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Classification loss for generalized additive model (GAM)

returns the Classification Loss (`L`

= loss(`Mdl`

,`Tbl`

,`ResponseVarName`

)`L`

),
a scalar representing how well the generalized additive model `Mdl`

classifies the predictor data in `Tbl`

compared to the true class labels
in `Tbl.ResponseVarName`

.

The interpretation of `L`

depends on the loss function
(`'LossFun'`

) and weighting scheme (`'Weights'`

). In
general, better classifiers yield smaller classification loss values. The default
`'LossFun'`

value is `'classiferror'`

(misclassification rate in decimal).

specifies options using one or more name-value arguments in addition to any of the input
argument combinations in previous syntaxes. For example,
`L`

= loss(___,`Name,Value`

)`'LossFun','mincost'`

sets the loss function to the minimal expected
misclassification cost function.

Determine the test sample classification error (loss) of a generalized additive model. When you compare the same type of loss among many models, a lower loss indicates a better predictive model.

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`

Randomly partition observations into a training set and a test set with stratification, using the class information in `Y`

. Specify a 30% holdout sample for testing.

rng('default') % For reproducibility cv = cvpartition(Y,'HoldOut',0.30);

Extract the training and test indices.

trainInds = training(cv); testInds = test(cv);

Specify the training and test data sets.

XTrain = X(trainInds,:); YTrain = Y(trainInds); XTest = X(testInds,:); YTest = Y(testInds);

Train a GAM using the predictors `XTrain`

and class labels `YTrain`

. A recommended practice is to specify the class names.

Mdl = fitcgam(XTrain,YTrain,'ClassNames',{'b','g'});

`Mdl`

is a `ClassificationGAM`

model object.

Determine how well the algorithm generalizes by estimating the test sample classification error. By default, the `loss`

function of `ClassificationGAM`

estimates classification error by using the `'classiferror'`

loss (misclassification rate in decimal).

L = loss(Mdl,XTest,YTest)

L = 0.1052

The trained classifier misclassifies approximately 11% of the test sample.

Train a generalized additive model (GAM) that contains both linear and interaction terms for predictors, and estimate the classification loss with and without interaction terms. Specify whether to include interaction terms when estimating the classification loss for training and test data.

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`

Partition the data set into two sets: one containing training data, and the other containing new, unobserved test data. Reserve 50 observations for the new test data set.

rng('default') % For reproducibility n = size(X,1); newInds = randsample(n,50); inds = ~ismember(1:n,newInds); XNew = X(newInds,:); YNew = Y(newInds);

Train a GAM using the predictors `X`

and class labels `Y`

. A recommended practice is to specify the class names. Specify to include the 10 most important interaction terms.

Mdl = fitcgam(X(inds,:),Y(inds),'ClassNames',{'b','g'},'Interactions',10)

Mdl = ClassificationGAM ResponseName: 'Y' CategoricalPredictors: [] ClassNames: {'b' 'g'} ScoreTransform: 'logit' Intercept: 2.0026 Interactions: [10x2 double] NumObservations: 301 Properties, Methods

`Mdl`

is a `ClassificationGAM`

model object.

Compute the resubstitution classification loss both with and without interaction terms in `Mdl`

. To exclude interaction terms, specify `'IncludeInteractions',false`

.

resubl = resubLoss(Mdl)

resubl = 0

`resubl_nointeraction = resubLoss(Mdl,'IncludeInteractions',false)`

resubl_nointeraction = 0

Estimate the classification loss both with and without interaction terms in `Mdl`

.

l = loss(Mdl,XNew,YNew)

l = 0.0615

`l_nointeraction = loss(Mdl,XNew,YNew,'IncludeInteractions',false)`

l_nointeraction = 0.0615

Including interaction terms does not change the classification loss for `Mdl`

. The trained model classifies all training samples correctly and misclassifies approximately 6% of the test samples.

