# loss

Regression loss for generalized additive model (GAM)

## Syntax

``L = loss(Mdl,Tbl,ResponseVarName)``
``L = loss(Mdl,Tbl,Y)``
``L = loss(Mdl,X,Y)``
``L = loss(___,Name,Value)``

## Description

example

````L = loss(Mdl,Tbl,ResponseVarName)` returns the regression loss (`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).```
````L = loss(Mdl,Tbl,Y)` uses the predictor data in table `Tbl` and the true response values in `Y`.```
````L = loss(Mdl,X,Y)` uses the predictor data in matrix `X` and the true response values in `Y`.```

example

````L = loss(___,Name,Value)` 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.```

## Examples

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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 ```

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

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Generalized additive model, specified as a `RegressionGAM` or `CompactRegressionGAM` model object.

• If you trained `Mdl` using sample data contained in a table, then the input data for `loss` must also be in a table (`Tbl`).

• If you trained `Mdl` using sample data contained in a matrix, then the input data for `loss` must also be in a matrix (`X`).

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`

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`

Response data, specified as a numeric column vector. Each entry in `Y` is the response to the data in the corresponding row of `X` or `Tbl`.

Data Types: `single` | `double`

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`.

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`

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`

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`

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### 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:

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

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