# kfoldMargin

Classification margins for observations not used in training

## Description

returns
the cross-validated classification margins obtained
by `m`

= kfoldMargin(`CVMdl`

)`CVMdl`

, which is a cross-validated, error-correcting
output codes (ECOC) model composed of linear classification models.
That is, for every fold, `kfoldMargin`

estimates the
classification margins for observations that it holds out when it
trains using all other observations.

`m`

contains classification margins for each
regularization strength in the linear classification models that comprise `CVMdl`

.

uses
additional options specified by one or more `m`

= kfoldMargin(`CVMdl`

,`Name,Value`

)`Name,Value`

pair
arguments. For example, specify a decoding scheme or verbosity level.

## Input Arguments

`CVMdl`

— Cross-validated, ECOC model composed of linear classification models

`ClassificationPartitionedLinearECOC`

model
object

Cross-validated, ECOC model composed of linear classification
models, specified as a `ClassificationPartitionedLinearECOC`

model
object. You can create a `ClassificationPartitionedLinearECOC`

model
using `fitcecoc`

and by:

Specifying any one of the cross-validation, name-value pair arguments, for example,

`CrossVal`

Setting the name-value pair argument

`Learners`

to`'linear'`

or a linear classification model template returned by`templateLinear`

To obtain estimates, kfoldMargin applies the same data used
to cross-validate the ECOC model (`X`

and `Y`

).

### 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.*

`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 the binary losses such 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, e.g.,

`customFunction`

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

.`customFunction`

should have this formwhere: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.

By default, if all binary learners are linear classification models using:

SVM, then

`BinaryLoss`

is`'hinge'`

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 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`

## Output Arguments

`m`

— Cross-validated classification margins

numeric vector | numeric matrix

Cross-validated classification margins, returned as a numeric vector or matrix.

`m`

is *n*-by-*L*,
where *n* is the number of observations in `X`

and *L* is
the number of regularization strengths in `Mdl`

(that
is, `numel(Mdl.Lambda)`

).

`m(`

is
the cross-validated classification margin of observation * i*,

*)*

`j`

*i*using the ECOC model, composed of linear classification models, that has regularization strength

`Mdl.Lambda(``j`

)

.## Examples

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

Load the NLP data set.

`load nlpdata`

`X`

is a sparse matrix of predictor data, and `Y`

is a categorical vector of class labels.

For simplicity, use the label 'others' for all observations in `Y`

that are not `'simulink'`

, `'dsp'`

, or `'comm'`

.

Y(~(ismember(Y,{'simulink','dsp','comm'}))) = 'others';

Cross-validate a multiclass, linear classification model.

rng(1); % For reproducibility CVMdl = fitcecoc(X,Y,'Learner','linear','CrossVal','on');

`CVMdl`

is a `ClassificationPartitionedLinearECOC`

model. By default, the software implements 10-fold cross validation. You can alter the number of folds using the `'KFold'`

name-value pair argument.

Estimate the *k*-fold margins.

m = kfoldMargin(CVMdl); size(m)

`ans = `*1×2*
31572 1

`m`

is a 31572-by-1 vector. `m(j)`

is the average of the out-of-fold margins for observation `j`

.

Plot the *k*-fold margins using box plots.

```
figure;
boxplot(m);
h = gca;
h.YLim = [-5 5];
title('Distribution of Cross-Validated Margins')
```

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

One way to perform feature selection is to compare *k*-fold margins from multiple models. Based solely on this criterion, the classifier with the larger margins is the better classifier.

Load the NLP data set. Preprocess the data as in Estimate k-Fold Cross-Validation Margins, and orient the predictor data so that observations correspond to columns.

load nlpdata Y(~(ismember(Y,{'simulink','dsp','comm'}))) = 'others'; X = X';

Create these two data sets:

`fullX`

contains all predictors.`partX`

contains 1/2 of the predictors chosen at random.

rng(1); % For reproducibility p = size(X,1); % Number of predictors halfPredIdx = randsample(p,ceil(0.5*p)); fullX = X; partX = X(halfPredIdx,:);

Create a linear classification model template that specifies optimizing the objective function using SpaRSA.

t = templateLinear('Solver','sparsa');

Cross-validate two ECOC models composed of binary, linear classification models: one that uses the all of the predictors and one that uses half of the predictors. Indicate that observations correspond to columns.

