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Generalized additive model (GAM) for binary classification

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.

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.

`BinEdges`

— Bin edges for numeric predictorscell array of numeric vectors |

`[]`

This property is read-only.

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`

`Interactions`

— Interaction term indicestwo-column matrix of positive integers |

`[]`

This property is read-only.

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`

— Intercept term of modelnumeric scalar

This property is read-only.

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`

`ModelParameters`

— Parameters used to train modelmodel parameter object

This property is read-only.

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`

.

`PairDetectionBinEdges`

— Bin edges for interaction term detectioncell array of numeric vectors

This property is read-only.

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`

`ReasonForTermination`

— Reason training stopsstructure

This property is read-only.

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`

`CategoricalPredictors`

— Categorical predictor indicesvector of positive integers |

`[]`

This property is read-only.

Categorical predictor
indices, specified as a vector of positive integers. `CategoricalPredictors`

contains index values corresponding to the columns of the predictor data that contain
categorical predictors. If none of the predictors are categorical, then this property is empty
(`[]`

).

**Data Types: **`double`

`ClassNames`

— Unique class labelscategorical array | character array | logical vector | numeric vector | cell array of character vectors

This property is read-only.

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`

`Cost`

— Misclassification costs2-by-2 numeric matrix

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

`Cost(`

is the cost of classifying a point into class * i*,

`j`

`j`

`i`

`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`

`ExpandedPredictorNames`

— Expanded predictor namescell array of character vectors

This property is read-only.

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`

`NumObservations`

— Number of observationsnumeric scalar

This property is read-only.

Number of observations in the training data stored in `X`

and `Y`

, specified as a numeric scalar.

**Data Types: **`double`

`PredictorNames`

— Predictor variable namescell array of character vectors

This property is read-only.

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`

— Prior class probabilitiesnumeric vector

This property is read-only.

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`

`ResponseName`

— Response variable namecharacter vector

This property is read-only.

Response variable name, specified as a character vector.

**Data Types: **`char`

`RowsUsed`

— Rows used in fitting`[]`

| logical vectorThis property is read-only.

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`

`ScoreTransform`

— Score transformationcharacter vector | function handle

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.

Value Description `'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 + *e*^{–x})`'none'`

or`'identity'`

*x*(no transformation)`'sign'`

–1 for *x*< 0

0 for*x*= 0

1 for*x*> 0`'symmetric'`

2 *x*– 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 + *e*^{–x}) – 1For a MATLAB

^{®}function or a function that you define, enter its function handle.Mdl.ScoreTransform = @

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

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`

`W`

— Observation weightsnumeric vector

This property is read-only.

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`

`X`

— Predictorsnumeric matrix | table

This property is read-only.

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`

`Y`

— Class labelscategorical array | character array | logical vector | numeric vector | cell array of character vectors

This property is read-only.

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`

`CompactClassificationGAM`

`compact` | Reduce size of machine learning model |

`ClassificationPartitionedGAM`

`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 |

`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 |

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.

A generalized additive model (GAM) is an interpretable model that explains class scores (the logit of class probabilities) using a sum of univariate and bivariate shape functions of predictors.

`fitcgam`

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 (classification score) are well separated, the model is
easy to interpret.

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

$$\begin{array}{l}y~Binomial(n,\mu )\\ g(\mu )=\mathrm{log}\frac{\mu}{1-\mu}=c+\text{}{f}_{1}({x}_{1})+\text{}{f}_{2}({x}_{2})+\cdots +{f}_{p}({x}_{p}),\end{array}$$

where *y* is a response variable that follows the
binomial distribution with the probability of success (probability of positive class)
*μ* in *n* observations. *g*(*μ*) is a logit link function, and *c* is an intercept
(constant) term.
*f _{i}*(

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

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

where
*f _{ij}*(

`fitcgam`

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

[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.

`addInteractions`

| `ClassificationPartitionedGAM`

| `CompactClassificationGAM`

| `fitcgam`

| `resume`

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