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Classification edge for observations not used for training



e = kfoldEdge(CVMdl) returns the cross-validated classification edges obtained by the cross-validated, error-correcting output codes (ECOC) model composed of linear classification models CVMdl. That is, for every fold, kfoldEdge estimates the classification edge for observations that it holds out when it trains using all other observations.

e contains a classification edge for each regularization strength in the linear classification models that comprise CVMdl.


e = kfoldEdge(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 edge 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, kfoldEdge 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)]
    '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 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)

    • 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

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

'average'Returns classification edges averaged over all folds
'individual'Returns classification edges 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 edges, returned as a numeric scalar, vector, or matrix.

Let L be the number of regularization strengths in the cross-validated models (that is, L is numel(CVMdl.Trained{1}.BinaryLearners{1}.Lambda)) and F be the number of folds (stored in CVMdl.KFold).

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

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


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Load the NLP data set.

load nlpdata

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'.

Y(~(ismember(Y,{'simulink','dsp','comm'}))) = 'others';

Cross-validate a multiclass, linear classification model.

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. You can alter the number of folds using the 'KFold' name-value pair argument.

Estimate the average of the out-of-fold edges.

e = kfoldEdge(CVMdl)
e = 1.4464

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

One way to perform feature selection is to compare k-fold edges from multiple models. Based solely on this criterion, the classifier with the highest edge is the best classifier.

Load the NLP data set. Preprocess the data as in Estimate k-Fold Cross-Validation Edge, and orient the predictor data so that observations correspond to columns.

load nlpdata
Y(~(ismember(Y,{'simulink','dsp','comm'}))) = 'others';
X = X';

Create these two data sets:

  • fullX contains all predictors.

  • partX contains a 1/2 of the predictors chosen at random.

rng(1); % For reproducibility
p = size(X,1); % Number of predictors
halfPredIdx = randsample(p,ceil(0.5*p));
fullX = X;
partX = X(halfPredIdx,:);

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

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

Cross-validate two ECOC models composed of binary, linear classification models: one that uses the all of the predictors and one that uses half of the predictors. Indicate that observations correspond to columns.

CVMdl = fitcecoc(fullX,Y,'Learners',t,'CrossVal','on',...
PCVMdl = fitcecoc(partX,Y,'Learners',t,'CrossVal','on',...

CVMdl and PCVMdl are ClassificationPartitionedLinearECOC models.

Estimate the k-fold edge for each classifier.

fullEdge = kfoldEdge(CVMdl)
fullEdge = 0.6181
partEdge = kfoldEdge(PCVMdl)
partEdge = 0.5235

Based on the k-fold edges, the classifier that uses all of the predictors is the better model.

To determine a good lasso-penalty strength for a linear classification model that uses a logistic regression learner, compare k-fold edges.

Load the NLP data set. Preprocess the data as in Feature Selection Using k-fold Edges.

load nlpdata
Y(~(ismember(Y,{'simulink','dsp','comm'}))) = 'others';
X = X';

Create a set of 8 logarithmically-spaced regularization strengths from 10-8 through 101.

Lambda = logspace(-8,1,8);

Create a linear classification model template that specifies to use logistic regression with a lasso penalty, use each of the regularization strengths, optimize the objective function using SpaRSA, and reduce the tolerance on the gradient of the objective function to 1e-8.

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

Cross-validate an ECOC model composed of binary, linear classification models using 5-fold cross-validation and that

rng(10) % For reproducibility
CVMdl = fitcecoc(X,Y,'Learners',t,'ObservationsIn','columns','KFold',5)
CVMdl = 
    CrossValidatedModel: 'LinearECOC'
           ResponseName: 'Y'
        NumObservations: 31572
                  KFold: 5
              Partition: [1x1 cvpartition]
             ClassNames: [comm    dsp    simulink    others]
         ScoreTransform: 'none'

  Properties, Methods

CVMdl is a ClassificationPartitionedLinearECOC model.

Estimate the edges for each fold and regularization strength.

eFolds = kfoldEdge(CVMdl,'Mode','individual')
eFolds = 5×8

    0.5513    0.5519    0.5520    0.5505    0.4949    0.2949    0.1034    0.0853
    0.5255    0.5262    0.5265    0.5266    0.4781    0.2954    0.1051    0.0867
    0.5299    0.5307    0.5313    0.5317    0.4776    0.2901    0.1027    0.0868
    0.5401    0.5413    0.5416    0.5381    0.4848    0.2930    0.1016    0.0856
    0.5502    0.5559    0.5585    0.5586    0.4921    0.2930    0.1030    0.0849

eFolds is a 5-by-8 matrix of edges. Rows correspond to folds and columns correspond to regularization strengths in Lambda. You can use eFolds to identify ill-performing folds, that is, unusually low edges.

Estimate the average edge over all folds for each regularization strength.

e = kfoldEdge(CVMdl)
e = 1×8

    0.5394    0.5412    0.5420    0.5411    0.4855    0.2933    0.1031    0.0858

Determine how well the models generalize by plotting the averages of the 5-fold edge for each regularization strength. Identify the regularization strength that maximizes the 5-fold edge over the grid.

[~, maxEIdx] = max(e);
maxLambda = Lambda(maxEIdx);
hold on
ylabel('log_{10} 5-fold edge')
xlabel('log_{10} Lambda')
legend('Edge','Max edge')
hold off

Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent Edge, Max edge.

Several values of Lambda yield similarly high edges. Greater regularization strength values lead to predictor variable sparsity, which is a good quality of a classifier.

Choose the regularization strength that occurs just before the edge starts decreasing.

LambdaFinal = Lambda(4);

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

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.

More About

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[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.

Extended Capabilities

Version History

Introduced in R2016a