# kfoldLoss

Classification loss for observations not used in training

## Description

returns
the cross-validated classification
error rates estimated by the cross-validated, error-correcting
output codes (ECOC) model composed of linear classification models `L`

= kfoldLoss(`CVMdl`

)`CVMdl`

.
That is, for every fold, `kfoldLoss`

estimates the
classification error rate for observations that it holds out when
it trains using all other observations. `kfoldLoss`

applies
the same data used create `CVMdl`

(see `fitcecoc`

).

`L`

contains a classification loss for each
regularization strength in the linear classification models that compose `CVMdl`

.

uses
additional options specified by one or more `L`

= kfoldLoss(`CVMdl`

,`Name,Value`

)`Name,Value`

pair
arguments. For example, specify a decoding scheme, which folds to
use for the loss 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, kfoldLoss 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`

`LossFun`

— Loss function

`'classiferror'`

(default) | `'classifcost'`

| function handle

Loss function, specified as `'classiferror'`

,
`'classifcost'`

, or a function handle.

You can:

Specify the built-in function

`'classiferror'`

, then the loss function is the classification error.Specify the built-in function

`'classifcost'`

. In this case, the loss function is the observed misclassification cost. If you use the default cost matrix (whose element value is 0 for correct classification and 1 for incorrect classification), then the loss values for`'classifcost'`

and`'classiferror'`

are identical.Specify your own function using function handle notation.

For what follows,

`n`

is the number of observations in the training data (`CVMdl.NumObservations`

) and`K`

is the number of classes (`numel(CVMdl.ClassNames)`

). Your function needs the signature`lossvalue =`

, where:(C,S,W,Cost)`lossfun`

The output argument

`lossvalue`

is a scalar.You choose the function name (

).`lossfun`

`C`

is an`n`

-by-`K`

logical matrix with rows indicating which class the corresponding observation belongs. The column order corresponds to the class order in`CVMdl.ClassNames`

.Construct

`C`

by setting`C(p,q) = 1`

if observation`p`

is in class`q`

, for each row. Set every element of row`p`

to`0`

.`S`

is an`n`

-by-`K`

numeric matrix of negated loss values for classes. Each row corresponds to an observation. The column order corresponds to the class order in`CVMdl.ClassNames`

.`S`

resembles the output argument`NegLoss`

of`kfoldPredict`

.`W`

is an`n`

-by-1 numeric vector of observation weights. If you pass`W`

, the software normalizes its elements to sum to`1`

.`Cost`

is a`K`

-by-`K`

numeric matrix of misclassification costs. For example,`Cost`

=`ones(K) -eye(K)`

specifies a cost of 0 for correct classification, and 1 for misclassification.

Specify your function using

`'LossFun',@lossfun`

.

**Data Types: **`function_handle`

| `char`

| `string`

`Mode`

— Loss aggregation level

`'average'`

(default) | `'individual'`

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

and `'average'`

or `'individual'`

.

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

`'average'` | Returns losses averaged over all folds |

`'individual'` | Returns losses 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

`L`

— Cross-validated classification losses

numeric scalar | numeric vector | numeric matrix

Cross-validated classification losses, returned
as a numeric scalar, vector, or matrix. The interpretation of `L`

depends
on `LossFun`

.

Let * R* be the number of regularizations
strengths is the cross-validated models (

`CVMdl.Trained{1}.BinaryLearners{1}.Lambda`

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

`F`

`CVMdl.KFold`

).If

`Mode`

is`'average'`

, then`L`

is a 1-by-vector.`R`

`L(`

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

.`j`

Otherwise,

`L`

is a-by-`F`

matrix.`R`

`L(`

is the classification loss for fold,`i`

)`j`

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

.`j`

## Examples

### Estimate *k*-Fold Cross-Validation Classification Error

Load the NLP data set.

`load nlpdata`

`X`

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

is a categorical vector of class labels.

Cross-validate an ECOC model of linear classification models.

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.

Estimate the average of the out-of-fold classification error rates.

ce = kfoldLoss(CVMdl)

ce = 0.0958

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

in `kfoldLoss`

.

### Specify Custom Classification Loss

Load the NLP data set. Transpose the predictor data.

```
load nlpdata
X = X';
```

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';

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

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

Cross-validate an ECOC model of linear classification models using 5-fold cross-validation. Optimize the objective function using SpaRSA. Specify that the predictor observations correspond to columns.

rng(1); % For reproducibility CVMdl = fitcecoc(X,Y,'Learners',t,'KFold',5,'ObservationsIn','columns'); CMdl1 = CVMdl.Trained{1}

CMdl1 = CompactClassificationECOC ResponseName: 'Y' ClassNames: [comm dsp simulink others] ScoreTransform: 'none' BinaryLearners: {6x1 cell} CodingMatrix: [4x6 double] Properties, Methods

`CVMdl`

is a `ClassificationPartitionedLinearECOC`

model. It contains the property `Trained`

, which is a 5-by-1 cell array holding a `CompactClassificationECOC`

models that the software trained using the training set of each fold.

