kfoldLoss

Classification loss for cross-validated kernel ECOC model

Description

example

loss = kfoldLoss(CVMdl) returns the classification loss obtained by the cross-validated kernel ECOC model (ClassificationPartitionedKernelECOC) CVMdl. For every fold, kfoldLoss computes the classification loss for validation-fold observations using a model trained on training-fold observations. kfoldLoss applies the same data used to create CVMdl (see fitcecoc).

By default, kfoldLoss returns the classification error.

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 classification loss function, number of folds, decoding scheme, or verbosity level.

Examples

collapse all

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 = 
  classreg.learning.partition.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 loss. By default, the software computes the classification error.

loss = kfoldLoss(CVMdl)
loss = 0.0333

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

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. 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.

rng(1) % For reproducibility
CVMdl = fitcecoc(X,Y,'Learners','kernel','CrossVal','on')
CVMdl = 
  classreg.learning.partition.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'.

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 = 0.0199

The average minimal binary loss for the validation-fold observations is about 0.02.

Input Arguments

collapse all

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: 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 contains names and descriptions of 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 such 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.

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 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

collapse all

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.

More About

collapse all

Classification Error

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

L=j=1nwjejj=1nwj,

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.

  • 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^=argminkj=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^=argminkj=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.

Introduced in R2018b