# predict

Predict responses using generalized additive model (GAM)

## Description

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

= predict(`Mdl`

,`X`

,`Name,Value`

)`'IncludeInteractions',true`

specifies to include interaction terms in
computations.

`[`

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

,`ySD`

,`yInt`

] = predict(___)`X`

, using any of the
input argument combinations in the previous syntaxes. This syntax is valid only when you
specify `'FitStandardDeviation'`

of `fitrgam`

as
`true`

for training `Mdl`

and the `IsStandardDeviationFit`

property of `Mdl`

is
`true`

.

## Examples

### Predict Test Sample Response

Train a generalized additive model using training samples, and then predict the test sample responses.

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')`

Mdl = RegressionGAM PredictorNames: {1x6 cell} ResponseName: 'Systolic' CategoricalPredictors: [3 5 6] ResponseTransform: 'none' Intercept: 122.7444 IsStandardDeviationFit: 0 NumObservations: 90 Properties, Methods

`Mdl`

is a `RegressionGAM`

model object.

Predict responses for the test set.

yFit = predict(Mdl,tbl(testInds,:));

Create a table containing the observed response values and the predicted response values.

table(tbl.Systolic(testInds),yFit, ... 'VariableNames',{'Observed Value','Predicted Value'})

`ans=`*10×2 table*
Observed Value Predicted Value
______________ _______________
124 126.58
121 123.95
130 116.72
115 117.35
121 117.45
116 118.5
123 126.16
132 124.14
125 127.36
124 115.99

### Compare Predicted Responses

Predict responses for new observations using a generalized additive model that contains both linear and interaction terms for predictors. Use a memory-efficient model object, and 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;

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')
n = size(X,1);
newInds = randsample(n,10);
inds = ~ismember(1:n,newInds);
XNew = X(newInds,:);
YNew = Y(newInds);
```

Train a GAM 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.

Conserve memory by reducing the size of the trained model.

CMdl = compact(Mdl); whos('Mdl','CMdl')

Name Size Bytes Class Attributes CMdl 1x1 1228131 classreg.learning.regr.CompactRegressionGAM Mdl 1x1 1262153 RegressionGAM

`CMdl`

is a `CompactRegressionGAM`

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 = predict(CMdl,XNew);
yFit_nointeraction = predict(CMdl,XNew,'IncludeInteractions',false);
```

Create a table containing the observed response values and the predicted response values.

t = table(YNew,yFit,yFit_nointeraction, ... 'VariableNames',{'Observed Response', ... 'Predicted Response','Predicted Response Without Interactions'})

`t=`*10×3 table*
Observed Response Predicted Response Predicted Response Without Interactions
_________________ __________________ _______________________________________
27.9 23.04 23.649
NaN 37.163 35.779
NaN 25.876 21.978
13 12.786 14.141
36 28.889 27.281
19.9 22.199 18.451
24.2 23.995 24.885
12 14.247 13.982
38 33.797 33.528
13 12.225 11.127

### Plot Prediction Intervals

Train a generalized additive model (GAM), and then compute and plot the prediction intervals of response values.

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`

. Specify the `FitStandardDeviation`

name-value argument as `true`

so that you can use the trained model to compute prediction intervals. A recommended practice is to use optimal hyperparameters when you fit the standard deviation model for the accuracy of the standard deviation estimates. Specify `'OptimizeHyperparameters'`

as `'all-univariate'`

. For reproducibility, use the `'expected-improvement-plus'`

acquisition function. Specify `'ShowPlots'`

as `false`

and `'Verbose'`

as 0 to disable plot and message displays, respectively.

rng('default') % For reproducibility Mdl = fitrgam(tbl,'Systolic','FitStandardDeviation',true, ... 'OptimizeHyperparameters','all-univariate', ... 'HyperparameterOptimizationOptions',struct('AcquisitionFunctionName','expected-improvement-plus', ... 'ShowPlots',false,'Verbose',0))

Mdl = RegressionGAM PredictorNames: {1x6 cell} ResponseName: 'Systolic' CategoricalPredictors: [3 5 6] ResponseTransform: 'none' Intercept: 122.7800 IsStandardDeviationFit: 1 NumObservations: 100 HyperparameterOptimizationResults: [1x1 BayesianOptimization] Properties, Methods

`Mdl`

is a `RegressionGAM`

model object that uses the best estimated feasible point. The best estimated feasible point indicates the set of hyperparameters that minimizes the upper confidence bound of the objective function value based on the underlying objective function model of the Bayesian optimization process. For more details on the optimization process, see Optimize GAM Using OptimizeHyperparameters.

