loss
Regression loss for Gaussian kernel regression model
Description
returns the MSE for the model L
= loss(Mdl
,Tbl
,ResponseVarName
)Mdl
using the predictor data
in Tbl
and the true responses in
Tbl.ResponseVarName
.
specifies options using one or more name-value arguments in addition to any of
the input argument combinations in previous syntaxes. For example, you can
specify a regression loss function and observation weights. Then,
L
= loss(___,Name,Value
)loss
returns the weighted regression loss using the
specified loss function.
Examples
Calculate Sample Loss for Gaussian Kernel Regression Model
Train a Gaussian kernel regression model for a tall array, then calculate the resubstitution mean squared error and epsilon-insensitive error.
When you perform calculations on tall arrays, MATLAB® uses either a parallel pool (default if you have Parallel Computing Toolbox™) or the local MATLAB session. To run the example using the local MATLAB session when you have Parallel Computing Toolbox, change the global execution environment by using the mapreducer
function.
mapreducer(0)
Create a datastore that references the folder location with the data. The data can be contained in a single file, a collection of files, or an entire folder. Treat 'NA'
values as missing data so that datastore
replaces them with NaN
values. Select a subset of the variables to use. Create a tall table on top of the datastore.
varnames = {'ArrTime','DepTime','ActualElapsedTime'}; ds = datastore('airlinesmall.csv','TreatAsMissing','NA',... 'SelectedVariableNames',varnames); t = tall(ds);
Specify DepTime
and ArrTime
as the predictor variables (X
) and ActualElapsedTime
as the response variable (Y
). Select the observations for which ArrTime
is later than DepTime
.
daytime = t.ArrTime>t.DepTime; Y = t.ActualElapsedTime(daytime); % Response data X = t{daytime,{'DepTime' 'ArrTime'}}; % Predictor data
Standardize the predictor variables.
Z = zscore(X); % Standardize the data
Train a default Gaussian kernel regression model with the standardized predictors. Set 'Verbose',0
to suppress diagnostic messages.
[Mdl,FitInfo] = fitrkernel(Z,Y,'Verbose',0)
Mdl = RegressionKernel ResponseName: 'Y' Learner: 'svm' NumExpansionDimensions: 64 KernelScale: 1 Lambda: 8.5385e-06 BoxConstraint: 1 Epsilon: 5.9303
FitInfo = struct with fields:
Solver: 'LBFGS-tall'
LossFunction: 'epsiloninsensitive'
Lambda: 8.5385e-06
BetaTolerance: 1.0000e-03
GradientTolerance: 1.0000e-05
ObjectiveValue: 26.1409
GradientMagnitude: 0.0023
RelativeChangeInBeta: 0.0150
FitTime: 25.1844
History: []
Mdl
is a trained RegressionKernel
model, and the structure array FitInfo
contains optimization details.
Determine how well the trained model generalizes to new predictor values by estimating the resubstitution mean squared error and epsilon-insensitive error.
lossMSE = loss(Mdl,Z,Y) % Resubstitution mean squared error
lossMSE = MxNx... tall array ? ? ? ... ? ? ? ... ? ? ? ... : : : : : :
lossEI = loss(Mdl,Z,Y,'LossFun','epsiloninsensitive') % Resubstitution epsilon-insensitive error
lossEI = MxNx... tall array ? ? ? ... ? ? ? ... ? ? ? ... : : : : : :
Evaluate the tall arrays and bring the results into memory by using gather
.
[lossMSE,lossEI] = gather(lossMSE,lossEI)
Evaluating tall expression using the Local MATLAB Session: - Pass 1 of 1: Completed in 0.79 sec Evaluation completed in 1.1 sec
lossMSE = 2.5141e+03
lossEI = 25.5148
Specify Custom Regression Loss
Specify a custom regression loss (Huber loss) for a Gaussian kernel regression model.
Load the carbig
data set.
load carbig
Specify the predictor variables (X
) and the response variable (Y
).
X = [Weight,Cylinders,Horsepower,Model_Year]; Y = MPG;
Delete rows of X
and Y
where either array has NaN
values. Removing rows with NaN
values before passing data to fitrkernel
can speed up training and reduce memory usage.
