kfoldEdge

Classification edge for cross-validated ECOC model

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Syntax

edge = kfoldEdge(CVMdl)
edge = kfoldEdge(CVMdl,Name,Value)

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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Open Live Script

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 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.

Open Live Script

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.

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 classification edge for each fold.

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

    0.4792
    0.4872
    0.4259
    0.5302
    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.

Open Live Script

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.

load fisheriris
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 = 0.4825

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 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: 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 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

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.

More About

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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

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.

  • 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,

k^=argmink∑j=1L|mkj|g(mkj,sj).

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,

k^=argmink∑j=1L|mkj|g(mkj,sj)∑j=1L|mkj|.

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.

Extended Capabilities

See Also

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Topics

Introduced in R2014b

Statistics and Machine Learning Toolbox Documentation

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