# resubEdge

Resubstitution classification edge for multiclass error-correcting output codes (ECOC) model

## Syntax

``e = resubEdge(Mdl)``
``e = resubEdge(Mdl,Name,Value)``

## Description

example

````e = resubEdge(Mdl)` returns the resubstitution classification edge (`e`) for the multiclass error-correcting output codes (ECOC) model `Mdl` using the training data stored in `Mdl.X` and the corresponding class labels stored in `Mdl.Y`.The classification edge is a scalar value that represents the weighted mean of the classification margins.```

example

````e = resubEdge(Mdl,Name,Value)` computes the resubstitution classification edge with additional options specified by one or more name-value pair arguments. For example, you can specify a decoding scheme, binary learner loss function, and verbosity level.```

## Examples

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Compute the resubstitution edge for an ECOC model with SVM binary learners.

Load Fisher's iris data set. Specify the predictor data `X` and the response data `Y`.

```load fisheriris X = meas; Y = species;```

Train an ECOC model using SVM binary classifiers. Standardize the predictors using an SVM template, and specify the class order.

```t = templateSVM('Standardize',true); classOrder = unique(Y)```
```classOrder = 3x1 cell {'setosa' } {'versicolor'} {'virginica' } ```
`Mdl = fitcecoc(X,Y,'Learners',t,'ClassNames',classOrder);`

`t` is an SVM template object. During training, the software uses default values for empty properties in `t`. `Mdl` is a `ClassificationECOC` model.

Compute the resubstitution edge, which is the mean of the training-sample margins.

`e = resubEdge(Mdl)`
```e = 0.7440 ```

Perform feature selection by comparing training-sample edges from multiple models. Based solely on this comparison, the classifier with the greatest edge is the best classifier.

Load Fisher's iris data set. Define two data sets:

• `fullX` contains all four predictors.

• `partX` contains the sepal measurements only.

```load fisheriris X = meas; fullX = X; partX = X(:,1:2); Y = species;```

Train an ECOC model using SVM binary learners for each predictor set. Standardize the predictors using an SVM template, specify the class order, and compute the posterior probabilities.

```t = templateSVM('Standardize',true); classOrder = unique(Y)```
```classOrder = 3x1 cell {'setosa' } {'versicolor'} {'virginica' } ```
```FullMdl = fitcecoc(fullX,Y,'Learners',t,'ClassNames',classOrder,... 'FitPosterior',true); PartMdl = fitcecoc(partX,Y,'Learners',t,'ClassNames',classOrder,... 'FitPosterior',true);```

The default SVM score is the distance from the decision boundary. If you specify to compute posterior probabilities, then the software uses posterior probabilities as scores.

Compute the resubstitution edge for each classifier. The quadratic loss function operates on scores in the domain [0,1]. Specify to use quadratic loss when aggregating the binary learners for both models.

`fullEdge = resubEdge(FullMdl,'BinaryLoss','quadratic')`
```fullEdge = 0.9896 ```
`partEdge = resubEdge(PartMdl,'BinaryLoss','quadratic')`
```partEdge = 0.5059 ```

The edge for the classifier trained on the complete data set is greater, suggesting that the classifier trained with all the predictors has a better training-sample fit.

## Input Arguments

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Full, trained multiclass ECOC model, specified as a `ClassificationECOC` model trained with `fitcecoc`.

### 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: `resubEdge(Mdl,'BinaryLoss','quadratic')` specifies a quadratic binary learner loss function.

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

• This table describes the built-in functions, where yj is the class label for a particular binary learner (in the set {–1,1,0}), sj is the score for observation j, and g(yj,sj) is the binary loss formula.

ValueDescriptionScore Domaing(yj,sj)
`'binodeviance'`Binomial deviance(–∞,∞)log[1 + exp(–2yjsj)]/[2log(2)]
`'exponential'`Exponential(–∞,∞)exp(–yjsj)/2
`'hamming'`Hamming[0,1] or (–∞,∞)[1 – sign(yjsj)]/2
`'hinge'`Hinge(–∞,∞)max(0,1 – yjsj)/2
`'linear'`Linear(–∞,∞)(1 – yjsj)/2
`'logit'`Logistic(–∞,∞)log[1 + exp(–yjsj)]/[2log(2)]
`'quadratic'`Quadratic[0,1][1 – yj(2sj – 1)]2/2

The software normalizes binary losses so that the loss is 0.5 when yj = 0. Also, the software calculates the mean binary loss for each class.

• For a custom binary loss function, for example `customFunction`, specify its function handle `'BinaryLoss',@customFunction`.

`customFunction` has this form:

`bLoss = customFunction(M,s)`

• `M` is the K-by-B coding matrix stored in `Mdl.CodingMatrix`.

• `s` is the 1-by-B row vector of classification scores.

• `bLoss` is the classification loss. This scalar aggregates the binary losses for every learner in a particular class. For example, you can use the mean binary loss to aggregate the loss over the learners for each class.

• K is the number of classes.

• B is the number of binary learners.

For an example of passing a custom binary loss function, see Predict Test-Sample Labels of ECOC Model Using Custom Binary Loss Function.

The default `BinaryLoss` value depends on the score ranges returned by the binary learners. This table identifies what some default `BinaryLoss` values are when you use the default score transform (`ScoreTransform` property of the model is `'none'`).

