resubLoss

Find classification loss for support vector machine (SVM) classifier by resubstitution

Syntax

``L = resubLoss(SVMModel)``
``L = resubLoss(SVMModel,'LossFun',lossFun)``

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 the corresponding class labels stored in `SVMModel.Y`. The classification loss (`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 classification loss values.```

example

````L = resubLoss(SVMModel,'LossFun',lossFun)` returns the classification loss by resubstitution using the loss function supplied in `LossFun`.```

Examples

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Load the `ionosphere` data set.

`load ionosphere`

Train an 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 resubstitution loss (that is, 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. 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.

Input Arguments

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

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'`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 by using function handle notation.

Suppose that `n` is the number of observations in `X`, and `K` is the number of distinct classes (`numel(SVMModel.ClassNames)`) used to create the input model (`SVMModel`). 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 the class to which 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, similar to the output of `predict`. The column order corresponds to the class order in `SVMModel.ClassNames`.

• `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.

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

For more details on loss functions, see Classification Loss.

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, 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 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'`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{log}\left\{1+\mathrm{exp}\left[-2{m}_{j}\right]\right\}.$
Exponential loss`'exponential'`$L=\sum _{j=1}^{n}{w}_{j}\mathrm{exp}\left(-{m}_{j}\right).$
Classification error`'classiferror'`

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

The classification error is the weighted fraction of misclassified observations where ${\stackrel{^}{y}}_{j}$ is the class label corresponding to the class with the maximal posterior probability. I{x} 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 cost`'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 stored by the input model in the `Cost` property.

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

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

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

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

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 negative class classification score for x, or 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.

For more details, see Understanding Support Vector Machines.

References

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

Introduced in R2014a