Cross-validated multiclass ECOC model for support vector machines (SVMs) and other classifiers

`ClassificationPartitionedECOC`

is a set of
error-correcting output codes (ECOC) models trained on cross-validated folds. Estimate
the quality of the cross-validated classification by using one or more
“kfold” functions: `kfoldPredict`

, `kfoldLoss`

, `kfoldMargin`

, `kfoldEdge`

, and `kfoldfun`

.

Every “kfold” method uses models trained on training-fold (in-fold)
observations to predict the response for validation-fold (out-of-fold) observations. For
example, suppose you cross-validate using five folds. In this case, the software
randomly assigns each observation into five groups of equal size (roughly). The
*training fold* contains four of the groups (roughly 4/5 of the
data), and the *validation fold* contains the other group (roughly
1/5 of the data). In this case, cross-validation proceeds as follows:

The software trains the first model (stored in

`CVMdl.Trained{1}`

) by using the observations in the last four groups and reserves the observations in the first group for validation.The software trains the second model (stored in

`CVMdl.Trained{2}`

) by using the observations in the first group and the last three groups. The software reserves the observations in the second group for validation.The software proceeds in a similar fashion for the third, fourth, and fifth models.

If you validate by using `kfoldPredict`

, the software computes
predictions for the observations in group *i* by using the
*i*th model. In short, the software estimates a response for every
observation by using the model trained without that observation.

You can create a `ClassificationPartitionedECOC`

model in two
ways:

`CrossValidatedModel`

— Cross-validated model namecharacter vector

Cross-validated model name, specified as a character vector.

For example, `'ECOC'`

specifies a cross-validated
ECOC model.

**Data Types: **`char`

`KFold`

— Number of cross-validated foldspositive integer

Number of cross-validated folds, specified as a positive integer.

**Data Types: **`double`

`ModelParameters`

— Cross-validation parameter valuesobject

Cross-validation parameter values, specified as an object. The
parameter values correspond to the name-value pair argument values used
to cross-validate the ECOC classifier.
`ModelParameters`

does not contain estimated
parameters.

You can access the properties of `ModelParameters`

using dot notation.

`NumObservations`

— Number of observationspositive numeric scalar

Number of observations in the training data, specified as a positive numeric scalar.

**Data Types: **`double`

`Partition`

— Data partition`cvpartition`

modelData partition indicating how the software splits the data into
cross-validation folds, specified as a `cvpartition`

model.

`Trained`

— Compact classifiers trained on cross-validation foldscell array of

`CompactClassificationECOC`

modelsCompact classifiers trained on cross-validation folds, specified as a
cell array of `CompactClassificationECOC`

models. `Trained`

has *k* cells,
where *k* is the number of folds.

**Data Types: **`cell`

`W`

— Observation weightsnumeric vector

Observation weights used to cross-validate the model, specified as a numeric vector.
`W`

has `NumObservations`

elements.

The software normalizes the weights used for training so that `nansum(W)`

is `1`

.

**Data Types: **`single`

| `double`

`X`

— Unstandardized predictor datanumeric matrix | table

Unstandardized predictor data used to cross-validate the classifier, specified as a numeric matrix or table.

Each row of `X`

corresponds to one observation, and
each column corresponds to one variable.

**Data Types: **`single`

| `double`

| `table`

`Y`

— Observed class labelscategorical array | character array | logical vector | numeric vector | cell array of character vectors

Observed class labels used to cross-validate the model, specified as a
categorical or character array, logical or numeric vector, or cell array
of character vectors. `Y`

has
`NumObservations`

elements and has the same data
type as the input argument `Y`

that you pass to
`fitcecoc`

to cross-validate
the model. (The software treats string arrays as cell arrays of character
vectors.)

Each row of `Y`

represents the observed
classification of the corresponding row of
`X`

.

**Data Types: **`categorical`

| `char`

| `logical`

| `single`

| `double`

| `cell`

`BinaryLoss`

— Binary learner loss function`'binodeviance'`

| `'exponential'`

| `'hamming'`

| `'hinge'`

| `'linear'`

| `'logit'`

| `'quadratic'`

Binary learner loss function, specified as a character vector representing the loss function name.

If you train using binary learners that use different loss functions,
then the software sets `BinaryLoss`

to
`'hamming'`

. To potentially increase accuracy,
specify a binary loss function other than the default during a
prediction or loss computation by using the
`'BinaryLoss'`

name-value pair argument of
`kfoldPredict`

or `kfoldLoss`

.

