# kfoldLoss

Classification loss for cross-validated kernel ECOC model

## Description

returns the classification loss obtained by the cross-validated kernel ECOC model (`loss`

= kfoldLoss(`CVMdl`

)`ClassificationPartitionedKernelECOC`

) `CVMdl`

. For every fold,
`kfoldLoss`

computes the classification loss for validation-fold
observations using a model trained on training-fold observations.
`kfoldLoss`

applies the same data used to create
`CVMdl`

(see `fitcecoc`

).

By default, `kfoldLoss`

returns the classification
error.

returns the classification loss with additional options specified by one or more name-value
pair arguments. For example, specify the classification loss function, number of folds,
decoding scheme, or verbosity level.`loss`

= kfoldLoss(`CVMdl`

,`Name,Value`

)

## Examples

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

Load Fisher's iris data set. `X`

contains flower measurements, and `Y`

contains the names of flower species.

```
load fisheriris
X = meas;
Y = species;
```

Cross-validate an ECOC model composed of kernel binary learners.

CVMdl = fitcecoc(X,Y,'Learners','kernel','CrossVal','on')

CVMdl = ClassificationPartitionedKernelECOC CrossValidatedModel: 'KernelECOC' ResponseName: 'Y' NumObservations: 150 KFold: 10 Partition: [1x1 cvpartition] ClassNames: {'setosa' 'versicolor' 'virginica'} ScoreTransform: 'none'

`CVMdl`

is a `ClassificationPartitionedKernelECOC`

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 cross-validated classification loss. By default, the software computes the classification error.

loss = kfoldLoss(CVMdl)

loss = 0.0333

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

in `kfoldLoss`

.

### Determine Model Quality Using Custom Cross-Validation Loss

In addition to knowing whether a model generally classifies observations correctly, you can determine how well the model classifies an observation into its predicted class. One way to determine this type of model quality is to pass a custom loss function to `kfoldLoss`

.

Load Fisher's iris data set. `X`

contains flower measurements, and `Y`

contains the names of flower species.

```
load fisheriris
X = meas;
Y = species;
```

Cross-validate an ECOC model composed of kernel binary learners.

rng(1) % For reproducibility CVMdl = fitcecoc(X,Y,'Learners','kernel','CrossVal','on')

CVMdl = ClassificationPartitionedKernelECOC CrossValidatedModel: 'KernelECOC' ResponseName: 'Y' NumObservations: 150 KFold: 10 Partition: [1x1 cvpartition] ClassNames: {'setosa' 'versicolor' 'virginica'} ScoreTransform: 'none'

`CVMdl`

is a `ClassificationPartitionedKernelECOC`

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'`

.

Create a custom function that takes the minimal loss for each observation, then averages the minimal losses for all observations. `S`

corresponds to the `NegLoss`

output of `kfoldPredict`

.

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

Compute the cross-validated custom loss.

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

ans = 0.0299

The average minimal binary loss for the validation-fold observations is about 0.02.

## Input Arguments

`CVMdl`

— Cross-validated kernel ECOC model

`ClassificationPartitionedKernelECOC`

model

Cross-validated kernel ECOC model, specified as a `ClassificationPartitionedKernelECOC`

model. You can create a
`ClassificationPartitionedKernelECOC`

model by training an ECOC model
using `fitcecoc`

and specifying these name-value
pair arguments:

`'Learners'`

– Set the value to`'kernel'`

, a template object returned by`templateKernel`

, or a cell array of such template objects.One of the arguments

`'CrossVal'`

,`'CVPartition'`

,`'Holdout'`

,`'KFold'`

, or`'Leaveout'`

.

### 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.*

**Example: **`kfoldLoss(CVMdl,'Folds',[1 3 5])`

specifies to use only the
first, third, and fifth folds to calculate the classification loss.

`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 binary losses so that the loss is 0.5 when

*y*= 0. Also, the software calculates the mean binary loss for each class [1]._{j}For a custom binary loss function, for example,

`customFunction`

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

.`customFunction`

has this form: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.

By default, if all binary learners are kernel classification models using SVM, then
`BinaryLoss`

is `'hinge'`

. If all binary
learners are kernel classification models using 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 for prediction

`1:CVMdl.KFold`

(default) | numeric vector of positive integers

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

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

must be within the range from `1`

to
`CVMdl.KFold`

.

The software uses only the folds specified in `Folds`

for
prediction.

**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.

Specify the built-in function

`'classiferror'`

. In this case, 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.Or, specify your own function using function handle notation.

Assume that

*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 specify the function name (

).`lossfun`

`C`

is an*n*-by-*K*logical matrix with rows indicating the class to which 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 the classes. Each row corresponds to an observation. The column order corresponds to the class order in`CVMdl.ClassNames`

. The input`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: **`char`

| `string`

| `function_handle`

`Mode`

— Aggregation level for output

`'average'`

(default) | `'individual'`

Aggregation level for the output, specified as the comma-separated pair consisting of
`'Mode'`

and `'average'`

or
`'individual'`

.

This table describes the values.

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

`'average'` | The output is a scalar average over all folds. |

`'individual'` | The output is a vector of length k containing one value per
fold, where k is the number of folds. |

**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

`loss`

— Classification loss

numeric scalar | numeric column vector

Classification loss, returned as a numeric scalar or numeric column vector.

If `Mode`

is `'average'`

, then
`loss`

is the average classification loss over all folds.
Otherwise, `loss`

is a *k*-by-1 numeric column
vector containing the classification loss for each fold, where *k* is
the number of folds.

## 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. 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.

## 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 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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