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# loss

Class: CompactClassificationSVM

Classification error for support vector machine classifiers

## Syntax

• ``L = loss(SVMModel,TBL,ResponseVarName)``
• ``L = loss(SVMModel,TBL,Y)``
• ``L = loss(SVMModel,X,Y)``
example
• ``L = loss(___,Name,Value)``
example

## Description

````L = loss(SVMModel,TBL,ResponseVarName)` returns the classification error (see Classification Loss), a scalar representing how well the trained support vector machine (SVM) classifer `SVMModel` classifies the predictor data in table `TBL` as compared to the true class labels in `TBL.ResponseVarName`.`loss` normalizes the class probabilities in `TBL.ResponseVarName` to the prior class probabilities `fitcsvm` used for training, stored in the `Prior` property of `SVMModel`.```
````L = loss(SVMModel,TBL,Y)` returns the classification error for the predictor data in table `TBL` and the true class labels in `Y`.`loss` normalizes the class probabilities in `Y` to the prior class probabilities `fitcsvm` used for training, stored in the `Prior` property of `SVMModel`.```

example

````L = loss(SVMModel,X,Y)` returns the classification error based on the predictor data in matrix `X` as compared to the true class labels in `Y`.```

example

````L = loss(___,Name,Value)` returns the classification error with additional options specified by one or more `Name,Value` pair arguments, using any of the previous syntaxes. For example, you can specify the loss function or classification weights.```

## Input Arguments

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SVM classification model, specified as a `ClassificationSVM` model object or `CompactClassificationSVM` model object returned by `fitcsvm` or `compact`, respectively.

Sample data, 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 of the predictors used to train `SVMModel`. Multi-column variables and cell arrays other than cell arrays of character vectors are not allowed.

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

If you trained `SVMModel` using sample data contained in a `table`, then the input data for this method must also be in a table.

Data Types: `table`

Response variable name, specified as the name of a variable in `TBL`.

You must specify `ResponseVarName` as a character vector. For example, if the response variable `Y` is stored as `TBL.Y`, then specify it as `'Y'`. Otherwise, the software treats all columns of `TBL`, including `Y`, as predictors when training the model.

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

Predictor data, specified as a numeric matrix.

Each row of `X` corresponds to one observation (also known as an instance or example), and each column corresponds to one variable (also known as a feature). The variables making up the columns of `X` must be the same as the variables that trained the `SVMModel` classifier.

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

If you set `'Standardize',true` in `fitcsvm` to train `SVMModel`, then the software standardizes the columns of `X` using the corresponding means in `SVMModel.Mu` and standard deviations in `SVMModel.Sigma`.

Data Types: `double` | `single`

Class labels, specified as a categorical or character array, logical or numeric vector, or cell array of character vectors. `Y` must be the same as the data type of `SVMModel.ClassNames`.

The length of `Y` must equal the number of rows of `TBL` or `X` must be equal.

### Name-Value Pair 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 single quotes (`' '`). You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

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

• The following lists available loss functions. Specify one using its corresponding character vector.

ValueDescription
`'binodeviance'`Binomial deviance
`'classiferror'`Classification error
`'exponential'`Exponential
`'hinge'`Hinge
`'logit'`Logistic
`'mincost'`Minimal expected misclassification cost (for classification scores that are posterior probabilities)
`'quadratic'`Quadratic

`'mincost'` is appropriate for classification scores that are posterior probabilities. You can specify to use posterior probabilities as classification scores for SVM models by setting `'FitPosterior',true` when you cross-validate the model using `fitcsvm`.

• Specify your own function using function handle notation.

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

``lossvalue = lossfun(C,S,W,Cost)``
where:

• The output argument `lossvalue` is a scalar.

• You choose the function name (`lossfun`).

• `C` is an `n`-by-`K` logical matrix with rows indicating which class the corresponding observation belongs. The column order corresponds to the class order in `SVMModel.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 `SVMModel.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. If you pass `W`, the software normalizes them 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.

Specify your function using `'LossFun',@lossfun`.

For more details on loss functions, see Classification Loss.

Data Types: `char` | `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`. The software weighs the observations in each row of `X` or `TBL` with the corresponding weight in `Weights`.

If you specify `Weights` as a vector, then the size of `Weights` must be equal to the number of rows of `X` or `TBL`.

If you specify `Weights` as the name of a variable in `TBL`, you must do so as a character vector. For example, if the weights are stored as `TBL.W`, then specify it as `'W'`. Otherwise, the software treats all columns of `TBL`, including `TBL.W`, as predictors.

If you do not specify your own loss function, then the software normalizes `Weights` to sum up to the value of the prior probability in the respective class.

Data Types: `single` | `double`

## Output Arguments

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Classification loss, returned as a scalar. `L` is a generalization or resubstitution quality measure. Its interpretation depends on the loss function and weighting scheme, but, in general, better classifiers yield smaller loss values.

## Definitions

### Classification Loss

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

Suppose that:

• 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, respectively.

• f(Xj) is the raw 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 to the average loss.

