resubLoss

Resubstitution classification loss for classification ensemble model

Syntax

``L = resubLoss(ens)``
``L = resubLoss(ens,Name=Value)``

Description

example

````L = resubLoss(ens)` returns the Classification Loss `L` by resubstitution for the trained classification ensemble model `ens` using the training data stored in `ens.X` and the corresponding true class labels stored in `ens.Y`. By default, `resubLoss` uses the loss, meaning the loss computed for the data used by `fitcensemble` to create `ens`.The classification loss (`L`) is a resubstitution quality measure. Its interpretation depends on the loss function (`LossFun`) and weighting scheme, but in general, better classifiers yield smaller classification loss values. The default `LossFun` value is `"classiferror"` (misclassification rate in decimal).```
````L = resubLoss(ens,Name=Value)` specifies additional options using one or more name-value arguments. For example, you can specify the loss function, the indices of the weak learners to use for calculating the loss, and the aggregation level for the output.```

Examples

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`load fisheriris`

Train a classification ensemble of 100 decision trees using AdaBoostM2. Specify tree stumps as the weak learners.

```t = templateTree(MaxNumSplits=1); ens = fitcensemble(meas,species,Method="AdaBoostM2",Learners=t);```

Estimate the resubstitution classification error.

`loss = resubLoss(ens)`
```loss = 0.0333 ```

Input Arguments

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Classification ensemble model, specified as a `ClassificationEnsemble` model object trained with `fitcensemble`.

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: `resubLoss(ens,LossFun="exponential",UseParallel=true)` specifies to use an exponential loss function, and to perform computations in parallel.

Indices of weak learners in the ensemble to use in `resubLoss`, specified as a vector of positive integers in the range [1:`ens.NumTrained`]. By default, all learners are used.

Example: `Learners=[1 2 4]`

Data Types: `single` | `double`

Loss function, specified as a built-in loss function name or a function handle.

The following table describes the values for the built-in loss functions.

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.

• Bagged and subspace ensembles return posterior probabilities by default (`ens.Method` is `"Bag"` or `"Subspace"`).

• To use posterior probabilities as classification scores when the ensemble method is `"AdaBoostM1"`, `"AdaBoostM2"`, `"GentleBoost"`, or `"LogitBoost"`, you must specify the double-logit score transform by entering the following:

`ens.ScoreTransform = "doublelogit";`

• For all other ensemble methods, the software does not support posterior probabilities as classification scores.

You can specify your own function using function handle notation. Suppose that `n` is the number of observations in `X`, and `K` is the number of distinct classes (`numel(ens.ClassNames)`, where `ens` is the input model). Your function must have the signature

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

• 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 `ens.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 `ens.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 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.

For more details on loss functions, see Classification Loss.

Example: `LossFun="binodeviance"`

Example: `LossFun=@Lossfun`

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

Aggregation level for the output, specified as `"ensemble"`, `"individual"`, or `"cumulative"`.

ValueDescription
`"ensemble"`The output is a scalar value, the loss for the entire ensemble.
`"individual"`The output is a vector with one element per trained learner.
`"cumulative"`The output is a vector in which element `J` is obtained by using learners `1:J` from the input list of learners.

Example: `Mode="individual"`

Data Types: `char` | `string`

Flag to run in parallel, specified as a numeric or logical `1` (`true`) or `0` (`false`). If you specify `UseParallel=true`, the `resubLoss` function executes `for`-loop iterations by using `parfor`. The loop runs in parallel when you have Parallel Computing Toolbox™.

Example: `UseParallel=true`

Data Types: `logical`

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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 (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.

• 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 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 R2011a

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