CVMdl = fitcecoc(fullX,Y,'Learners',t,'CrossVal','on',... 'ObservationsIn','columns'); PCVMdl = fitcecoc(partX,Y,'Learners',t,'CrossVal','on',... 'ObservationsIn','columns');

`CVMdl`

and `PCVMdl`

are `ClassificationPartitionedLinearECOC`

models.

Estimate the *k*-fold margins for each classifier. Plot the distribution of the *k*-fold margins sets using box plots.

fullMargins = kfoldMargin(CVMdl); partMargins = kfoldMargin(PCVMdl); figure; boxplot([fullMargins partMargins],'Labels',... {'All Predictors','Half of the Predictors'}); h = gca; h.YLim = [-1 1]; title('Distribution of Cross-Validated Margins')

The distributions of the *k*-fold margins of the two classifiers are similar.

### Find Good Lasso Penalty Using *k*-fold Margins

To determine a good lasso-penalty strength for a linear classification model that uses a logistic regression learner, compare distributions of *k*-fold margins.

Load the NLP data set. Preprocess the data as in Feature Selection Using k-fold Margins.

load nlpdata Y(~(ismember(Y,{'simulink','dsp','comm'}))) = 'others'; X = X';

Create a set of 11 logarithmically-spaced regularization strengths from $$1{0}^{-8}$$ through $$1{0}^{1}$$.

Lambda = logspace(-8,1,11);

Create a linear classification model template that specifies using logistic regression with a lasso penalty, using each of the regularization strengths, optimizing the objective function using SpaRSA, and reducing the tolerance on the gradient of the objective function to `1e-8`

.

t = templateLinear('Learner','logistic','Solver','sparsa',... 'Regularization','lasso','Lambda',Lambda,'GradientTolerance',1e-8);

Cross-validate an ECOC model composed of binary, linear classification models using 5-fold cross-validation and that

rng(10); % For reproducibility CVMdl = fitcecoc(X,Y,'Learners',t,'ObservationsIn','columns','KFold',5)

CVMdl = ClassificationPartitionedLinearECOC CrossValidatedModel: 'LinearECOC' ResponseName: 'Y' NumObservations: 31572 KFold: 5 Partition: [1x1 cvpartition] ClassNames: [comm dsp simulink others] ScoreTransform: 'none' Properties, Methods

`CVMdl`

is a `ClassificationPartitionedLinearECOC`

model.

Estimate the *k*-fold margins for each regularization strength. The scores for logistic regression are in [0,1]. Apply the quadratic binary loss.

m = kfoldMargin(CVMdl,'BinaryLoss','quadratic'); size(m)

`ans = `*1×2*
31572 11

`m`

is a 31572-by-11 matrix of cross-validated margins for each observation. The columns correspond to the regularization strengths.

Plot the *k*-fold margins for each regularization strength.

figure; boxplot(m) ylabel('Cross-validated margins') xlabel('Lambda indices')

Several values of `Lambda`

yield similarly high margin distribution centers with low spreads. Higher values of `Lambda`

lead to predictor variable sparsity, which is a good quality of a classifier.

Choose the regularization strength that occurs just before the margin distribution center starts decreasing and spread starts increasing.

LambdaFinal = Lambda(5);

Train an ECOC model composed of linear classification model using the entire data set and specify the regularization strength `LambdaFinal`

.

t = templateLinear('Learner','logistic','Solver','sparsa',... 'Regularization','lasso','Lambda',Lambda(5),'GradientTolerance',1e-8); MdlFinal = fitcecoc(X,Y,'Learners',t,'ObservationsIn','columns');

To estimate labels for new observations, pass `MdlFinal`

and the new data to `predict`

.

## More About

### 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.

### 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.

## 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 R2016a**

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