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resubLoss

Class: ClassificationSVM

Classification loss for support vector machine classifiers by resubstitution

Syntax

• ``L = resubLoss(SVMModel)``
example
• ``L = resubLoss(SVMModel,Name,Value)``
example

Description

example

````L = resubLoss(SVMModel)` returns the classification loss by resubstitution (`L`), the in-sample classification loss, for the support vector machine (SVM) classifier `SVMModel` using the training data stored in `SVMModel.X` and corresponding class labels stored in `SVMModel.Y`.```

example

````L = resubLoss(SVMModel,Name,Value)` returns the classification loss by resubstitution with additional options specified by one or more `Name,Value` pair arguments.```

Input Arguments

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Full, trained SVM classifier, specified as a `ClassificationSVM` model trained using `fitcsvm`.

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`

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 ```

Train an SVM classifier. It is good practice to standardize the data.

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

`SVMModel` is a trained `ClassificationSVM` classifier. The negative class is `'b'` and the positive class is `'g'`.

Estimate the resubstitution loss (i.e., the in-sample classification error).

```L = resubLoss(SVMModel) ```
```L = 0.0570 ```

The SVM classifier misclassifies 5.7% of the training sample radar returns.

Load the `ionosphere` data set.

```load ionosphere ```

Train an SVM classifier. It is good practice to standardize the data.

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

`SVMModel` is a trained `ClassificationSVM` classifier. The negative class is `'b'` and the positive class is `'g'`.

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 desirable.

References

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