# resubPredict

Predict responses for training data using trained regression model

## Syntax

## Description

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

= resubPredict(`Mdl`

,`Name,Value`

)`'IncludeInteractions',true`

specifies to include interaction terms in
computations for generalized additive models. This syntax applies only to generalized
additive models and Gaussian process regression models.

`[`

also returns the standard deviations and prediction intervals of the response variable,
evaluated at each observation in the predictor data `yFit`

,`ySD`

,`yInt`

] = resubPredict(___)`Mdl.X`

, using any of
the input argument combinations in the previous syntaxes. This syntax applies only to
generalized additive models for which `IsStandardDeviationFit`

is `true`

, and to Gaussian process
regression models for which the `PredictMethod`

is not
`'bcd'`

.

## Examples

### Resubstitution Predictions

Train a generalized additive model (GAM), then predict responses for the training data.

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.

Predict responses for the training set.

yFit = resubPredict(Mdl);

Create a table containing the observed response values and the predicted response values. Display the first eight rows of the table.

t = table(tbl.Systolic,yFit, ... 'VariableNames',{'Observed Value','Predicted Value'}); head(t)

`ans=`*8×2 table*
Observed Value Predicted Value
______________ _______________
124 124.75
109 109.48
125 122.89
117 115.87
122 121.61
121 122.02
130 126.39
115 115.95

### Compute Prediction Intervals

Train a Gaussian process regression (GPR) model by using the `fitrgp`

function. Then predict responses for the training data and estimate prediction intervals of the responses at each observation in the training data by using the `resubPredict`

function.

Generate a training data set.

```
rng(1) % For reproducibility
n = 100000;
X = linspace(0,1,n)';
X = [X,X.^2];
y = 1 + X*[1;2] + sin(20*X*[1;-2]) + 0.2*randn(n,1);
```

Train a GPR model using the squared exponential kernel function. Estimate parameters by using the subset of regressors (`'sr'`

) approximation method, and make predictions using the subset of data (`'sd'`

) method. Use 50 points in the active set, and specify `'sgma'`

(sparse greedy matrix approximation) method for active set selection. Because the scales of the first and second predictors are different, standardize the data set.

gprMdl = fitrgp(X,y,'KernelFunction','squaredExponential', ... 'FitMethod','sr','PredictMethod','sd', ... 'ActiveSetSize',50,'ActiveSetMethod','sgma','Standardize',true);

`fitrgp`

accepts any combination of fitting, prediction, and active set selection methods. However, if you train a model using the block coordinate descent prediction method (`'PredictMethod','bcd'`

), you cannot use the model to compute the standard deviations of the predicted responses; therefore, you also cannot use the model to compute the prediction intervals. For more details, see Tips.

Use the trained model to predict responses for the training data and to estimate the prediction intervals of the predicted responses.

[ypred,~,yci] = resubPredict(gprMdl);

Plot the true responses, predicted responses, and prediction intervals.

figure plot(y,'r') hold on plot(ypred,'b') plot(yci(:,1),'k--') plot(yci(:,2),'k--') legend('True responses','GPR predictions','95% prediction limits','Location','Best') xlabel('X') ylabel('y') hold off

Compute the mean squared error loss on the training data using the trained GPR model.

L = resubLoss(gprMdl)

L = 0.0523

### Compare GAMs by Examining Resubstitution Predictions

Predict responses for a training data set using a generalized additive model (GAM) that contains both linear and interaction terms for predictors. Specify whether to include interaction terms when predicting responses.

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;

Train a generalized additive model that contains all the available linear and interaction terms in `X`

.

Mdl = fitrgam(X,Y,'Interactions','all');

`Mdl`

is a `RegressionGAM`

model object.

Predict the responses using both linear and interaction terms, and then using only linear terms. To exclude interaction terms, specify `'IncludeInteractions',false`

.

```
yFit = resubPredict(Mdl);
yFit_nointeraction = resubPredict(Mdl,'IncludeInteractions',false);
```

Create a table containing the observed response values and the predicted response values. Display the first eight rows of the table.

t = table(Mdl.Y,yFit,yFit_nointeraction, ... 'VariableNames',{'Observed Response', ... 'Predicted Response','Predicted Response Without Interactions'}); head(t)

`ans=`*8×3 table*
Observed Response Predicted Response Predicted Response Without Interactions
_________________ __________________ _______________________________________
18 18.026 17.22
15 15.003 15.791
18 17.663 16.18
16 16.178 15.536
17 17.107 17.361
15 14.943 14.424
14 14.119 14.981
14 13.864 13.498

## 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: **`'Alpha',0.01,'IncludeInteractions',false`

specifies the
confidence level as 99% and excludes interaction terms from computations for a generalized
additive model.