**Data Types: **`char`

`BinaryY`

— Binary learner class labelsnumeric matrix |

`[]`

Binary learner class labels, specified as a numeric matrix or
`[]`

.

If the coding matrix is the same across all folds, then

`BinaryY`

is a`NumObservations`

-by-*L*matrix, where*L*is the number of binary learners (`size(CodingMatrix,2)`

).The elements of

`BinaryY`

are`–1`

,`0`

, or`1`

, and the values correspond to dichotomous class assignments. This table describes how learner`j`

assigns observation`k`

to a dichotomous class corresponding to the value of`BinaryY(k,j)`

.Value Dichotomous Class Assignment `–1`

Learner `j`

assigns observation`k`

to a negative class.`0`

Before training, learner `j`

removes observation`k`

from the data set.`1`

Learner `j`

assigns observation`k`

to a positive class.If the coding matrix varies across folds, then

`BinaryY`

is empty (`[]`

).

**Data Types: **`double`

`CodingMatrix`

— Codes specifying class assignmentsnumeric matrix |

`[]`

Codes specifying class assignments for the binary learners, specified
as a numeric matrix or `[]`

.

If the coding matrix is the same across all folds, then

`CodingMatrix`

is a*K*-by-*L*matrix, where*K*is the number of classes and*L*is the number of binary learners.The elements of

`CodingMatrix`

are`–1`

,`0`

, or`1`

, and the values correspond to dichotomous class assignments. This table describes how learner`j`

assigns observations in class`i`

to a dichotomous class corresponding to the value of`CodingMatrix(i,j)`

.Value Dichotomous Class Assignment `–1`

Learner `j`

assigns observations in class`i`

to a negative class.`0`

Before training, learner `j`

removes observations in class`i`

from the data set.`1`

Learner `j`

assigns observations in class`i`

to a positive class.If the coding matrix varies across folds, then

`CodingMatrix`

is empty (`[]`

). You can obtain the coding matrix for each fold by using the`Trained`

property. For example,`CVMdl.Trained{1}.CodingMatrix`

is the coding matrix in the first fold of the cross-validated ECOC model`CVMdl`

.

**Data Types: **`double`

| `single`

| `int8`

| `int16`

| `int32`

| `int64`

`CategoricalPredictors`

— Categorical predictor indicesvector of positive integers |

`[]`

Categorical predictor
indices, specified as a vector of positive integers. `CategoricalPredictors`

contains index values corresponding to the columns of the predictor data that contain
categorical predictors. If none of the predictors are categorical, then this property is empty
(`[]`

).

**Data Types: **`single`

| `double`

`ClassNames`

— Unique class labelscategorical array | character array | logical vector | numeric vector | cell array of character vectors

Unique class labels used in training, specified as a categorical or
character array, logical or numeric vector, or cell array of
character vectors. `ClassNames`

has the same
data type as the class labels `Y`

.
(The software treats string arrays as cell arrays of character
vectors.)
`ClassNames`

also determines the class
order.

**Data Types: **`categorical`

| `char`

| `logical`

| `single`

| `double`

| `cell`

`Cost`

— Misclassification costssquare numeric matrix

This property is read-only.

Misclassification costs, specified as a square numeric matrix. `Cost`

has
*K* rows and columns, where *K* is the number of
classes.

`Cost(i,j)`

is the cost of classifying a point into class
`j`

if its true class is `i`

. The order of the
rows and columns of `Cost`

corresponds to the order of the classes in
`ClassNames`

.

`fitcecoc`

incorporates misclassification
costs differently among different types of binary learners.

**Data Types: **`double`

`PredictorNames`

— Predictor namescell array of character vectors

Predictor names in order of their appearance in the predictor data
`X`

, specified as a cell array of
character vectors. The length of
`PredictorNames`

is equal to the
number of columns in `X`

.

**Data Types: **`cell`

`Prior`

— Prior class probabilitiesnumeric vector

This property is read-only.

Prior class probabilities, specified as a numeric vector. `Prior`

has as
many elements as the number of classes in
`ClassNames`

, and the order of
the elements corresponds to the order of the classes in
`ClassNames`

.

`fitcecoc`

incorporates misclassification
costs differently among different types of binary learners.

**Data Types: **`double`

`ResponseName`

— Response variable namecharacter vector

Response variable name, specified as a character vector.