R = rmmissing([X Y]); X = R(:,1:4); Y = R(:,end);
Reserve 10% of the observations as a holdout sample. Extract the training and test indices from the partition definition.
rng(10) % For reproducibility N = length(Y); cvp = cvpartition(N,'Holdout',0.1); idxTrn = training(cvp); % Training set indices idxTest = test(cvp); % Test set indices
Train the regression kernel model. Standardize the training data.
Xtrain = X(idxTrn,:);
Ytrain = Y(idxTrn);
Mdl = fitrkernel(Xtrain,Ytrain,'Standardize',true)
Mdl = RegressionKernel ResponseName: 'Y' Learner: 'svm' NumExpansionDimensions: 128 KernelScale: 1 Lambda: 0.0028 BoxConstraint: 1 Epsilon: 0.8617
Mdl
is a RegressionKernel
model.
Create an anonymous function that measures Huber loss , 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 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 training set regression loss using the Huber loss function.
eTrain = loss(Mdl,Xtrain,Ytrain,'LossFun',huberloss)
eTrain = 1.7210
Estimate the test set regression loss using the Huber loss function.
Xtest = X(idxTest,:);
Ytest = Y(idxTest);
eTest = loss(Mdl,Xtest,Ytest,'LossFun',huberloss)
eTest = 1.3062
Input Arguments
Mdl
— Kernel regression model
RegressionKernel
model object
Kernel regression model, specified as a RegressionKernel
model object. You can create a
RegressionKernel
model object using fitrkernel
.
X
— Predictor data
n-by-p numeric matrix
Predictor data, specified as an
n-by-p numeric matrix, where
n is the number of observations and
p is the number of predictors. p
must be equal to the number of predictors used to train
Mdl
.
Data Types: single
| double
Tbl
— Sample data
table
Sample data used to train the model, specified as a table. Each row of
Tbl
corresponds to one observation, and each column corresponds
to one predictor variable. Optionally, Tbl
can contain additional
columns for the response variable and observation weights. Tbl
must
contain all the predictors used to train Mdl
. Multicolumn variables
and cell arrays other than cell arrays of character vectors are not allowed.
If Tbl
contains the response variable used to train Mdl
, then you do not need to specify ResponseVarName
or Y
.
If you train Mdl
using sample data contained in a table, then the input
data for loss
must also be in a table.
ResponseVarName
— Response variable name
name of variable in Tbl
Response variable name, specified as the name of a variable in
Tbl
. The response variable must be a numeric
vector. If Tbl
contains the response variable used to
train Mdl
, then you do not need to specify
ResponseVarName
.
If you specify ResponseVarName
, then you must specify
it as a character vector or string scalar. For example, if the response
variable is stored as Tbl.Y
, then specify
ResponseVarName
as 'Y'
.
Otherwise, the software treats all columns of Tbl
,
including Tbl.Y
, as predictors.
Data Types: char
| string
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: L =
loss(Mdl,X,Y,'LossFun','epsiloninsensitive','Weights',weights)
returns
the weighted regression loss using the epsilon-insensitive loss
function.
LossFun
— Loss function
'mse'
(default) | 'epsiloninsensitive'
| function handle
Loss function, specified as the comma-separated pair consisting of
'LossFun'
and a built-in loss function name or a
function handle.
The following table lists the available loss functions. Specify one using its corresponding character vector or string scalar. Also, in the table,
x is an observation (row vector) from p predictor variables.
is a transformation of an observation (row vector) for feature expansion. T(x) maps x in to a high-dimensional space ().
β is a vector of m coefficients.
b is the scalar bias.
Value Description 'epsiloninsensitive'
Epsilon-insensitive loss: 'mse'
MSE: 'epsiloninsensitive'
is appropriate for SVM learners only.Specify your own function by using function handle notation.
