kfoldLoss

Regression loss for cross-validated linear regression model

Syntax

L = kfoldLoss(CVMdl)

L = kfoldLoss(CVMdl,Name=Value)

Description

L = kfoldLoss(CVMdl) returns the cross-validated mean squared error (MSE) obtained by the cross-validated, linear regression model CVMdl. That is, for every fold, kfoldLoss estimates the regression loss for observations that it holds out when it trains using all other observations.

L contains a regression loss for each regularization strength in the linear regression models that compose CVMdl.

L = kfoldLoss(CVMdl,Name=Value) uses additional options specified by one or more name-value arguments. For example, indicate which folds to use for the loss calculation, or specify the regression loss function.

Examples

Estimate k-Fold Mean Squared Error

Simulate 10000 observations from this model

$$y=x_{100}+2x_{200}+e.$$

• $$X=\{x_1,...,x_{1000}\}$$ is a 10000-by-1000 sparse matrix with 10% nonzero standard normal elements.

• e is random normal error with mean 0 and standard deviation 0.3.

rng(1) % For reproducibility
n = 1e4;
d = 1e3;
nz = 0.1;
X = sprandn(n,d,nz);
Y = X(:,100) + 2*X(:,200) + 0.3*randn(n,1);

Cross-validate a linear regression model using SVM learners.

rng(1); % For reproducibility 
CVMdl = fitrlinear(X,Y,'CrossVal','on');

CVMdl is a RegressionPartitionedLinear model. By default, the software implements 10-fold cross validation. You can alter the number of folds using the 'KFold' name-value pair argument.

Estimate the average of the test-sample MSEs.

mse = kfoldLoss(CVMdl)

mse = 
0.1735

Alternatively, you can obtain the per-fold MSEs by specifying the name-value pair 'Mode','individual' in kfoldLoss.

Specify Custom Regression Loss

Simulate data as in Estimate k-Fold Mean Squared Error.

rng(1) % For reproducibility
n = 1e4;
d = 1e3;
nz = 0.1;
X = sprandn(n,d,nz); 
Y = X(:,100) + 2*X(:,200) + 0.3*randn(n,1);
X = X'; % Put observations in columns for faster training 

Cross-validate a linear regression model using 10-fold cross-validation. Optimize the objective function using SpaRSA.

CVMdl = fitrlinear(X,Y,'CrossVal','on','ObservationsIn','columns',...
    'Solver','sparsa');

CVMdl is a RegressionPartitionedLinear model. It contains the property Trained, which is a 10-by-1 cell array holding RegressionLinear models that the software trained using the training set.

Create an anonymous function that measures Huber loss ($$\delta$$ = 1), that is,

$$L=\frac{1}{\sum w_j}\sum _{j=1}^n{w}_j{\ell }_j,$$

where

$$\begin{array}{l}\\ {\ell }_j=\{\begin{array}{c}0.5{\underset{}{\overset{ˆ}{e_j}}}^2;\\ \left|\underset{}{\overset{ˆ}{e_j}}\right|-0.5;\end{array}\begin{array}{c}\left|\underset{}{\overset{ˆ}{e_j}}\right|\le 1\\ \left|\underset{}{\overset{ˆ}{e_j}}\right|>1\end{array}.\end{array}$$

$\underset{}{\overset{ˆ}{e_j}}$ is the residual for observation j. Custom loss functions must be written in a particular form. For rules on writing a custom loss function, see the 'LossFun' name-value pair argument.

huberloss = @(Y,Yhat,W)sum(W.*((0.5*(abs(Y-Yhat)<=1).*(Y-Yhat).^2) + ...
    ((abs(Y-Yhat)>1).*abs(Y-Yhat)-0.5)))/sum(W);

Estimate the average Huber loss over the folds. Also, obtain the Huber loss for each fold.

mseAve = kfoldLoss(CVMdl,'LossFun',huberloss)

mseAve = 
-0.4448

mseFold = kfoldLoss(CVMdl,'LossFun',huberloss,'Mode','individual')

mseFold = 10×1

   -0.4454
   -0.4473
   -0.4453
   -0.4469
   -0.4434
   -0.4434
   -0.4465
   -0.4430
   -0.4438
   -0.4426

Find Good Lasso Penalty Using Cross-Validation

To determine a good lasso-penalty strength for a linear regression model that uses least squares, implement 5-fold cross-validation.

