# ClassificationPartitionedGAM

Cross-validated generalized additive model (GAM) for classification

## Description

`ClassificationPartitionedGAM`

is a set of generalized additive
models trained on cross-validated folds. Estimate the quality of the cross-validated
classification by using one or more *kfold* functions:
`kfoldPredict`

, `kfoldLoss`

,
`kfoldMargin`

, `kfoldEdge`

, and
`kfoldfun`

.

Every *kfold* object function uses models trained on training-fold
(in-fold) observations to predict the response for validation-fold (out-of-fold) observations.
For example, suppose you cross-validate using five folds. The software randomly assigns each
observation into five groups of equal size (roughly). The *training fold*
contains four of the groups (roughly 4/5 of the data), and the *validation
fold* contains the other group (roughly 1/5 of the data). In this case,
cross-validation proceeds as follows:

The software trains the first model (stored in

`CVMdl.Trained{1}`

) by using the observations in the last four groups, and reserves the observations in the first group for validation.The software trains the second model (stored in

`CVMdl.Trained{2}`

) by using the observations in the first group and the last three groups. The software reserves the observations in the second group for validation.The software proceeds in a similar manner for the third, fourth, and fifth models.

If you validate by using `kfoldPredict`

, the software computes
predictions for the observations in group *i* by using the
*i*th model. In short, the software estimates a response for every
observation by using the model trained without that observation.

## Creation

You can create a `ClassificationPartitionedGAM`

model in two ways:

Create a cross-validated model from a GAM object

`ClassificationGAM`

by using the`crossval`

object function.Create a cross-validated model by using the

`fitcgam`

function and specifying one of the name-value arguments`'CrossVal'`

,`'CVPartition'`

,`'Holdout'`

,`'KFold'`

, or`'Leaveout'`

.

## Properties

### Cross-Validation Properties

`CrossValidatedModel`

— Cross-validated model name

`'GAM'`

This property is read-only.

Cross-validated model name, specified as `'GAM'`

.

`KFold`

— Number of cross-validated folds

positive integer

This property is read-only.

Number of cross-validated folds, specified as a positive integer.

**Data Types: **`double`

`ModelParameters`

— Cross-validation parameter values

object

This property is read-only.

Cross-validation parameter values, specified as an object. The parameter values
correspond to the values of the name-value arguments used to cross-validate the
generalized additive model. `ModelParameters`

does not contain
estimated parameters.

You can access the properties of `ModelParameters`

using dot
notation.

`Partition`

— Data partition

`cvpartition`

model

This property is read-only.

Data partition indicating how the software splits the data into cross-validation folds, specified as a `cvpartition`

model.

`Trained`

— Compact classifiers trained on cross-validation folds

cell array of `CompactClassificationGAM`

models

This property is read-only.

Compact classifiers trained on cross-validation folds, specified as a cell array
of `CompactClassificationGAM`

model objects. `Trained`

has
*k* cells, where *k* is the number of
folds.

**Data Types: **`cell`

### Other Classification Properties

`CategoricalPredictors`

— Categorical predictor indices

vector of positive integers | `[]`

This property is read-only.

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`

`ClassNames`

— Unique class labels

categorical 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 costs

2-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`

*if its true class is*

`j`

*. The order of the rows and columns of*

`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`

`NumObservations`

— Number of observations

numeric 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 names

cell 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 probabilities

numeric 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 name

character vector

This property is read-only.

Response variable name, specified as a character vector.

**Data Types: **`char`

`ScoreTransform`

— Score transformation

character 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
`kfoldPredict`

, `kfoldMargin`

, and `kfoldEdge`

. Use `'logit'`

to compute posterior
probabilities, and use `'none'`

to compute the logit of posterior
probabilities.

**Data Types: **`char`

| `function_handle`

`W`

— Observation weights

numeric 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`

— Predictors

numeric matrix | table

This property is read-only.

Predictors used to cross-validate 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 labels

categorical array | character array | logical vector | numeric vector | cell array of character vectors

This property is read-only.

