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# loss

Classification loss for Gaussian kernel classification model

## Syntax

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

## Description

example

````L = loss(Mdl,X,Y)` returns the classification loss for the binary Gaussian kernel classification model `Mdl` using the predictor data in `X` and the corresponding class labels in `Y`.```
````L = loss(Mdl,Tbl,ResponseVarName)` returns the classification loss for the model `Mdl` using the predictor data in `Tbl` and the true class labels in `Tbl.ResponseVarName`.```
````L = loss(Mdl,Tbl,Y)` returns the classification loss for the model `Mdl` using the predictor data in table `Tbl` and the true class labels in `Y`.```

example

````L = loss(___,Name,Value)` specifies options using one or more name-value pair arguments in addition to any of the input argument combinations in previous syntaxes. For example, you can specify a classification loss function and observation weights. Then, `loss` returns the weighted classification loss using the specified loss function. NoteIf the predictor data in `X` or `Tbl` contains any missing values and `LossFun` is not set to `"classifcost"`, `"classiferror"`, or `"mincost"`, the `loss` function can return NaN. For more details, see loss can return NaN for predictor data with missing values. ```

## Examples

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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 training and test sets. Specify a 15% holdout sample for the test set.

```rng('default') % For reproducibility Partition = cvpartition(Y,'Holdout',0.15); trainingInds = training(Partition); % Indices for the training set testInds = test(Partition); % Indices for the test set```

Train a binary kernel classification model using the training set.

`Mdl = fitckernel(X(trainingInds,:),Y(trainingInds));`

Estimate the training-set classification error and the test-set classification error.

`ceTrain = loss(Mdl,X(trainingInds,:),Y(trainingInds))`
```ceTrain = 0.0067 ```
`ceTest = loss(Mdl,X(testInds,:),Y(testInds))`
```ceTest = 0.1140 ```

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 training and test sets. Specify a 15% holdout sample for the test set.

```rng('default') % For reproducibility Partition = cvpartition(Y,'Holdout',0.15); trainingInds = training(Partition); % Indices for the training set testInds = test(Partition); % Indices for the test set```

Train a binary kernel classification model using the training set.

`Mdl = fitckernel(X(trainingInds,:),Y(trainingInds));`

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 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 training-set classification loss and the test-set classification loss using the linear loss function.

`ceTrain = loss(Mdl,X(trainingInds,:),Y(trainingInds),'LossFun',linearloss)`
```ceTrain = -1.0851 ```
`ceTest = loss(Mdl,X(testInds,:),Y(testInds),'LossFun',linearloss)`
```ceTest = -0.7821 ```

## Input Arguments

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Binary kernel classification model, specified as a `ClassificationKernel` model object. You can create a `ClassificationKernel` model object using `fitckernel`.

Predictor data, specified as an n-by-p numeric matrix, where n is the number of observations and p is the number of predictors used to train `Mdl`.

The length of `Y` and the number of observations in `X` must be equal.

Data Types: `single` | `double`

Class labels, specified as a categorical, character, or string array; logical or numeric vector; or cell array of character vectors.

• The data type of `Y` must be the same as the data type of `Mdl.ClassNames`. (The software treats string arrays as cell arrays of character vectors.)

• The distinct classes in `Y` must be a subset of `Mdl.ClassNames`.

• If `Y` is a character array, then each element must correspond to one row of the array.

• The length of `Y` must be equal to the number of observations in `X` or `Tbl`.

Data Types: `categorical` | `char` | `string` | `logical` | `single` | `double` | `cell`

Sample data used to train the model, specified as a table. Each row of `Tbl` corresponds to one observation, and each column corresponds to one predictor variable. Optionally, `Tbl` can contain additional columns for the response variable and observation weights. `Tbl` must contain all the predictors used to train `Mdl`. Multicolumn variables and cell arrays other than cell arrays of character vectors are not allowed.

If `Tbl` contains the response variable used to train `Mdl`, then you do not need to specify `ResponseVarName` or `Y`.

If you train `Mdl` using sample data contained in a table, then the input data for `loss` must also be in a table.

Response variable name, specified as the name of a variable in `Tbl`. If `Tbl` contains the response variable used to train `Mdl`, then you do not need to specify `ResponseVarName`.

If you specify `ResponseVarName`, then you must specify it as a character vector or string scalar. For example, if the response variable is stored as `Tbl.Y`, then specify `ResponseVarName` as `'Y'`. Otherwise, the software treats all columns of `Tbl`, including `Tbl.Y`, as predictors.

The response variable must be a categorical, character, or string array; a logical or numeric vector; or a cell array of character vectors. If the response variable is a character array, then each element must correspond to one row of the array.

Data Types: `char` | `string`

### 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: ```L = loss(Mdl,X,Y,'LossFun','quadratic','Weights',weights)``` returns the weighted classification loss using the quadratic loss function.

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
`'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

`'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 `predict`).

• 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',@lossfun`

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

Observation weights, specified as the comma-separated pair consisting of `'Weights'` and a numeric vector or the name of a variable in `Tbl`.

• If `Weights` is a numeric vector, then the size of `Weights` must be equal to the number of rows in `X` or `Tbl`.

• If `Weights` is the name of a variable in `Tbl`, you must specify `Weights` as a character vector or string scalar. For example, if the weights are stored as `Tbl.W`, then specify `Weights` as `'W'`. Otherwise, the software treats all columns of `Tbl`, including `Tbl.W`, as predictors.

If you supply weights, `loss` computes the weighted classification loss and normalizes the weights to sum up to the value of the prior probability in the respective class.

Data Types: `double` | `single` | `char` | `string`

## Output Arguments

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Classification loss, returned as a numeric scalar. The interpretation of `L` depends on `Weights` and `LossFun`.

## 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 R2017b

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