# resubLoss

Resubstitution regression loss

## Description

returns the regression loss by resubstitution (L), or the in-sample regression loss, for the
trained regression model `L`

= resubLoss(`Mdl`

)`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).

specifies additional options using one or more name-value arguments. For example,
`L`

= resubLoss(`Mdl`

,`Name,Value`

)`'IncludeInteractions',false`

specifies to exclude interaction terms from
a generalized additive model `Mdl`

.

## Examples

### Resubstitution Loss

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: {'Age' 'Diastolic' 'Smoker' 'Weight' 'Gender' 'SelfAssessedHealthStatus'} ResponseName: 'Systolic' CategoricalPredictors: [3 5 6] ResponseTransform: 'none' Intercept: 122.7800 IsStandardDeviationFit: 0 NumObservations: 100

`Mdl`

is a `RegressionGAM`

model object.

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

L = resubLoss(Mdl)

L = 4.1957

### Compute Custom Resubstitution Loss

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

### 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`

— Regression machine learning model

full regression model object

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

Model | Regression 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`

.

`IncludeInteractions`

— Flag to include interaction terms

`true`

| `false`

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`

`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`

`PredictionForMissingValue`

— Predicted response value to use for observations with missing predictor values

`"median"`

| `"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. This argument is valid only for a
Gaussian process regression or neural network model. That is, you can specify this
argument only when `Mdl`

is a `RegressionGP`

or `RegressionNeuralNetwork`

object.

Value | Description |
---|---|

`"median"` |
This value is the
default when |

`"mean"` | `resubLoss` uses the mean of the observed response
values in the training data as the predicted response value for observations
with missing predictor values. |

`"omitted"` | `resubLoss` excludes observations with missing
predictor values from the loss computation. |

Numeric scalar | `resubLoss` uses this value as the predicted
response value for observations with missing predictor values. |

If an observation is missing all predictor values, an observed response value, or
an observation weight, then `resubLoss`

does not use the
observation in the loss computation.

**Example: **`"PredictionForMissingValue","omitted"`

**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.

## 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.

Model | Regression 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 R2015b**

### R2023b: 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.

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