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# loss

Classification loss for generalized additive model (GAM)

## Syntax

``L = loss(Mdl,Tbl,ResponseVarName)``
``L = loss(Mdl,Tbl,Y)``
``L = loss(Mdl,X,Y)``
``L = loss(___,Name,Value)``

## Description

````L = loss(Mdl,Tbl,ResponseVarName)` returns the Classification Loss (`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).```
````L = loss(Mdl,Tbl,Y)` uses the predictor data in table `Tbl` and the true class labels in `Y`.```

example

````L = loss(Mdl,X,Y)` uses the predictor data in matrix `X` and the true class labels in `Y`.```

example

````L = loss(___,Name,Value)` specifies options using one or more name-value arguments in addition to any of the input argument combinations in previous syntaxes. For example, `'LossFun','mincost'` sets the loss function to the minimal expected misclassification cost function.```

## Examples

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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.

## Input Arguments

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Generalized additive model, specified as a `ClassificationGAM` or `CompactClassificationGAM` model object.

• If you trained `Mdl` using sample data contained in a table, then the input data for `loss` must also be in a table (`Tbl`).

• If you trained `Mdl` using sample data contained in a matrix, then the input data for `loss` must also be in a matrix (`X`).

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`

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`

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`

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`

### Name-Value Arguments

Specify optional pairs of arguments as `Name1=Value1,...,NameN=ValueN`, where `Name` is the argument name and `Value` is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose `Name` in quotes.

Example: `'IncludeInteractions',false,'Weights',w` specifies to exclude interaction terms from the model and to use the observation weights `w`.

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`

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

The default value is `'mincost'` if the `ScoreTransform` property of the input model object (`Mdl.ScoreTransform`) is `'logit'`; otherwise, the default value is `'classiferror'`.

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

ValueDescription
`'binodeviance'`Binomial deviance
`'classifcost'`Observed misclassification cost
`'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 = lossfun(C,S,W,Cost)``

• 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`

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`

## More About

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### Classification Loss

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.

Consider the following scenario.

• L is the weighted average classification loss.

• n is the sample size.

• yj 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 `ClassNames` property), respectively.

• f(Xj) is the positive-class classification score for observation (row) j of the predictor data X.

• mj = yjf(Xj) is the classification score for classifying observation j into the class corresponding to yj. Positive values of mj indicate correct classification and do not contribute much to the average loss. Negative values of mj indicate incorrect classification and contribute significantly to the average loss.

• The weight for observation j is wj. The software normalizes the observation weights so that they sum to the corresponding prior class probability stored in the `Prior` property. Therefore,

`$\sum _{j=1}^{n}{w}_{j}=1.$`

Given this scenario, the following table describes the supported loss functions that you can specify by using the `LossFun` name-value argument.

Loss FunctionValue of `LossFun`Equation
Binomial deviance`'binodeviance'`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}.$
Observed misclassification cost`'classifcost'`

$L=\sum _{j=1}^{n}{w}_{j}{c}_{{y}_{j}{\stackrel{^}{y}}_{j}},$

where ${\stackrel{^}{y}}_{j}$ is the class label corresponding to the class with the maximal score, and ${c}_{{y}_{j}{\stackrel{^}{y}}_{j}}$ is the user-specified cost of classifying an observation into class ${\stackrel{^}{y}}_{j}$ when its true class is yj.

Misclassified rate in decimal`'classiferror'`

$L=\sum _{j=1}^{n}{w}_{j}I\left\{{\stackrel{^}{y}}_{j}\ne {y}_{j}\right\},$

where I{·} is the indicator function.

Cross-entropy loss`'crossentropy'`

`'crossentropy'` is appropriate only for neural network models.

The weighted cross-entropy loss is

`$L=-\sum _{j=1}^{n}\frac{{\stackrel{˜}{w}}_{j}\mathrm{log}\left({m}_{j}\right)}{Kn},$`

where the weights ${\stackrel{˜}{w}}_{j}$ are normalized to sum to n instead of 1.

Exponential loss`'exponential'`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right).$
Hinge loss`'hinge'`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{max}\left\{0,1-{m}_{j}\right\}.$
Logit loss`'logit'`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left(1+\mathrm{exp}\left(-{m}_{j}\right)\right).$
Minimal expected misclassification cost`'mincost'`

`'mincost'` is appropriate only if classification scores are posterior probabilities.

The software computes the weighted minimal expected classification cost using this procedure for observations j = 1,...,n.

1. Estimate the expected misclassification cost of classifying the observation Xj into the class k:

`${\gamma }_{jk}={\left(f{\left({X}_{j}\right)}^{\prime }C\right)}_{k}.$`

f(Xj) is the column vector of class posterior probabilities for the observation Xj. C is the cost matrix stored in the `Cost` property of the model.

2. For observation j, predict the class label corresponding to the minimal expected misclassification cost:

`${\stackrel{^}{y}}_{j}=\underset{k=1,...,K}{\text{argmin}}{\gamma }_{jk}.$`

3. Using C, identify the cost incurred (cj) for making the prediction.

The weighted average of the minimal expected misclassification cost loss is

`$L=\sum _{j=1}^{n}{w}_{j}{c}_{j}.$`

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

If you use the default cost matrix (whose element value is 0 for correct classification and 1 for incorrect classification), then the loss values for `'classifcost'`, `'classiferror'`, and `'mincost'` are identical. For a model with a nondefault cost matrix, the `'classifcost'` loss is equivalent to the `'mincost'` loss most of the time. These losses can be different if prediction into the class with maximal posterior probability is different from prediction into the class with minimal expected cost. Note that `'mincost'` is appropriate only if classification scores are posterior probabilities.

This figure compares the loss functions (except `'classifcost'`, `'crossentropy'`, and `'mincost'`) over the score m for one observation. Some functions are normalized to pass through the point (0,1).

## Version History

Introduced in R2021a

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Behavior changed in R2022a