loss
Description
returns the regression loss (L
= loss(Mdl
,Tbl
,ResponseVarName
)L
), a scalar representing how well the
generalized additive model Mdl
predicts the predictor data in
Tbl
compared to the true response values in
Tbl.ResponseVarName
.
The interpretation of L
depends on the loss function
('LossFun'
) and weighting scheme ('Weights'
). In
general, better models yield smaller loss values. The default 'LossFun'
value is 'mse'
(mean squared error).
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 the loss function
and the observation weights.L
= loss(___,Name,Value
)
Examples
Determine Test Sample Regression Loss
Determine the test sample regression loss (mean squared error) of a generalized additive model. When you compare the same type of loss among many models, a lower loss indicates a better predictive model.
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);
Randomly partition observations into a training set and a test set. Specify a 10% holdout sample for testing.
rng('default') % For reproducibility cv = cvpartition(size(tbl,1),'HoldOut',0.10);
Extract the training and test indices.
trainInds = training(cv); testInds = test(cv);
Train a univariate GAM that contains the linear terms for the predictors in tbl
.
Mdl = fitrgam(tbl(trainInds,:),"Systolic");
Determine how well the algorithm generalizes by estimating the test sample regression loss. By default, the loss
function of RegressionGAM
estimates the mean squared error.
L = loss(Mdl,tbl(testInds,:))
L = 35.7540
Compare GAMs by Examining Regression Loss
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
Mdl
— Generalized additive model
RegressionGAM
model object | CompactRegressionGAM
model object
Generalized additive model, specified as a RegressionGAM
or CompactRegressionGAM
model object.
Tbl
— Sample data
table
Sample data, specified as a table. Each row of Tbl
corresponds
to one observation, and each column corresponds to one predictor variable. Multicolumn
variables and cell arrays other than cell arrays of character vectors are not
allowed.
Tbl
must contain all of the predictors used to train
Mdl
. Optionally, Tbl
can contain a column
for the response variable and a column for the observation weights.
The response variable must be a numeric vector. If the response variable in
Tbl
has the same name as the response variable used to trainMdl
, then you do not need to specifyResponseVarName
.The weight values must be a numeric vector. You must specify the observation weights in
Tbl
by using'Weights'
.
If you trained Mdl
using sample data contained in a table, then
the input data for loss
must also be in a table.
Data Types: table
ResponseVarName
— Response variable name
name of variable in Tbl
Response variable name, specified as a character vector or string scalar containing the name
of the response variable in Tbl
. For example, if the response
variable Y
is stored in Tbl.Y
, then specify it as
'Y'
.
Data Types: char
| string
X
— Predictor data
numeric matrix
Predictor data, specified as a numeric matrix. Each row of X
corresponds to one observation, and each column corresponds to one predictor variable.
If you trained Mdl
using sample data contained in a matrix, then the input data for loss
must also be in a matrix.
Data Types: single
| double
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: 'IncludeInteractions',false,'Weights',w
specifies to exclude
interaction terms from the model and to use the observation weights
w
.
IncludeInteractions
— Flag to include interaction terms
true
| false
Flag to include interaction terms of the model, specified as true
or
false
.
The default 'IncludeInteractions'
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
LossFun
— Loss function
'mse'
(default) | function handle
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 inTbl
orX
.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
Weights
— Observation weights
ones(size(X,1),1)
(default) | vector of scalar values | name of variable in Tbl
Observation weights, specified as a vector of scalar values or the name of a variable in Tbl
. The software weights the observations in each row of X
or Tbl
with the corresponding value in Weights
. The size of Weights
must equal the number of rows in X
or Tbl
.
If you specify the input data as a table Tbl
, then Weights
can be the name of a variable in Tbl
that contains a numeric vector. In this case, you must specify Weights
as a character vector or string scalar. For example, if weights vector W
is stored as Tbl.W
, then specify it as 'W'
.
loss
normalizes the values of Weights
to sum to 1.
Data Types: single
| double
| char
| string
More About
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:
where:
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.
Version History
Introduced in R2021a
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