# fitrgam

Fit generalized additive model (GAM) for regression

Since R2021a

## Description

example

Mdl = fitrgam(Tbl,ResponseVarName) returns a generalized additive model Mdl trained using the sample data contained in the table Tbl. The input argument ResponseVarName is the name of the variable in Tbl that contains the response values for regression.

example

Mdl = fitrgam(Tbl,formula) uses the model specification argument formula to specify the response variable and predictor variables in Tbl. You can specify a subset of predictor variables and interaction terms for predictor variables by using formula.

Mdl = fitrgam(Tbl,Y) uses the predictor variables in the table Tbl and the response values in the vector Y.

example

Mdl = fitrgam(X,Y) uses the predictors in the matrix X and the response values in the vector Y.

example

Mdl = fitrgam(___,Name,Value) specifies options using one or more name-value arguments in addition to any of the input argument combinations in the previous syntaxes. For example, 'Interactions',5 specifies to include five interaction terms in the model. You can also specify a list of interaction terms using the 'Interactions' name-value argument.

## Examples

collapse all

Train a univariate GAM, which contains linear terms for predictors. Then, interpret the prediction for a specified data instance by using the plotLocalEffects function.

The data set includes 10 variables with information on the sales of properties in New York City in 2015. This example uses these variables to analyze the sale prices (SALEPRICE).

Preprocess the data set. Remove outliers, convert the datetime array (SALEDATE) to the month numbers, and move the response variable (SALEPRICE) to the last column.

idx = isoutlier(NYCHousing2015.SALEPRICE);
NYCHousing2015(idx,:) = [];
NYCHousing2015.SALEDATE = month(NYCHousing2015.SALEDATE);
NYCHousing2015 = movevars(NYCHousing2015,'SALEPRICE','After','SALEDATE');

Display the first three rows of the table.

BOROUGH    NEIGHBORHOOD       BUILDINGCLASSCATEGORY        RESIDENTIALUNITS    COMMERCIALUNITS    LANDSQUAREFEET    GROSSSQUAREFEET    YEARBUILT    SALEDATE    SALEPRICE
_______    ____________    ____________________________    ________________    _______________    ______________    _______________    _________    ________    _________

2       {'BATHGATE'}    {'01  ONE FAMILY DWELLINGS'}           1                   0                4750              2619            1899           8           0
2       {'BATHGATE'}    {'01  ONE FAMILY DWELLINGS'}           1                   0                4750              2619            1899           8           0
2       {'BATHGATE'}    {'01  ONE FAMILY DWELLINGS'}           1                   1                1287              2528            1899          12           0

Train a univariate GAM for the sale prices. Specify the variables for BOROUGH, NEIGHBORHOOD, BUILDINGCLASSCATEGORY, and SALEDATE as categorical predictors.

Mdl = fitrgam(NYCHousing2015,'SALEPRICE','CategoricalPredictors',[1 2 3 9])
Mdl =
RegressionGAM
PredictorNames: {'BOROUGH'  'NEIGHBORHOOD'  'BUILDINGCLASSCATEGORY'  'RESIDENTIALUNITS'  'COMMERCIALUNITS'  'LANDSQUAREFEET'  'GROSSSQUAREFEET'  'YEARBUILT'  'SALEDATE'}
ResponseName: 'SALEPRICE'
CategoricalPredictors: [1 2 3 9]
ResponseTransform: 'none'
Intercept: 3.7518e+05
IsStandardDeviationFit: 0
NumObservations: 83517

Properties, Methods

Mdl is a RegressionGAM model object. The model display shows a partial list of the model properties. To view the full list of properties, double-click the variable name Mdl in the Workspace. The Variables editor opens for Mdl. Alternatively, you can display the properties in the Command Window by using dot notation. For example, display the estimated intercept (constant) term of Mdl.

Mdl.Intercept
ans = 3.7518e+05

Predict the sale price for the first observation of the training data, and plot the local effects of the terms in Mdl on the prediction.

yFit = predict(Mdl,NYCHousing2015(1,:))
yFit = 4.4421e+05
plotLocalEffects(Mdl,NYCHousing2015(1,:))

The predict function predicts the sale price for the first observation as 4.4421e5. The plotLocalEffects function creates a horizontal bar graph that shows the local effects of the terms in Mdl on the prediction. Each local effect value shows the contribution of each term to the predicted sale price.

Train a generalized additive model that contains linear and interaction terms for predictors in three different ways:

• Specify the interaction terms using the formula input argument.

• Specify the 'Interactions' name-value argument.

• Build a model with linear terms first and add interaction terms to the model by using the addInteractions function.

Load the carbig data set, which contains measurements of cars made in the 1970s and early 1980s.

Create a table that contains the predictor variables (Acceleration, Displacement, Horsepower, and Weight) and the response variable (MPG).

tbl = table(Acceleration,Displacement,Horsepower,Weight,MPG);

Specify formula

Train a GAM that contains the four linear terms (Acceleration, Displacement, Horsepower, and Weight) and two interaction terms (Acceleration*Displacement and Displacement*Horsepower). Specify the terms using a formula in the form 'Y ~ terms'.

Mdl1 = fitrgam(tbl,'MPG ~ Acceleration + Displacement + Horsepower + Weight + Acceleration:Displacement + Displacement:Horsepower');

The function adds interaction terms to the model in the order of importance. You can use the Interactions property to check the interaction terms in the model and the order in which fitrgam adds them to the model. Display the Interactions property.

Mdl1.Interactions
ans = 2×2

2     3
1     2

Each row of Interactions represents one interaction term and contains the column indexes of the predictor variables for the interaction term.

Specify 'Interactions'

Pass the training data (tbl) and the name of the response variable in tbl to fitrgam, so that the function includes the linear terms for all the other variables as predictors. Specify the 'Interactions' name-value argument using a logical matrix to include the two interaction terms, x1*x2 and x2*x3.

Mdl2 = fitrgam(tbl,'MPG','Interactions',logical([1 1 0 0; 0 1 1 0]));
Mdl2.Interactions
ans = 2×2

2     3
1     2

You can also specify 'Interactions' as the number of interaction terms or as 'all' to include all available interaction terms. Among the specified interaction terms, fitrgam identifies those whose p-values are not greater than the 'MaxPValue' value and adds them to the model. The default 'MaxPValue' is 1 so that the function adds all specified interaction terms to the model.

Specify 'Interactions','all' and set the 'MaxPValue' name-value argument to 0.05.

Mdl3 = fitrgam(tbl,'MPG','Interactions','all','MaxPValue',0.05);
Warning: Model does not include interaction terms because all interaction terms have p-values greater than the 'MaxPValue' value, or the software was unable to improve the model fit.
Mdl3.Interactions
ans =

0x2 empty double matrix

Mdl3 includes no interaction terms, which implies one of the following: all interaction terms have p-values greater than 0.05, or adding the interaction terms does not improve the model fit.

