# loss

Classification loss for multiclass error-correcting output codes (ECOC) model

## Description

L = loss(Mdl,tbl,ResponseVarName) returns the classification loss (L), a scalar representing how well the trained multiclass error-correcting output codes (ECOC) model Mdl classifies the predictor data in tbl compared to the true class labels in tbl.ResponseVarName. By default, loss uses the classification error to compute L.

L = loss(Mdl,tbl,Y) returns the classification loss for the predictor data in table tbl and the true class labels in Y.

example

L = loss(Mdl,X,Y) returns the classification loss for the predictor data in matrix X and the true class labels in Y.

example

L = loss(___,Name,Value) specifies options using one or more name-value pair arguments in addition to any of the input argument combinations in previous syntaxes. For example, you can specify a decoding scheme, classification loss function, and verbosity level.

## Examples

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

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

Train an ECOC model using SVM binary classifiers. Specify a 15% holdout sample, standardize the predictors using an SVM template, and specify the class order.

t = templateSVM('Standardize',true);
PMdl = fitcecoc(X,Y,'Holdout',0.15,'Learners',t,'ClassNames',classOrder);
Mdl = PMdl.Trained{1};           % Extract trained, compact classifier

PMdl is a ClassificationPartitionedECOC model. It has the property Trained, a 1-by-1 cell array containing the CompactClassificationECOC model that the software trained using the training set.

Estimate the test-sample classification error, which is the default classification loss.

testInds = test(PMdl.Partition);  % Extract the test indices
XTest = X(testInds,:);
YTest = Y(testInds,:);
L = loss(Mdl,XTest,YTest)
L = 0

The ECOC model correctly classifies all irises in the test sample.

Determine the quality of an ECOC model by using a custom loss function that considers the minimal binary loss for each observation.

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

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

Train an ECOC model using SVM binary classifiers. Specify a 15% holdout sample, standardize the predictors using an SVM template, and define the class order.

t = templateSVM('Standardize',true);
PMdl = fitcecoc(X,Y,'Holdout',0.15,'Learners',t,'ClassNames',classOrder);
Mdl = PMdl.Trained{1};           % Extract trained, compact classifier

PMdl is a ClassificationPartitionedECOC model. It has the property Trained, a 1-by-1 cell array containing the CompactClassificationECOC model that the software trained using the training set.

Create a function that takes the minimal loss for each observation, then averages the minimal losses for all observations. S corresponds to the NegLoss output of predict.

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

Compute the test-sample custom loss.

testInds = test(PMdl.Partition);  % Extract the test indices
XTest = X(testInds,:);
YTest = Y(testInds,:);
loss(Mdl,XTest,YTest,'LossFun',lossfun)
ans = 0.0033

The average minimal binary loss for the test-sample observations is 0.0033.

## Input Arguments

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Full or compact multiclass ECOC model, specified as a ClassificationECOC or CompactClassificationECOC model object.

To create a full or compact ECOC model, see ClassificationECOC or CompactClassificationECOC.

Sample data, specified as a table. Each row of tbl corresponds to one observation, and each column corresponds to one predictor variable. Optionally, tbl can contain additional columns for the response variable and observation weights. tbl must contain all the predictors used to train Mdl. Multicolumn variables and cell arrays other than cell arrays of character vectors are not allowed.

If you train Mdl using sample data contained in a table, then the input data for loss must also be in a table.

When training Mdl, assume that you set 'Standardize',true for a template object specified in the 'Learners' name-value pair argument of fitcecoc. In this case, for the corresponding binary learner j, the software standardizes the columns of the new predictor data using the corresponding means in Mdl.BinaryLearner{j}.Mu and standard deviations in Mdl.BinaryLearner{j}.Sigma.

Data Types: table

Response variable name, specified as the name of a variable in tbl. If tbl contains the response variable used to train Mdl, then you do not need to specify ResponseVarName.

If you specify ResponseVarName, then you must do so as a character vector or string scalar. For example, if the response variable is stored as tbl.y, then specify ResponseVarName as 'y'. Otherwise, the software treats all columns of tbl, including tbl.y, as predictors.

The response variable must be a categorical, character, or string array, a logical or numeric vector, or a cell array of character vectors. If the response variable is a character array, then each element must correspond to one row of the array.

Data Types: char | string

Predictor data, specified as a numeric matrix.

Each row of X corresponds to one observation, and each column corresponds to one variable. The variables in the columns of X must be the same as the variables that trained the classifier Mdl.

