# incrementalClassificationECOC

Multiclass classification model using binary learners for incremental learning

*Since R2022a*

## Description

The `incrementalClassificationECOC`

function creates an
`incrementalClassificationECOC`

model object, which represents a multiclass error-correcting output codes (ECOC)
classification model that uses binary learners for incremental learning.

Unlike other Statistics and Machine Learning Toolbox™ model objects, `incrementalClassificationECOC`

can be called directly. Also,
you can specify learning options, such as performance metrics configurations and prior class
probabilities, before fitting the model to data. After you create an
`incrementalClassificationECOC`

object, it is prepared for incremental learning.

`incrementalClassificationECOC`

is best suited for incremental learning. For a traditional
approach to training a multiclass classification model (such as creating a model by fitting it
to data, performing cross-validation, tuning hyperparameters, and so on), see `fitcecoc`

.

## Creation

You can create an `incrementalClassificationECOC`

model object in several ways:

**Call the function directly**— Configure incremental learning options, or specify learner-specific options, by calling`incrementalClassificationECOC`

directly. This approach is best when you do not have data yet or you want to start incremental learning immediately. You must specify the maximum number of classes or all class names expected in the response data during incremental learning.**Convert a traditionally trained model**— To initialize a multiclass ECOC classification model for incremental learning using the model parameters of a trained model object (`ClassificationECOC`

or`CompactClassificationECOC`

), you can convert the traditionally trained model to an`incrementalClassificationECOC`

model object by passing it to the`incrementalLearner`

function.**Call an incremental learning function**—`fit`

,`updateMetrics`

, and`updateMetricsAndFit`

accept a configured`incrementalClassificationECOC`

model object and data as input, and return an`incrementalClassificationECOC`

model object updated with information learned from the input model and data.

### Syntax

### Description

returns a default incremental learning model object for multiclass ECOC classification,
`Mdl`

= incrementalClassificationECOC(`MaxNumClasses`

=maxNumClasses)`Mdl`

, where `MaxNumClasses`

is the maximum number
of classes expected in the response data during incremental learning. Properties of a
default model contain placeholders for unknown model parameters. You must train a default
model before you can track its performance or generate predictions from it.

specifies all class names `Mdl`

= incrementalClassificationECOC(`ClassNames`

=classNames)`ClassNames`

expected in the response data
during incremental learning, and sets the `ClassNames`

property.

uses either of the previous syntaxes to set properties and additional
options using name-value arguments. For example,
`Mdl`

= incrementalClassificationECOC(___,`Name=Value`

)`incrementalClassificationECOC(MaxNumClasses=5,Coding="onevsone",MetricsWarmupPeriod=100)`

sets the maximum number of classes expected in the response data to `5`

,
specifies to use a one-versus-one coding design, and sets the metrics warm-up period to
`100`

.

### Input Arguments

`MaxNumClasses`

— Maximum number of classes

positive integer

Maximum number of classes expected in the response data during incremental learning, specified as a positive integer.

`MaxNumClasses`

sets the number of class names in the `ClassNames`

property.

If you do not specify `MaxNumClasses`

, you must specify the
`ClassNames`

argument.

**Example: **`MaxNumClasses=5`

**Data Types: **`single`

| `double`

`ClassNames`

— All unique class labels

categorical array | character array | string array | logical vector | numeric vector | cell array of character vectors

All unique class labels expected in the response data during incremental learning,
specified as a categorical, character, or string array; logical or numeric vector; or
cell array of character vectors. `ClassNames`

and the response data
must have the same data type. This argument sets the `ClassNames`

property.

`ClassNames`

specifies the order of any input or output
argument dimension that corresponds to the class order. For example, set
`ClassNames`

to
specify the column
order of classification scores returned by `predict`

.

If you do not specify `ClassNames`

, you must specify the
`MaxNumClasses`

argument. In that case, the software infers the
`ClassNames`

property from the data during incremental
learning.

**Example: **`ClassNames=["virginica","setosa","versicolor"]`

**Data Types: **`single`

| `double`

| `logical`

| `string`

| `char`

| `cell`

| `categorical`

**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.

**Example: **`NumPredictors=4,Prior=[0.3 0.3 0.4]`

specifies the number of
predictor variables as `4`

and sets the prior class probability
distribution to `[0.3 0.3 0.4]`

.

`Coding`

— Coding design

`"onevsone"`

(default) | `"allpairs"`

| `"binarycomplete"`

| `"denserandom"`

| `"onevsall"`

| `"ordinal"`

| `"sparserandom"`

| `"ternarycomplete"`

| numeric matrix

Coding design name, specified as a numeric matrix or a value in this table.