Simulate 10000 observations from this model

$$y=x_{100}+2x_{200}+e.$$

• $$X=\{x_1,...,x_{1000}\}$$ is a 10000-by-1000 sparse matrix with 10% nonzero standard normal elements.

• e is random normal error with mean 0 and standard deviation 0.3.

rng(1) % For reproducibility
n = 1e4;
d = 1e3;
nz = 0.1;
X = sprandn(n,d,nz);
Y = X(:,100) + 2*X(:,200) + 0.3*randn(n,1);

Create a set of 15 logarithmically-spaced regularization strengths from $$10^{-5}$$ through $$10^{-1}$$.

Lambda = logspace(-5,-1,15);

Cross-validate the models. To increase execution speed, transpose the predictor data and specify that the observations are in columns. Optimize the objective function using SpaRSA.

X = X'; 
CVMdl = fitrlinear(X,Y,'ObservationsIn','columns','KFold',5,'Lambda',Lambda,...
    'Learner','leastsquares','Solver','sparsa','Regularization','lasso');

numCLModels = numel(CVMdl.Trained)

numCLModels = 
5

CVMdl is a RegressionPartitionedLinear model. Because fitrlinear implements 5-fold cross-validation, CVMdl contains 5 RegressionLinear models that the software trains on each fold.

Display the first trained linear regression model.

Mdl1 = CVMdl.Trained{1}

Mdl1 = 
  RegressionLinear
         ResponseName: 'Y'
    ResponseTransform: 'none'
                 Beta: [1000×15 double]
                 Bias: [-0.0049 -0.0049 -0.0049 -0.0049 -0.0049 -0.0048 -0.0044 -0.0037 -0.0030 -0.0031 -0.0033 -0.0036 -0.0041 -0.0051 -0.0071]
               Lambda: [1.0000e-05 1.9307e-05 3.7276e-05 7.1969e-05 1.3895e-04 2.6827e-04 5.1795e-04 1.0000e-03 0.0019 0.0037 0.0072 0.0139 0.0268 0.0518 0.1000]
              Learner: 'leastsquares'


  Properties, Methods

Mdl1 is a RegressionLinear model object. fitrlinear constructed Mdl1 by training on the first four folds. Because Lambda is a sequence of regularization strengths, you can think of Mdl1 as 15 models, one for each regularization strength in Lambda.

Estimate the cross-validated MSE.

mse = kfoldLoss(CVMdl);

Higher values of Lambda lead to predictor variable sparsity, which is a good quality of a regression model. For each regularization strength, train a linear regression model using the entire data set and the same options as when you cross-validated the models. Determine the number of nonzero coefficients per model.

Mdl = fitrlinear(X,Y,'ObservationsIn','columns','Lambda',Lambda,...
    'Learner','leastsquares','Solver','sparsa','Regularization','lasso');
numNZCoeff = sum(Mdl.Beta~=0);

In the same figure, plot the cross-validated MSE and frequency of nonzero coefficients for each regularization strength. Plot all variables on the log scale.

figure
[h,hL1,hL2] = plotyy(log10(Lambda),log10(mse),...
    log10(Lambda),log10(numNZCoeff)); 
hL1.Marker = 'o';
hL2.Marker = 'o';
ylabel(h(1),'log_{10} MSE')
ylabel(h(2),'log_{10} nonzero-coefficient frequency')
xlabel('log_{10} Lambda')
hold off

Choose the index of the regularization strength that balances predictor variable sparsity and low MSE (for example, Lambda(10)).

idxFinal = 10;

Extract the model with corresponding to the minimal MSE.

MdlFinal = selectModels(Mdl,idxFinal)

MdlFinal = 
  RegressionLinear
         ResponseName: 'Y'
    ResponseTransform: 'none'
                 Beta: [1000×1 double]
                 Bias: -0.0050
               Lambda: 0.0037
              Learner: 'leastsquares'


  Properties, Methods

idxNZCoeff = find(MdlFinal.Beta~=0)

idxNZCoeff = 2×1

   100
   200

EstCoeff = Mdl.Beta(idxNZCoeff)

EstCoeff = 2×1

    1.0051
    1.9965

MdlFinal is a RegressionLinear model with one regularization strength. The nonzero coefficients EstCoeff are close to the coefficients that simulated the data.