**Data Types: **`char`

`ScoreTransform`

— Score transformation function to apply to predicted scores`'doublelogit'`

| `'invlogit'`

| `'ismax'`

| `'logit'`

| `'none'`

| function handle | ...Score transformation function to apply to predicted scores, specified as a function name or function handle.

To change the score transformation function to * function*, for
example, use dot notation.

For a built-in function, enter this code and replace

with a value in the table.`function`

Mdl.ScoreTransform = '

*function*';Value Description `'doublelogit'`

1/(1 + *e*^{–2x})`'invlogit'`

log( *x*/ (1 –*x*))`'ismax'`

Sets the score for the class with the largest score to `1`

, and sets the scores for all other classes to`0`

`'logit'`

1/(1 + *e*^{–x})`'none'`

or`'identity'`

*x*(no transformation)`'sign'`

–1 for *x*< 0

0 for*x*= 0

1 for*x*> 0`'symmetric'`

2 *x*– 1`'symmetricismax'`

Sets the score for the class with the largest score to `1`

, and sets the scores for all other classes to`–1`

`'symmetriclogit'`

2/(1 + *e*^{–x}) – 1For a MATLAB

^{®}function or a function that you define, enter its function handle.Mdl.ScoreTransform = @

*function*;must accept a matrix (the original scores) and return a matrix of the same size (the transformed scores).`function`

**Data Types: **`char`

| `function_handle`

`kfoldEdge` | Classification edge for cross-validated ECOC model |

`kfoldLoss` | Classification loss for cross-validated ECOC model |

`kfoldMargin` | Classification margins for cross-validated ECOC model |

`kfoldPredict` | Classify observations in cross-validated ECOC model |

`kfoldfun` | Cross-validate function using cross-validated ECOC model |

Cross-validate an ECOC classifier with SVM binary learners, and estimate the generalized classification error.

Load Fisher's iris data set. Specify the predictor data `X`

and the response data `Y`

.

load fisheriris X = meas; Y = species; rng(1); % For reproducibility

Create an SVM template, and standardize the predictors.

`t = templateSVM('Standardize',true)`

t = Fit template for classification SVM. Alpha: [0x1 double] BoxConstraint: [] CacheSize: [] CachingMethod: '' ClipAlphas: [] DeltaGradientTolerance: [] Epsilon: [] GapTolerance: [] KKTTolerance: [] IterationLimit: [] KernelFunction: '' KernelScale: [] KernelOffset: [] KernelPolynomialOrder: [] NumPrint: [] Nu: [] OutlierFraction: [] RemoveDuplicates: [] ShrinkagePeriod: [] Solver: '' StandardizeData: 1 SaveSupportVectors: [] VerbosityLevel: [] Version: 2 Method: 'SVM' Type: 'classification'

`t`

is an SVM template. Most of the template object properties are empty. When training the ECOC classifier, the software sets the applicable properties to their default values.

Train the ECOC classifier, and specify the class order.

Mdl = fitcecoc(X,Y,'Learners',t,... 'ClassNames',{'setosa','versicolor','virginica'});

`Mdl`

is a `ClassificationECOC`

classifier. You can access its properties using dot notation.

Cross-validate `Mdl`

using 10-fold cross-validation.

CVMdl = crossval(Mdl);

`CVMdl`

is a `ClassificationPartitionedECOC`

cross-validated ECOC classifier.

Estimate the generalized classification error.

genError = kfoldLoss(CVMdl)

genError = 0.0400

The generalized classification error is 4%, which indicates that the ECOC classifier generalizes fairly well.

Train a one-versus-all ECOC classifier using a `GentleBoost`

ensemble of decision trees with surrogate splits. To speed up training, bin numeric predictors and use parallel computing. Binning is valid only when `fitcecoc`

uses a tree learner. After training, estimate the classification error using 10-fold cross-validation. Note that parallel computing requires Parallel Computing Toolbox™.

**Load Sample Data**

Load and inspect the `arrhythmia`

data set.

```
load arrhythmia
[n,p] = size(X)
```

n = 452

p = 279

isLabels = unique(Y); nLabels = numel(isLabels)

nLabels = 13

tabulate(categorical(Y))

Value Count Percent 1 245 54.20% 2 44 9.73% 3 15 3.32% 4 15 3.32% 5 13 2.88% 6 25 5.53% 7 3 0.66% 8 2 0.44% 9 9 1.99% 10 50 11.06% 14 4 0.88% 15 5 1.11% 16 22 4.87%

The data set contains `279`

predictors, and the sample size of `452`

is relatively small. Of the 16 distinct labels, only 13 are represented in the response (`Y`

). Each label describes various degrees of arrhythmia, and 54.20% of the observations are in class `1`

.

