# kfoldLoss

Classification loss for observations not used in training

## Description

example

L = kfoldLoss(CVMdl) returns the cross-validated classification error rates estimated by the cross-validated, error-correcting output codes (ECOC) model composed of linear classification models CVMdl. That is, for every fold, kfoldLoss estimates the classification error rate for observations that it holds out when it trains using all other observations. kfoldLoss applies the same data used create CVMdl (see fitcecoc).

L contains a classification loss for each regularization strength in the linear classification models that compose CVMdl.

example

L = kfoldLoss(CVMdl,Name,Value) uses additional options specified by one or more Name,Value pair arguments. For example, specify a decoding scheme, which folds to use for the loss calculation, or verbosity level.

## Input Arguments

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Cross-validated, ECOC model composed of linear classification models, specified as a ClassificationPartitionedLinearECOC model object. You can create a ClassificationPartitionedLinearECOC model using fitcecoc and by:

1. Specifying any one of the cross-validation, name-value pair arguments, for example, CrossVal

2. Setting the name-value pair argument Learners to 'linear' or a linear classification model template returned by templateLinear

To obtain estimates, kfoldLoss applies the same data used to cross-validate the ECOC model (X and Y).

### 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.

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 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 the 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, e.g., customFunction, specify its function handle 'BinaryLoss',@customFunction.

customFunction should have this form

bLoss = customFunction(M,s)
where:

• 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.

By default, if all binary learners are linear classification models using:

• SVM, then BinaryLoss is 'hinge'

• 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 to use for classification-score prediction, specified as the comma-separated pair consisting of 'Folds' and a numeric vector of positive integers. The elements of Folds must range from 1 through CVMdl.KFold.

Example: 'Folds',[1 4 10]

Data Types: single | double

Loss function, specified as 'classiferror', 'classifcost', or a function handle.

You can:

• Specify the built-in function 'classiferror', then the loss function is the classification error.

• Specify the built-in function 'classifcost'. In this case, the loss function is the observed misclassification cost. If you use the default cost matrix (whose element value is 0 for correct classification and 1 for incorrect classification), then the loss values for 'classifcost' and 'classiferror' are identical.

• Specify your own function using function handle notation.

For what follows, 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 choose the function name (lossfun).

• C is an n-by-K logical matrix with rows indicating which class 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 classes. Each row corresponds to an observation. The column order corresponds to the class order in CVMdl.ClassNames. 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.

Data Types: function_handle | char | string

Loss aggregation level, specified as the comma-separated pair consisting of 'Mode' and 'average' or 'individual'.

ValueDescription
'average'Returns losses averaged over all folds
'individual'Returns losses for each fold

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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Cross-validated classification losses, returned as a numeric scalar, vector, or matrix. The interpretation of L depends on LossFun.

Let R be the number of regularizations strengths is the cross-validated models (CVMdl.Trained{1}.BinaryLearners{1}.Lambda) and F be the number of folds (stored in CVMdl.KFold).

• If Mode is 'average', then L is a 1-by-R vector. L(j) is the average classification loss over all folds of the cross-validated model that uses regularization strength j.

• Otherwise, L is a F-by-R matrix. L(i,j) is the classification loss for fold i of the cross-validated model that uses regularization strength j.

## Examples

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X is a sparse matrix of predictor data, and Y is a categorical vector of class labels.

Cross-validate an ECOC model of linear classification models.

rng(1); % For reproducibility
CVMdl = fitcecoc(X,Y,'Learner','linear','CrossVal','on');

CVMdl is a ClassificationPartitionedLinearECOC model. By default, the software implements 10-fold cross validation.

Estimate the average of the out-of-fold classification error rates.

ce = kfoldLoss(CVMdl)
ce = 0.0958

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

Load the NLP data set. Transpose the predictor data.

X = X';

For simplicity, use the label 'others' for all observations in Y that are not 'simulink', 'dsp', or 'comm'.

Create a linear classification model template that specifies optimizing the objective function using SpaRSA.

t = templateLinear('Solver','sparsa');

Cross-validate an ECOC model of linear classification models using 5-fold cross-validation. Optimize the objective function using SpaRSA. Specify that the predictor observations correspond to columns.

rng(1); % For reproducibility
CVMdl = fitcecoc(X,Y,'Learners',t,'KFold',5,'ObservationsIn','columns');
CMdl1 = CVMdl.Trained{1}
CMdl1 =
CompactClassificationECOC
ResponseName: 'Y'
ScoreTransform: 'none'
BinaryLearners: {6x1 cell}
CodingMatrix: [4x6 double]

Properties, Methods

CVMdl is a ClassificationPartitionedLinearECOC model. It contains the property Trained, which is a 5-by-1 cell array holding a CompactClassificationECOC models that the software trained using the training set of each fold.

