# loss

Regression loss for generalized additive model (GAM)

## 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 train`Mdl`

, then you do not need to specify`ResponseVarName`

.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 in`Tbl`

or`X`

.`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:

$$\text{mse}=\frac{{\displaystyle \sum _{j=1}^{n}{w}_{j}{\left(f\left({x}_{j}\right)-{y}_{j}\right)}^{2}}}{{\displaystyle \sum _{j=1}^{n}{w}_{j}}}\text{\hspace{0.17em}},$$

where:

*n*is the number of rows of data.*x*is the_{j}*j*th row of data.*y*is the true response to_{j}*x*._{j}*f*(*x*) is the response prediction of the model_{j}`Mdl`

to*x*._{j}*w*is the vector of observation weights.

## Version History

**Introduced in R2021a**

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