Train a univariate GAM that contains linear terms for predictors, and then add interaction terms to the trained model by using the addInteractions function. Specify the second input argument of addInteractions in the same way you specify the 'Interactions' name-value argument of fitrgam. You can specify the list of interaction terms using a logical matrix, the number of interaction terms, or 'all'.

Specify the number of interaction terms as 3 to add the three most important interaction terms to the trained model.

Mdl4 = fitrgam(tbl,'MPG');
UpdatedMdl4.Interactions
ans = 3×2

2     3
1     2
3     4

Mdl4 is a univariate GAM, and UpdatedMdl4 is an updated GAM that contains all the terms in Mdl4 and three additional interaction terms.

Train a cross-validated GAM with 10 folds, which is the default cross-validation option, by using fitrgam. Then, use kfoldPredict to predict responses for validation-fold observations using a model trained on training-fold observations.

Load the carbig data set, which contains measurements of cars made in the 1970s and early 1980s.

Create a table that contains the predictor variables (Acceleration, Displacement, Horsepower, and Weight) and the response variable (MPG).

tbl = table(Acceleration,Displacement,Horsepower,Weight,MPG);

Create a cross-validated GAM by using the default cross-validation option. Specify the 'CrossVal' name-value argument as 'on'.

rng('default') % For reproducibility
CVMdl = fitrgam(tbl,'MPG','CrossVal','on')
CVMdl =
RegressionPartitionedGAM
CrossValidatedModel: 'GAM'
PredictorNames: {'Acceleration'  'Displacement'  'Horsepower'  'Weight'}
ResponseName: 'MPG'
NumObservations: 398
KFold: 10
Partition: [1x1 cvpartition]
NumTrainedPerFold: [1x1 struct]
ResponseTransform: 'none'
IsStandardDeviationFit: 0

Properties, Methods

The fitrgam function creates a RegressionPartitionedGAM model object CVMdl with 10 folds. During cross-validation, the software completes these steps:

1. Randomly partition the data into 10 sets.

2. For each set, reserve the set as validation data, and train the model using the other 9 sets.

3. Store the 10 compact, trained models a in a 10-by-1 cell vector in the Trained property of the cross-validated model object RegressionPartitionedGAM.

You can override the default cross-validation setting by using the 'CVPartition', 'Holdout', 'KFold', or 'Leaveout' name-value argument.

Predict responses for the observations in tbl by using kfoldPredict. The function predicts responses for every observation using the model trained without that observation.

yHat = kfoldPredict(CVMdl);

yHat is a numeric vector. Display the first five predicted responses.

yHat(1:5)
ans = 5×1

19.4848
15.7203
15.5742
15.3185
17.8223

Compute the regression loss (mean squared error).

L = kfoldLoss(CVMdl)
L = 17.7248

kfoldLoss returns the average mean squared error over 10 folds.

Optimize the hyperparameters of a GAM with respect to cross-validation by using the OptimizeHyperparameters name-value argument.

Load the carbig data set, which contains measurements of cars made in the 1970s and early 1980s.

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 into training and test sets. Use approximately 80% of the observations to train a model, and 20% of the observations to test the performance of the trained model on new data. Use cvpartition to partition the data.

rng('default') % For reproducibility
cvp = cvpartition(length(MPG),'Holdout',0.20);
XTrain = X(training(cvp),:);
YTrain = Y(training(cvp));
XTest = X(test(cvp),:);
YTest = Y(test(cvp));

Train a GAM for regression by passing the training data to the fitrgam function, and include the OptimizeHyperparameters argument. Specify 'OptimizeHyperparameters' as 'auto' so that fitrgam finds optimal values of InitialLearnRateForPredictors, NumTreesPerPredictor, Interactions, InitialLearnRateForInteractions, and NumTreesPerInteraction. For reproducibility, choose the 'expected-improvement-plus' acquisition function. The default acquisition function depends on run time and, therefore, can give varying results.