Value | Number of Binary Learners | Description |
---|---|---|

`"allpairs"` and `"onevsone"` | K(K – 1)/2 | For each binary learner, one class is positive, another is negative, and the software ignores the rest. This design exhausts all combinations of class pair assignments. |

`"binarycomplete"` | $${2}^{(K-1)}-1$$ | This design partitions the classes into all binary combinations, and
does not ignore any classes. For each binary learner, all class assignments
are `–1` and `1` with at least one
positive class and one negative class in the assignment. |

`"denserandom"` | Random, but approximately 10
log_{2}K | For each binary learner, the software randomly assigns classes into positive or negative classes, with at least one of each type. For more details, see Random Coding Design Matrices. |

`"onevsall"` | K | For each binary learner, one class is positive and the rest are negative. This design exhausts all combinations of positive class assignments. |

`"ordinal"` | K – 1 | For the first binary learner, the first class is negative and the rest are positive. For the second binary learner, the first two classes are negative and the rest are positive, and so on. |

`"sparserandom"` | Random, but approximately 15
log_{2}K | For each binary learner, the software randomly assigns classes as positive or negative with probability 0.25 for each, and ignores classes with probability 0.5. For more details, see Random Coding Design Matrices. |

`"ternarycomplete"` | $$\left({3}^{K}-{2}^{(K+1)}+1\right)/2$$ | This design partitions the classes into all ternary combinations. All
class assignments are `0` , `–1` , and
`1` with at least one positive class and one negative
class in each assignment. |

You can also specify a coding design using a custom coding matrix, which is a
*K*-by-*L* matrix. Each row corresponds to a
class and each column corresponds to a binary learner. The class order (rows)
corresponds to the order in the `ClassNames`

property. Create the matrix by following these guidelines:

Every element of the custom coding matrix must be

`–1`

,`0`

, or`1`

, and the value must correspond to a dichotomous class assignment. Consider`Coding(i,j)`

, the class that learner`j`

assigns to observations in class`i`

.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.Every column must contain at least one

`–1`

and one`1`

.For all column indices

`i`

,`j`

where`i`

≠`j`

,`Coding(:,i)`

cannot equal`Coding(:,j)`

, and`Coding(:,i)`

cannot equal`–Coding(:,j)`

.All rows of the custom coding matrix must be different.

For more details on the form of custom coding design matrices, see Custom Coding Design Matrices.

**Example: **`Coding="ternarycomplete"`

**Data Types: **`char`

| `string`

| `double`

| `single`

| `int16`

| `int32`

| `int64`

| `int8`

`Metrics`

— Model performance metrics to track during incremental learning

`"classiferror"`

(default) | function handle | cell vector | structure array

Model performance metrics to track during incremental learning, specified as
`"classiferror"`

(classification error, or
misclassification error rate), a function handle (for example,
`@metricName`

), a structure array of function handles, or a cell
vector of names, function handles, or structure arrays.

When `Mdl`

is *warm* (see `IsWarm`

), `updateMetrics`

and `updateMetricsAndFit`

track performance metrics in the `Metrics`

property of
`Mdl`

.

To specify a custom function that returns a performance metric, use function handle notation. The function must have this form.

metric = customMetric(C,S)

The output argument

`metric`

is an*n*-by-1 numeric vector, where each element is the loss of the corresponding observation in the data processed by the incremental learning functions during a learning cycle.You specify the function name (here,

`customMetric`

).`C`

is an*n*-by-*K*logical matrix with rows indicating the class to which the corresponding observation belongs, where*K*is the number of classes. The column order corresponds to the class order in the`ClassNames`

property. Create`C`

by setting`C(`

=,`p`

)`q`

`1`

, if observation

is in class`p`

, for each observation in the specified data. Set the other element in row`q`

to`p`

`0`

.`S`

is an*n*-by-*K*numeric matrix of predicted classification scores.`S`

is similar to the`NegLoss`

output of`predict`

, where rows correspond to observations in the data and the column order corresponds to the class order in the`ClassNames`

property.`S(`

is the classification score of observation,`p`

)`q`

being classified in class`p`

.`q`

To specify multiple custom metrics and assign a custom name to each, use a structure array. To specify a combination of built-in and custom metrics, use a cell vector.

`updateMetrics`

and `updateMetricsAndFit`

store
specified metrics in a table in the `Metrics`

property. The data type of `Metrics`

determines the
row names of the table.

`Metrics` Value Data Type | Description of `Metrics` Property Row Name | Example |
---|---|---|

String or character vector | Name of corresponding built-in metric | Row name for `"classiferror"` is
`"ClassificationError"` |

Structure array | Field name | Row name for `struct(Metric1=@customMetric1)` is
`"Metric1"` |

Function handle to function stored in a program file | Name of function | Row name for `@customMetric` is
`"customMetric"` |

Anonymous function | `CustomMetric_` , where
is metric
in
`Metrics` | Row name for `@(C,S)customMetric(C,S)...` is
`CustomMetric_1` |

For more details on performance metrics options, see Performance Metrics.

