Documentation

# kfoldLoss

Classification loss for cross-validated ECOC model

## Syntax

``loss = kfoldLoss(CVMdl)``
``loss = kfoldLoss(CVMdl,Name,Value)``

## Description

example

````loss = kfoldLoss(CVMdl)` returns the classification loss obtained by the cross-validated ECOC model (`ClassificationPartitionedECOC`) `CVMdl`. For every fold, `kfoldLoss` computes the classification loss for validation-fold observations using a model trained on training-fold observations. `CVMdl.X` contains both sets of observations.```

example

````loss = kfoldLoss(CVMdl,Name,Value)` returns the classification loss with additional options specified by one or more name-value pair arguments. For example, specify the number of folds, decoding scheme, or verbosity level.```

## Examples

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Load Fisher's iris data set. Specify the predictor data `X`, the response data `Y`, and the order of the classes in `Y`.

```load fisheriris X = meas; Y = categorical(species); classOrder = unique(Y); % Class order rng(1); % For reproducibility```

Train and cross-validate an ECOC model using support vector machine (SVM) binary classifiers. Standardize the predictors using an SVM template, and specify the class order.

```t = templateSVM('Standardize',1); CVMdl = fitcecoc(X,Y,'CrossVal','on','Learners',t,'ClassNames',classOrder);```

`CVMdl` is a `ClassificationPartitionedECOC` model. By default, the software implements 10-fold cross-validation. You can specify a different number of folds using the `'KFold'` name-value pair argument.

Estimate the average classification error.

`L = kfoldLoss(CVMdl)`
```L = 0.0400 ```

The average classification error for the folds is 4%.

Alternatively, you can obtain the per-fold losses by specifying the name-value pair `'Mode','individual'` in `kfoldLoss`.

The classification loss is a measure of classifier quality. To determine which folds perform poorly, display the losses for each fold.

Load Fisher's iris data set. Specify the predictor data `X`, the response data `Y`, and the order of the classes in `Y`.

```load fisheriris X = meas; Y = categorical(species); classOrder = unique(Y); rng(1); % For reproducibility```

Train an ECOC model using SVM binary classifiers. Use 8-fold cross-validation, standardize the predictors using an SVM template, and specify the class order.

```t = templateSVM('Standardize',1); CVMdl = fitcecoc(X,Y,'KFold',8,'Learners',t,'ClassNames',classOrder);```

Estimate the average classification loss across all folds and the losses for each fold.

`loss = kfoldLoss(CVMdl)`
```loss = 0.0333 ```
`losses = kfoldLoss(CVMdl,'Mode','individual')`
```losses = 8×1 0.0556 0.0526 0.1579 0 0 0 0 0 ```

The third fold misclassifies a much higher percentage of observations than any other fold.

Return the average classification loss for the folds that perform well by specifying the `'Folds'` name-value pair argument.

`newloss = kfoldLoss(CVMdl,'Folds',[1:2 4:8])`
```newloss = 0.0153 ```

The total classification loss decreases by approximately half its original size.

Consider adjusting parameters of the binary classifiers or the coding design to see if performance for all folds improves.

In addition to knowing whether a model generally classifies observations correctly, you can determine how well the model classifies an observation into its predicted class. One way to determine this type of model quality is to pass a custom loss function to `kfoldLoss`.

Load Fisher's iris data set. Specify the predictor data `X`, the response data `Y`, and the order of the classes in `Y`.

```load fisheriris X = meas; Y = categorical(species); classOrder = unique(Y) % Class order```
```classOrder = 3x1 categorical array setosa versicolor virginica ```
`rng(1) % For reproducibility`

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

```t = templateSVM('Standardize',1); CVMdl = fitcecoc(X,Y,'CrossVal','on','Learners',t,'ClassNames',classOrder);```

`CVMdl` is a `ClassificationPartitionedECOC` model. By default, the software implements 10-fold cross-validation. You can specify a different number of folds using the `'KFold'` name-value pair argument.

Create a custom function that takes the minimal loss for each observation, then averages the minimal losses for all observations. `S` corresponds to the `NegLoss` output of `kfoldPredict`.

`lossfun = @(~,S,~,~)mean(min(-S,[],2));`

Compute the cross-validated custom loss.

`kfoldLoss(CVMdl,'LossFun',lossfun)`
```ans = 0.0101 ```

The average minimal binary loss for the validation-fold observations is `0.0101`.