rng('default')
Mdl = fitrgam(XTrain,YTrain,'OptimizeHyperparameters','auto', ...
'HyperparameterOptimizationOptions', ...
struct('AcquisitionFunctionName','expected-improvement-plus'))
|==========================================================================================================================================================|
| Iter | Eval   | Objective:  | Objective   | BestSoFar   | BestSoFar   | InitialLearnRate-| NumTreesPerP-| Interactions | InitialLearnRate-| NumTreesPerI-|
|      | result | log(1+loss) | runtime     | (observed)  | (estim.)    | ForPredictors    | redictor     |              | ForInteractions  | nteraction   |
|==========================================================================================================================================================|
|    1 | Best   |       2.874 |      4.6069 |       2.874 |       2.874 |          0.21533 |          500 |            1 |          0.35042 |           13 |
|    2 | Accept |        2.89 |     0.20809 |       2.874 |      2.8748 |         0.062841 |           14 |            1 |         0.014907 |           10 |
|    3 | Accept |      3.3298 |       1.796 |       2.874 |      2.8746 |         0.001387 |          222 |            0 |                - |            - |
|    4 | Best   |      2.8562 |      5.8182 |      2.8562 |      2.8564 |          0.08216 |          434 |            4 |          0.14875 |          283 |
|    5 | Accept |       2.976 |      1.8052 |      2.8562 |      2.8564 |          0.99942 |          217 |            1 |        0.0017491 |           34 |
|    6 | Best   |      2.8195 |       1.382 |      2.8195 |      2.8198 |          0.13778 |          152 |            6 |         0.012566 |           13 |
|    7 | Best   |      2.7519 |     0.90985 |      2.7519 |       2.752 |          0.12531 |           42 |            4 |          0.27647 |           53 |
|    8 | Best   |      2.7301 |       3.565 |      2.7301 |      2.7301 |          0.18671 |           10 |            3 |        0.0063418 |          487 |
|    9 | Best   |      2.7196 |     0.46532 |      2.7196 |      2.7196 |          0.13792 |           10 |            5 |           0.1663 |           27 |
|   10 | Accept |      2.8281 |      2.9027 |      2.7196 |      2.7196 |          0.23324 |           10 |            4 |          0.75904 |          314 |
|   11 | Accept |      2.7864 |     0.25131 |      2.7196 |      2.7196 |          0.13035 |           10 |            1 |          0.30171 |          476 |
|   12 | Accept |      2.7993 |     0.61803 |      2.7196 |      2.7647 |          0.16476 |           10 |            6 |         0.015498 |           32 |
|   13 | Accept |      2.7847 |      4.5171 |      2.7196 |      2.7197 |        0.0090953 |          499 |            5 |         0.027878 |           40 |
|   14 | Accept |      3.5847 |     0.27508 |      2.7196 |      2.7592 |        0.0035123 |           11 |            3 |         0.011127 |           11 |
|   15 | Accept |      2.7237 |      4.9018 |      2.7196 |       2.759 |         0.015848 |          498 |            3 |          0.14359 |          238 |
|   16 | Accept |       2.779 |       1.569 |      2.7196 |      2.7588 |         0.012829 |           10 |            3 |         0.028814 |          217 |
|   17 | Accept |      2.7761 |      4.7776 |      2.7196 |      2.7272 |         0.023165 |          488 |            1 |          0.32642 |          302 |
|   18 | Accept |      2.8604 |      4.1417 |      2.7196 |      2.7677 |         0.013548 |          495 |            2 |          0.97963 |          141 |
|   19 | Accept |      3.5466 |     0.12735 |      2.7196 |      2.7196 |         0.019794 |           10 |            0 |                - |            - |
|   20 | Accept |      2.7513 |      7.3431 |      2.7196 |      2.7196 |          0.02408 |           62 |            6 |         0.023502 |          490 |
|==========================================================================================================================================================|
| Iter | Eval   | Objective:  | Objective   | BestSoFar   | BestSoFar   | InitialLearnRate-| NumTreesPerP-| Interactions | InitialLearnRate-| NumTreesPerI-|
|      | result | log(1+loss) | runtime     | (observed)  | (estim.)    | ForPredictors    | redictor     |              | ForInteractions  | nteraction   |
|==========================================================================================================================================================|
|   21 | Accept |      2.7243 |     0.92354 |      2.7196 |      2.7196 |         0.040761 |           11 |            3 |          0.10556 |          120 |
|   22 | Best   |      2.6969 |      5.0161 |      2.6969 |       2.697 |        0.0032557 |          494 |            2 |         0.039381 |          487 |
|   23 | Accept |      2.8184 |      3.8034 |      2.6969 |       2.697 |        0.0072249 |           19 |            3 |          0.27653 |          494 |
|   24 | Accept |      2.7788 |      4.3989 |      2.6969 |       2.697 |        0.0064015 |          482 |            1 |         0.013479 |          479 |
|   25 | Accept |      2.7646 |      4.4343 |      2.6969 |      2.6971 |        0.0013222 |          473 |            2 |          0.17272 |          436 |
|   26 | Accept |      2.8368 |     0.28304 |      2.6969 |      2.6971 |          0.93418 |           11 |            5 |          0.16983 |           11 |
|   27 | Accept |      2.7724 |      1.7205 |      2.6969 |      2.6971 |         0.039216 |           11 |            2 |         0.037865 |          480 |
|   28 | Accept |      2.8795 |     0.87918 |      2.6969 |      2.6971 |          0.73103 |           11 |            1 |         0.014567 |          480 |
|   29 | Accept |       2.782 |      4.0221 |      2.6969 |      2.7267 |        0.0047632 |          493 |            1 |         0.069346 |          247 |
|   30 | Accept |      2.7734 |     0.98578 |      2.6969 |      2.7297 |         0.038679 |          103 |            1 |         0.052986 |           68 |

__________________________________________________________
Optimization completed.
MaxObjectiveEvaluations of 30 reached.
Total function evaluations: 30
Total elapsed time: 88.0979 seconds
Total objective function evaluation time: 78.4482

Best observed feasible point:
InitialLearnRateForPredictors    NumTreesPerPredictor    Interactions    InitialLearnRateForInteractions    NumTreesPerInteraction
_____________________________    ____________________    ____________    _______________________________    ______________________

0.0032557                      494                  2                     0.039381                         487

Observed objective function value = 2.6969
Estimated objective function value = 2.7297
Function evaluation time = 5.0161

Best estimated feasible point (according to models):
InitialLearnRateForPredictors    NumTreesPerPredictor    Interactions    InitialLearnRateForInteractions    NumTreesPerInteraction
_____________________________    ____________________    ____________    _______________________________    ______________________

0.0032557                      494                  2                     0.039381                         487

Estimated objective function value = 2.7297
Estimated function evaluation time = 5.009
Mdl =
RegressionGAM
ResponseName: 'Y'
CategoricalPredictors: []
ResponseTransform: 'none'
Intercept: 23.7405
Interactions: [2×2 double]
IsStandardDeviationFit: 0
NumObservations: 318
HyperparameterOptimizationResults: [1×1 BayesianOptimization]

Properties, Methods

fitrgam returns a RegressionGAM model object that uses the best estimated feasible point. The best estimated feasible point is the set of hyperparameters that minimizes the upper confidence bound of the cross-validation loss (mean squared error, MSE) based on the underlying Gaussian process model of the Bayesian optimization process.

The Bayesian optimization process internally maintains a Gaussian process model of the objective function. The objective function is log(1 + cross-validation MSE) for regression. For each iteration, the optimization process updates the Gaussian process model and uses the model to find a new set of hyperparameters. Each line of the iterative display shows the new set of hyperparameters and these column values:

• Objective — Objective function value computed at the new set of hyperparameters.

• Objective runtime — Objective function evaluation time.

• Eval result — Result report, specified as Accept, Best, or Error. Accept indicates that the objective function returns a finite value, and Error indicates that the objective function returns a value that is not a finite real scalar. Best indicates that the objective function returns a finite value that is lower than previously computed objective function values.

• BestSoFar(observed) — The minimum objective function value computed so far. This value is either the objective function value of the current iteration (if the Eval result value for the current iteration is Best) or the value of the previous Best iteration.

• BestSoFar(estim.) — At each iteration, the software estimates the upper confidence bounds of the objective function values, using the updated Gaussian process model, at all the sets of hyperparameters tried so far. Then the software chooses the point with the minimum upper confidence bound. The BestSoFar(estim.) value is the objective function value returned by the predictObjective function at the minimum point.

The plot below the iterative display shows the BestSoFar(observed) and BestSoFar(estim.) values in blue and green, respectively.

The returned object Mdl uses the best estimated feasible point, that is, the set of hyperparameters that produces the BestSoFar(estim.) value in the final iteration based on the final Gaussian process model.

Obtain the best estimated feasible point from Mdl in the HyperparameterOptimizationResults property.

Mdl.HyperparameterOptimizationResults.XAtMinEstimatedObjective
ans=1×5 table
InitialLearnRateForPredictors    NumTreesPerPredictor    Interactions    InitialLearnRateForInteractions    NumTreesPerInteraction
_____________________________    ____________________    ____________    _______________________________    ______________________

0.0032557                      494                  2                     0.039381                         487

Alternatively, you can use the bestPoint function. By default, the bestPoint function uses the 'min-visited-upper-confidence-interval' criterion.

