# resubLoss

Resubstitution regression loss

## Syntax

``L = resubLoss(Mdl)``
``L = resubLoss(Mdl,Name,Value)``

## Description

example

````L = resubLoss(Mdl)` returns the regression loss by resubstitution (L), or the in-sample regression loss, for the trained regression model `Mdl` using the training data stored in `Mdl.X` and the corresponding responses stored in `Mdl.Y`. The interpretation of `L` depends on the loss function (`'LossFun'`) and weighting scheme (`Mdl.W`). In general, better models yield smaller loss values. The default `'LossFun'` value is `'mse'` (mean squared error).```

example

````L = resubLoss(Mdl,Name,Value)` specifies additional options using one or more name-value arguments. For example, `'IncludeInteractions',false` specifies to exclude interaction terms from a generalized additive model `Mdl`.```

## Examples

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Train a generalized additive model (GAM), then calculate the resubstitution loss using the mean squared error (MSE).

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);`

Train a univariate GAM that contains the linear terms for the predictors in `tbl`.

`Mdl = fitrgam(tbl,"Systolic")`
```Mdl = RegressionGAM PredictorNames: {1x6 cell} ResponseName: 'Systolic' CategoricalPredictors: [3 5 6] ResponseTransform: 'none' Intercept: 122.7800 IsStandardDeviationFit: 0 NumObservations: 100 Properties, Methods ```

`Mdl` is a `RegressionGAM` model object.

Calculate the resubstitution loss using the mean squared error (MSE).

`L = resubLoss(Mdl)`
```L = 4.1957 ```

Load the sample data and store in a `table`.

```load fisheriris tbl = table(meas(:,1),meas(:,2),meas(:,3),meas(:,4),species,... 'VariableNames',{'meas1','meas2','meas3','meas4','species'});```

Fit a GPR model using the first measurement as the response and the other variables as the predictors.

`mdl = fitrgp(tbl,'meas1');`

Predict the responses using the trained model.

`ypred = predict(mdl,tbl);`

Compute the mean absolute error.

```n = height(tbl); y = tbl.meas1; fun = @(y,ypred,w) sum(abs(y-ypred))/n; L = resubLoss(mdl,'lossfun',fun)```
```L = 0.2345 ```

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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Regression machine learning model, specified as a full regression model object, as given in the following table of supported models.

ModelRegression Model Object
Gaussian process regression model`RegressionGP`
Generalized additive model (GAM)`RegressionGAM`
Neural network model`RegressionNeuralNetwork`

### 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: `resubLoss(Mdl,'IncludeInteractions',false)` excludes interaction terms from a generalized additive model `Mdl`.

Flag to include interaction terms of the model, specified as `true` or `false`. This argument is valid only for a generalized additive model. That is, you can specify this argument only when `Mdl` is `RegressionGAM`.

The default 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`

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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.

## Algorithms

`resubLoss` computes the regression loss according to the corresponding `loss` function of the object (`Mdl`). For a model-specific description, see the `loss` function reference pages in the following table.

ModelRegression Model Object (`Mdl`)`loss` Object Function
Gaussian process regression model`RegressionGP``loss`
Generalized additive model`RegressionGAM``loss`
Neural network model`RegressionNeuralNetwork``loss`

## Alternative Functionality

To compute the response loss for new predictor data, use the corresponding `loss` function of the object (`Mdl`).

## Version History

Introduced in R2021a