## Input Arguments

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Cross-validated ECOC model, specified as a `ClassificationPartitionedECOC` model. You can create a `ClassificationPartitionedECOC` model in two ways:

### 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 quotes. You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

Example: `kfoldLoss(CVMdl,'Folds',[1 3 5])` specifies to use only the first, third, and fifth folds to calculate the classification loss.

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 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 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)`
where:

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

• `s` is the 1-by-L 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.

• L 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 describes some default `BinaryLoss` values based on the given assumptions.

AssumptionDefault Value
All binary learners are SVMs or either 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'`
All binary learners are linear or kernel classification models of logistic regression learners. Or, you specify to predict class posterior probabilities by setting `'FitPosterior',true` in `fitcecoc`.`'quadratic'`

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'`

Fold indices for prediction, specified as the comma-separated pair consisting of `'Folds'` and a numeric vector of positive integers. The elements of `Folds` must be within the range from `1` to `CVMdl.KFold`.

The software uses only the folds specified in `Folds` for prediction.

Example: `'Folds',[1 4 10]`

Data Types: `single` | `double`

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

• Specify the built-in function `'classiferror'`. In this case, the loss function is the classification error.

• Or, specify your own function using function handle notation.

Assume that n is the number of observations in the training data (`CVMdl.NumObservations`) and K is the number of classes (`numel(CVMdl.ClassNames)`). Your function needs 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 `CVMdl.ClassNames`.

Construct `C` by setting `C(p,q) = 1` if observation `p` is in class `q`, for each row. Set every element of row `p` to `0`.

• `S` is an n-by-K numeric matrix of negated loss values for the classes. Each row corresponds to an observation. The column order corresponds to the class order in `CVMdl.ClassNames`. The input `S` resembles the output argument `NegLoss` of `kfoldPredict`.

• `W` is an n-by-1 numeric vector of observation weights. If you pass `W`, the software normalizes its elements 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`.

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

Aggregation level for the output, specified as the comma-separated pair consisting of `'Mode'` and `'average'` or `'individual'`.

This table describes the values.

ValueDescription
`'average'`The output is a scalar average over all folds.
`'individual'`The output is a vector of length k containing one value per fold, where k is the number of folds.

Example: `'Mode','individual'`

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`

## Output Arguments

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Classification loss, returned as a numeric scalar or numeric column vector.

If `Mode` is `'average'`, then `loss` is the average classification loss over all folds. Otherwise, `loss` is a k-by-1 numeric column vector containing the classification loss for each fold, where k is the number of folds.

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### Classification Error

The classification error is a binary classification error measure that has the form

`$L=\frac{\sum _{j=1}^{n}{w}_{j}{e}_{j}}{\sum _{j=1}^{n}{w}_{j}},$`

where:

• wj is the weight for observation j. The software renormalizes the weights to sum to 1.

• ej = 1 if the predicted class of observation j differs from its true class, and 0 otherwise.

In other words, the classification error is the proportion of observations misclassified by the classifier.

### Binary Loss

A 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).

• 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.

In loss-based decoding [Escalera et al.], the class producing the minimum sum of the binary losses over binary learners determines the predicted class of an observation, that is,

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

In loss-weighted decoding [Escalera et al.], the class producing the minimum average of the binary losses over binary learners determines the predicted class of an observation, that is,

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

Allwein et al. suggest that loss-weighted decoding improves classification accuracy by keeping loss values for all classes in the same dynamic range.

This table summarizes the supported 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).

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 such that the loss is 0.5 when yj = 0, and aggregates using the average of the binary learners [Allwein et al.].

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

## 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. “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.

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