0,1]. The confidence level of yInt is equal to 100(1 – Alpha)%.

This argument is valid only for a generalized additive model object that includes the standard deviation fit, or a Gaussian process regression model that does not use the block coordinate descent method for prediction. That is, you can specify this argument only in one of these situations:

• Mdl is RegressionGAM and the IsStandardDeviationFit property of Mdl is true.

• Mdl is RegressionGP and the PredictMethod property of Mdl is not 'bcd'.

Example: 'Alpha',0.01

Data Types: single | double

IncludeInteractions — Flag to include interaction terms

true | false

Flag to include interaction terms of the model, specified as true or false. This argument is valid only for a generalized additive model. 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.

Data Types: logical

PredictionForMissingValue — Predicted response value to use for observations with missing predictor values

"median" | "mean" | numeric scalar

Since R2023b

Predicted response value to use for observations with missing predictor values, specified as "median", "mean", or a numeric scalar. This argument is valid only for a Gaussian process regression or neural network model. That is, you can specify this argument only when Mdl is a RegressionGP or RegressionNeuralNetwork object.

Value	Description
"median"	resubPredict uses the median of the observed response values in the training data as the predicted response value for observations with missing predictor values. This value is the default when Mdl is a RegressionGP or RegressionNeuralNetwork object.
"mean"	resubPredict uses the mean of the observed response values in the training data as the predicted response value for observations with missing predictor values.
Numeric scalar	resubPredict uses this value as the predicted response value for observations with missing predictor values.

Example: "PredictionForMissingValue","mean"

Example: "PredictionForMissingValue",NaN

Data Types: single | double | char | string

Output Arguments

yFit — Predicted responses

vector

Predicted responses, returned as a vector of length n, where n is the number of observations in the predictor data (Mdl.X).

ySD — Standard deviations of response variable

column vector

Standard deviations of the response variable, evaluated at each observation in the predictor data Mdl.X, returned as a column vector of length n, where n is the number of observations in Mdl.X. The ith element ySD(i) contains the standard deviation of the ith response for the ith observation Mdl.X(i,:), estimated using the trained standard deviation model in Mdl.

This argument is valid only for a generalized additive model object that includes the standard deviation fit, or a Gaussian process regression model that does not use the block coordinate descent method for prediction. That is, resubPredict can return this argument only in one of these situations:

• Mdl is RegressionGAM and the IsStandardDeviationFit property of Mdl is true.

• Mdl is RegressionGP and the PredictMethod property of Mdl is not 'bcd'.

yInt — Prediction intervals of response variable

two-column matrix

Prediction intervals of the response variable, evaluated at each observation in the predictor data Mdl.X, returned as an n-by-2 matrix, where n is the number of observations in Mdl.X. The ith row yInt(i,:) contains the 100(1 – Alpha)% prediction interval of the ith response for the ith observation Mdl.X(i,:). The Alpha value is the probability that the prediction interval does not contain the true response value Mdl.Y(i). The first column of yInt contains the lower limits of the prediction intervals, and the second column contains the upper limits.

This argument is valid only for a generalized additive model object that includes the standard deviation fit, or a Gaussian process regression model that does not use the block coordinate descent method for prediction. That is, resubPredict can return this argument only in one of these situations:

• Mdl is RegressionGAM and the IsStandardDeviationFit property of Mdl is true.

• Mdl is RegressionGP and the PredictMethod property of Mdl is not 'bcd'.

Algorithms

resubPredict predicts responses according to the corresponding predict function of the object (Mdl). For a model-specific description, see the predict function reference pages in the following table.

Model	Regression Model Object (Mdl)	predict Object Function
Gaussian process regression model	RegressionGP	predict
Generalized additive model	RegressionGAM	predict
Neural network model	RegressionNeuralNetwork	predict

Alternative Functionality

To compute the predicted responses for new predictor data, use the corresponding predict function of the object (Mdl).

Version History

Introduced in R2015b

R2023b: Specify predicted response value to use for observations with missing predictor values

Starting in R2023b, when you predict or compute the loss, some regression models allow you to specify the predicted response value for observations with missing predictor values. Specify the PredictionForMissingValue name-value argument to use a numeric scalar, the training set median, or the training set mean as the predicted value. When computing the loss, you can also specify to omit observations with missing predictor values.

This table lists the object functions that support the PredictionForMissingValue name-value argument. By default, the functions use the training set median as the predicted response value for observations with missing predictor values.

Model Type	Model Objects	Object Functions
Gaussian process regression (GPR) model	RegressionGP, CompactRegressionGP	loss, predict, resubLoss, resubPredict
	RegressionPartitionedGP	kfoldLoss, kfoldPredict
Gaussian kernel regression model	RegressionKernel	loss, predict
	RegressionPartitionedKernel	kfoldLoss, kfoldPredict
Linear regression model	RegressionLinear	loss, predict
	RegressionPartitionedLinear	kfoldLoss, kfoldPredict
Neural network regression model	RegressionNeuralNetwork, CompactRegressionNeuralNetwork	loss, predict, resubLoss, resubPredict
	RegressionPartitionedNeuralNetwork	kfoldLoss, kfoldPredict
Support vector machine (SVM) regression model	RegressionSVM, CompactRegressionSVM	loss, predict, resubLoss, resubPredict
	RegressionPartitionedSVM	kfoldLoss, kfoldPredict

In previous releases, the regression model loss and predict functions listed above used NaN predicted response values for observations with missing predictor values. The software omitted observations with missing predictor values from the resubstitution ("resub") and cross-validation ("kfold") computations for prediction and loss.