**Train One-Versus-All ECOC Classifier**

Create an ensemble template. You must specify at least three arguments: a method, a number of learners, and the type of learner. For this example, specify `'GentleBoost'`

for the method, `100`

for the number of learners, and a decision tree template that uses surrogate splits because there are missing observations.

tTree = templateTree('surrogate','on'); tEnsemble = templateEnsemble('GentleBoost',100,tTree);

`tEnsemble`

is a template object. Most of its properties are empty, but the software fills them with their default values during training.

Train a one-versus-all ECOC classifier using the ensembles of decision trees as binary learners. To speed up training, use binning and parallel computing.

Binning (

`'NumBins',50`

) — When you have a large training data set, you can speed up training (a potential decrease in accuracy) by using the`'NumBins'`

name-value pair argument. This argument is valid only when`fitcecoc`

uses a tree learner. If you specify the`'NumBins'`

value, then the software bins every numeric predictor into a specified number of equiprobable bins, and then grows trees on the bin indices instead of the original data. You can try`'NumBins',50`

first, and then change the`'NumBins'`

value depending on the accuracy and training speed.Parallel computing (

`'Options',statset('UseParallel',true)`

) — With a Parallel Computing Toolbox license, you can speed up the computation by using parallel computing, which sends each binary learner to a worker in the pool. The number of workers depends on your system configuration. When you use decision trees for binary learners,`fitcecoc`

parallelizes training using Intel® Threading Building Blocks (TBB) for dual-core systems and above. Therefore, specifying the`'UseParallel'`

option is not helpful on a single computer. Use this option on a cluster.

Additionally, specify that the prior probabilities are 1/*K*, where *K* = 13 is the number of distinct classes.

options = statset('UseParallel',true); Mdl = fitcecoc(X,Y,'Coding','onevsall','Learners',tEnsemble,... 'Prior','uniform','NumBins',50,'Options',options);

Starting parallel pool (parpool) using the 'local' profile ... Connected to the parallel pool (number of workers: 6).

`Mdl`

is a `ClassificationECOC`

model.

**Cross-Validation**

Cross-validate the ECOC classifier using 10-fold cross-validation.

`CVMdl = crossval(Mdl,'Options',options);`

Warning: One or more folds do not contain points from all the groups.

`CVMdl`

is a `ClassificationPartitionedECOC`

model. The warning indicates that some classes are not represented while the software trains at least one fold. Therefore, those folds cannot predict labels for the missing classes. You can inspect the results of a fold using cell indexing and dot notation. For example, access the results of the first fold by entering `CVMdl.Trained{1}`

.

Use the cross-validated ECOC classifier to predict validation-fold labels. You can compute the confusion matrix by using `confusionchart`

. Move and resize the chart by changing the inner position property to ensure that the percentages appear in the row summary.

oofLabel = kfoldPredict(CVMdl,'Options',options); ConfMat = confusionchart(Y,oofLabel,'RowSummary','total-normalized'); ConfMat.InnerPosition = [0.10 0.12 0.85 0.85];

**Reproduce Binned Data**

Reproduce binned predictor data by using the `BinEdges`

property of the trained model and the `discretize`

function.

X = Mdl.X; % Predictor data Xbinned = zeros(size(X)); edges = Mdl.BinEdges; % Find indices of binned predictors. idxNumeric = find(~cellfun(@isempty,edges)); if iscolumn(idxNumeric) idxNumeric = idxNumeric'; end for j = idxNumeric x = X(:,j); % Convert x to array if x is a table. if istable(x) x = table2array(x); end % Group x into bins by using the discretize function. xbinned = discretize(x,[-inf; edges{j}; inf]); Xbinned(:,j) = xbinned; end

`Xbinned`

contains the bin indices, ranging from 1 to the number of bins, for numeric predictors. `Xbinned`

values are `0`

for categorical predictors. If `X`

contains `NaN`

s, then the corresponding `Xbinned`

values are `NaN`

s.

`ClassificationECOC`

| `CompactClassificationECOC`

| `crossval`

| `cvpartiton`

| `fitcecoc`

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