[x,CriterionValue,iteration] = bestPoint(Mdl.HyperparameterOptimizationResults)
x=1×5 table
InitialLearnRateForPredictors    NumTreesPerPredictor    Interactions    InitialLearnRateForInteractions    NumTreesPerInteraction
_____________________________    ____________________    ____________    _______________________________    ______________________

0.0032557                      494                  2                     0.039381                         487

CriterionValue = 2.7908
iteration = 22

You can also extract the best observed feasible point (that is, the last Best point in the iterative display) from the HyperparameterOptimizationResults property or by specifying Criterion as 'min-observed'.

Mdl.HyperparameterOptimizationResults.XAtMinObjective
ans=1×5 table
InitialLearnRateForPredictors    NumTreesPerPredictor    Interactions    InitialLearnRateForInteractions    NumTreesPerInteraction
_____________________________    ____________________    ____________    _______________________________    ______________________

0.0032557                      494                  2                     0.039381                         487

[x_observed,CriterionValue_observed,iteration_observed] = bestPoint(Mdl.HyperparameterOptimizationResults,'Criterion','min-observed')
x_observed=1×5 table
InitialLearnRateForPredictors    NumTreesPerPredictor    Interactions    InitialLearnRateForInteractions    NumTreesPerInteraction
_____________________________    ____________________    ____________    _______________________________    ______________________

0.0032557                      494                  2                     0.039381                         487

CriterionValue_observed = 2.6969
iteration_observed = 22

In this example, the two criteria choose the same set (22nd iteration) of hyperparameters as the best point. The criterion value of each is different because CriterionValue is the upper bound of the objective function value computed by the final Gaussian process model, and CriterionValue_observed is the actual objective function value computed using the selected hyperparameters. For more information, see the Criterion name-value argument of bestPoint.

Evaluate the performance of the regression model on the training set and test set by computing the mean squared errors (MSEs). Smaller MSE values indicate better performance.

LTraining = resubLoss(Mdl)
LTraining = 6.2224
LTest = loss(Mdl,XTest,YTest)
LTest = 18.5724

Optimize the parameters of a GAM with respect to cross-validation by using the bayesopt function.

Alternatively, you can find optimal values of fitrgam name-value arguments by using the OptimizeHyperparameters name-value argument. For an example, see Optimize GAM Using OptimizeHyperparameters.

Load the carbig data set, which contains measurements of cars made in the 1970s and early 1980s.

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;

You must remove the observations with missing response values to fix the cross-validation sets for the optimization process. Remove missing values from the response variable, and remove the corresponding observations in the predictor variables.

[Y,TF] = rmmissing(Y);
X = X(~TF);

Set up a partition for cross-validation. This step fixes the cross-validation sets that the optimization uses at each step.

c = cvpartition(length(Y),'KFold',5);

Prepare optimizableVariable objects for the name-value arguments that you want to optimize using Bayesian optimization. This example finds optimal values for the MaxNumSplitsPerPredictor and NumTreesPerPredictor arguments of fitrgam.

maxNumSplits = optimizableVariable('maxNumSplits',[1,10],'Type','integer');
numTrees = optimizableVariable('numTrees',[1,500],'Type','integer');

Create an objective function that takes an input z = [maxNumSplits,numTrees] and returns the cross-validated loss value of z.

minfun = @(z)kfoldLoss(fitrgam(X,Y,'CVPartition',c, ...
'MaxNumSplitsPerPredictor',z.maxNumSplits, ...
'NumTreesPerPredictor',z.numTrees));

If you specify a cross-validation option, then the fitrgam function returns a cross-validated model object RegressionPartitionedGAM. The kfoldLoss function returns the regression loss (mean squared error) obtained by the cross-validated model. Therefore, the function handle minfun computes the cross-validation loss at the parameters in z.

Search for the best parameters [maxNumSplits,numTrees] using bayesopt. For reproducibility, choose the 'expected-improvement-plus' acquisition function. The default acquisition function depends on run time and, therefore, can give varying results.

rng('default')
results = bayesopt(minfun,[maxNumSplits,numTrees],'Verbose',0, ...
'IsObjectiveDeterministic',true, ...
'AcquisitionFunctionName','expected-improvement-plus');

Obtain the best point from results.

zbest = bestPoint(results)
zbest=1×2 table
maxNumSplits    numTrees
____________    ________

1             8

Train an optimized GAM using the zbest values.

Mdl = fitrgam(X,Y, ...
'MaxNumSplitsPerPredictor',zbest.maxNumSplits, ...
'NumTreesPerPredictor',zbest.numTrees);

## Input Arguments

collapse all

Sample data used to train the model, 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.

• Optionally, Tbl can contain a column for the response variable and a column for the observation weights. The response variable and the weight values must be numeric vectors.

You must specify the response variable in Tbl by using ResponseVarName or formula and specify the observation weights in Tbl by using 'Weights'.

• Specify the response variable by using ResponseVarNamefitrgam uses the remaining variables as predictors. To use a subset of the remaining variables in Tbl as predictors, specify predictor variables by using 'PredictorNames'.

• Define a model specification by using formulafitrgam uses a subset of the variables in Tbl as predictor variables and the response variable, as specified in formula.

• If Tbl does not contain the response variable, then specify a response variable by using Y. The length of the response variable Y and the number of rows in Tbl must be equal. To use a subset of the variables in Tbl as predictors, specify predictor variables by using 'PredictorNames'.

fitrgam considers NaN, '' (empty character vector), "" (empty string), <missing>, and <undefined> values in Tbl to be missing values.

• fitrgam does not use observations with all missing values in the fit.

• fitrgam does not use observations with missing response values in the fit.

• fitrgam uses observations with some missing values for predictors to find splits on variables for which these observations have valid values.

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

Model specification, specified as a character vector or string scalar in the form 'Y ~ terms'. The formula argument specifies a response variable and linear and interaction terms for predictor variables. Use formula to specify a subset of variables in Tbl as predictors for training the model. If you specify a formula, then the software does not use any variables in Tbl that do not appear in formula.

For example, specify 'Y~x1+x2+x3+x1:x2'. In this form, Y represents the response variable, and x1, x2, and x3 represent the linear terms for the predictor variables. x1:x2 represents the interaction term for x1 and x2.

The variable names in the formula must be both variable names in Tbl (Tbl.Properties.VariableNames) and valid MATLAB® identifiers. You can verify the variable names in Tbl by using the isvarname function. If the variable names are not valid, then you can convert them by using the matlab.lang.makeValidName function.

