# kfoldEdge

Classification edge for cross-validated ECOC model

## Description

example

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

example

edge = kfoldEdge(CVMdl,Name,Value) returns the classification edge 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.

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

Train and cross-validate an ECOC model using support vector machine (SVM) binary classifiers. Standardize the predictor data 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 of the edges.

edge = kfoldEdge(CVMdl)
edge = 0.4825

Alternatively, you can obtain the per-fold edges by specifying the name-value pair 'Mode','individual' in kfoldEdge.

The classification edge is a relative measure of classifier quality. To determine which folds perform poorly, display the edges 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.

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 classification edge for each fold.

edges = kfoldEdge(CVMdl,'Mode','individual')
edges = 8×1

0.4791
0.4872
0.4260
0.5301
0.5064
0.4575
0.4860
0.4687

The edges have similar magnitudes across folds. Folds that perform poorly have small edges relative to the other folds.

To return the average classification edge across the folds that perform well, specify the 'Folds' name-value pair argument.

The classifier edge measures the average of the classifier margins. One way to perform feature selection is to compare cross-validation edges from multiple models. Based solely on this criterion, the classifier with the greatest edge is the best classifier.

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

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

Define the following two data sets.

• fullX contains all the predictors.

• partX contains the petal dimensions.

fullX = X;
partX = X(:,3:4);

For each predictor set, 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(fullX,Y,'CrossVal','on','Learners',t,...
'ClassNames',classOrder);
PCVMdl = fitcecoc(partX,Y,'CrossVal','on','Learners',t,...
'ClassNames',classOrder);

CVMdl and PCVMdl are ClassificationPartitionedECOC models. By default, the software implements 10-fold cross-validation.

Estimate the edge for each classifier.

fullEdge = kfoldEdge(CVMdl)
fullEdge = 0.4826
partEdge = kfoldEdge(PCVMdl)
partEdge = 0.4951

The two models have comparable edges.

## Input Arguments

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

• Pass a trained ECOC model (ClassificationECOC) to crossval.

• Train an ECOC model using fitcecoc and specify any one of these cross-validation name-value pair arguments: 'CrossVal', 'CVPartition', 'Holdout', 'KFold', or 'Leaveout'.

### 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: kfoldEdge(CVMdl,'BinaryLoss','hinge') specifies 'hinge' as the 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)]

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

• Naive Bayes models

All binary learners are SVMs.'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'

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

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 edge, returned as a numeric scalar or numeric column vector.

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

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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)]

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 kfoldLoss and kfoldPredict 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. “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