# ClassificationGAM

Generalized additive model (GAM) for binary classification

## Description

A `ClassificationGAM` object is a generalized additive model (GAM) object for binary classification. It is an interpretable model that explains class scores (the logit of class probabilities) using a sum of univariate and bivariate shape functions.

You can classify new observations by using the `predict` function, and plot the effect of each shape function on the prediction (class score) for an observation by using the `plotLocalEffects` function. For the full list of object functions for `ClassificationGAM`, see Object Functions.

## Creation

Create a `ClassificationGAM` object by using `fitcgam`. 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 `NaN`s, then the corresponding `Xbinned` values are `NaN`s.

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`

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 `fitcgam` function terminates training before adding the specified number of trees, this property contains the reason for the termination.

Data Types: `struct`

### Other Classification 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`

Unique class labels used in training, specified as a categorical or character array, logical or numeric vector, or cell array of character vectors. `ClassNames` has the same data type as the class labels `Y`. (The software treats string arrays as cell arrays of character vectors.) `ClassNames` also determines the class order.

Data Types: `single` | `double` | `logical` | `char` | `cell` | `categorical`

Misclassification costs, specified as a 2-by-2 numeric matrix.

`Cost(i,j)` is the cost of classifying a point into class `j` if its true class is `i`. The order of the rows and columns of `Cost` corresponds to the order of the classes in `ClassNames`.

The software uses the `Cost` value for prediction, but not training. You can change the value by using dot notation.

Example: `Mdl.Cost = C;`

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`

Prior class probabilities, specified as a numeric vector with two elements. The order of the elements corresponds to the order of the elements in `ClassNames`.

Data Types: `double`

Response variable name, specified as a character vector.

Data Types: `char`

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

Data Types: `logical`

Score transformation, specified as a character vector or function handle. `ScoreTransform` represents a built-in transformation function or a function handle for transforming predicted classification scores.

To change the score transformation function to `function`, for example, use dot notation.

• For a built-in function, enter a character vector.

`Mdl.ScoreTransform = 'function';`

This table describes the available built-in functions.

ValueDescription
`'doublelogit'`1/(1 + e–2x)
`'invlogit'`log(x / (1 – x))
`'ismax'`Sets the score for the class with the largest score to 1, and sets the scores for all other classes to 0
`'logit'`1/(1 + ex)
`'none'` or `'identity'`x (no transformation)
`'sign'`–1 for x < 0
0 for x = 0
1 for x > 0
`'symmetric'`2x – 1
`'symmetricismax'`Sets the score for the class with the largest score to 1, and sets the scores for all other classes to –1
`'symmetriclogit'`2/(1 + ex) – 1

• For a MATLAB® function or a function that you define, enter its function handle.

`Mdl.ScoreTransform = @function;`

`function` must accept a matrix (the original scores) and return a matrix of the same size (the transformed scores).

This property determines the output score computation for object functions such as `predict`, `margin`, and `edge`. Use `'logit'` to compute posterior probabilities, and use `'none'` to compute the logit of posterior probabilities.

Data Types: `char` | `function_handle`

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` within a particular class sum up to the prior probability of that class.

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`

Class labels used to train the model, specified as a categorical or character array, logical or numeric vector, or cell array of character vectors. `Y` has the same data type as the response variable used to train the model. (The software treats string arrays as cell arrays of character vectors.)

Each row of `Y` represents the observed classification of the corresponding row of `X`.

Data Types: `single` | `double` | `logical` | `char` | `cell` | `categorical`

### 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 `fitcgam` 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 `fitcgam` 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
 `addInteractions` Add interaction terms to univariate generalized additive model (GAM) `resume` Resume training of generalized additive model (GAM)
 `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` Classify observations using generalized additive model (GAM) `loss` Classification loss for generalized additive model (GAM) `margin` Classification margins for generalized additive model (GAM) `edge` Classification edge for generalized additive model (GAM)
 `resubPredict` Classify training data using trained classifier `resubLoss` Resubstitution classification loss `resubMargin` Resubstitution classification margin `resubEdge` Resubstitution classification edge
 `compareHoldout` Compare accuracies of two classification models using new data `testckfold` Compare accuracies of two classification models by repeated cross-validation

## Examples

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

Load the `ionosphere` data set. This data set has 34 predictors and 351 binary responses for radar returns, either bad (`'b'`) or good (`'g'`).

`load ionosphere`

Train a univariate GAM that identifies whether the radar return is bad (`'b'`) or good (`'g'`).