Predict responses for the training data in `tbl`

, and compute the 99% prediction intervals of the response variable. Specify the significance level (`'Alpha'`

) as 0.01 to set the confidence level of the prediction intervals to 99%.

`[yFit,~,yInt] = predict(Mdl,tbl,'Alpha',0.01);`

Plot the sorted true responses together with the predicted responses and prediction intervals.

figure yTrue = tbl.Systolic; [sortedYTrue,I] = sort(yTrue); plot(sortedYTrue,'o') hold on plot(yFit(I)) plot(yInt(I,1),'k:') plot(yInt(I,2),'k:') legend('True responses','Predicted responses', ... 'Prediction interval limits','Location','best') hold off

## Input Arguments

`Mdl`

— Generalized additive model

`RegressionGAM`

model object | `CompactRegressionGAM`

model object

Generalized additive model, specified as a `RegressionGAM`

or a `CompactRegressionGAM`

model object.

`X`

— Predictor data

numeric matrix | table

Predictor data, specified as a numeric matrix or table.

Each row of `X`

corresponds to one observation, and each column corresponds to one variable.

For a numeric matrix:

The variables that make up the columns of

`X`

must have the same order as the predictor variables that trained`Mdl`

.If you trained

`Mdl`

using a table, then`X`

can be a numeric matrix if the table contains all numeric predictor variables.

For a table:

If you trained

`Mdl`

using a table (for example,`Tbl`

), then all predictor variables in`X`

must have the same variable names and data types as those in`Tbl`

. However, the column order of`X`

does not need to correspond to the column order of`Tbl`

.If you trained

`Mdl`

using a numeric matrix, then the predictor names in`Mdl.PredictorNames`

and the corresponding predictor variable names in`X`

must be the same. To specify predictor names during training, use the`'PredictorNames'`

name-value argument. All predictor variables in`X`

must be numeric vectors.`X`

can contain additional variables (response variables, observation weights, and so on), but`predict`

ignores them.`predict`

does not support multicolumn variables or cell arrays other than cell arrays of character vectors.

**Data Types: **`table`

| `double`

| `single`

### Name-Value Arguments

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

**Example:**

`'Alpha',0.01,'IncludeInteractions',false`

specifies the
confidence level as 99% and excludes interaction terms from computations.`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 when the `IsStandardDeviationFit`

property of `Mdl`

is
`true`

. Specify the `'FitStandardDeviation'`

name-value argument of
`fitrgam`

as `true`

to fit the model for the
standard deviation.

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

specifies to return 99% prediction
intervals.

**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`

.

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`

## Output Arguments

`yFit`

— Predicted responses

column vector

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

.

`ySD`

— Standard deviations of response variable

column vector

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

, returned as a column vector of length
*n*, where *n* is the number of observations in
`X`

. The `i`

th element `ySD(i)`

contains the standard deviation of the `i`

th response for the
`i`

th observation

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

(i,:)`Mdl`

.

This argument is valid only when the `IsStandardDeviationFit`

property of `Mdl`

is
`true`

. Specify the `'FitStandardDeviation'`

name-value argument of `fitrgam`

as `true`

to fit the model for the standard deviation.

`yInt`

— Prediction intervals of response variable

two-column matrix

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

, returned as an *n*-by-2 matrix,
where *n* is the number of observations in `X`

. The
`i`

th row `yInt(i,:)`

contains the
`100(1–`

prediction interval of the
`Alpha`

)%`i`

th response for the `i`

th observation

. The `X`

(i,:)`Alpha`

value
is the probability that the prediction interval does not contain the true response value
for

. The first column of
`X`

(i,:)`yInt`

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

This argument is valid only when the `IsStandardDeviationFit`

property of `Mdl`

is
`true`

. Specify the `'FitStandardDeviation'`

name-value argument of `fitrgam`

as `true`

to fit the model for the standard deviation.

## Algorithms

### Standard Deviation and Prediction Interval

`predict`

returns the predicted responses (`yFit`

)
and, optionally, the standard deviations (`ySD`

) and prediction intervals
(`yInt`

) of the response variable, estimated at each observation in
`X`

.

A Generalized Additive Model (GAM) for Regression
assumes that the response variable *y* follows the normal distribution with
mean *μ* and standard deviation *σ*. If you specify
`'FitStandardDeviation'`

of `fitrgam`

as
`false`

(default), then `fitrgam`

trains a model for
*μ*. If you specify `'FitStandardDeviation'`

as
`true`

, then `fitrgam`

trains an additional model for
*σ* and sets the `IsStandardDeviationFit`

property of
the GAM object to `true`

. The outputs `yFit`

and
`ySD`

correspond to the estimated mean *μ* and
standard deviation *σ*, respectively.

**Introduced in R2021a**

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