Create a function that takes the minimal loss for each observation, and then averages the minimal losses across all observations. Because the function does not use the class-identifier matrix (`C`

), observation weights (`W`

), and classification cost (`Cost`

), use `~`

to have `kfoldLoss`

ignore its their positions.

lossfun = @(~,S,~,~)mean(min(-S,[],2));

Estimate the average cross-validated classification loss using the minimal loss per observation function. Also, obtain the loss for each fold.

`ce = kfoldLoss(CVMdl,'LossFun',lossfun)`

ce = 0.0485

ceFold = kfoldLoss(CVMdl,'LossFun',lossfun,'Mode','individual')

`ceFold = `*5×1*
0.0488
0.0511
0.0496
0.0479
0.0452

### Find Good Lasso Penalty Using Cross-Validation

To determine a good lasso-penalty strength for an ECOC model composed of linear classification models that use logistic regression learners, implement 5-fold cross-validation.

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';

Create a set of 11 logarithmically-spaced regularization strengths from $$1{0}^{-7}$$ through $$1{0}^{-2}$$.

Lambda = logspace(-7,-2,11);

Create a linear classification model template that specifies to use logistic regression learners, use lasso penalties with strengths in `Lambda`

, train using SpaRSA, and lower 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 the models. To increase execution speed, transpose the predictor data and specify that the observations are in columns.

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

`CVMdl`

is a `ClassificationPartitionedLinearECOC`

model.

Dissect `CVMdl`

, and each model within it.

numECOCModels = numel(CVMdl.Trained)

numECOCModels = 5

ECOCMdl1 = CVMdl.Trained{1}

ECOCMdl1 = CompactClassificationECOC ResponseName: 'Y' ClassNames: [comm dsp simulink others] ScoreTransform: 'none' BinaryLearners: {6×1 cell} CodingMatrix: [4×6 double] Properties, Methods

numCLModels = numel(ECOCMdl1.BinaryLearners)

numCLModels = 6

CLMdl1 = ECOCMdl1.BinaryLearners{1}

CLMdl1 = ClassificationLinear ResponseName: 'Y' ClassNames: [-1 1] ScoreTransform: 'logit' Beta: [34023×11 double] Bias: [-0.3169 -0.3169 -0.3168 -0.3168 -0.3168 -0.3167 -0.1725 -0.0805 -0.1762 -0.3450 -0.5174] Lambda: [1.0000e-07 3.1623e-07 1.0000e-06 3.1623e-06 1.0000e-05 3.1623e-05 1.0000e-04 3.1623e-04 1.0000e-03 0.0032 0.0100] Learner: 'logistic' Properties, Methods

Because `fitcecoc`

implements 5-fold cross-validation, `CVMdl`

contains a 5-by-1 cell array of `CompactClassificationECOC`

models that the software trains on each fold. The `BinaryLearners`

property of each `CompactClassificationECOC`

model contains the `ClassificationLinear`

models. The number of `ClassificationLinear`

models within each compact ECOC model depends on the number of distinct labels and coding design. Because `Lambda`

is a sequence of regularization strengths, you can think of `CLMdl1`

as 11 models, one for each regularization strength in `Lambda`

.

Determine how well the models generalize by plotting the averages of the 5-fold classification error for each regularization strength. Identify the regularization strength that minimizes the generalization error over the grid.

ce = kfoldLoss(CVMdl); figure; plot(log10(Lambda),log10(ce)) [~,minCEIdx] = min(ce); minLambda = Lambda(minCEIdx); hold on plot(log10(minLambda),log10(ce(minCEIdx)),'ro'); ylabel('log_{10} 5-fold classification error') xlabel('log_{10} Lambda') legend('MSE','Min classification error') hold off

Train an ECOC model composed of linear classification model using the entire data set, and specify the minimal regularization strength.

t = templateLinear('Learner','logistic','Solver','sparsa',... 'Regularization','lasso','Lambda',minLambda,'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

### Classification Error

The *classification error* has the form

$$L={\displaystyle \sum _{j=1}^{n}{w}_{j}{e}_{j}},$$

where:

*w*is the weight for observation_{j}*j*. The software renormalizes the weights to sum to 1.*e*= 1 if the predicted class of observation_{j}*j*differs from its true class, and 0 otherwise.

In other words, the classification error is the proportion of observations misclassified by the classifier.

### Observed Misclassification Cost

The *observed misclassification cost* has the form

$$L={\displaystyle \sum _{j=1}^{n}{w}_{j}}{c}_{{y}_{j}{\widehat{y}}_{j}},$$

where:

*w*is the weight for observation_{j}*j*. The software renormalizes the weights to sum to 1.$${c}_{{y}_{j}{\widehat{y}}_{j}}$$ is the user-specified cost of classifying an observation into class $${\widehat{y}}_{j}$$ when its true class is

*y*._{j}

### 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.

## 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`

| `loss`

| `kfoldPredict`

| `fitcecoc`

| `statset`

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