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

• yj* is a vector of K – 1 zeros, and a 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 y*2 = [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,

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

The supported loss functions are:

• Binomial deviance, specified using `'LossFun','binodeviance'`. Its equation is

`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}.$`
• Exponential loss, specified using `'LossFun','exponential'`. Its equation is

`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right).$`
• Classification error, specified using `'LossFun','classiferror'`. It is the weighted fraction of misclassified observations, with equation

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

${\stackrel{^}{y}}_{j}$ is the class label corresponding to the class with the maximal posterior probability. I{x} is the indicator function.

• Hinge loss, specified using `'LossFun','hinge'`. Its equation is

`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{max}\left\{0,1-{m}_{j}\right\}.$`
• Logit loss, specified using `'LossFun','logit'`. Its equation is

`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left(1+\mathrm{exp}\left(-{m}_{j}\right)\right).$`
• Minimal cost, specified using `'LossFun','mincost'`. The software computes the weighted minimal cost using this procedure for observations j = 1,...,n:

1. Estimate the 1-by-K vector of expected classification costs for observation j

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

f(Xj) is the column vector of class posterior probabilities for binary and multiclass classification. C is the cost matrix the input model stores in the property `Cost`.

2. For observation j, predict the class label corresponding to the minimum, expected classification cost:

`${\stackrel{^}{y}}_{j}=\underset{j=1,...,K}{\mathrm{min}}\left({\gamma }_{j}\right).$`
3. Using C, identify the cost incurred (cj) for making the prediction.

The weighted, average, minimum cost loss is

`$L=\sum _{j=1}^{n}{w}_{j}{c}_{j}.$`
• Quadratic loss, specified using `'LossFun','quadratic'`. Its equation is

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

This figure compares some of the loss functions for one observation over m (some functions are normalized to pass through [0,1]).

### Score

The SVM classification score for classifying observation x is the signed distance from x to the decision boundary ranging from -∞ to +∞. A positive score for a class indicates that x is predicted to be in that class, a negative score indicates otherwise.

The score for predicting x into the positive class, also the numerical, predicted response for x, $f\left(x\right)$, is the trained SVM classification function

`$f\left(x\right)=\sum _{j=1}^{n}{\alpha }_{j}{y}_{j}G\left({x}_{j},x\right)+b,$`

where $\left({\alpha }_{1},...,{\alpha }_{n},b\right)$ are the estimated SVM parameters, $G\left({x}_{j},x\right)$ is the dot product in the predictor space between x and the support vectors, and the sum includes the training set observations. The score for predicting x into the negative class is –f(x).

If G(xj,x) = xjx (the linear kernel), then the score function reduces to

`$f\left(x\right)=\left(x/s\right)\prime \beta +b.$`

s is the kernel scale and β is the vector of fitted linear coefficients.

## Examples

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Load the `ionosphere` data set.

```load ionosphere rng(1); % For reproducibility ```

Train an SVM classifier. Specify a 15% holdout sample for testing. It is good practice to specify the class order and standardize the data.

```CVSVMModel = fitcsvm(X,Y,'Holdout',0.15,'ClassNames',{'b','g'},... 'Standardize',true); CompactSVMModel = CVSVMModel.Trained{1}; % Extract the trained, compact classifier testInds = test(CVSVMModel.Partition); % Extract the test indices XTest = X(testInds,:); YTest = Y(testInds,:); ```

`CVSVMModel` is a `ClassificationPartitionedModel` classifier. It contains the property `Trained`, which is a 1-by-1 cell array holding a `CompactClassificationSVM` classifier that the software trained using the training set.

Determine how well the algorithm generalizes by estimating the test sample classification error.

```L = loss(CompactSVMModel,XTest,YTest) ```
```L = 0.0787 ```

The SVM classifier misclassifies approximately 8% of the test sample radar returns.

Load the `ionosphere` data set.

```load ionosphere rng(1); % For reproducibility ```

Train an SVM classifier. Specify a 15% holdout sample for testing. It is good practice to specify the class order and standardize the data.

```CVSVMModel = fitcsvm(X,Y,'Holdout',0.15,'ClassNames',{'b','g'},... 'Standardize',true); CompactSVMModel = CVSVMModel.Trained{1}; % Extract the trained, compact classifier testInds = test(CVSVMModel.Partition); % Extract the test indices XTest = X(testInds,:); YTest = Y(testInds,:); ```

`CVSVMModel` is a `ClassificationPartitionedModel` classifier. It contains the property `Trained`, which is a 1-by-1 cell array holding a `CompactClassificationSVM` classifier that the software trained using the training set.

Determine how well the algorithm generalizes by estimating the test sample hinge loss.

```L = loss(CompactSVMModel,XTest,YTest,'LossFun','Hinge') ```
```L = 0.2998 ```

The hinge loss is approximately 0.3. Classifiers with hinge losses close to 0 are desirable.

## References

[1] Hastie, T., R. Tibshirani, and J. Friedman. The Elements of Statistical Learning, second edition. Springer, New York, 2008.