`Alpha`

— Significance level

0.05 (default) | numeric scalar in `[0,1]`

Significance level for the confidence level of the prediction intervals
`yInt`

, specified as a numeric scalar in the range
`[0,1]`

. The confidence level of `yInt`

is equal
to `100(1 – Alpha)%`

.

This argument is valid only for a generalized additive model object that includes the standard deviation fit, or a Gaussian process regression model that does not use the block coordinate descent method for prediction. That is, you can specify this argument only in one of these situations:

`Mdl`

is`RegressionGAM`

and the`IsStandardDeviationFit`

property of`Mdl`

is`true`

.`Mdl`

is`RegressionGP`

and the`PredictMethod`

property of`Mdl`

is not`'bcd'`

.

**Example: **`'Alpha',0.01`

**Data Types: **`single`

| `double`

`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.

**Data Types: **`logical`

## Output Arguments

`yFit`

— Predicted responses

vector

Predicted responses, returned as a vector of length *n*, where
*n* is the number of observations in the predictor data
(`Mdl.X`

).

`ySD`

— Standard deviations of response variable

column vector

Standard deviations of the response variable, evaluated at each observation in the
predictor data

, returned as a column
vector of length `Mdl`

.X*n*, where *n* is the number of
observations in

. The
`Mdl`

.X`i`

th element `ySD(i)`

contains the standard deviation
of the `i`

th response for the `i`

th observation
`Mdl.X(i,:)`

, estimated using the trained standard deviation model in
`Mdl`

.

This argument is valid only for a generalized additive model object that includes
the standard deviation fit, or a Gaussian process regression model that does not use the
block coordinate descent method for prediction. That is,
`resubPredict`

can return this argument only in one of these situations:

`Mdl`

is`RegressionGAM`

and the`IsStandardDeviationFit`

property of`Mdl`

is`true`

.`Mdl`

is`RegressionGP`

and the`PredictMethod`

property of`Mdl`

is not`'bcd'`

.

`yInt`

— Prediction intervals of response variable

two-column matrix

Prediction intervals of the response variable, evaluated at each observation in the
predictor data

, returned as an
`Mdl`

.X*n*-by-2 matrix, where *n* is the number of
observations in

. The
`Mdl`

.X`i`

th row `yInt(i,:)`

contains the
`100(1 – `

prediction
interval of the `Alpha`

)%`i`

th response for the `i`

th
observation `Mdl.X(i,:)`

. The `Alpha`

value is the
probability that the prediction interval does not contain the true response value
`Mdl.Y(i)`

. The first column of `yInt`

contains
the lower limits of the prediction intervals, and the second column contains the upper
limits.

This argument is valid only for a generalized additive model object that includes
the standard deviation fit, or a Gaussian process regression model that does not use the
block coordinate descent method for prediction. That is,
`resubPredict`

can return this argument only in one of these
situations:

`Mdl`

is`RegressionGAM`

and the`IsStandardDeviationFit`

property of`Mdl`

is`true`

.`Mdl`

is`RegressionGP`

and the`PredictMethod`

property of`Mdl`

is not`'bcd'`

.

## Algorithms

`resubPredict`

predicts responses according to the corresponding
`predict`

function of the object (`Mdl`

). For a
model-specific description, see the `predict`

function reference pages in
the following table.

Model | Regression Model Object (`Mdl` ) | `predict` Object Function |
---|---|---|

Gaussian process regression model | `RegressionGP` | `predict` |

Generalized additive model | `RegressionGAM` | `predict` |

Neural network model | `RegressionNeuralNetwork` | `predict` |

## Alternative Functionality

To compute the predicted responses for new predictor data, use the corresponding
`predict`

function of the object (`Mdl`

).

## Version History

**Introduced in R2021a**

## See Also

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