`Mdl = fitcgam(X,Y)`
```Mdl = ClassificationGAM ResponseName: 'Y' CategoricalPredictors: [] ClassNames: {'b' 'g'} ScoreTransform: 'logit' Intercept: 2.2715 NumObservations: 351 Properties, Methods ```

`Mdl` is a `ClassificationGAM` 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 class order of `Mdl`.

`classOrder = Mdl.ClassNames`
```classOrder = 2x1 cell {'b'} {'g'} ```

Classify the first observation of the training data, and plot the local effects of the terms in `Mdl` on the prediction.

`label = predict(Mdl,X(1,:))`
```label = 1x1 cell array {'g'} ```
`plotLocalEffects(Mdl,X(1,:))`

The `predict` function classifies the first observation `X(1,:)` as `'g'`. The `plotLocalEffects` function creates a horizontal bar graph that shows the local effects of the 10 most important terms on the prediction. Each local effect value shows the contribution of each term to the classification score for `'g'`, which is the logit of the posterior probability that the classification is `'g'` for the observation.

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 Fisher's iris data set. Create a table that contains observations for versicolor and virginica.

```load fisheriris inds = strcmp(species,'versicolor') | strcmp(species,'virginica'); tbl = array2table(meas(inds,:),'VariableNames',["x1","x2","x3","x4"]); tbl.Y = species(inds,:);```

Specify `formula`

Train a GAM that contains the four linear terms (`x1`, `x2`, `x3`, and `x4`) and two interaction terms (`x1*x2` and `x2*x3`). Specify the terms using a formula in the form `'Y ~ terms'`.

`Mdl1 = fitcgam(tbl,'Y ~ x1 + x2 + x3 + x4 + x1:x2 + x2:x3');`

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 `fitcgam` 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 `fitcgam`, 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 = fitcgam(tbl,'Y','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, `fitcgam` 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 0.01.

```Mdl3 = fitcgam(tbl,'Y','Interactions','all','MaxPValue',0.01); Mdl3.Interactions```
```ans = 5×2 3 4 2 4 1 4 2 3 1 3 ```

`Mdl3` includes five of the six available pairs of interaction terms.

Use `addInteractions` Function

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 `fitcgam`. 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 5 to add the five most important interaction terms to the trained model.

```Mdl4 = fitcgam(tbl,'Y'); UpdatedMdl4 = addInteractions(Mdl4,5); UpdatedMdl4.Interactions```
```ans = 5×2 3 4 2 4 1 4 2 3 1 3 ```

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

Train a univariate classification GAM (which contains only linear terms) for a small number of iterations. After training the model for more iterations, compare the resubstitution loss.

Load the `ionosphere` data set. This data set has 34 predictors and 351 binary responses for radar returns, either bad (`'b'`) or good (`'g'`).

`load ionosphere`

Train a univariate GAM that identifies whether the radar return is bad (`'b'`) or good (`'g'`). Specify the number of trees per linear term as 2. `fitcgam` iterates the boosting algorithm for the specified number of iterations. For each boosting iteration, the function adds one tree per linear term. Specify `'Verbose'` as 2 to display diagnostic messages at every iteration.

`Mdl = fitcgam(X,Y,'NumTreesPerPredictor',2,'Verbose',2);`
```|========================================================| | Type | NumTrees | Deviance | RelTol | LearnRate | |========================================================| | 1D| 0| 486.59| - | - | | 1D| 1| 166.71| Inf| 1| | 1D| 2| 78.336| 0.58205| 1| ```

To check whether `fitcgam` trains the specified number of trees, display the `ReasonForTermination` property of the trained model and view the displayed message.

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

Compute the classification loss for the training data.

`resubLoss(Mdl)`
```ans = 0.0142 ```

Resume training the model for another 100 iterations. Because `Mdl` contains only linear terms, the `resume` function resumes training for the linear terms and adds more trees for them (predictor trees). Specify `'Verbose'` and `'NumPrint'` to display diagnostic messages at every 10 iterations.

`UpdatedMdl = resume(Mdl,100,'Verbose',1,'NumPrint',10);`
```|========================================================| | Type | NumTrees | Deviance | RelTol | LearnRate | |========================================================| | 1D| 0| 78.336| - | - | | 1D| 1| 38.364| 0.17429| 1| | 1D| 10| 0.16311| 0.011894| 1| | 1D| 20| 0.00035693| 0.0025178| 1| | 1D| 30| 8.1191e-07| 0.0011006| 1| | 1D| 40| 1.7978e-09| 0.00074607| 1| | 1D| 50| 3.6113e-12| 0.00034404| 1| | 1D| 60| 1.7497e-13| 0.00016541| 1| ```
`UpdatedMdl.ReasonForTermination`
```ans = struct with fields: PredictorTrees: 'Unable to improve the model fit.' InteractionTrees: '' ```

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

Compute the classification loss using the updated model.

`resubLoss(UpdatedMdl)`
```ans = 0 ```

The classification 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