# kfoldLoss

Classification loss for cross-validated kernel classification model

## Syntax

``loss = kfoldLoss(CVMdl)``
``loss = kfoldLoss(CVMdl,Name,Value)``

## Description

example

````loss = kfoldLoss(CVMdl)` returns the classification loss obtained by the cross-validated, binary kernel model (`ClassificationPartitionedKernel`) `CVMdl`. For every fold, `kfoldLoss` computes the classification loss for validation-fold observations using a model trained on training-fold observations.By default, `kfoldLoss` returns the classification error.```

example

````loss = kfoldLoss(CVMdl,Name,Value)` returns the classification loss with additional options specified by one or more name-value pair arguments. For example, specify the classification loss function, number of folds, or aggregation level.```

## Examples

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Load the `ionosphere` data set. This data set has 34 predictors and 351 binary responses for radar returns, which are labeled either bad (`'b'`) or good (`'g'`).

`load ionosphere`

Cross-validate a binary kernel classification model using the data.

`CVMdl = fitckernel(X,Y,'Crossval','on')`
```CVMdl = ClassificationPartitionedKernel CrossValidatedModel: 'Kernel' ResponseName: 'Y' NumObservations: 351 KFold: 10 Partition: [1x1 cvpartition] ClassNames: {'b' 'g'} ScoreTransform: 'none' Properties, Methods ```

`CVMdl` is a `ClassificationPartitionedKernel` model. By default, the software implements 10-fold cross-validation. To specify a different number of folds, use the `'KFold'` name-value pair argument instead of `'Crossval'`.

Estimate the cross-validated classification loss. By default, the software computes the classification error.

`loss = kfoldLoss(CVMdl)`
```loss = 0.0940 ```

Alternatively, you can obtain the per-fold classification errors by specifying the name-value pair `'Mode','individual'` in `kfoldLoss`.

Load the `ionosphere` data set. This data set has 34 predictors and 351 binary responses for radar returns, which are labeled either bad (`'b'`) or good (`'g'`).

`load ionosphere`

Cross-validate a binary kernel classification model using the data.

`CVMdl = fitckernel(X,Y,'Crossval','on')`
```CVMdl = ClassificationPartitionedKernel CrossValidatedModel: 'Kernel' ResponseName: 'Y' NumObservations: 351 KFold: 10 Partition: [1x1 cvpartition] ClassNames: {'b' 'g'} ScoreTransform: 'none' Properties, Methods ```

`CVMdl` is a `ClassificationPartitionedKernel` model. By default, the software implements 10-fold cross-validation. To specify a different number of folds, use the `'KFold'` name-value pair argument instead of `'Crossval'`.

Create an anonymous function that measures linear loss, that is,

`$L=\frac{\sum _{j}-{w}_{j}{y}_{j}{f}_{j}}{\sum _{j}{w}_{j}}.$`

${w}_{j}$ is the weight for observation j, ${y}_{j}$ is the response j (–1 for the negative class and 1 otherwise), and ${f}_{j}$ is the raw classification score of observation j.

`linearloss = @(C,S,W,Cost)sum(-W.*sum(S.*C,2))/sum(W);`

Custom loss functions must be written in a particular form. For rules on writing a custom loss function, see the `'LossFun'` name-value pair argument.

Estimate the cross-validated classification loss using the linear loss function.

`loss = kfoldLoss(CVMdl,'LossFun',linearloss)`
```loss = -0.7792 ```

## Input Arguments

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Cross-validated, binary kernel classification model, specified as a `ClassificationPartitionedKernel` model object. You can create a `ClassificationPartitionedKernel` model by using `fitckernel` and specifying any one of the cross-validation name-value pair arguments.

To obtain estimates, `kfoldLoss` applies the same data used to cross-validate the kernel classification model (`X` and `Y`).

### 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: `kfoldLoss(CVMdl,'Folds',[1 3 5])` specifies to use only the first, third, and fifth folds to calculate the classification loss.

Fold indices for prediction, specified as the comma-separated pair consisting of `'Folds'` and a numeric vector of positive integers. The elements of `Folds` must be within the range from `1` to `CVMdl.KFold`.

The software uses only the folds specified in `Folds` for prediction.

Example: `'Folds',[1 4 10]`

Data Types: `single` | `double`

Loss function, specified as the comma-separated pair consisting of `'LossFun'` and a built-in loss function name or a function handle.

• This table lists the available loss functions. Specify one using its corresponding value.

ValueDescription
`'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

`'mincost'` is appropriate for classification scores that are posterior probabilities. For kernel classification models, logistic regression learners return posterior probabilities as classification scores by default, but SVM learners do not (see `kfoldPredict`).

• Specify your own function by using function handle notation.

Assume that `n` is the number of observations in `X`, and `K` is the number of distinct classes (`numel(CVMdl.ClassNames)`, where `CVMdl` is the input model). Your function must have this signature:

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

Construct `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 `CVMdl.ClassNames`. `S` is a matrix of classification scores, similar to the output of `kfoldPredict`.

• `W` is an `n`-by-1 numeric vector of observation weights. If you pass `W`, the software normalizes the weights to sum to `1`.

• `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',@lossfun`

Data Types: `char` | `string` | `function_handle`

Aggregation level for the output, specified as the comma-separated pair consisting of `'Mode'` and `'average'` or `'individual'`.

This table describes the values.

ValueDescription
`'average'`The output is a scalar average over all folds.
`'individual'`The output is a vector of length k containing one value per fold, where k is the number of folds.

Example: `'Mode','individual'`

## Output Arguments

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Classification loss, returned as a numeric scalar or numeric column vector.

If `Mode` is `'average'`, then `loss` is the average classification loss over all folds. Otherwise, `loss` is a k-by-1 numeric column vector containing the classification loss for each fold, where k is the number of folds.

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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.

Suppose the following:

• 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. The software also normalizes the prior probabilities so that they sum to 1. Therefore,

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

This table describes the supported loss functions that you can specify by using the `'LossFun'` name-value argument.

Loss FunctionValue of `LossFun`Equation
Binomial deviance`'binodeviance'`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}.$
Exponential loss`'exponential'`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right).$
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.

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