**Example: **`Metrics=struct(Metric1=@customMetric1,Metric2=@customMetric2)`

**Example: **`Metrics={@customMetric1,@customMetric2,"classiferror",struct(Metric3=@customMetric3)}`

**Data Types: **`char`

| `string`

| `struct`

| `cell`

| `function_handle`

`Learners`

— Binary learner templates

`"linear"`

(default) | `"kernel"`

| incremental learning object | template object | cell array of incremental learning objects and template objects

Binary learner templates, specified as `"linear"`

,
`"kernel"`

, an incremental learning object, a template object, or
a cell array of supported incremental learning objects and template objects.

`"linear"`

or`"kernel"`

— Specify the`Learners`

value as a string scalar or character vector to use the default linear learners or default kernel learners (default`incrementalClassificationLinear`

or`incrementalClassificationKernel`

objects, respectively).Incremental learning object (

`incrementalClassificationLinear`

or`incrementalClassificationKernel`

object) — Configure binary learner properties (both model-specific properties and incremental learning properties) when you create an incremental learning object, and pass the object to`incrementalClassificationECOC`

as the`Learners`

value.Template object returned by the

`templateLinear`

,`templateSVM`

, or`templateKernel`

function — Configure model-specific properties when you create a template object, and pass the object to`incrementalClassificationECOC`

as the`Learners`

value. Use this approach to specify model properties with a template object and to use the default incremental learning options.Cell array of supported incremental learning objects and template objects — Use this approach to customize each learner individually.

You cannot specify the `ClassNames`

(class names) and `Prior`

(prior class probabilities) properties for an
`incrementalClassificationECOC`

object by using the binary
learners. Instead, specify the properties by using the corresponding name-value
arguments of `incrementalClassificationECOC`

.

**Example: **`Learners="kernel"`

`UpdateBinaryLearnerMetrics`

— Flag for updating metrics of binary learners

`false`

or `0`

(default) | `true`

or `1`

Flag for updating the metrics of binary learners, specified as logical `0`

(`false`

) or `1`

(`true`

).

If the value is `true`

, the software tracks the performance metrics
of binary learners using the `Metrics`

property of the binary learners,
stored in the `BinaryLearners`

property. For an example, see Configure Incremental Model to Track Performance Metrics for Model and Binary Learners.

**Example: **`UpdateBinaryLearnerMetrics=true`

**Data Types: **`logical`

## Properties

You can set most properties by using name-value argument syntax when you call
`incrementalClassificationECOC`

directly. You cannot set the properties
`BinaryLearners`

, `CodingMatrix`

,
`CodingName`

, `NumTrainingObservations`

, and
`IsWarm`

using name-value argument syntax with the arguments of the same
names. However, you can set `CodingMatrix`

and
`CodingName`

by using the `Coding`

name-value
argument, and you can set `BinaryLearners`

by using the
`Learners`

name-value argument.

You can set some properties when you call `incrementalLearner`

to convert a traditionally trained model.

### Classification Model Parameters

`BinaryLearners`

— Trained binary learners

cell array of model objects

This property is read-only.

Trained binary learners, specified as a cell array of `incrementalClassificationLinear`

or `incrementalClassificationKernel`

model objects. The number of binary
learners depends on the coding design.

The software trains `BinaryLearner{j}`

according to the binary
problem specified by `CodingMatrix`

`(:,j)`

.

The default `BinaryLearners`

value depends on how you create
the model:

If you convert a traditionally trained model (for example,

`TTMdl`

) to create`Mdl`

,`BinaryLearners`

contains incremental learners converted from the binary learners in`TTMdl`

.When you train

`TTMdl`

, you must specify the`Learners`

name-value argument of`fitcecoc`

to use support vector machine (SVM) binary learner templates (`templateSVM`

) or linear classification model binary learner templates (`templateLinear`

).Otherwise, the

`Learners`

name-value argument sets this property. The default value of the argument is`"linear"`

, which uses`incrementalClassificationLinear`

model objects with SVM learners.

**Data Types: **`cell`

`BinaryLoss`

— Binary learner loss function

`"hamming"`

| `"linear"`

| `"logit"`

| `"exponential"`

| `"binodeviance"`

| `"hinge"`

| `"quadratic"`

| function handle

This property is read-only.

Binary learner loss function, specified as a built-in loss function name or
function handle. `incrementalClassificationECOC`

stores the
`BinaryLoss`

value as a character vector or function
handle.

This table describes 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.

For an example of a custom binary loss function, see Predict Test-Sample Labels of ECOC Model Using Custom Binary Loss Function. This example is for a traditionally trained model. You can define a custom loss function for incremental learning as shown in the example.

