kfoldLoss

Classification loss for cross-validated ECOC model

Syntax

loss = kfoldLoss(CVMdl)

loss = kfoldLoss(CVMdl,Name,Value)

Description

loss = kfoldLoss(CVMdl) returns the classification loss obtained by the cross-validated ECOC model (ClassificationPartitionedECOC) CVMdl. For every fold, kfoldLoss computes the classification loss for validation-fold observations using a model trained on training-fold observations. CVMdl.X contains both sets of observations.

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 number of folds, decoding scheme, or verbosity level.

Examples

Determine k-Fold Cross-Validation Loss

Load Fisher's iris data set. Specify the predictor data X, the response data Y, and the order of the classes in Y.

load fisheriris
X = meas;
Y = categorical(species);
classOrder = unique(Y); % Class order
rng(1); % For reproducibility

Train and cross-validate an ECOC model using support vector machine (SVM) binary classifiers. Standardize the predictors using an SVM template, and specify the class order.

t = templateSVM('Standardize',1);
CVMdl = fitcecoc(X,Y,'CrossVal','on','Learners',t,'ClassNames',classOrder);

CVMdl is a ClassificationPartitionedECOC model. By default, the software implements 10-fold cross-validation. You can specify a different number of folds using the 'KFold' name-value pair argument.

Estimate the average classification error.

L = kfoldLoss(CVMdl)

L = 0.0400

The average classification error for the folds is 4%.

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

Display Individual Losses for Each Cross-Validation Fold

The classification loss is a measure of classifier quality. To determine which folds perform poorly, display the losses for each fold.

Load Fisher's iris data set. Specify the predictor data X, the response data Y, and the order of the classes in Y.

load fisheriris
X = meas;
Y = categorical(species);
classOrder = unique(Y);
rng(1); % For reproducibility

Train an ECOC model using SVM binary classifiers. Use 8-fold cross-validation, standardize the predictors using an SVM template, and specify the class order.

t = templateSVM('Standardize',1);
CVMdl = fitcecoc(X,Y,'KFold',8,'Learners',t,'ClassNames',classOrder);

Estimate the average classification loss across all folds and the losses for each fold.

loss = kfoldLoss(CVMdl)

loss = 0.0333

losses = kfoldLoss(CVMdl,'Mode','individual')

losses = 8×1

    0.0556
    0.0526
    0.1579
         0
         0
         0
         0
         0

The third fold misclassifies a much higher percentage of observations than any other fold.

Return the average classification loss for the folds that perform well by specifying the 'Folds' name-value pair argument.

newloss = kfoldLoss(CVMdl,'Folds',[1:2 4:8])

newloss = 0.0153

The total classification loss decreases by approximately half its original size.

Consider adjusting parameters of the binary classifiers or the coding design to see if performance for all folds improves.

Determine ECOC Model Quality Using Custom Cross-Validation Loss

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. Specify the predictor data X, the response data Y, and the order of the classes in Y.

load fisheriris
X = meas;
Y = categorical(species);
classOrder = unique(Y)  % Class order

classOrder = 3x1 categorical
     setosa 
     versicolor 
     virginica 

rng(1) % For reproducibility

Train and cross-validate an ECOC model using SVM binary classifiers. Standardize the predictors using an SVM template, and specify the class order.

t = templateSVM('Standardize',1);
CVMdl = fitcecoc(X,Y,'CrossVal','on','Learners',t,'ClassNames',classOrder);

CVMdl is a ClassificationPartitionedECOC model. By default, the software implements 10-fold cross-validation. You can specify a different number of folds using the 'KFold' name-value pair argument.

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 = 0.0152

The average minimal binary loss for the validation-fold observations is 0.0101.

Input Arguments

CVMdl — Cross-validated ECOC model

ClassificationPartitionedECOC model

Cross-validated ECOC model, specified as a ClassificationPartitionedECOC model. You can create a ClassificationPartitionedECOC model in two ways:

• Pass a trained ECOC model (ClassificationECOC) to crossval.

• Train an ECOC model using fitcecoc and specify any one of these cross-validation name-value pair arguments: 'CrossVal', 'CVPartition', 'Holdout', 'KFold', or 'Leaveout'.

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.

Example: kfoldLoss(CVMdl,'Folds',[1 3 5]) specifies to use only the first, third, and fifth folds to calculate the classification loss.

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 describes the built-in functions, where y_j is the 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 so that the loss is 0.5 when y_j = 0. Also, the software calculates the mean binary loss for each class [1].

• For a custom binary loss function, for example customFunction, specify its function handle 'BinaryLoss',@customFunction.

  customFunction has 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.

This table identifies the default BinaryLoss value, which depends on the score ranges returned by the binary learners.

Assumption	Default Value
All binary learners are any of the following: Classification decision trees Discriminant analysis models k-nearest neighbor models Naive Bayes models	'quadratic'
All binary learners are SVMs.	'hinge'
All binary learners are ensembles trained by AdaboostM1 or GentleBoost.	'exponential'
All binary learners are ensembles trained by LogitBoost.	'binodeviance'
You specify to predict class posterior probabilities by setting 'FitPosterior',true in fitcecoc.	'quadratic'
Binary learners are heterogeneous and use different loss functions.	'hamming'

To check the default value, use dot notation to display the BinaryLoss property of the trained model at the command line.

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

LossFun — Loss function

'classiferror' (default) | 'classifcost' | function handle

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

• Specify the built-in function 'classiferror'. In this case, 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.

• 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

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

loss — Classification loss

numeric scalar | numeric column vector

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

Classification Error

The classification error has the form

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

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.

Observed Misclassification Cost

The observed misclassification cost has the form

$$L={\displaystyle \sum _{j=1}^n{w}_j}{c}_{y_j{\widehat{y}}_j},$$

where:

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

• $$c_{y_j{\widehat{y}}_j}$$ is the user-specified cost of classifying an observation into class $${\widehat{y}}_j$$ when its true class is y_j.

Binary Loss

The binary loss is a function of the class and classification score that determines how well a binary learner classifies an observation into the class. The decoding scheme of an ECOC model specifies how the software aggregates the binary losses and determines the predicted class for each observation.

Assume 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. M is a K-by-B matrix, where K is the number of classes, and B is the number of binary learners.

• 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.

The software supports two decoding schemes:

• Loss-based decoding [2] (Decoding is 'lossbased') — The predicted class of an observation corresponds to the class that produces the minimum average of the binary losses over all binary learners.

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

• Loss-weighted decoding [3] (Decoding is 'lossweighted') — The predicted class of an observation corresponds to the class that produces the minimum average of the binary losses over the binary learners for the corresponding class.

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

  The denominator corresponds to the number of binary learners for class k. [1] suggests that loss-weighted decoding improves classification accuracy by keeping loss values for all classes in the same dynamic range.

The predict, resubPredict, and kfoldPredict functions return the negated value of the objective function of argmin as the second output argument (NegLoss) for each observation and class.

This table summarizes the supported binary 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) is the binary loss function.

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 so that the loss is 0.5 when y_j = 0, and aggregates using the average of the binary learners [1].

Do not confuse the binary loss with the overall classification loss (specified by the LossFun name-value argument of the kfoldLoss and kfoldPredict 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. “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

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:

Options=statset(UseParallel=true)

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

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2014b