# RegressionGAM

Generalized additive model (GAM) for regression

## Description

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

You can predict responses for new observations by using the predict function, and plot the effect of each shape function on the prediction (response value) for an observation by using the plotLocalEffects function. For the full list of object functions for RegressionGAM, see Object Functions.

## Creation

Create a RegressionGAM object by using fitrgam. You can specify both linear terms and interaction terms for predictors to include univariate shape functions (predictor trees) and bivariate shape functions (interaction trees) in a trained model, respectively.

You can update a trained model by using resume or addInteractions.

• The resume function resumes training for the existing terms in a model.

• The addInteractions function adds interaction terms to a model that contains only linear terms.

## Properties

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### GAM Properties

Bin edges for numeric predictors, specified as a cell array of p numeric vectors, where p is the number of predictors. Each vector includes the bin edges for a numeric predictor. The element in the cell array for a categorical predictor is empty because the software does not bin categorical predictors.

The software bins numeric predictors only if you specify the 'NumBins' name-value argument as a positive integer scalar when training a model with tree learners. The BinEdges property is empty if the 'NumBins' value is empty (default).

You can reproduce the binned predictor data Xbinned by using the BinEdges property of the trained model mdl.

X = mdl.X; % Predictor data
Xbinned = zeros(size(X));
edges = mdl.BinEdges;
% Find indices of binned predictors.
idxNumeric = find(~cellfun(@isempty,edges));
if iscolumn(idxNumeric)
idxNumeric = idxNumeric';
end
for j = idxNumeric
x = X(:,j);
% Convert x to array if x is a table.
if istable(x)
x = table2array(x);
end
% Group x into bins by using the discretize function.
xbinned = discretize(x,[-inf; edges{j}; inf]);
Xbinned(:,j) = xbinned;
end
Xbinned contains the bin indices, ranging from 1 to the number of bins, for numeric predictors. Xbinned values are 0 for categorical predictors. If X contains NaNs, then the corresponding Xbinned values are NaNs.

Data Types: cell

Interaction term indices, specified as a t-by-2 matrix of positive integers, where t is the number of interaction terms in the model. Each row of the matrix represents one interaction term and contains the column indexes of the predictor data X for the interaction term. If the model does not include an interaction term, then this property is empty ([]).

The software adds interaction terms to the model in the order of importance based on the p-values. Use this property to check the order of the interaction terms added to the model.

Data Types: double

Intercept (constant) term of the model, which is the sum of the intercept terms in the predictor trees and interaction trees, specified as a numeric scalar.

Data Types: single | double

Flag indicating whether a model for the standard deviation of the response variable is fit, specified as false or true. Specify the 'FitStandardDeviation' name-value argument of fitrgam as true to fit the model for the standard deviation.

If IsStandardDeviationFit is true, then you can evaluate the standard deviation at a new observation or at a training observation of predictor values by using predict or resubPredict, respectively. These functions also return the prediction intervals of the response variable, evaluated at given observations.

Data Types: logical

Parameters used to train the model, specified as a model parameter object. ModelParameters contains parameter values such as those for the name-value arguments used to train the model. ModelParameters does not contain estimated parameters.

Access the fields of ModelParameters by using dot notation. For example, access the maximum number of decision splits per interaction tree by using Mdl.ModelParameters.MaxNumSplitsPerInteraction.

Bin edges for interaction term detection for numeric predictors, specified as a cell array of p numeric vectors, where p is the number of predictors. Each vector includes the bin edges for a numeric predictor. The element in the cell array for a categorical predictor is empty because the software does not bin categorical predictors.

To speed up the interaction term detection process, the software 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.

Data Types: cell

Reason training the model stops, specified as a structure with two fields, PredictorTrees and InteractionTrees.

Use this property to check if the model contains the specified number of trees for each linear term ('NumTreesPerPredictor') and for each interaction term ('NumTreesPerInteraction'). If the fitrgam function terminates training before adding the specified number of trees, this property contains the reason for the termination.

