kfoldLoss
Regression loss for cross-validated linear regression model
Description
returns
the cross-validated mean squared error (MSE) obtained by the cross-validated, linear
regression model L
= kfoldLoss(CVMdl
)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
.
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.L
= kfoldLoss(CVMdl
,Name=Value
)
Examples
Estimate k-Fold Mean Squared Error
Simulate 10000 observations from this model
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 ( = 1), that is,
where
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
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 through .
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: [1000x15 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'
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: [1000x1 double] Bias: -0.0050 Lambda: 0.0037 Learner: 'leastsquares'
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,
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: "mse"
MSE: "epsiloninsensitive"
is appropriate for SVM learners only.Specify your own function using function handle notation.
Let
n
be the number of observations inX
. Your function must have this signaturewhere:lossvalue =
lossfun
(Y,Yhat,W)The output argument
lossvalue
is a scalar.You choose the function name (
lossfun
).Y
is ann
-dimensional vector of observed responses.kfoldLoss
passes the input argumentY
in forY
.Yhat
is ann
-dimensional vector of predicted responses, which is similar to the output ofpredict
.W
is ann
-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'
, thenL
is a 1-by-R
vector.L(
is the average regression loss over all folds of the cross-validated model that uses regularization strengthj
)j
.Otherwise,
L
is anF
-by-R
matrix.L(
is the regression loss for foldi
,j
)i
of the cross-validated model that uses regularization strengthj
.
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 R2016aR2024a: 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.
See Also
RegressionPartitionedLinear
| RegressionLinear
| kfoldPredict
| loss
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