# kfoldEdge

Classification edge for observations not used for training

## Description

returns
the cross-validated classification
edges obtained by the cross-validated, binary, linear classification
model `e`

= kfoldEdge(`CVMdl`

)`CVMdl`

. That is, for every fold, `kfoldEdge`

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

`e`

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

.

uses
additional options specified by one or more `e`

= kfoldEdge(`CVMdl`

,`Name,Value`

)`Name,Value`

pair
arguments. For example, indicate which folds to use for the edge calculation.

## Input Arguments

`CVMdl`

— Cross-validated, binary, linear classification model

`ClassificationPartitionedLinear`

model object

Cross-validated, binary, linear classification model, specified as a `ClassificationPartitionedLinear`

model object. You can create a
`ClassificationPartitionedLinear`

model using `fitclinear`

and specifying any one of the cross-validation, name-value
pair arguments, for example, `CrossVal`

.

To obtain estimates, kfoldEdge applies the same data used to cross-validate the linear
classification 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.*

`Folds`

— Fold indices to use for classification-score prediction

`1:CVMdl.KFold`

(default) | numeric vector of positive integers

Fold indices to use for classification-score prediction, specified
as the comma-separated pair consisting of `'Folds'`

and
a numeric vector of positive integers. The elements of `Folds`

must
range from `1`

through `CVMdl.KFold`

.

**Example: **`'Folds',[1 4 10]`

**Data Types: **`single`

| `double`

`Mode`

— Edge aggregation level

`'average'`

(default) | `'individual'`

Edge aggregation level, specified as the comma-separated pair
consisting of `'Mode'`

and `'average'`

or `'individual'`

.

Value | Description |
---|---|

`'average'` | Returns classification edges averaged over all folds |

`'individual'` | Returns classification edges for each fold |

**Example: **`'Mode','individual'`

## Output Arguments

`e`

— Cross-validated classification edges

numeric scalar | numeric vector | numeric matrix

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

Let * L* be the number of regularization
strengths in the cross-validated models (that is,

*L*is

`numel(CVMdl.Trained{1}.Lambda)`

)
and *be the number of folds (stored in*

`F`

`CVMdl.KFold`

).If

`Mode`

is`'average'`

, then`e`

is a 1-by-vector.`L`

`e(`

is the average classification edge over all folds of the cross-validated model that uses regularization strength)`j`

.`j`

Otherwise,

`e`

is an-by-`F`

matrix.`L`

`e(`

is the classification edge for fold,`i`

)`j`

of the cross-validated model that uses regularization strength`i`

.`j`

To estimate `e`

, `kfoldEdge`

uses the data that created
`CVMdl`

(see `X`

and `Y`

).

## Examples

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

Load the NLP data set.

`load nlpdata`

`X`

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

is a categorical vector of class labels. There are more than two classes in the data.

The models should identify whether the word counts in a web page are from the Statistics and Machine Learning Toolbox™ documentation. So, identify the labels that correspond to the Statistics and Machine Learning Toolbox™ documentation web pages.

`Ystats = Y == 'stats';`

Cross-validate a binary, linear classification model that can identify whether the word counts in a documentation web page are from the Statistics and Machine Learning Toolbox™ documentation.

rng(1); % For reproducibility CVMdl = fitclinear(X,Ystats,'CrossVal','on');

`CVMdl`

is a `ClassificationPartitionedLinear`

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 average of the out-of-fold edges.

e = kfoldEdge(CVMdl)

e = 8.1243

Alternatively, you can obtain the per-fold edges by specifying the name-value pair `'Mode','individual'`

in `kfoldEdge`

.

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

One way to perform feature selection is to compare *k*-fold edges from multiple models. Based solely on this criterion, the classifier with the highest edge is the best classifier.

Load the NLP data set. Preprocess the data as in Estimate k-Fold Cross-Validation Edge.

load nlpdata Ystats = Y == 'stats'; 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,:);

Cross-validate two binary, linear classification models: one that uses the all of the predictors and one that uses half of the predictors. Optimize the objective function using SpaRSA, and indicate that observations correspond to columns.

CVMdl = fitclinear(fullX,Ystats,'CrossVal','on','Solver','sparsa',... 'ObservationsIn','columns'); PCVMdl = fitclinear(partX,Ystats,'CrossVal','on','Solver','sparsa',... 'ObservationsIn','columns');

`CVMdl`

and `PCVMdl`

are `ClassificationPartitionedLinear`

models.

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

fullEdge = kfoldEdge(CVMdl)

fullEdge = 16.5629

partEdge = kfoldEdge(PCVMdl)

partEdge = 13.9030

Based on the *k*-fold edges, the classifier that uses all of the predictors is the better model.

