kfoldEdge

Classification edge for cross-validated kernel ECOC model

Syntax

edge = kfoldEdge(CVMdl)

edge = kfoldEdge(CVMdl,Name,Value)

Description

edge = kfoldEdge(CVMdl) returns the classification edge obtained by the cross-validated kernel ECOC model (ClassificationPartitionedKernelECOC) CVMdl. For every fold, kfoldEdge computes the classification edge for validation-fold observations using a model trained on training-fold observations.

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

Estimate k-Fold Cross-Validation Edge

Load Fisher's iris data set. X contains flower measurements, and Y contains the names of flower species.

load fisheriris
X = meas;
Y = species;

Cross-validate an ECOC model composed of kernel binary learners.

CVMdl = fitcecoc(X,Y,'Learners','kernel','CrossVal','on')

CVMdl = 
  ClassificationPartitionedKernelECOC
    CrossValidatedModel: 'KernelECOC'
           ResponseName: 'Y'
        NumObservations: 150
                  KFold: 10
              Partition: [1x1 cvpartition]
             ClassNames: {'setosa'  'versicolor'  'virginica'}
         ScoreTransform: 'none'


  Properties, Methods

CVMdl is a ClassificationPartitionedKernelECOC model. By default, the software implements 10-fold cross-validation. To specify a different number of folds, use the 'KFold' name-value pair argument instead of 'Crossval'.

Estimate the cross-validated classification edges.

edge = kfoldEdge(CVMdl)

edge = 0.4145

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

Feature Selection Using k-Fold Edges

Perform feature selection by comparing k-fold 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. X contains flower measurements, and Y contains the names of flower species.

load fisheriris
X = meas;
Y = species;

Randomly choose half of the predictor variables.

rng(1); % For reproducibility
p = size(X,2); % Number of predictors
idxPart = randsample(p,ceil(0.5*p));

Cross-validate two ECOC models composed of kernel classification models: one that uses all of the predictors, and one that uses half of the predictors.

CVMdl = fitcecoc(X,Y,'Learners','kernel','CrossVal','on');
PCVMdl = fitcecoc(X(:,idxPart),Y,'Learners','kernel','CrossVal','on');

CVMdl and PCVMdl are ClassificationPartitionedKernelECOC models. By default, the software implements 10-fold cross-validation. To specify a different number of folds, use the 'KFold' name-value pair argument instead of 'Crossval'.

Estimate the k-fold edge for each classifier.

fullEdge = kfoldEdge(CVMdl)

fullEdge = 0.4092

partEdge = kfoldEdge(PCVMdl)

partEdge = 0.4161

Based on the k-fold edges, the two classifiers are comparable.

Input Arguments

CVMdl — Cross-validated kernel ECOC model

ClassificationPartitionedKernelECOC model

Cross-validated kernel ECOC model, specified as a ClassificationPartitionedKernelECOC model. You can create a ClassificationPartitionedKernelECOC model by training an ECOC model using fitcecoc and specifying these name-value pair arguments:

• 'Learners'– Set the value to 'kernel', a template object returned by templateKernel, or a cell array of such template objects.

• One of the 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.

'BinaryLoss' — Binary learner loss function

'hamming' | 'linear' | 'logit' | 'exponential' | 'binodeviance' | 'hinge' | 'quadratic' | function handle

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 contains names and descriptions of the built-in functions, where y_j is a class label for a particular binary learner (in the set {–1,1,0}), s_j is the score for observation j, and g(y_j,s_j) is the binary loss formula.

  Value	Description	Score Domain	g(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 y_j = 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.

By default, if all binary learners are kernel classification models using SVM, then BinaryLoss is 'hinge'. If all binary learners are kernel classification models using logistic regression, then BinaryLoss is 'quadratic'.

Example: 'BinaryLoss','binodeviance'

Data Types: char | string | function_handle

'Decoding' — Decoding scheme

'lossweighted' (default) | 'lossbased'

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'

'Folds' — Fold indices for prediction

1:CVMdl.KFold (default) | numeric vector of positive integers

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

'Mode' — Aggregation level for output

'average' (default) | 'individual'

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

This table describes the values.

Value	Description
'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'

'Options' — Estimation options

[] (default) | structure array returned by statset

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).

'Verbose' — Verbosity level

0 (default) | 1

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

edge — Classification edge

numeric scalar | numeric column vector

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

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:

• m_kj is element (k,j) of the coding design matrix M (that is, the code corresponding to class k of binary learner j).

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

• g is the binary loss function.

• $$\widehat{k}$$ 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,

$$\widehat{k}=\underset{k}{\text{argmin}}{\displaystyle \sum _{j=1}^L\left|m_{kj}\right|g}(m_{kj},s_j).$$

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,

$$\widehat{k}=\underset{k}{\text{argmin}}\frac{{\displaystyle \sum _{j=1}^L\left|m_{kj}\right|g}(m_{kj},s_j)}{{\displaystyle \sum }_{j=1}^L\left|m_{kj}\right|}.$$

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 y_j is a class label for a particular binary learner (in the set {–1,1,0}), s_j is the score for observation j, and g(y_j,s_j).

Value	Description	Score Domain	g(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 y_j = 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.