Alternatively, you can specify a response variable and linear terms for predictors using formula, and specify interaction terms for predictors using 'Interactions'.

fitrgam builds a set of interaction trees using only the terms whose p-values are not greater than the 'MaxPValue' value.

Example: 'Y~x1+x2+x3+x1:x2'

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.

The software considers NaN values in Y to be missing values. fitrgam does not use observations with missing response values in the fit.

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.

fitrgam considers NaN values in X as missing values. The function does not use observations with all missing values in the fit. fitrgam uses observations with some missing values for X to find splits on variables for which these observations have valid values.

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: 'Interactions','all','MaxPValue',0.05 specifies to include all available interaction terms whose p-values are not greater than 0.05.

GAM Options

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Flag to fit a model for the standard deviation of the response variable, specified as logical 0 (false) or 1 (true).

If you specify 'FitStandardDeviation' as true, then fitrgam trains an additional model for the standard deviation of the response variable, and sets the IsStandardDeviationFit property of the output GAM object Mdl to true.

To compute the standard deviation values for given observations, use predict, resubPredict, or kfoldPredict. These functions also return the prediction intervals of the response variable.

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 a univariate GAM) or 'all' (for a bivariate GAM) together with 'FitStandardDeviation',true.

Example: 'FitStandardDeviation',true

Data Types: logical

Learning rate of the gradient boosting for interaction terms, specified as a numeric scalar in the interval (0,1]. fitrgam uses this rate throughout the training for interaction terms.

Training a model using a small learning rate requires more learning iterations, but often achieves better accuracy.

Example: 'InitialLearnRateForInteractions',0.1

Data Types: single | double

Learning rate of the gradient boosting for linear terms, specified as a numeric scalar in the interval (0,1]. fitrgam uses this rate throughout the training for linear terms.

Training a model using a small learning rate requires more learning iterations, but often achieves better accuracy.

Example: 'InitialLearnRateForPredictors',0.1

Data Types: single | double

Number or list of interaction terms to include in the candidate set S, specified as a nonnegative integer scalar, a logical matrix, or 'all'.

• Number of interaction terms, specified as a nonnegative integer — S includes the specified number of important interaction terms, selected based on the p-values of the terms.

• List of interaction terms, specified as a logical matrix — S includes the terms specified by a t-by-p logical matrix, where t is the number of interaction terms, and p is the number of predictors used to train the model. For example, logical([1 1 0; 0 1 1]) represents two pairs of interaction terms: a pair of the first and second predictors, and a pair of the second and third predictors.

If fitrgam uses a subset of input variables as predictors, then the function indexes the predictors using only the subset. That is, the column indexes of the logical matrix do not count the response and observation weight variables. The indexes also do not count any variables not used by the function.

• 'all'S includes all possible pairs of interaction terms, which is p*(p – 1)/2 number of terms in total.

Among the interaction terms in S, the fitrgam function identifies those whose p-values are not greater than the 'MaxPValue' value and uses them to build a set of interaction trees. Use the default value ('MaxPValue',1) to build interaction trees using all terms in S.

Example: 'Interactions','all'

Data Types: single | double | logical | char | string

Maximum number of decision splits (or branch nodes) for each interaction tree (boosted tree for an interaction term), specified as a positive integer scalar.

Example: 'MaxNumSplitsPerInteraction',5

Data Types: single | double

Maximum number of decision splits (or branch nodes) for each predictor tree (boosted tree for a linear term), specified as a positive integer scalar. By default, fitrgam uses a tree stump for a predictor tree.

Example: 'MaxNumSplitsPerPredictor',5

Data Types: single | double

Maximum p-value for detecting interaction terms, specified as a numeric scalar in the interval [0,1].

fitrgam first finds the candidate set S of interaction terms from formula or 'Interactions'. Then the function identifies the interaction terms whose p-values are not greater than the 'MaxPValue' value and uses them to build a set of interaction trees.

The default value ('MaxPValue',1) builds interaction trees for all interaction terms in the candidate set S.

For more details about detecting interaction terms, see Interaction Term Detection.

Example: 'MaxPValue',0.05

Data Types: single | double

Number of bins for numeric predictors, specified as a positive integer scalar or [] (empty).

• If you specify the 'NumBins' value as a positive integer scalar (numBins), then fitrgam bins every numeric predictor into at most numBins equiprobable bins, and then grows trees on the bin indices instead of the original data.

• The number of bins can be less than numBins if a predictor has fewer than numBins unique values.

• fitrgam does not bin categorical predictors.

• If the 'NumBins' value is empty ([]), then fitrgam does not bin any predictors.

When you use a large training data set, this binning option speeds up training but might cause a decrease in accuracy. You can first use the default value of 'NumBins', and then change the value depending on the accuracy and training speed.

The trained model Mdl stores the bin edges in the BinEdges property.

Example: 'NumBins',50

Data Types: single | double

Number of trees per interaction term, specified as a positive integer scalar.

The 'NumTreesPerInteraction' value is equivalent to the number of gradient boosting iterations for the interaction terms for predictors. For each iteration, fitrgam adds a set of interaction trees to the model, one tree for each interaction term. To learn about the gradient boosting algorithm, see Gradient Boosting Algorithm.

You can determine whether the fitted model has the specified number of trees by viewing the diagnostic message displayed when 'Verbose' is 1 or 2, or by checking the ReasonForTermination property value of the model Mdl.

Example: 'NumTreesPerInteraction',500

Data Types: single | double

Number of trees per linear term, specified as a positive integer scalar.

The 'NumTreesPerPredictor' value is equivalent to the number of gradient boosting iterations for the linear terms for predictors. For each iteration, fitrgam adds a set of predictor trees to the model, one tree for each predictor. To learn about the gradient boosting algorithm, see Gradient Boosting Algorithm.

You can determine whether the fitted model has the specified number of trees by viewing the diagnostic message displayed when 'Verbose' is 1 or 2, or by checking the ReasonForTermination property value of the model Mdl.

Example: 'NumTreesPerPredictor',500

Data Types: single | double

Other Regression Options

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Categorical predictors list, specified as one of the values in this table.

ValueDescription
Vector of positive integers

Each entry in the vector is an index value indicating that the corresponding predictor is categorical. The index values are between 1 and p, where p is the number of predictors used to train the model.

If fitrgam uses a subset of input variables as predictors, then the function indexes the predictors using only the subset. The CategoricalPredictors values do not count the response variable, observation weights variable, or any other variables that the function does not use.

Logical vector

A true entry means that the corresponding predictor is categorical. The length of the vector is p.