Create a function that takes the minimal loss for each observation, and then averages the minimal losses across all observations. Because the function does not use the class-identifier matrix (C), observation weights (W), and classification cost (Cost), use ~ to have kfoldLoss ignore its their positions.

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

Estimate the average cross-validated classification loss using the minimal loss per observation function. Also, obtain the loss for each fold.

ce = kfoldLoss(CVMdl,'LossFun',lossfun)
ce = 0.0485
ceFold = kfoldLoss(CVMdl,'LossFun',lossfun,'Mode','individual')
ceFold = 5×1

0.0488
0.0511
0.0496
0.0479
0.0452

To determine a good lasso-penalty strength for an ECOC model composed of linear classification models that use logistic regression learners, implement 5-fold cross-validation.

X is a sparse matrix of predictor data, and Y is a categorical vector of class labels.

For simplicity, use the label 'others' for all observations in Y that are not 'simulink', 'dsp', or 'comm'.

Create a set of 11 logarithmically-spaced regularization strengths from $1{0}^{-7}$ through $1{0}^{-2}$.

Lambda = logspace(-7,-2,11);

Create a linear classification model template that specifies to use logistic regression learners, use lasso penalties with strengths in Lambda, train using SpaRSA, and lower the tolerance on the gradient of the objective function to 1e-8.

t = templateLinear('Learner','logistic','Solver','sparsa',...

Cross-validate the models. To increase execution speed, transpose the predictor data and specify that the observations are in columns.

X = X';
rng(10); % For reproducibility
CVMdl = fitcecoc(X,Y,'Learners',t,'ObservationsIn','columns','KFold',5);

CVMdl is a ClassificationPartitionedLinearECOC model.

Dissect CVMdl, and each model within it.

numECOCModels = numel(CVMdl.Trained)
numECOCModels = 5
ECOCMdl1 = CVMdl.Trained{1}
ECOCMdl1 =
CompactClassificationECOC
ResponseName: 'Y'
ScoreTransform: 'none'
BinaryLearners: {6×1 cell}
CodingMatrix: [4×6 double]

Properties, Methods

numCLModels = numel(ECOCMdl1.BinaryLearners)
numCLModels = 6
CLMdl1 = ECOCMdl1.BinaryLearners{1}
CLMdl1 =
ClassificationLinear
ResponseName: 'Y'
ClassNames: [-1 1]
ScoreTransform: 'logit'
Beta: [34023×11 double]
Bias: [-0.3169 -0.3169 -0.3168 -0.3168 -0.3168 -0.3167 -0.1725 -0.0805 -0.1762 -0.3450 -0.5174]
Lambda: [1.0000e-07 3.1623e-07 1.0000e-06 3.1623e-06 1.0000e-05 3.1623e-05 1.0000e-04 3.1623e-04 1.0000e-03 0.0032 0.0100]
Learner: 'logistic'

Properties, Methods

Because fitcecoc implements 5-fold cross-validation, CVMdl contains a 5-by-1 cell array of CompactClassificationECOC models that the software trains on each fold. The BinaryLearners property of each CompactClassificationECOC model contains the ClassificationLinear models. The number of ClassificationLinear models within each compact ECOC model depends on the number of distinct labels and coding design. Because Lambda is a sequence of regularization strengths, you can think of CLMdl1 as 11 models, one for each regularization strength in Lambda.

Determine how well the models generalize by plotting the averages of the 5-fold classification error for each regularization strength. Identify the regularization strength that minimizes the generalization error over the grid.

ce = kfoldLoss(CVMdl);
figure;
plot(log10(Lambda),log10(ce))
[~,minCEIdx] = min(ce);
minLambda = Lambda(minCEIdx);
hold on
plot(log10(minLambda),log10(ce(minCEIdx)),'ro');
ylabel('log_{10} 5-fold classification error')
xlabel('log_{10} Lambda')
legend('MSE','Min classification error')
hold off

Train an ECOC model composed of linear classification model using the entire data set, and specify the minimal regularization strength.

t = templateLinear('Learner','logistic','Solver','sparsa',...
MdlFinal = fitcecoc(X,Y,'Learners',t,'ObservationsIn','columns');

To estimate labels for new observations, pass MdlFinal and the new data to predict.

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## 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 R2016a