Class labels used to cross-validate 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`

## Object Functions

`kfoldPredict` | Classify observations in cross-validated classification model |

`kfoldLoss` | Classification loss for cross-validated classification model |

`kfoldMargin` | Classification margins for cross-validated classification model |

`kfoldEdge` | Classification edge for cross-validated classification model |

`kfoldfun` | Cross-validate function for classification |

## Examples

### Create Cross-Validated GAM Using `fitcgam`

Train a cross-validated GAM with 10 folds, which is the default cross-validation option, by using `fitcgam`

. Then, use `kfoldPredict`

to predict class labels for validation-fold observations using a model trained on training-fold observations.

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`

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 = fitcgam(X,Y,'CrossVal','on')

CVMdl = ClassificationPartitionedGAM CrossValidatedModel: 'GAM' PredictorNames: {1x34 cell} ResponseName: 'Y' NumObservations: 351 KFold: 10 Partition: [1x1 cvpartition] NumTrainedPerFold: [1x1 struct] ClassNames: {'b' 'g'} ScoreTransform: 'logit' Properties, Methods

The `fitcgam`

function creates a `ClassificationPartitionedGAM`

model object `CVMdl`

with 10 folds. During cross-validation, the software completes these steps:

Randomly partition the data into 10 sets.

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

Store the 10 compact, trained models in a 10-by-1 cell vector in the

`Trained`

property of the cross-validated model object`ClassificationPartitionedGAM`

.

You can override the default cross-validation setting by using the `'CVPartition'`

, `'Holdout'`

, `'KFold'`

, or `'Leaveout' `

name-value argument.

Classify the observations in `X`

by using `kfoldPredict`

. The function predicts class labels for every observation using the model trained without that observation.

label = kfoldPredict(CVMdl);

Create a confusion matrix to compare the true classes of the observations to their predicted labels.

C = confusionchart(Y,label);

Compute the classification error.

L = kfoldLoss(CVMdl)

L = 0.0712

The average misclassification rate over 10 folds is about 7%.

### Create Cross-Validated GAM Using `crossval`

Train a GAM by using `fitcgam`

, and create a cross-validated GAM by using `crossval`

and the holdout option. Then, use `kfoldPredict`

to predict responses for validation-fold observations using a model trained on training-fold observations.

Load the 1994 census data stored in `census1994.mat`

. The data set consists of demographic data from the US Census Bureau to predict whether an individual makes over $50,000 per year. The classification task is to fit a model that predicts the salary category of people given their age, working class, education level, marital status, race, and so on.

`load census1994`

`census1994`

contains the training data set `adultdata`

and the test data set `adulttest`

. To reduce the running time for this example, subsample 500 training observations from `adultdata`

by using the `datasample`

function.

rng('default') NumSamples = 5e2; adultdata = datasample(adultdata,NumSamples,'Replace',false);

Train a GAM that contains both linear and interaction terms for predictors. Specify to include all available interaction terms whose *p*-values are not greater than 0.05.

Mdl = fitcgam(adultdata,'salary','Interactions','all','MaxPValue',0.05);

`Mdl`

is a `ClassificationGAM`

model object.

Cross-validate the model by specifying a 30% holdout sample.

`CVMdl = crossval(Mdl,'Holdout',0.3)`

CVMdl = ClassificationPartitionedGAM CrossValidatedModel: 'GAM' PredictorNames: {1x14 cell} CategoricalPredictors: [2 4 6 7 8 9 10 14] ResponseName: 'salary' NumObservations: 500 KFold: 1 Partition: [1x1 cvpartition] NumTrainedPerFold: [1x1 struct] ClassNames: [<=50K >50K] ScoreTransform: 'logit' Properties, Methods

The `crossval`

function creates a `ClassificationPartitionedGAM`

model object `CVMdl`

with the holdout option. During cross-validation, the software completes these steps:

Randomly select and reserve 30% of the data as validation data, and train the model using the rest of the data.

Store the compact, trained model in the

`Trained`

property of the cross-validated model object`ClassificationPartitionedGAM`

.