AssumptionDefault Value

All binary learners are any of the following:

• Classification decision trees

• Discriminant analysis models

• k-nearest neighbor models

• Linear or kernel classification models of logistic regression learners

• Naive Bayes models

`'quadratic'`
All binary learners are SVMs or linear or kernel classification models of SVM learners.`'hinge'`
All binary learners are ensembles trained by `AdaboostM1` or `GentleBoost`.`'exponential'`
All binary learners are ensembles trained by `LogitBoost`.`'binodeviance'`
You specify to predict class posterior probabilities by setting `'FitPosterior',true` in `fitcecoc`.`'quadratic'`
Binary learners are heterogeneous and use different loss functions.`'hamming'`

To check the default value, use dot notation to display the `BinaryLoss` property of the trained model at the command line.

Example: `'BinaryLoss','binodeviance'`

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

Decoding scheme that aggregates the binary losses, specified as the comma-separated pair consisting of `'Decoding'` and `'lossweighted'` or `'lossbased'`. For more information, see Binary Loss.

Example: `'Decoding','lossbased'`

Estimation options, specified as the comma-separated pair consisting of `'Options'` and a structure array returned by `statset`.

To invoke parallel computing:

• You need a Parallel Computing Toolbox™ license.

• Specify `'Options',statset('UseParallel',true)`.

Verbosity level, specified as the comma-separated pair consisting of `'Verbose'` and `0` or `1`. `Verbose` controls the number of diagnostic messages that the software displays in the Command Window.

If `Verbose` is `0`, then the software does not display diagnostic messages. Otherwise, the software displays diagnostic messages.

Example: `'Verbose',1`

Data Types: `single` | `double`

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### Classification Edge

The classification edge is the weighted mean of the classification margins.

One way to choose among multiple classifiers, for example to perform feature selection, is to choose the classifier that yields the greatest edge.

### Classification Margin

The classification margin is, for each observation, the difference between the negative loss for the true class and the maximal negative loss among the false classes. If the margins are on the same scale, then they serve as a classification confidence measure. Among multiple classifiers, those that yield greater margins are better.

### Binary Loss

The binary loss is a function of the class and classification score that determines how well a binary learner classifies an observation into the class.

Suppose the following:

• mkj is element (k,j) of the coding design matrix M—that is, the code corresponding to class k of binary learner j. M is a K-by-B matrix, where K is the number of classes, and B is the number of binary learners.

• sj is the score of binary learner j for an observation.

• g is the binary loss function.

• $\stackrel{^}{k}$ is the predicted class for the observation.

The decoding scheme of an ECOC model specifies how the software aggregates the binary losses and determines the predicted class for each observation. The software supports two decoding schemes:

• Loss-based decoding [2] (`Decoding` is `'lossbased'`) — The predicted class of an observation corresponds to the class that produces the minimum average of the binary losses over all binary learners.

`$\stackrel{^}{k}=\underset{k}{\text{argmin}}\frac{1}{B}\sum _{j=1}^{B}|{m}_{kj}|g\left({m}_{kj},{s}_{j}\right).$`

• Loss-weighted decoding [3] (`Decoding` is `'lossweighted'`) — The predicted class of an observation corresponds to the class that produces the minimum average of the binary losses over the binary learners for the corresponding class.

`$\stackrel{^}{k}=\underset{k}{\text{argmin}}\frac{\sum _{j=1}^{B}|{m}_{kj}|g\left({m}_{kj},{s}_{j}\right)}{\sum _{j=1}^{B}|{m}_{kj}|}.$`

The denominator corresponds to the number of binary learners for class k. [1] suggests that loss-weighted decoding improves classification accuracy by keeping loss values for all classes in the same dynamic range.

The `predict`, `resubPredict`, and `kfoldPredict` functions return the negated value of the objective function of `argmin` as the second output argument (`NegLoss`) for each observation and class.

This table summarizes the supported binary loss functions, where yj is a class label for a particular binary learner (in the set {–1,1,0}), sj is the score for observation j, and g(yj,sj) is the binary loss function.

ValueDescriptionScore Domaing(yj,sj)
`"binodeviance"`Binomial deviance(–∞,∞)log[1 + exp(–2yjsj)]/[2log(2)]
`"exponential"`Exponential(–∞,∞)exp(–yjsj)/2
`"hamming"`Hamming[0,1] or (–∞,∞)[1 – sign(yjsj)]/2
`"hinge"`Hinge(–∞,∞)max(0,1 – yjsj)/2
`"linear"`Linear(–∞,∞)(1 – yjsj)/2
`"logit"`Logistic(–∞,∞)log[1 + exp(–yjsj)]/[2log(2)]
`"quadratic"`Quadratic[0,1][1 – yj(2sj – 1)]2/2

The software normalizes binary losses so that the loss is 0.5 when yj = 0, and aggregates using the average of the binary learners.

Do not confuse the binary loss with the overall classification loss (specified by the `LossFun` name-value argument of the `resubLoss` and `resubPredict` object functions), which measures how well an ECOC classifier performs as a whole.

## Tips

• To compare the margins or edges of several ECOC classifiers, use template objects to specify a common score transform function among the classifiers during training.

## References

[1] Allwein, E., R. Schapire, and Y. Singer. “Reducing multiclass to binary: A unifying approach for margin classiﬁers.” Journal of Machine Learning Research. Vol. 1, 2000, pp. 113–141.

[2] Escalera, S., O. Pujol, and P. Radeva. “Separability of ternary codes for sparse designs of error-correcting output codes.” Pattern Recog. Lett., Vol. 30, Issue 3, 2009, pp. 285–297.

[3] Escalera, S., O. Pujol, and P. Radeva. “On the decoding process in ternary error-correcting output codes.” IEEE Transactions on Pattern Analysis and Machine Intelligence. Vol. 32, Issue 7, 2010, pp. 120–134.

## Version History

Introduced in R2014b