Character matrixEach row of the matrix is the name of a predictor variable. The names must match the entries in PredictorNames. Pad the names with extra blanks so each row of the character matrix has the same length.
String array or cell array of character vectorsEach element in the array is the name of a predictor variable. The names must match the entries in PredictorNames.
"all"All predictors are categorical.

By default, if the predictor data is a table (Tbl), fitrgam assumes that a variable is categorical if it is a logical vector, unordered categorical vector, character array, string array, or cell array of character vectors. If the predictor data is a matrix (X), fitrgam assumes that all predictors are continuous. To identify any other predictors as categorical predictors, specify them by using the CategoricalPredictors name-value argument.

Example: 'CategoricalPredictors','all'

Data Types: single | double | logical | char | string | cell

Number of iterations between diagnostic message printouts, specified as a nonnegative integer scalar. This argument is valid only when you specify 'Verbose' as 1.

If you specify 'Verbose',1 and 'NumPrint',numPrint, then the software displays diagnostic messages every numPrint iterations in the Command Window.

Example: 'NumPrint',500

Data Types: single | double

Predictor variable names, specified as a string array of unique names or cell array of unique character vectors. The functionality of PredictorNames depends on the way you supply the training data.

• If you supply X and Y, then you can use PredictorNames to assign names to the predictor variables in X.

• The order of the names in PredictorNames must correspond to the column order of X. That is, PredictorNames{1} is the name of X(:,1), PredictorNames{2} is the name of X(:,2), and so on. Also, size(X,2) and numel(PredictorNames) must be equal.

• By default, PredictorNames is {'x1','x2',...}.

• If you supply Tbl, then you can use PredictorNames to choose which predictor variables to use in training. That is, fitrgam uses only the predictor variables in PredictorNames and the response variable during training.

• PredictorNames must be a subset of Tbl.Properties.VariableNames and cannot include the name of the response variable.

• By default, PredictorNames contains the names of all predictor variables.

• A good practice is to specify the predictors for training using either PredictorNames or formula, but not both.

Example: "PredictorNames",["SepalLength","SepalWidth","PetalLength","PetalWidth"]

Data Types: string | cell

Response variable name, specified as a character vector or string scalar.

• If you supply Y, then you can use ResponseName to specify a name for the response variable.

• If you supply ResponseVarName or formula, then you cannot use ResponseName.

Example: "ResponseName","response"

Data Types: char | string

Response transformation, specified as either 'none' or a function handle. The default is 'none', which means @(y)y, or no transformation. For a MATLAB function or a function you define, use its function handle for the response transformation. The function handle must accept a vector (the original response values) and return a vector of the same size (the transformed response values).

Example: Suppose you create a function handle that applies an exponential transformation to an input vector by using myfunction = @(y)exp(y). Then, you can specify the response transformation as 'ResponseTransform',myfunction.

Data Types: char | string | function_handle

Verbosity level, specified as 0, 1, or 2. The Verbose value controls the amount of information that the software displays in the Command Window.

This table summarizes the available verbosity level options.

ValueDescription
0The software displays no information.
1The software displays diagnostic messages every numPrint iterations, where numPrint is the 'NumPrint' value.
2The software displays diagnostic messages at every iteration.

Each line of the diagnostic messages shows the information about each boosting iteration and includes the following columns:

• Type — Type of trained trees, 1D (predictor trees, or boosted trees for linear terms for predictors) or 2D (interaction trees, or boosted trees for interaction terms for predictors)

• NumTrees — Number of trees per linear term or interaction term that fitrgam added to the model so far

• DevianceDeviance of the model

• RelTol — Relative change of model predictions: ${\left({\stackrel{^}{y}}_{k}-{\stackrel{^}{y}}_{k-1}\right)}^{\prime }\left({\stackrel{^}{y}}_{k}-{\stackrel{^}{y}}_{k-1}\right)/{\stackrel{^}{y}}_{k}{}^{\prime }{\stackrel{^}{y}}_{k}$, where ${\stackrel{^}{y}}_{k}$ is a column vector of model predictions at iteration k

• LearnRate — Learning rate used for the current iteration

Example: 'Verbose',1

Data Types: single | double

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'.

fitrgam normalizes the values of Weights to sum to 1.

Data Types: single | double | char | string

Note

You cannot use any cross-validation name-value argument together with the 'OptimizeHyperparameters' name-value argument. You can modify the cross-validation for 'OptimizeHyperparameters' only by using the 'HyperparameterOptimizationOptions' name-value argument.

Cross-Validation Options

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Flag to train a cross-validated model, specified as 'on' or 'off'.

If you specify 'on', then the software trains a cross-validated model with 10 folds.

You can override this cross-validation setting using the 'CVPartition', 'Holdout', 'KFold', or 'Leaveout' name-value argument. You can use only one cross-validation name-value argument at a time to create a cross-validated model.

Alternatively, cross-validate after creating a model by passing Mdl to crossval.

Example: 'Crossval','on'

Cross-validation partition, specified as a cvpartition partition object created by cvpartition. The partition object specifies the type of cross-validation and the indexing for the training and validation sets.

To create a cross-validated model, you can specify only one of these four name-value arguments: CVPartition, Holdout, KFold, or Leaveout.

Example: Suppose you create a random partition for 5-fold cross-validation on 500 observations by using cvp = cvpartition(500,'KFold',5). Then, you can specify the cross-validated model by using 'CVPartition',cvp.

Fraction of the data used for holdout validation, specified as a scalar value in the range (0,1). If you specify 'Holdout',p, then the software completes these steps:

1. Randomly select and reserve p*100% of the data as validation data, and train the model using the rest of the data.

2. Store the compact, trained model in the Trained property of the cross-validated model.

To create a cross-validated model, you can specify only one of these four name-value arguments: CVPartition, Holdout, KFold, or Leaveout.

Example: 'Holdout',0.1

Data Types: double | single

Number of folds to use in a cross-validated model, specified as a positive integer value greater than 1. If you specify 'KFold',k, then the software completes these steps:

1. Randomly partition the data into k sets.

2. For each set, reserve the set as validation data, and train the model using the other k – 1 sets.

3. Store the k compact, trained models in a k-by-1 cell vector in the Trained property of the cross-validated model.

To create a cross-validated model, you can specify only one of these four name-value arguments: CVPartition, Holdout, KFold, or Leaveout.

Example: 'KFold',5

Data Types: single | double

Leave-one-out cross-validation flag, specified as 'on' or 'off'. If you specify 'Leaveout','on', then for each of the n observations (where n is the number of observations, excluding missing observations, specified in the NumObservations property of the model), the software completes these steps:

1. Reserve the one observation as validation data, and train the model using the other n – 1 observations.

2. Store the n compact, trained models in an n-by-1 cell vector in the Trained property of the cross-validated model.

To create a cross-validated model, you can specify only one of these four name-value arguments: CVPartition, Holdout, KFold, or Leaveout.