You can choose a different cross-validation setting by using the `'CrossVal'`

, `'CVPartition'`

, `'KFold'`

, or `'Leaveout' `

name-value argument.

Classify the validation-fold observations by using `kfoldPredict`

. The function predicts class labels for the validation-fold observations by using the model trained on the training-fold observations. The function assigns the most frequently predicted label to the training-fold observations.

[labels,scores] = kfoldPredict(CVMdl);

Find the validation-fold observations. `kfoldPredict`

returns 0 scores for both classes for the training-fold observations. Therefore, you can identify the validation-fold observations by finding the observations whose scores are all zeros.

idx = find(sum(abs(scores),2)~=0);

Create a confusion matrix to compare the true classes of the observations to their predicted labels, and compute the classification error for the validation-fold observations.

C = confusionchart(adultdata.salary(idx),labels(idx));

L = kfoldLoss(CVMdl)

L = 0.1800

### Find Optimal Number of Trees for GAM Using `kfoldLoss`

Train a cross-validated generalized additive model (GAM) with 10 folds. Then, use `kfoldLoss`

to compute cumulative cross-validation classification errors (misclassification rate in decimal). Use the errors to determine the optimal number of trees per predictor (linear term for predictor) and the optimal number of trees per interaction term.

Alternatively, you can find optimal values of `fitcgam`

name-value arguments by using the OptimizeHyperparameters name-value argument. For an example, see Optimize GAM Using OptimizeHyperparameters.

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`

Create a cross-validated GAM by using the default cross-validation option. Specify the `'CrossVal'`

name-value argument as `'on'`

. Specify to include all available interaction terms whose *p*-values are not greater than 0.05.

rng('default') % For reproducibility CVMdl = fitcgam(X,Y,'CrossVal','on','Interactions','all','MaxPValue',0.05);

If you specify `'Mode'`

as `'cumulative'`

for `kfoldLoss`

, then the function returns cumulative errors, which are the average errors across all folds obtained using the same number of trees for each fold. Display the number of trees for each fold.

CVMdl.NumTrainedPerFold

`ans = `*struct with fields:*
PredictorTrees: [65 64 59 61 60 66 65 62 64 61]
InteractionTrees: [1 2 2 2 2 1 2 2 2 2]

`kfoldLoss`

can compute cumulative errors using up to 59 predictor trees and one interaction tree.

Plot the cumulative, 10-fold cross-validated, classification error (misclassification rate in decimal). Specify `'IncludeInteractions'`

as `false`

to exclude interaction terms from the computation.

L_noInteractions = kfoldLoss(CVMdl,'Mode','cumulative','IncludeInteractions',false); figure plot(0:min(CVMdl.NumTrainedPerFold.PredictorTrees),L_noInteractions)

The first element of `L_noInteractions`

is the average error over all folds obtained using only the intercept (constant) term. The (`J+1`

)th element of `L_noInteractions`

is the average error obtained using the intercept term and the first `J`

predictor trees per linear term. Plotting the cumulative loss allows you to monitor how the error changes as the number of predictor trees in GAM increases.

Find the minimum error and the number of predictor trees used to achieve the minimum error.

[M,I] = min(L_noInteractions)

M = 0.0655

I = 23

The GAM achieves the minimum error when it includes 22 predictor trees.

Compute the cumulative classification error using both linear terms and interaction terms.

L = kfoldLoss(CVMdl,'Mode','cumulative')

`L = `*2×1*
0.0712
0.0712

The first element of `L`

is the average error over all folds obtained using the intercept (constant) term and all predictor trees per linear term. The second element of `L`

is the average error obtained using the intercept term, all predictor trees per linear term, and one interaction tree per interaction term. The error does not decrease when interaction terms are added.

If you are satisfied with the error when the number of predictor trees is 22, you can create a predictive model by training the univariate GAM again and specifying `'NumTreesPerPredictor',22`

without cross-validation.

## Version History

**Introduced in R2021a**

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