# loss

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

## Syntax

``L = loss(Mdl,tbl,ResponseVarName)``
``L = loss(Mdl,tbl,Y)``
``L = loss(Mdl,X,Y)``
``L = loss(___,Name,Value)``

## 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`.

```load fisheriris 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`.

```load fisheriris 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 = 0.0049 ```

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 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: `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 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)]
`'quadratic'`Quadratic[0,1][1 – yj(2sj – 1)]2/2

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)`

• `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.

The default `BinaryLoss` value depends on the score ranges returned by the binary learners. This table identifies what some default `BinaryLoss` values are when you use the default score transform (`ScoreTransform` property of the model is `'none'`).

AssumptionDefault Value

All binary learners are any of the following:

• Classification decision trees

• Discriminant analysis models

• k-nearest neighbor models

• Linear or kernel classification models of logistic regression learners

• Naive Bayes models

`'quadratic'`
All binary learners are SVMs or 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'`
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 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 `'classiferror'`, `'classifcost'`, 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.

• 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 = 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.

Specify your function using `'LossFun',@lossfun`.

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 has the form

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

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.

### Observed Misclassification Cost

The observed misclassification cost has the form

`$L=\sum _{j=1}^{n}{w}_{j}{c}_{{y}_{j}{\stackrel{^}{y}}_{j}},$`

where:

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

• ${c}_{{y}_{j}{\stackrel{^}{y}}_{j}}$ is the user-specified cost of classifying an observation into class ${\stackrel{^}{y}}_{j}$ when its true class is yj.

### 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.

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

• 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.

The decoding scheme of an ECOC model specifies how the software aggregates the binary losses and determines the predicted class for each 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.

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

• 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.

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

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 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 function.

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)]
`"quadratic"`Quadratic[0,1][1 – yj(2sj – 1)]2/2

The software normalizes binary losses so that the loss is 0.5 when yj = 0, and aggregates using the average of the binary learners.

Do not confuse the binary loss with the overall classification loss (specified by the `LossFun` name-value 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. “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 R2014b