kfoldLoss

Classification loss for observations not used in training

Syntax

L = kfoldLoss(CVMdl)

L = kfoldLoss(CVMdl,Name,Value)

Description

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.

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

CVMdl — Cross-validated, ECOC model composed of linear classification models

ClassificationPartitionedLinearECOC model object

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 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.

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

  customFunction should have 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.

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' — 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 to use for classification-score prediction

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

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

'LossFun' — Loss function

'classiferror' (default) | function handle

Loss function, specified as the comma-separated pair consisting of 'LossFun' and a function handle or 'classiferror'.

You can:

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

• 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.

  Specify your function using 'LossFun',@lossfun.

Data Types: function_handle | char | string

'Mode' — Loss aggregation level

'average' (default) | 'individual'

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

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

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

L — Cross-validated classification losses

numeric scalar | numeric vector | numeric matrix

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

Estimate k-Fold Cross-Validation Classification Error

Load the NLP data set.

load nlpdata

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.

Specify Custom Classification Loss

Load the NLP data set. Transpose the predictor data.

load nlpdata
X = X';

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';

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'
        ClassNames: [comm    dsp    simulink    others]
    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 = 0.0243

ceFold = kfoldLoss(CVMdl,'LossFun',lossfun,'Mode','individual')

ceFold = 5×1

    0.0244
    0.0255
    0.0248
    0.0240
    0.0226

Find Good Lasso Penalty Using Cross-Validation

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.

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';

Create a set of 11 logarithmically-spaced regularization strengths from $$10^{-7}$$ through $$10^{-2}$$.

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',...
    'Regularization','lasso','Lambda',Lambda,'GradientTolerance',1e-8);

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'
        ClassNames: [comm    dsp    simulink    others]
    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',...
    'Regularization','lasso','Lambda',minLambda,'GradientTolerance',1e-8);
MdlFinal = fitcecoc(X,Y,'Learners',t,'ObservationsIn','columns');

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

More About

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.

Classification Error

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

$$L=\frac{{\displaystyle \sum _{j=1}^n{w}_j{e}_j}}{{\displaystyle \sum _{j=1}^n{w}_j}},$$

where:

• w_j is the weight for observation j. The software renormalizes the weights to sum to 1.

• e_j = 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.

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

Automatic Parallel Support

Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.

To run in parallel, specify the 'Options' name-value argument in the call to this function and set the 'UseParallel' field of the options structure to true using statset.

For example: 'Options',statset('UseParallel',true)

For more information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).