# kfoldEdge

Classification edge for observations not used for training

## Description

returns
the cross-validated classification
edges obtained by the cross-validated, error-correcting output
codes (ECOC) model composed of linear classification models `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, specify a decoding scheme, which folds to
use for the edge calculation, 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, kfoldEdge 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'`

`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'`

`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

`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}.BinaryLearners{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 a-by-`F`

matrix.`L`

`e(`

is the classification edge for fold,`i`

)`j`

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

.`j`

## 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.

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 average of the out-of-fold edges.

e = kfoldEdge(CVMdl)

e = 1.4464

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, 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 a 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 to optimize 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 edge for each classifier.

fullEdge = kfoldEdge(CVMdl)

fullEdge = 0.6181

partEdge = kfoldEdge(PCVMdl)

partEdge = 0.5235

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 Feature Selection Using k-fold Edges.

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

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

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

Create a linear classification model template that specifies to use logistic regression with a lasso penalty, use each of the regularization strengths, optimize the objective function using SpaRSA, and reduce 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 edges for each fold and regularization strength.

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

`eFolds = `*5×8*
0.5520 0.5524 0.5528 0.5508 0.4936 0.2933 0.1029 0.0853
0.5241 0.5255 0.5259 0.5260 0.4773 0.2945 0.1052 0.0868
0.5281 0.5298 0.5301 0.5302 0.4786 0.2880 0.1034 0.0868
0.5389 0.5404 0.5408 0.5373 0.4833 0.2910 0.1022 0.0854
0.5497 0.5552 0.5586 0.5571 0.4936 0.2949 0.1027 0.0849

`eFolds`

is a 5-by-8 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 = `*1×8*
0.5386 0.5407 0.5417 0.5403 0.4853 0.2923 0.1033 0.0858

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. Greater regularization strength values 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(4);

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',LambdaFinal,'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.

Suppose 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 decoding scheme of an ECOC model specifies how the software aggregates the binary losses and determines the predicted class for each 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.

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 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* 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`

.

For example: `'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**

## See Also

`ClassificationPartitionedLinearECOC`

| `ClassificationECOC`

| `ClassificationLinear`

| `kfoldMargin`

| `edge`

| `kfoldPredict`

| `fitcecoc`

| `statset`

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