`Mdl`

— Generalized additive model`ClassificationGAM`

model object | `CompactClassificationGAM`

model objectGeneralized additive model, specified as a `ClassificationGAM`

or `CompactClassificationGAM`

model object.

`Tbl`

— Sample datatable

Sample data, 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.

`Tbl`

must contain all the predictors used to train
`Mdl`

. Optionally, `Tbl`

can contain a column
for the response variable and a column for the observation weights.

The response variable must have the same data type as

`Mdl.Y`

. (The software treats string arrays as cell arrays of character vectors.) If the response variable in`Tbl`

has the same name as the response variable used to train`Mdl`

, then you do not need to specify`ResponseVarName`

.The weight values must be a numeric vector. You must specify the observation weights in

`Tbl`

by using`'Weights'`

.

If you trained `Mdl`

using sample data contained in a table, then the input data for `loss`

must also be in a table.

**Data Types: **`table`

`ResponseVarName`

— Response variable namename of variable in

`Tbl`

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`

`Y`

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

Class labels, specified as a categorical, character, or string array, a logical or
numeric vector, or a cell array of character vectors. Each row of `Y`

represents the classification of the corresponding row of `X`

or
`Tbl`

.

`Y`

must have the same data type as `Mdl.Y`

. (The software treats string arrays as cell arrays of character
vectors.)

**Data Types: **`single`

| `double`

| `categorical`

| `logical`

| `char`

| `string`

| `cell`

`X`

— Predictor datanumeric matrix

Predictor data, specified as a numeric matrix. Each row of `X`

corresponds to one observation, and each column corresponds to one predictor variable.

If you trained `Mdl`

using sample data contained in a matrix, then the input data for `loss`

must also be in a matrix.

**Data Types: **`single`

| `double`

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

`'IncludeInteractions',false,'Weights',w`

specifies to exclude
interaction terms from the model and to use the observation weights
`w`

.`'IncludeInteractions'`

— Flag to include interaction terms`true`

| `false`

Flag to include interaction terms of the model, specified as `true`

or
`false`

.

The default `'IncludeInteractions'`

value is `true`

if `Mdl`

contains interaction terms. The value must be `false`

if the model does not contain interaction terms.

**Example: **`'IncludeInteractions',false`

**Data Types: **`logical`

`'LossFun'`

— Loss function`'classiferror'`

(default) | `'binodeviance'`

| `'exponential'`

| `'hinge'`

| `'logit'`

| `'mincost'`

| `'quadratic'`

| function handleLoss function, specified as a built-in loss function name or a function handle.

This table lists the available loss functions. Specify one using its corresponding character vector or string scalar.

Value Description `'binodeviance'`

Binomial deviance `'classiferror'`

Misclassified rate in decimal `'exponential'`

Exponential loss `'hinge'`

Hinge loss `'logit'`

Logistic loss `'mincost'`

Minimal expected misclassification cost (for classification scores that are posterior probabilities) `'quadratic'`

Quadratic loss For more details on loss functions, see Classification Loss.

To specify a custom loss function, use function handle notation. The function must have this form:

`lossvalue =`

(C,S,W,Cost)`lossfun`

The output argument

`lossvalue`

is a scalar.You specify the function name (

).`lossfun`

`C`

is an`n`

-by-`K`

logical matrix with rows indicating the class to which the corresponding observation belongs.`n`

is the number of observations in`Tbl`

or`X`

, and`K`

is the number of distinct classes (`numel(Mdl.ClassNames)`

. The column order corresponds to the class order in`Mdl.ClassNames`

. Create`C`

by setting`C(p,q) = 1`

, if observation`p`

is in class`q`

, for each row. Set all other elements of row`p`

to`0`

.`S`

is an`n`

-by-`K`

numeric matrix of classification scores. The column order corresponds to the class order in`Mdl.ClassNames`

.`S`

is a matrix of classification scores, similar to the output of`predict`

.`W`

is an`n`

-by-1 numeric vector of observation weights.`Cost`

is a`K`

-by-`K`

numeric matrix of misclassification costs. For example,`Cost = ones(K) – eye(K)`

specifies a cost of`0`

for correct classification and`1`

for misclassification.