Example: 'Leaveout','on'

Hyperparameter Optimization Options

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Parameters to optimize, specified as one of these values:

• 'none' — Do not optimize.

• 'auto-univariate' — Optimize InitialLearnRateForPredictors and NumTreesPerPredictor.

• 'auto-bivariate' — Optimize Interactions, InitialLearnRateForInteractions, and NumTreesPerInteraction.

• 'all' — Optimize all eligible parameters.

• 'all-univariate' — Optimize all eligible univariate parameters.

• 'all-bivariate' — Optimize all eligible bivariate parameters.

• String array or cell array of eligible parameter names.

• Vector of optimizableVariable objects, typically the output of hyperparameters.

The eligible parameters for fitrgam are:

• Univariate hyperparameters

• InitialLearnRateForPredictorsfitrgam searches among real values, log-scaled in the range [1e-3,1].

• MaxNumSplitsPerPredictorfitrgam searches among integers in the range [1,maxNumSplits], where maxNumSplits is min(30,max(2,NumObservations–1)). NumObservations is the number of observations, excluding missing observations, stored in the NumObservations property of the returned model Mdl.

• NumTreesPerPredictorfitrgam searches among integers, log-scaled in the range [10,500].

• Bivariate hyperparameters

• Interactionsfitrgam searches among integers, log-scaled in the range [0,MaxNumInteractions]t, where MaxNumInteractions is NumPredictors*(NumPredictors – 1)/2, and NumPredictors is the number of predictors used to train the model.

• InitialLearnRateForInteractionsfitrgam searches among real values, log-scaled in the range [1e-3,1].

• MaxNumSplitsPerInteractionfitrgam searches among integers in the range [1,maxNumSplits].

• NumTreesPerInteractionfitrgam searches among integers, log-scaled in the range [10,500].

Use 'auto' or 'all' to find optimal hyperparameter values for both univariate and bivariate parameters. Alternatively, you can find optimal values for univariate parameters using 'auto-univariate' or 'all-univariate' and then find optimal values for bivariate parameters using 'auto-bivariate' or 'all-bivariate'. For examples, see Optimize GAM Using OptimizeHyperparameters and Train Generalized Additive Model for Regression.

The optimization attempts to minimize the cross-validation loss (error) for fitrgam by varying the parameters. To control the cross-validation type and other aspects of the optimization, use the HyperparameterOptimizationOptions name-value argument.

Note

The values of 'OptimizeHyperparameters' override any values you specify using other name-value arguments. For example, setting 'OptimizeHyperparameters' to 'auto' causes fitrgam to optimize hyperparameters corresponding to the 'auto' option and to ignore any specified values for the hyperparameters.

Set nondefault parameters by passing a vector of optimizableVariable objects that have nondefault values. For example:

params = hyperparameters('fitrgam',[Horsepower,Weight],MPG);
params(1).Range = [1e-4,1e6];

Pass params as the value of OptimizeHyperparameters.

By default, the iterative display appears at the command line, and plots appear according to the number of hyperparameters in the optimization. For the optimization and plots, the objective function is log(1 + cross-validation loss). To control the iterative display, set the Verbose field of the 'HyperparameterOptimizationOptions' name-value argument. To control the plots, set the ShowPlots field of the 'HyperparameterOptimizationOptions' name-value argument.

Example: 'OptimizeHyperparameters','auto'

Options for optimization, specified as a structure. This argument modifies the effect of the OptimizeHyperparameters name-value argument. All fields in the structure are optional.

Field NameValuesDefault
Optimizer
• 'bayesopt' — Use Bayesian optimization. Internally, this setting calls bayesopt.

• 'gridsearch' — Use grid search with NumGridDivisions values per dimension.

• 'randomsearch' — Search at random among MaxObjectiveEvaluations points.

'gridsearch' searches in a random order, using uniform sampling without replacement from the grid. After optimization, you can get a table in grid order by using the command sortrows(Mdl.HyperparameterOptimizationResults).

'bayesopt'
AcquisitionFunctionName

• 'expected-improvement-per-second-plus'

• 'expected-improvement'

• 'expected-improvement-plus'

• 'expected-improvement-per-second'

• 'lower-confidence-bound'

• 'probability-of-improvement'

Acquisition functions whose names include per-second do not yield reproducible results because the optimization depends on the runtime of the objective function. Acquisition functions whose names include plus modify their behavior when they are overexploiting an area. For more details, see Acquisition Function Types.

'expected-improvement-per-second-plus'
MaxObjectiveEvaluationsMaximum number of objective function evaluations.30 for 'bayesopt' and 'randomsearch', and the entire grid for 'gridsearch'
MaxTime

Time limit, specified as a positive real scalar. The time limit is in seconds, as measured by tic and toc. The run time can exceed MaxTime because MaxTime does not interrupt function evaluations.

Inf
NumGridDivisionsFor 'gridsearch', the number of values in each dimension. The value can be a vector of positive integers giving the number of values for each dimension, or a scalar that applies to all dimensions. This field is ignored for categorical variables.10
ShowPlotsLogical value indicating whether to show plots. If true, this field plots the best observed objective function value against the iteration number. If you use Bayesian optimization (Optimizer is 'bayesopt'), then this field also plots the best estimated objective function value. The best observed objective function values and best estimated objective function values correspond to the values in the BestSoFar (observed) and BestSoFar (estim.) columns of the iterative display, respectively. You can find these values in the properties ObjectiveMinimumTrace and EstimatedObjectiveMinimumTrace of Mdl.HyperparameterOptimizationResults. If the problem includes one or two optimization parameters for Bayesian optimization, then ShowPlots also plots a model of the objective function against the parameters.true
SaveIntermediateResultsLogical value indicating whether to save results when Optimizer is 'bayesopt'. If true, this field overwrites a workspace variable named 'BayesoptResults' at each iteration. The variable is a BayesianOptimization object.false
Verbose

Display at the command line:

• 0 — No iterative display

• 1 — Iterative display

• 2 — Iterative display with extra information

For details, see the bayesopt Verbose name-value argument and the example Optimize Classifier Fit Using Bayesian Optimization.

1
UseParallelLogical value indicating whether to run Bayesian optimization in parallel, which requires Parallel Computing Toolbox™. Due to the nonreproducibility of parallel timing, parallel Bayesian optimization does not necessarily yield reproducible results. For details, see Parallel Bayesian Optimization.false
Repartition

Logical value indicating whether to repartition the cross-validation at every iteration. If this field is false, the optimizer uses a single partition for the optimization.