Input Arguments

CVMdl — Cross-validated, linear regression model

RegressionPartitionedLinear model object

Cross-validated, linear regression model, specified as a RegressionPartitionedLinear model object. You can create a RegressionPartitionedLinear model using fitrlinear and specifying any of the one of the cross-validation, name-value pair arguments, for example, CrossVal.

To obtain estimates, kfoldLoss applies the same data used to cross-validate the linear regression model (X and Y).

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: kfoldLoss(CVMdl,Folds=[1 2 3 5]) specifies to use the first, second, third, and fifth folds to compute the regression loss, but to exclude the fourth fold.

Folds — Fold indices to use for response prediction

1:CVMdl.KFold (default) | numeric vector of positive integers

Fold indices to use for response prediction, specified as a numeric vector of positive integers. The elements of Folds must range from 1 through CVMdl.KFold.

Example: Folds=[1 4 10]

Data Types: single | double

LossFun — Loss function

"mse" (default) | "epsiloninsensitive" | function handle

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

• The following table lists the available loss functions. Specify one using its corresponding character vector or string scalar. Also, in the table, $$f\left(x\right)=x\beta +b.$$

  • β is a vector of p coefficients.

  • x is an observation from p predictor variables.

  • b is the scalar bias.

  Value	Description
  "epsiloninsensitive"	Epsilon-insensitive loss: $$\ell \left[y,f\left(x\right)\right]=\max \left[0,\left|y-f\left(x\right)\right|-\epsilon \right]$$
  "mse"	MSE: $$\ell \left[y,f\left(x\right)\right]={\left[y-f\left(x\right)\right]}^2$$

  "epsiloninsensitive" is appropriate for SVM learners only.

• Specify your own function using function handle notation.

  Let n be the number of observations in X. Your function must have this signature

  lossvalue = lossfun(Y,Yhat,W)

  where:

  • The output argument lossvalue is a scalar.

  • You choose the function name (lossfun).

  • Y is an n-dimensional vector of observed responses. kfoldLoss passes the input argument Y in for Y.

  • Yhat is an n-dimensional vector of predicted responses, which is similar to the output of predict.

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

  Specify your function using LossFun=@lossfun.

Example: LossFun="epsiloninsensitive"

Data Types: char | string | function_handle

Mode — Loss aggregation level

"average" (default) | "individual"

Loss aggregation level, specified as "average" or "individual".

Value	Description
"average"	Returns losses averaged over all folds
"individual"	Returns losses for each fold

Example: Mode="individual"

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

"median" | "mean" | "omitted" | numeric scalar

Since R2023b

Predicted response value to use for observations with missing predictor values, specified as "median", "mean", "omitted", or a numeric scalar.

Value	Description
"median"	kfoldLoss uses the median of the observed response values in the training-fold data as the predicted response value for observations with missing predictor values.
"mean"	kfoldLoss uses the mean of the observed response values in the training-fold data as the predicted response value for observations with missing predictor values.
"omitted"	kfoldLoss excludes observations with missing predictor values from the loss computation.
Numeric scalar	kfoldLoss uses this value as the predicted response value for observations with missing predictor values.

If an observation is missing an observed response value or an observation weight, then kfoldLoss does not use the observation in the loss computation.

Example: PredictionForMissingValue="omitted"

Data Types: single | double | char | string

Output Arguments

L — Cross-validated regression losses

numeric scalar | numeric vector | numeric matrix

Cross-validated regression losses, returned as a numeric scalar, vector, or matrix. The interpretation of L depends on LossFun.

Let R be the number of regularizations strengths is the cross-validated models (stored in numel(CVMdl.Trained{1}.Lambda)) and F be the number of folds (stored in CVMdl.KFold).

• If Mode is 'average', then L is a 1-by-R vector. L(j) is the average regression loss over all folds of the cross-validated model that uses regularization strength j.

• Otherwise, L is an F-by-R matrix. L(i,j) is the regression loss for fold i of the cross-validated model that uses regularization strength j.

To estimate L, kfoldLoss uses the data that created CVMdl (see X and Y).

Extended Capabilities

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2016a

R2024a: Specify GPU arrays (requires Parallel Computing Toolbox)

kfoldLoss fully supports GPU arrays.

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.