# resubMargin

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

## Description

returns the resubstitution classification margins
(`m`

= resubMargin(`Mdl`

)`m`

) 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`

.

`m`

is returned as a numeric column vector with the same length as
`Mdl.Y`

. The software estimates each entry of `m`

using the trained ECOC model `Mdl`

, the corresponding row of
`Mdl.X`

, and the true class label `Mdl.Y`

.

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

= resubMargin(`Mdl`

,`Name,Value`

)

## Examples

### Resubstitution Classification Margins of ECOC Model

Calculate the resubstitution classification margins 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.

Calculate the classification margins for the observations used to train `Mdl`

. Display the distribution of the margins using a boxplot.

```
m = resubMargin(Mdl);
boxplot(m)
title('In-Sample Margins')
```

The classification margin of an observation is the positive-class negated loss minus the maximum negative-class negated loss. Choose classifiers that yield relatively large margins.

### Select ECOC Model Features by Examining Training-Sample Margins

Perform feature selection by comparing training-sample margins from multiple models. Based solely on this comparison, the model with the greatest margins is the best model.

Load Fisher's iris data set. Define two data sets:

`fullX`

contains all four predictors.`partX`

contains the sepal measurements only.

```
load fisheriris
X = meas;
fullX = X;
partX = X(:,1:2);
Y = species;
```

Train an ECOC model using SVM binary learners for each predictor set. Standardize the predictors using an SVM template, specify the class order, and compute posterior probabilities.

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

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

FullMdl = fitcecoc(fullX,Y,'Learners',t,'ClassNames',classOrder,... 'FitPosterior',true); PartMdl = fitcecoc(partX,Y,'Learners',t,'ClassNames',classOrder,... 'FitPosterior',true);

Compute the resubstitution margins for each classifier. For each model, display the distribution of the margins using a boxplot.

fullMargins = resubMargin(FullMdl); partMargins = resubMargin(PartMdl); boxplot([fullMargins partMargins],'Labels',{'All Predictors','Two Predictors'}) title('Training-Sample Margins')

The margin distribution of `FullMdl`

is situated higher and has less variability than the margin distribution of `PartMdl`

. This result suggests that the model trained with all the predictors fits the training data better.

## 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 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: **`resubMargin(Mdl,'Verbose',1)`

specifies to display diagnostic
messages in the Command Window.

`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 the 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 [1]._{j}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 modelsLinear 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`

— 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'`

`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 Margin

The *classification margin* is, for each observation,
the difference between the negative loss for the true class and the maximal negative loss
among the false classes. If the margins are on the same scale, then they serve as a
classification confidence measure. Among multiple classifiers, those that yield greater
margins are better.

### 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*is element (_{kj}*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*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.

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*is the score for observation

_{j}*j*, and

*g*(

*y*,

_{j}*s*) is the binary loss function.

_{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 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 `resubLoss`

and
`resubPredict`

object functions), which measures how well an ECOC
classifier performs as a whole.

## Tips

To compare the margins or edges of several ECOC classifiers, use template objects to specify a common score transform function among the classifiers during training.

## 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**

## See Also

`ClassificationECOC`

| `resubEdge`

| `margin`

| `predict`

| `resubPredict`

| `fitcecoc`

| `resubLoss`

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