Data Types: struct

### Other Regression Properties

Categorical predictor indices, specified as a vector of positive integers. CategoricalPredictors contains index values indicating that the corresponding predictors are categorical. The index values are between 1 and p, where p is the number of predictors used to train the model. If none of the predictors are categorical, then this property is empty ([]).

Data Types: double

Expanded predictor names, specified as a cell array of character vectors.

ExpandedPredictorNames is the same as PredictorNames for a generalized additive model.

Data Types: cell

Number of observations in the training data stored in X and Y, specified as a numeric scalar.

Data Types: double

Predictor variable names, specified as a cell array of character vectors. The order of the elements of PredictorNames corresponds to the order in which the predictor names appear in the training data.

Data Types: cell

Response variable name, specified as a character vector.

Data Types: char

Response transformation function, specified as 'none' or a function handle. ResponseTransform describes how the software transforms raw response values.

For a MATLAB® function or a function that you define, enter its function handle. For example, you can enter Mdl.ResponseTransform = @function, where function accepts a numeric vector of the original responses and returns a numeric vector of the same size containing the transformed responses.

Data Types: char | function_handle

Rows of the original training data used in fitting the RegressionGAM model, specified as a logical vector. This property is empty if all rows are used.

Data Types: logical

Observation weights used to train the model, specified as an n-by-1 numeric vector. n is the number of observations (NumObservations).

The software normalizes the observation weights specified in the 'Weights' name-value argument so that the elements of W sum up to 1.

Data Types: double

Predictors used to train the model, specified as a numeric matrix or table.

Each row of X corresponds to one observation, and each column corresponds to one variable.

Data Types: single | double | table

Response, specified as a numeric vector.

Each row of Y represents the observed response of the corresponding row of X.

Data Types: single | double

### Hyperparameter Optimization Properties

Description of the cross-validation optimization of hyperparameters, specified as a BayesianOptimization object or a table of hyperparameters and associated values. This property is nonempty when the 'OptimizeHyperparameters' name-value argument of fitrgam is not 'none' (default) when the object is created. The value of HyperparameterOptimizationResults depends on the setting of the Optimizer field in the HyperparameterOptimizationOptions structure of fitrgam when the object is created.

Value of Optimizer FieldValue of HyperparameterOptimizationResults
'bayesopt' (default)Object of class BayesianOptimization
'gridsearch' or 'randomsearch'Table of hyperparameters used, observed objective function values (cross-validation loss), and rank of observations from lowest (best) to highest (worst)

## Object Functions

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 compact Reduce size of machine learning model
 crossval Cross-validate machine learning model
 lime Local interpretable model-agnostic explanations (LIME) partialDependence Compute partial dependence plotLocalEffects Plot local effects of terms in generalized additive model (GAM) plotPartialDependence Create partial dependence plot (PDP) and individual conditional expectation (ICE) plots shapley Shapley values
 predict Predict responses using generalized additive model (GAM) loss Regression loss for generalized additive model (GAM)
 resubPredict Predict responses for training data using trained regression model resubLoss Resubstitution regression loss

## Examples

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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.

ans=3×10 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: {1x9 cell}
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 regression GAM that contains both linear and interaction terms. Specify to train the interaction terms for a small number of iterations. After training the interaction terms for more iterations, compare the resubstitution loss.

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;

Train a GAM that includes all available linear and interaction terms in X. Specify the number of trees per interaction term as 2. fitrgam iterates the boosting algorithm 300 times (default) for linear terms, and iterates the algorithm the specified number of iterations for interaction terms. For each boosting iteration, the function adds one tree per linear term or one tree per interaction term. Specify 'Verbose' as 1 to display diagnostic messages at every 10 iterations.

