# kfoldMargin

Classification margins for cross-validated kernel ECOC model

## Description

returns the classification
margins obtained by the cross-validated kernel ECOC model (`margin`

= kfoldMargin(`CVMdl`

)`ClassificationPartitionedKernelECOC`

) `CVMdl`

. For every fold,
`kfoldMargin`

computes the classification margins for validation-fold
observations using a model trained on training-fold observations.

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

= kfoldMargin(`CVMdl`

,`Name,Value`

)

## Examples

### Estimate *k*-Fold Cross-Validation Margins

Load Fisher's iris data set. `X`

contains flower measurements, and `Y`

contains the names of flower species.

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

Cross-validate an ECOC model composed of kernel binary learners.

CVMdl = fitcecoc(X,Y,'Learners','kernel','CrossVal','on')

CVMdl = ClassificationPartitionedKernelECOC CrossValidatedModel: 'KernelECOC' ResponseName: 'Y' NumObservations: 150 KFold: 10 Partition: [1x1 cvpartition] ClassNames: {'setosa' 'versicolor' 'virginica'} ScoreTransform: 'none'

`CVMdl`

is a `ClassificationPartitionedKernelECOC`

model. By default, the software implements 10-fold cross-validation. To specify a different number of folds, use the `'KFold'`

name-value pair argument instead of `'Crossval'`

.

Estimate the classification margins for validation-fold observations.

m = kfoldMargin(CVMdl); size(m)

`ans = `*1×2*
150 1

`m`

is a 150-by-1 vector. `m(j)`

is the classification margin for observation `j`

.

Plot the k-fold margins using a boxplot.

boxplot(m,'Labels','All Observations') title('Distribution of Margins')

### Feature Selection Using *k*-Fold Margins

Perform feature selection by comparing *k*-fold margins from multiple models. Based solely on this criterion, the classifier with the greatest margins is the best classifier.

Load Fisher's iris data set. `X`

contains flower measurements, and `Y`

contains the names of flower species.

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

Randomly choose half of the predictor variables.

rng(1); % For reproducibility p = size(X,2); % Number of predictors idxPart = randsample(p,ceil(0.5*p));

Cross-validate two ECOC models composed of kernel classification models: one that uses all of the predictors, and one that uses half of the predictors.

CVMdl = fitcecoc(X,Y,'Learners','kernel','CrossVal','on'); PCVMdl = fitcecoc(X(:,idxPart),Y,'Learners','kernel','CrossVal','on');

`CVMdl`

and `PCVMdl`

are `ClassificationPartitionedKernelECOC`

models. By default, the software implements 10-fold cross-validation. To specify a different number of folds, use the `'KFold'`

name-value pair argument instead of `'Crossval'`

.

Estimate the *k*-fold margins for each classifier.

fullMargins = kfoldMargin(CVMdl); partMargins = kfoldMargin(PCVMdl);

Plot the distribution of the margin sets using box plots.

boxplot([fullMargins partMargins], ... 'Labels',{'All Predictors','Half of the Predictors'}); title('Distribution of Margins')

The `PCVMdl`

margin distribution is similar to the `CVMdl`

margin distribution.

## Input Arguments

`CVMdl`

— Cross-validated kernel ECOC model

`ClassificationPartitionedKernelECOC`

model

Cross-validated kernel ECOC model, specified as a `ClassificationPartitionedKernelECOC`

model. You can create a
`ClassificationPartitionedKernelECOC`

model by training an ECOC model
using `fitcecoc`

and specifying these name-value
pair arguments:

`'Learners'`

– Set the value to`'kernel'`

, a template object returned by`templateKernel`

, or a cell array of such template objects.One of the 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: **`kfoldMargin(CVMdl,'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 contains names and descriptions of 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.

By default, if all binary learners are kernel classification models using SVM, then
`BinaryLoss`

is `'hinge'`

. If all binary
learners are kernel classification models using logistic regression, then
`BinaryLoss`

is `'quadratic'`

.

**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

Estimation options, specified as a structure array as returned by `statset`

.

To invoke parallel computing you need a Parallel Computing Toolbox™ license.

**Example: **`Options=statset(UseParallel=true)`

**Data Types: **`struct`

`Verbose`

— Verbosity level

`0`

(default) | `1`

Verbosity level, specified as `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

`margin`

— Classification margins

numeric vector

Classification
margins, returned as a numeric vector. `margin`

is an
*n*-by-1 vector, where each row is the margin of the corresponding
observation and *n* is the number of observations
(`size(CVMdl.Y,1)`

).

## 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 `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).

## Version History

**Introduced in R2018b**

### R2023b: Observations with missing predictor values are used in resubstitution and cross-validation computations

Starting in R2023b, the following classification model object functions use observations with missing predictor values as part of resubstitution ("resub") and cross-validation ("kfold") computations for classification edges, losses, margins, and predictions.

In previous releases, the software omitted observations with missing predictor values from the resubstitution and cross-validation computations.

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