The setting true usually gives the most robust results because it takes partitioning noise into account. However, for good results, true requires at least twice as many function evaluations.

false
Use no more than one of the following three options.
CVPartitionA cvpartition object, as created by cvpartition'Kfold',5 if you do not specify a cross-validation field
HoldoutA scalar in the range (0,1) representing the holdout fraction
KfoldAn integer greater than 1

Example: 'HyperparameterOptimizationOptions',struct('MaxObjectiveEvaluations',60)

Data Types: struct

## Output Arguments

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Trained generalized additive model, returned as one of the model objects in this table.

Model ObjectCross-Validation Options to Train Model ObjectWays to Predict Responses Using Model Object
RegressionGAMNoneUse predict to predict responses for new observations, and use resubPredict to predict responses for training observations.
RegressionPartitionedGAMSpecify the name-value argument KFold, Holdout, Leaveout, CrossVal, or CVPartitionUse kfoldPredict to predict responses for observations that fitrgam holds out during training. kfoldPredict predicts a response for every observation by using the model trained without that observation.

To reference properties of Mdl, use dot notation. For example, enter Mdl.Interactions in the Command Window to display the interaction terms in Mdl.

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### Generalized Additive Model (GAM) for Regression

A generalized additive model (GAM) is an interpretable model that explains a response variable using a sum of univariate and bivariate shape functions of predictors.

fitrgam uses a boosted tree as a shape function for each predictor and, optionally, each pair of predictors; therefore, the function can capture a nonlinear relation between a predictor and the response variable. Because contributions of individual shape functions to the prediction (response value) are well separated, the model is easy to interpret.

The standard GAM uses a univariate shape function for each predictor.

$\begin{array}{l}y~N\left(\mu ,{\sigma }^{2}\right)\\ g\left(\mu \right)=\mu =c+\text{​}{f}_{1}\left({x}_{1}\right)+\text{​}{f}_{2}\left({x}_{2}\right)+\cdots +{f}_{p}\left({x}_{p}\right),\end{array}$

where y is a response variable that follows the normal distribution with mean μ and standard deviation σ. g(μ) is an identity link function, and c is an intercept (constant) term. fi(xi) is a univariate shape function for the ith predictor, which is a boosted tree for a linear term for the predictor (predictor tree).

You can include interactions between predictors in a model by adding bivariate shape functions of important interaction terms to the model.

$\mu =c+\text{​}{f}_{1}\left({x}_{1}\right)+\text{​}{f}_{2}\left({x}_{2}\right)+\cdots +{f}_{p}\left({x}_{p}\right)+\sum _{i,j\in \left\{1,2,\cdots ,p\right\}}{f}_{ij}\left({x}_{i}{x}_{j}\right),$

where fij(xixj) is a bivariate shape function for the ith and jth predictors, which is a boosted tree for an interaction term for the predictors (interaction tree).

fitrgam finds important interaction terms based on the p-values of F-tests. For details, see Interaction Term Detection.

If you specify 'FitStandardDeviation' of fitrgam as false (default), then fitrgam trains a model for the mean μ. If you specify 'FitStandardDeviation' as true, then fitrgam trains an additional model for the standard deviation σ and sets the IsStandardDeviationFit property of the GAM object to true.

### Deviance

Deviance is a generalization of the residual sum of squares. It measures the goodness of fit compared to the saturated model.

The deviance of a fitted model is twice the difference between the loglikelihoods of the model and the saturated model:

-2(logL - logLs),

where L and Ls are the likelihoods of the fitted model and the saturated model, respectively. The saturated model is the model with the maximum number of parameters that you can estimate.

fitrgam uses the deviance to measure the goodness of model fit and finds a learning rate that reduces the deviance at each iteration. Specify 'Verbose' as 1 or 2 to display the deviance and learning rate in the Command Window.

## Algorithms

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fitrgam fits a generalized additive model using a gradient boosting algorithm (Least-Squares Boosting).

fitrgam first builds sets of predictor trees (boosted trees for linear terms for predictors) and then builds sets of interaction trees (boosted trees for interaction terms for predictors). The boosting algorithm iterates for at most 'NumTreesPerPredictor' times for predictor trees, and then iterates for at most 'NumTreesPerInteraction' times for interaction trees.

For each boosting iteration, fitrgam builds a set of predictor trees with the learning rate 'InitialLearnRateForPredictors', or builds a set of interaction trees with the learning rate 'InitialLearnRateForInteractions'.

• When building a set of trees, the function trains one tree at a time. It fits a tree to the residual that is the difference between the response and the aggregated prediction from all trees grown previously. To control the boosting learning speed, the function shrinks the tree by the learning rate and then adds the tree to the model and updates the residual.

• Updated model = current model + (learning rate)·(new tree)

• Updated residual = current residual – (learning rate)·(response explained by new tree)

• If adding the set of trees improves the model fit (that is, reduces the deviance of the fit by a value larger than the tolerance), then fitrgam moves to the next iteration.

• If adding the set of trees does not improve the model fit when fitrgam trains linear terms, then the function stops boosting iterations for linear terms and starts boosting iterations for interaction terms. If the model fit is not improved when the function trains interaction terms, then the function terminates the model fitting.

You can determine why training stopped by checking the ReasonForTermination property of the trained model.

### Interaction Term Detection

For each pairwise interaction term xixj (specified by formula or 'Interactions'), the software performs an F-test to examine whether the term is statistically significant.

To speed up the process, fitrgam bins numeric predictors into at most 8 equiprobable bins. The number of bins can be less than 8 if a predictor has fewer than 8 unique values. The F-test examines the null hypothesis that the bins created by xi and xj have equal responses versus the alternative that at least one bin has a different response value from the others. A small p-value indicates that differences are significant, which implies that the corresponding interaction term is significant and, therefore, including the term can improve the model fit.

fitrgam builds a set of interaction trees using the terms whose p-values are not greater than the 'MaxPValue' value. You can use the default 'MaxPValue' value 1 to build interaction trees using all terms specified by formula or 'Interactions'.

fitrgam adds interaction terms to the model in the order of importance based on the p-values. Use the Interactions property of the returned model to check the order of the interaction terms added to the model.

## References

[1] Lou, Yin, Rich Caruana, and Johannes Gehrke. "Intelligible Models for Classification and Regression." Proceedings of the 18th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD ’12). Beijing, China: ACM Press, 2012, pp. 150–158.

[2] Lou, Yin, Rich Caruana, Johannes Gehrke, and Giles Hooker. "Accurate Intelligible Models with Pairwise Interactions." Proceedings of the 19th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD ’13) Chicago, Illinois, USA: ACM Press, 2013, pp. 623–631.

## Version History

Introduced in R2021a