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

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

Load the NLP data set. Preprocess the data as in Estimate k-Fold Cross-Validation Edge.

load nlpdata Ystats = Y == 'stats'; X = X';

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

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

Cross-validate a binary, linear classification model using 5-fold cross-validation and that uses each of the regularization strengths. Optimize the objective function using SpaRSA. Lower the tolerance on the gradient of the objective function to `1e-8`

.

rng(10); % For reproducibility CVMdl = fitclinear(X,Ystats,'ObservationsIn','columns','KFold',5,... 'Learner','logistic','Solver','sparsa','Regularization','lasso',... 'Lambda',Lambda,'GradientTolerance',1e-8)

CVMdl = ClassificationPartitionedLinear CrossValidatedModel: 'Linear' ResponseName: 'Y' NumObservations: 31572 KFold: 5 Partition: [1x1 cvpartition] ClassNames: [0 1] ScoreTransform: 'none'

`CVMdl`

is a `ClassificationPartitionedLinear`

model. Because `fitclinear`

implements 5-fold cross-validation, `CVMdl`

contains 5 `ClassificationLinear`

models that the software trains on each fold.

Estimate the edges for each fold and regularization strength.

eFolds = kfoldEdge(CVMdl,'Mode','individual')

`eFolds = `*5×11*
0.9958 0.9958 0.9958 0.9958 0.9958 0.9923 0.9772 0.9231 0.8419 0.8127 0.8127
0.9991 0.9991 0.9991 0.9991 0.9991 0.9939 0.9780 0.9181 0.8257 0.8128 0.8128
0.9992 0.9992 0.9992 0.9992 0.9992 0.9942 0.9779 0.9103 0.8255 0.8128 0.8128
0.9974 0.9974 0.9974 0.9974 0.9974 0.9931 0.9772 0.9195 0.8486 0.8130 0.8130
0.9976 0.9976 0.9976 0.9976 0.9976 0.9942 0.9782 0.9194 0.8400 0.8127 0.8127

`eFolds`

is a 5-by-11 matrix of edges. Rows correspond to folds and columns correspond to regularization strengths in `Lambda`

. You can use `eFolds`

to identify ill-performing folds, that is, unusually low edges.

Estimate the average edge over all folds for each regularization strength.

e = kfoldEdge(CVMdl)

`e = `*1×11*
0.9978 0.9978 0.9978 0.9978 0.9978 0.9935 0.9777 0.9181 0.8364 0.8128 0.8128

Determine how well the models generalize by plotting the averages of the 5-fold edge for each regularization strength. Identify the regularization strength that maximizes the 5-fold edge over the grid.

figure; plot(log10(Lambda),log10(e),'-o') [~, maxEIdx] = max(e); maxLambda = Lambda(maxEIdx); hold on plot(log10(maxLambda),log10(e(maxEIdx)),'ro'); ylabel('log_{10} 5-fold edge') xlabel('log_{10} Lambda') legend('Edge','Max edge') hold off

Several values of `Lambda`

yield similarly high edges. 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 edge starts decreasing.

LambdaFinal = Lambda(5);

Train a linear classification model using the entire data set and specify the regularization strength `LambdaFinal`

.

MdlFinal = fitclinear(X,Ystats,'ObservationsIn','columns',... 'Learner','logistic','Solver','sparsa','Regularization','lasso',... 'Lambda',LambdaFinal);

To estimate labels for new observations, pass `MdlFinal`

and the new data to `predict`

.

## More About

### Classification Edge

The *classification edge* is the weighted mean of the
classification margins.

One way to choose among multiple classifiers, for example to perform feature selection, is to choose the classifier that yields the greatest edge.

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

$${f}_{j}(x)=x{\beta}_{j}+{b}_{j}.$$

For the model with regularization strength *j*, $${\beta}_{j}$$ is the estimated column vector of coefficients (the model property
`Beta(:,j)`

) and $${b}_{j}$$ is the estimated, scalar bias (the model property
`Bias(j)`

).

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

If the linear 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 R2016a**

### 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.

### R2022a: `kfoldEdge`

returns a different value for a model with a nondefault cost matrix

If you specify a nondefault cost matrix when you train the input model object, the `kfoldEdge`

function returns a different value compared to previous releases.

The `kfoldEdge`

function uses the
observation weights stored in the `W`

property. The way the function uses the
`W`

property value has not changed. However, the property value stored in the input model object has changed for a
model with a nondefault cost matrix, so the function might return a different value.

For details about the property value change, see Cost property stores the user-specified cost matrix.

If you want the software to handle the cost matrix, prior
probabilities, and observation weights in the same way as in previous releases, adjust the prior
probabilities and observation weights for the nondefault cost matrix, as described in Adjust Prior Probabilities and Observation Weights for Misclassification Cost Matrix. Then, when you train a
classification model, specify the adjusted prior probabilities and observation weights by using
the `Prior`

and `Weights`

name-value arguments, respectively,
and use the default cost matrix.

## See Also

`ClassificationPartitionedLinear`

| `kfoldMargin`

| `ClassificationLinear`

| `kfoldPredict`

| `edge`

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