Let
n
be the number of observations inX
. Your function must have this signature:lossvalue =
lossfun
(Y,Yhat,W)The output argument
lossvalue
is a scalar.You choose the function name (
lossfun
).Y
is an n-dimensional vector of observed responses.loss
passes the input argumentY
in forY
.Yhat
is an n-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
Data Types: char
| string
| function_handle
PredictionForMissingValue
— Predicted response value to use for observations with missing predictor values
"median"
(default) | "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" | loss uses the median of the observed
response values in the training data as the predicted response value for
observations with missing predictor values. |
"mean" | loss uses the mean of the observed
response values in the training data as the predicted response value for
observations with missing predictor values. |
"omitted" | loss excludes observations with missing
predictor values from the loss computation. |
Numeric scalar | loss 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
loss
does not use the observation in the loss
computation.
Example: PredictionForMissingValue="omitted"
Data Types: single
| double
| char
| string
Weights
— Observation weights
ones(size(X,1),1)
(default) | numeric vector | name of variable in Tbl
Observation weights, specified as the comma-separated pair consisting
of 'Weights'
and a numeric vector or the name of a
variable in Tbl
.
If
Weights
is a numeric vector, then the size ofWeights
must be equal to the number of rows inX
orTbl
.If
Weights
is the name of a variable inTbl
, you must specifyWeights
as a character vector or string scalar. For example, if the weights are stored asTbl.W
, then specifyWeights
as'W'
. Otherwise, the software treats all columns ofTbl
, includingTbl.W
, as predictors.
If you supply the observation weights, loss
computes the weighted regression loss, that is, the Weighted Mean Squared Error or
Epsilon-Insensitive Loss Function.
loss
normalizes Weights
to
sum to 1.
Data Types: double
| single
| char
| string
Output Arguments
More About
Weighted Mean Squared Error
The weighted mean squared error is calculated as follows:
where:
n is the number of observations.
xj is the jth observation (row of predictor data).
yj is the observed response to xj.
f(xj) is the response prediction of the Gaussian kernel regression model
Mdl
to xj.w is the vector of observation weights.
Each observation weight in w is equal to
ones(n,1)/n
by
default. You can specify different values for the observation weights by using the
'Weights'
name-value pair argument.
loss
normalizes Weights
to sum to
1.
Epsilon-Insensitive Loss Function
The epsilon-insensitive loss function ignores errors that are within the distance epsilon (ε) of the function value. The function is formally described as:
The mean epsilon-insensitive loss is calculated as follows:
where:
n is the number of observations.
xj is the jth observation (row of predictor data).
yj is the observed response to xj.
f(xj) is the response prediction of the Gaussian kernel regression model
Mdl
to xj.w is the vector of observation weights.
Each observation weight in w is equal to
ones(n,1)/n
by
default. You can specify different values for the observation weights by using the
'Weights'
name-value pair argument.
loss
normalizes Weights
to sum to
1.
Extended Capabilities
Tall Arrays
Calculate with arrays that have more rows than fit in memory.
The
loss
function supports tall arrays with the following usage
notes and limitations:
loss
does not support talltable
data.
For more information, see Tall Arrays.
Version History
Introduced in R2018aR2023b: 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.
R2022a: loss
can return NaN for predictor data with missing values
The loss
function no longer omits an observation with a
NaN prediction when computing the weighted average regression loss. Therefore,
loss
can now return NaN when the predictor data
X
or the predictor variables in Tbl
contain any missing values. In most cases, if the test set observations do not contain
missing predictors, the loss
function does not return
NaN.
This change improves the automatic selection of a regression model when you use
fitrauto
.
Before this change, the software might select a model (expected to best predict the
responses for new data) with few non-NaN predictors.
If loss
in your code returns NaN, you can update your code
to avoid this result. Remove or replace the missing values by using rmmissing
or fillmissing
, respectively.
The following table shows the regression models for which the
loss
object function might return NaN. For more details,
see the Compatibility Considerations for each loss
function.
Model Type | Full or Compact Model Object | loss Object Function |
---|---|---|
Gaussian process regression (GPR) model | RegressionGP , CompactRegressionGP | loss |
Gaussian kernel regression model | RegressionKernel | loss |
Linear regression model | RegressionLinear | loss |
Neural network regression model | RegressionNeuralNetwork , CompactRegressionNeuralNetwork | loss |
Support vector machine (SVM) regression model | RegressionSVM , CompactRegressionSVM | loss |
See Also
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