The number of rows in X must equal the number of rows in Y.

When training Mdl, assume that you set 'Standardize',true for a template object specified in the 'Learners' name-value pair argument of fitcecoc. In this case, for the corresponding binary learner j, the software standardizes the columns of the new predictor data using the corresponding means in Mdl.BinaryLearner{j}.Mu and standard deviations in Mdl.BinaryLearner{j}.Sigma.

Data Types: double | single

Class labels, specified as a categorical, character, or string array, a logical or numeric vector, or a cell array of character vectors. Y must have the same data type as Mdl.ClassNames. (The software treats string arrays as cell arrays of character vectors.)

The number of rows in Y must equal the number of rows in tbl or X.

Data Types: categorical | char | string | logical | single | double | cell

### Name-Value 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: loss(Mdl,X,Y,'BinaryLoss','hinge','LossFun',@lossfun) specifies 'hinge' as the binary learner loss function and the custom function handle @lossfun as the overall loss function.

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 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)]

The software normalizes binary losses so 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.

For an example of passing a custom binary loss function, see Predict Test-Sample Labels of ECOC Model Using Custom Binary Loss Function.

The default BinaryLoss value depends on the score ranges returned by the binary learners. This table describes some default BinaryLoss values based on the given assumptions.

AssumptionDefault Value
All binary learners are SVMs or either linear or kernel classification models of SVM learners.'hinge'
All binary learners are ensembles trained by AdaboostM1 or GentleBoost.'exponential'
All binary learners are ensembles trained by LogitBoost.'binodeviance'
All binary learners are linear or kernel classification models of logistic regression learners. Or, you specify to predict class posterior probabilities by setting 'FitPosterior',true in fitcecoc.'quadratic'

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 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'

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, which is the proportion of misclassified observations.

• Or, specify your own function using function handle notation.

Assume that n = size(X,1) is the sample size and K is the number of classes. Your function must have 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 Mdl.ClassNames.

Construct C by setting C(p,q) = 1 if observation p is in class q, for each row. Set all other elements 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 Mdl.ClassNames. The input S resembles the output argument NegLoss of predict.

• 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: char | string | function_handle

Predictor data observation dimension, specified as the comma-separated pair consisting of 'ObservationsIn' and 'columns' or 'rows'. Mdl.BinaryLearners must contain ClassificationLinear models.

Note

If you orient your predictor matrix so that observations correspond to columns and specify 'ObservationsIn','columns', you can experience a significant reduction in execution time. You cannot specify 'ObservationsIn','columns' for predictor data in a table.

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

Observation weights, specified as the comma-separated pair consisting of 'Weights' and a numeric vector or the name of a variable in tbl. If you supply weights, then loss computes the weighted loss.

If you specify Weights as a numeric vector, then the size of Weights must be equal to the number of rows in X or tbl.

If you specify Weights as the name of a variable in tbl, you must do so as a character vector or string scalar. For example, if the weights are stored as tbl.w, then specify Weights as 'w'. Otherwise, the software treats all columns of tbl, including tbl.w, as predictors.

If you do not specify your own loss function (using LossFun), then the software normalizes Weights to sum up to the value of the prior probability in the respective class.

Data Types: single | double | char | string

## Output Arguments

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Classification loss, returned as a numeric scalar or row vector. L is a generalization or resubstitution quality measure. Its interpretation depends on the loss function and weighting scheme, but in general, better classifiers yield smaller classification loss values.

If Mdl.BinaryLearners contains ClassificationLinear models, then L is a 1-by- vector, where is the number of regularization strengths in the linear classification models (numel(Mdl.BinaryLearners{1}.Lambda)). The value L(j) is the loss for the model trained using regularization strength Mdl.BinaryLearners{1}.Lambda(j).

Otherwise, L is a scalar value.

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### Classification Error

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

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

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.

• $\stackrel{^}{k}$ 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,

$\stackrel{^}{k}=\underset{k}{\text{argmin}}\sum _{j=1}^{L}|{m}_{kj}|g\left({m}_{kj},{s}_{j}\right).$

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,

$\stackrel{^}{k}=\underset{k}{\text{argmin}}\frac{\sum _{j=1}^{L}|{m}_{kj}|g\left({m}_{kj},{s}_{j}\right)}{\sum _{j=1}^{L}|{m}_{kj}|}.$

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)]

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.

## Extended Capabilities

Introduced in R2014b