# resubLoss

Resubstitution classification loss

## Syntax

``L = resubLoss(Mdl)``
``L = resubLoss(Mdl,Name,Value)``

## Description

example

````L = resubLoss(Mdl)` returns the Classification Loss by resubstitution (L), or the in-sample classification loss, for the trained classification model `Mdl` using the training data stored in `Mdl.X` and the corresponding class labels stored in `Mdl.Y`. The interpretation of `L` depends on the loss function (`'LossFun'`) and weighting scheme (`Mdl.W`). In general, better classifiers yield smaller classification loss values. The default `'LossFun'` value varies depending on the model object `Mdl`.```

example

````L = resubLoss(Mdl,Name,Value)` specifies additional options using one or more name-value arguments. For example, `'LossFun','binodeviance'` sets the loss function to the binomial deviance function.```

## Examples

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Determine the in-sample classification error (resubstitution loss) of a naive Bayes classifier. In general, a smaller loss indicates a better classifier.

Load the `fisheriris` data set. Create `X` as a numeric matrix that contains four measurements for 150 irises. Create `Y` as a cell array of character vectors that contains the corresponding iris species.

```load fisheriris X = meas; Y = species;```

Train a naive Bayes classifier using the predictors `X` and class labels `Y`. A recommended practice is to specify the class names. `fitcnb` assumes that each predictor is conditionally and normally distributed.

`Mdl = fitcnb(X,Y,'ClassNames',{'setosa','versicolor','virginica'})`
```Mdl = ClassificationNaiveBayes ResponseName: 'Y' CategoricalPredictors: [] ClassNames: {'setosa' 'versicolor' 'virginica'} ScoreTransform: 'none' NumObservations: 150 DistributionNames: {'normal' 'normal' 'normal' 'normal'} DistributionParameters: {3x4 cell} Properties, Methods ```

`Mdl` is a trained `ClassificationNaiveBayes` classifier.

Estimate the in-sample classification error.

`L = resubLoss(Mdl)`
```L = 0.0400 ```

The naive Bayes classifier misclassifies 4% of the training 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`

Train a support vector machine (SVM) classifier. Standardize the data and specify that `'g'` is the positive class.

`SVMModel = fitcsvm(X,Y,'ClassNames',{'b','g'},'Standardize',true);`

`SVMModel` is a trained `ClassificationSVM` classifier.

Estimate the in-sample hinge loss.

`L = resubLoss(SVMModel,'LossFun','hinge')`
```L = 0.1603 ```

The hinge loss is `0.1603`. Classifiers with hinge losses close to 0 are preferred.

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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Classification machine learning model, specified as a full classification model object, as given in the following table of supported models.

ModelClassification Model Object
Generalized additive model`ClassificationGAM`
k-nearest neighbor model`ClassificationKNN`
Naive Bayes model`ClassificationNaiveBayes`
Neural network model`ClassificationNeuralNetwork`
Support vector machine for one-class and binary classification`ClassificationSVM`

### Name-Value Arguments

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`.

Example: `resubLoss(Mdl,'LossFun','logit')` estimates the logit resubstitution loss.

Flag to include interaction terms of the model, specified as `true` or `false`. This argument is valid only for a generalized additive model (GAM). That is, you can specify this argument only when `Mdl` is `ClassificationGAM`.

The default value is `true` if `Mdl` contains interaction terms. The value must be `false` if the model does not contain interaction terms.

Data Types: `logical`

Loss 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.

ValueDescription
`'binodeviance'`Binomial deviance
`'classiferror'`Misclassified rate in decimal
`'crossentropy'`Cross-entropy loss (for neural networks only)
`'exponential'`Exponential loss
`'hinge'`Hinge loss
`'logit'`Logistic loss
`'mincost'`Minimal expected misclassification cost (for classification scores that are posterior probabilities)
`'quadratic'`Quadratic loss

The default value depends on the trained model (`Mdl`).

• The default value is `'classiferror'` if `Mdl` is a `ClassificationGAM`, `ClassificationNeuralNetwork`, or `ClassificationSVM` object.

• The default value is `'mincost'` if `Mdl` is a `ClassificationKNN` or `ClassificationNaiveBayes` object.

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`

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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.

• For binary classification:

• 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.

• For algorithms that support multiclass classification (that is, K ≥ 3):

• yj* is a vector of K – 1 zeros, with 1 in the position corresponding to the true, observed class yj. For example, if the true class of the second observation is the third class and K = 4, then y2* = [0 0 1 0]′. The order of the classes corresponds to the order in the `ClassNames` property of the input model.

• f(Xj) is the length K vector of class scores for observation j of the predictor data X. The order of the scores corresponds to the order of the classes in the `ClassNames` property of the input model.

• mj = yj*f(Xj). Therefore, mj is the scalar classification score that the model predicts for the true, observed class.

• The weight for observation j is wj. The software normalizes the observation weights so that they sum to the corresponding prior class probability. The software also normalizes the prior probabilities so they sum to 1. 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 pair argument.

Loss FunctionValue of `LossFun`Equation
Binomial deviance`'binodeviance'`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}.$
Misclassified rate in decimal`'classiferror'`

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

${\stackrel{^}{y}}_{j}$ is the class label corresponding to the class with the maximal score. 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 binary and multiclass classification 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

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

If you use the default cost matrix (whose element value is 0 for correct classification and 1 for incorrect classification), then the `'mincost'` loss is equivalent to the `'classiferror'` loss.

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

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

## Algorithms

`resubLoss` computes the classification loss according to the corresponding `loss` function of the object (`Mdl`). For a model-specific description, see the `loss` function reference pages in the following table.

ModelClassification Model Object (`Mdl`)`loss` Object Function
Generalized additive model`ClassificationGAM``loss`
k-nearest neighbor model`ClassificationKNN``loss`
Naive Bayes model`ClassificationNaiveBayes``loss`
Neural network model`ClassificationNeuralNetwork``loss`
Support vector machine for one-class and binary classification`ClassificationSVM``loss`