# resubLoss

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

## Description

returns the classification loss by resubstitution (`L`

= resubLoss(`Mdl`

)`L`

) for the
multiclass error-correcting output codes (ECOC) model `Mdl`

using the
training data stored in `Mdl.X`

and the corresponding class labels stored
in `Mdl.Y`

. By default, `resubLoss`

uses the classification error to compute `L`

.

The classification loss (`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.

returns the classification loss with additional options specified by one or more name-value
pair arguments. For example, you can specify the loss function, decoding scheme, and
verbosity level.`L`

= resubLoss(`Mdl`

,`Name,Value`

)

## Examples

### Resubstitution Loss of ECOC Model

Compute the resubstitution loss for an ECOC model with SVM binary learners.

Load Fisher's iris data set. Specify the predictor data `X`

and the response data `Y`

.

```
load fisheriris
X = meas;
Y = species;
```

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

```
t = templateSVM('Standardize',true);
classOrder = unique(Y)
```

`classOrder = `*3x1 cell*
{'setosa' }
{'versicolor'}
{'virginica' }

Mdl = fitcecoc(X,Y,'Learners',t,'ClassNames',classOrder);

`t`

is an SVM template object. During training, the software uses default values for empty properties in `t`

. `Mdl`

is a `ClassificationECOC`

model.

Estimate the resubstitution classification error, which is the default classification loss.

L = resubLoss(Mdl)

L = 0.0267

The ECOC model misclassifies 2.67% of the training-sample irises.

### Determine ECOC Model Quality Using Custom Resubstitution Loss

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

`classOrder = `*3x1 categorical*
setosa
versicolor
virginica

`rng(1); % For reproducibility`

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

t = templateSVM('Standardize',true); Mdl = fitcecoc(X,Y,'Learners',t,'ClassNames',classOrder);

`t`

is an SVM template object. During training, the software uses default values for empty properties in `t`

. `Mdl`

is a `ClassificationECOC`

model.

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 `resubPredict`

.

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

Compute the custom classification loss for the training data.

`resubLoss(Mdl,'LossFun',lossfun)`

ans = 0.0065

The average minimal binary loss for the training data is `0.0065`

.

## Input Arguments

`Mdl`

— Full, trained multiclass ECOC model

`ClassificationECOC`

model

Full, trained multiclass ECOC model, specified as a `ClassificationECOC`

model trained with `fitcecoc`

.

### 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:**

`resubLoss(Mdl,'BinaryLoss','hamming','LossFun',@lossfun)`

specifies `'hamming'`

as the binary learner loss function and the custom
function handle `@lossfun`

as the overall loss function.`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*is a class label for a particular binary learner (in the set {–1,1,0}),_{j}*s*is the score for observation_{j}*j*, and*g*(*y*,_{j}*s*) is the binary loss formula._{j}Value Description Score Domain *g*(*y*,_{j}*s*)_{j}`'binodeviance'`

Binomial deviance (–∞,∞) log[1 + exp(–2 *y*)]/[2log(2)]_{j}s_{j}`'exponential'`

Exponential (–∞,∞) exp(– *y*)/2_{j}s_{j}`'hamming'`

Hamming [0,1] or (–∞,∞) [1 – sign( *y*)]/2_{j}s_{j}`'hinge'`

Hinge (–∞,∞) max(0,1 – *y*)/2_{j}s_{j}`'linear'`

Linear (–∞,∞) (1 – *y*)/2_{j}s_{j}`'logit'`

Logistic (–∞,∞) log[1 + exp(– *y*)]/[2log(2)]_{j}s_{j}`'quadratic'`

Quadratic [0,1] [1 – *y*(2_{j}*s*– 1)]_{j}^{2}/2The software normalizes binary losses so that the loss is 0.5 when

*y*= 0. Also, the software calculates the mean binary loss for each class._{j}For a custom binary loss function, for example

`customFunction`

, specify its function handle`'BinaryLoss',@customFunction`

.`customFunction`

has this form:where:bLoss = customFunction(M,s)

`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.

Assumption | Default 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`

— 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'`

`LossFun`

— Loss function

`'classiferror'`

(default) | function handle

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`resubPredict`

.`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`

`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`

## More About

### 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*is the weight for observation_{j}*j*. The software renormalizes the weights to sum to 1.*e*= 1 if the predicted class of observation_{j}*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:

*m*is element (_{kj}*k*,*j*) of the coding design matrix*M*(that is, the code corresponding to class*k*of binary learner*j*).*s*is the score of binary learner_{j}*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*is the score for observation

_{j}*j*, and

*g*(

*y*,

_{j}*s*).

_{j}Value | Description | Score Domain | g(y,_{j}s)_{j} |
---|---|---|---|

`'binodeviance'` | Binomial deviance | (–∞,∞) | log[1 +
exp(–2y)]/[2log(2)]_{j}s_{j} |

`'exponential'` | Exponential | (–∞,∞) | exp(–y)/2_{j}s_{j} |

`'hamming'` | Hamming | [0,1] or (–∞,∞) | [1 – sign(y)]/2_{j}s_{j} |

`'hinge'` | Hinge | (–∞,∞) | max(0,1 – y)/2_{j}s_{j} |

`'linear'` | Linear | (–∞,∞) | (1 – y)/2_{j}s_{j} |

`'logit'` | Logistic | (–∞,∞) | log[1 +
exp(–y)]/[2log(2)]_{j}s_{j} |

`'quadratic'` | Quadratic | [0,1] | [1 – y(2_{j}s –
1)]_{j}^{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.

## 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).

### 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).

## See Also

`ClassificationECOC`

| `loss`

| `predict`

| `resubPredict`

| `fitcecoc`

**Introduced in R2014b**

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