Mdl = fitrgam(X,Y,'Interactions','all','NumTreesPerInteraction',2,'Verbose',1);
|========================================================|
| Type | NumTrees |  Deviance  |   RelTol   | LearnRate  |
|========================================================|
|    1D|         0|  2.4432e+05|      -     |      -     |
|    1D|         1|      9507.4|         Inf|           1|
|    1D|        10|      4470.6|  0.00025206|           1|
|    1D|        20|      3895.3|  0.00011448|           1|
|    1D|        30|      3617.7|  3.5365e-05|           1|
|    1D|        40|      3402.5|  3.7992e-05|           1|
|    1D|        50|      3257.1|  2.4983e-05|           1|
|    1D|        60|      3131.8|  2.3873e-05|           1|
|    1D|        70|      3019.8|  2.2967e-05|           1|
|    1D|        80|      2925.9|  2.8071e-05|           1|
|    1D|        90|      2845.3|  1.6811e-05|           1|
|    1D|       100|      2772.7|   1.852e-05|           1|
|    1D|       110|      2707.8|  1.6754e-05|           1|
|    1D|       120|      2649.8|   1.651e-05|           1|
|    1D|       130|      2596.6|  1.1723e-05|           1|
|    1D|       140|      2547.4|   1.813e-05|           1|
|    1D|       150|      2501.1|  1.8659e-05|           1|
|    1D|       160|      2455.7|   1.386e-05|           1|
|    1D|       170|      2416.9|  1.0615e-05|           1|
|    1D|       180|      2377.2|   8.534e-06|           1|
|    1D|       190|        2339|  7.6771e-06|           1|
|    1D|       200|      2303.3|  9.5866e-06|           1|
|    1D|       210|      2270.7|  8.4276e-06|           1|
|    1D|       220|      2240.1|  8.5778e-06|           1|
|    1D|       230|      2209.2|  9.6761e-06|           1|
|    1D|       240|      2178.7|  7.0622e-06|           1|
|    1D|       250|      2150.3|  8.3082e-06|           1|
|    1D|       260|      2122.3|  7.9542e-06|           1|
|    1D|       270|      2097.7|  7.6328e-06|           1|
|    1D|       280|      2070.4|  9.4322e-06|           1|
|    1D|       290|      2044.3|  7.5722e-06|           1|
|    1D|       300|      2019.7|  6.6719e-06|           1|
|========================================================|
| Type | NumTrees |  Deviance  |   RelTol   | LearnRate  |
|========================================================|
|    2D|         0|      2019.7|      -     |      -     |
|    2D|         1|      1795.5|   0.0005975|           1|
|    2D|         2|      1523.4|   0.0010079|           1|

To check whether fitrgam trains the specified number of trees, display the ReasonForTermination property of the trained model and view the displayed messages.

Mdl.ReasonForTermination
ans = struct with fields:
PredictorTrees: 'Terminated after training the requested number of trees.'
InteractionTrees: 'Terminated after training the requested number of trees.'

Compute the regression loss for the training data.

resubLoss(Mdl)
ans = 3.8277

Resume training the model for another 100 iterations. Because Mdl contains both linear and interaction terms, the resume function resumes training for the interaction terms and adds more trees for them (interaction trees).

UpdatedMdl = resume(Mdl,100);
|========================================================|
| Type | NumTrees |  Deviance  |   RelTol   | LearnRate  |
|========================================================|
|    2D|         0|      1523.4|      -     |      -     |
|    2D|         1|      1363.9|  0.00039695|           1|
|    2D|        10|      594.04|  8.0295e-05|           1|
|    2D|        20|      359.44|  4.3201e-05|           1|
|    2D|        30|      238.51|  2.6869e-05|           1|
|    2D|        40|      153.98|  2.6271e-05|           1|
|    2D|        50|      91.464|  8.0936e-06|           1|
|    2D|        60|      61.882|  3.8528e-06|           1|
|    2D|        70|      43.206|  5.9888e-06|           1|

UpdatedMdl.ReasonForTermination
ans = struct with fields:
PredictorTrees: 'Terminated after training the requested number of trees.'
InteractionTrees: 'Unable to improve the model fit.'

resume terminates training when adding more trees does not improve the deviance of the model fit.

Compute the regression loss using the updated model.

resubLoss(UpdatedMdl)
ans = 0.0944

The regression loss decreases after resume updates the model with more iterations.

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## 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