# kfoldMargin

Classification margins for cross-validated kernel classification model

## Description

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

= kfoldMargin(`CVMdl`

)`ClassificationPartitionedKernel`

) `CVMdl`

. For every fold,
`kfoldMargin`

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

## Examples

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

Load the `ionosphere`

data set. This data set has 34 predictors and 351 binary responses for radar returns, which are labeled as either bad (`'b'`

) or good (`'g'`

).

`load ionosphere`

Cross-validate a binary kernel classification model using the data.

CVMdl = fitckernel(X,Y,'Crossval','on')

CVMdl = ClassificationPartitionedKernel CrossValidatedModel: 'Kernel' ResponseName: 'Y' NumObservations: 351 KFold: 10 Partition: [1x1 cvpartition] ClassNames: {'b' 'g'} ScoreTransform: 'none'

`CVMdl`

is a `ClassificationPartitionedKernel`

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*
351 1

`m`

is a 351-by-1 vector. `m(j)`

is the classification margin for observation `j`

.

Plot the k-fold margins using a box plot.

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 the `ionosphere`

data set. This data set has 34 predictors and 351 binary responses for radar returns, which are labeled either bad (`'b'`

) or good (`'g'`

).

`load ionosphere`

Randomly choose 10% of the predictor variables.

rng(1); % For reproducibility p = size(X,2); % Number of predictors idxPart = randsample(p,ceil(0.1*p));

Cross-validate two binary kernel classification models: one that uses all of the predictors, and one that uses 10% of the predictors.

CVMdl = fitckernel(X,Y,'CrossVal','on'); PCVMdl = fitckernel(X(:,idxPart),Y,'CrossVal','on');

`CVMdl`

and `PCVMdl`

are `ClassificationPartitionedKernel`

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','10% of the Predictors'}); title('Distribution of Margins')

The quartiles of the `PCVMdl`

margin distribution are situated higher than the quartiles of the `CVMdl`

margin distribution, indicating that the `PCVMdl`

model is the better classifier.

## Input Arguments

`CVMdl`

— Cross-validated, binary kernel classification model

`ClassificationPartitionedKernel`

model object

Cross-validated, binary kernel classification model, specified as a `ClassificationPartitionedKernel`

model object. You can create a
`ClassificationPartitionedKernel`

model by using `fitckernel`

and specifying any one of the cross-validation name-value pair arguments.

To obtain estimates, `kfoldMargin`

applies the same data used to
cross-validate the kernel classification model (`X`

and
`Y`

).

## 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* for binary classification
is, for each observation, the difference between the classification score for the true class
and the classification score for the false class.

The software defines the classification margin for binary classification as

$$m=2yf\left(x\right).$$

*x* is an observation. If the true label of
*x* is the positive class, then *y* is 1, and –1
otherwise. *f*(*x*) is the positive-class classification
score for the observation *x*. The classification margin is commonly
defined as *m* =
*y**f*(*x*).

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.

### Classification Score

For kernel classification models, the raw *classification
score* for classifying the observation *x*, a row vector,
into the positive class is defined by

$$f\left(x\right)=T(x)\beta +b.$$

$$T(\xb7)$$ is a transformation of an observation for feature expansion.

*β*is the estimated column vector of coefficients.*b*is the estimated scalar bias.

The raw classification score for classifying *x* into the negative class is −*f*(*x*). The software classifies observations into the class that yields a
positive score.

If the kernel classification model consists of logistic regression learners, then the
software applies the `'logit'`

score transformation to the raw
classification scores (see `ScoreTransform`

).

## 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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