**Example: **`'LossFun','binodeviance'`

**Data Types: **`char`

| `string`

| `function_handle`

`'Weights'`

— Observation weights`ones(size(X,1),1)`

(default) | vector of scalar values | name of variable in `Tbl`

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
the weights vector `W`

is stored in `Tbl.W`

, then
specify it as `'W'`

.

`loss`

normalizes the weights in each class to add up to the value of the prior probability of the respective class.

**Data Types: **`single`

| `double`

| `char`

| `string`

*Classification loss* functions measure the predictive
inaccuracy of classification models. When you compare the same type of loss among many
models, a lower loss indicates a better predictive model.

Suppose the following:

*L*is the weighted average classification loss.*n*is the sample size.*y*is the observed class label. The software codes it as –1 or 1, indicating the negative or positive class (or the first or second class in the_{j}`ClassNames`

property), respectively.*f*(*X*) is the positive-class classification score for observation (row)_{j}*j*of the predictor data*X*.*m*=_{j}*y*_{j}*f*(*X*) is the classification score for classifying observation_{j}*j*into the class corresponding to*y*. Positive values of_{j}*m*indicate correct classification and do not contribute much to the average loss. Negative values of_{j}*m*indicate incorrect classification and contribute significantly to the average loss._{j}The weight for observation

*j*is*w*. The software normalizes the observation weights so that they sum to the corresponding prior class probability. The software also normalizes the prior probabilities so that they sum to 1. Therefore,_{j}$$\sum _{j=1}^{n}{w}_{j}}=1.$$

This table describes the supported loss functions that you can specify by using the
`'LossFun'`

name-value argument.

Loss Function | Value of `LossFun` | Equation |
---|---|---|

Binomial deviance | `'binodeviance'` | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}}.$$ |

Exponential loss | `'exponential'` | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right)}.$$ |

Misclassified rate in decimal | `'classiferror'` | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}}I\left\{{\widehat{y}}_{j}\ne {y}_{j}\right\}.$$ $${\widehat{y}}_{j}$$ is the class label corresponding to the class with the
maximal score. |

Hinge loss | `'hinge'` | $$L={\displaystyle \sum}_{j=1}^{n}{w}_{j}\mathrm{max}\left\{0,1-{m}_{j}\right\}.$$ |

Logit loss | `'logit'` | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{log}\left(1+\mathrm{exp}\left(-{m}_{j}\right)\right)}.$$ |

Minimal expected misclassification cost | `'mincost'` |
The software computes
the weighted minimal expected classification cost using this procedure
for observations Estimate the expected misclassification cost of classifying the observation *X*into the class_{j}*k*:$${\gamma}_{jk}={\left(f{\left({X}_{j}\right)}^{\prime}C\right)}_{k}.$$ *f*(*X*) is the column vector of class posterior probabilities for binary and multiclass classification for the observation_{j}*X*._{j}*C*is the cost matrix stored in the`Cost` property of the model.For observation *j*, predict the class label corresponding to the minimal expected misclassification cost:$${\widehat{y}}_{j}=\underset{k=1,\mathrm{...},K}{\text{argmin}}{\gamma}_{jk}.$$ Using *C*, identify the cost incurred (*c*) for making the prediction._{j}
The weighted average of the minimal expected misclassification cost loss is $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}{c}_{j}}.$$ If you use the default cost matrix (whose element
value is 0 for correct classification and 1 for incorrect
classification), then the |

Quadratic loss | `'quadratic'` | $$L={\displaystyle \sum _{j=1}^{n}{w}_{j}{\left(1-{m}_{j}\right)}^{2}}.$$ |

This figure compares the loss functions (except `'mincost'`

) over the
score *m* for one observation. Some functions are normalized to pass
through the point (0,1).

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