For more information, see Binary Loss.

The default `BinaryLoss`

value depends on how you create the model:

If you convert a traditionally trained model to create

`Mdl`

,`BinaryLoss`

is specified by the corresponding property of the traditionally trained model. You can also specify the`BinaryLoss`

value by using the`BinaryLoss`

name-value argument of`incrementalLearner`

.Otherwise, the default value of

`BinaryLoss`

is`"hinge"`

.

**Data Types: **`char`

| `string`

| `function_handle`

`ClassNames`

— All unique class labels

categorical array | character array | string array | logical vector | numeric vector | cell array of character vectors

This property is read-only.

All unique class labels expected in the response data during incremental learning, specified as a categorical or character array, a logical or numeric vector, or a cell array of character vectors.

You can set `ClassNames`

in one of three ways:

If you specify the

`MaxNumClasses`

argument, the software infers the`ClassNames`

property during incremental learning.If you specify the

`ClassNames`

argument,`incrementalClassificationECOC`

stores your specification in the`ClassNames`

property. (The software treats string arrays as cell arrays of character vectors.)If you convert a traditionally trained model to create

`Mdl`

, the`ClassNames`

property is specified by the corresponding property of the traditionally trained model.

**Data Types: **`single`

| `double`

| `logical`

| `char`

| `string`

| `cell`

| `categorical`

`CodingMatrix`

— Class assignment codes

numeric matrix

This property is read-only.

Class assignment codes for the binary learners, specified as a numeric matrix.
`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`

, and `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. |

For details, see Coding Design.

The default `CodingMatrix`

value depends on how you create the model:

If you convert a traditionally trained model to create

`Mdl`

,`CodingMatrix`

is specified by the corresponding property of the traditionally trained model.Otherwise, the

`Coding`

name-value argument sets this property. The default value of the argument uses the one-versus-one coding design.

**Data Types: **`double`

| `single`

| `int8`

| `int16`

| `int32`

| `int64`

`CodingName`

— Coding design name

character vector

This property is read-only.

Coding design name, specified as a character vector.

The default `CodingName`

value depends on how you create the model:

If you convert a full, traditionally trained model (

`ClassificationECOC`

) to create`Mdl`

,`CodingName`

is specified by the corresponding property of the traditionally trained model.If you convert a compact, traditionally trained model (

`CompactClassificationECOC`

) to create`Mdl`

,`CodingName`

is`"converted"`

.Otherwise, the

`Coding`

name-value argument sets this property. The default value of the argument is`"onevsone"`

. If you specify a custom coding matrix using`Coding`

,`CodingName`

is`"custom"`

.

For details, see Coding Design.

**Data Types: **`char`

`Decoding`

— Decoding scheme

`"lossweighted"`

| `"lossbased"`

This property is read-only.

Decoding scheme, specified as `"lossweighted"`

or
`"lossbased"`

. `incrementalClassificationECOC`

stores the
`Decoding`

value as a character vector.

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:

`"lossweighted"`

— The predicted class of an observation corresponds to the class that produces the minimum sum of the binary losses over binary learners.`"lossbased"`

— The predicted class of an observation corresponds to the class that produces the minimum average of the binary losses over binary learners.

For more information, see Binary Loss.

The default `Decoding`

value depends on how you create the model:

If you convert a traditionally trained model to create

`Mdl`

, the`Decoding`

name-value argument of`incrementalLearner`

sets this property. The default value of the argument is`"lossweighted"`

.Otherwise, the default value of

`Decoding`

is`"lossweighted"`

.

**Data Types: **`char`

| `string`

`NumPredictors`

— Number of predictor variables

nonnegative numeric scalar

This property is read-only.

Number of predictor variables, specified as a nonnegative numeric scalar.

The default `NumPredictors`

value depends on how you create the model:

If you convert a traditionally trained model to create

`Mdl`

,`NumPredictors`

is specified by the corresponding property of the traditionally trained model.If you create

`Mdl`

by calling`incrementalClassificationECOC`

directly, you can specify`NumPredictors`

by using name-value argument syntax. If you do not specify the value, then the default value is`0`

, and incremental fitting functions infer`NumPredictors`

from the predictor data during training.

**Data Types: **`double`

`NumTrainingObservations`

— Number of observations fit to incremental model

`0`

(default) | nonnegative numeric scalar

This property is read-only.

Number of observations fit to the incremental model `Mdl`

, specified as a nonnegative numeric scalar. `NumTrainingObservations`

increases when you pass `Mdl`

and training data to `fit`

or `updateMetricsAndFit`

.

**Note**

If you convert a traditionally trained model to create `Mdl`

, `incrementalClassificationECOC`

does not add the number of observations fit to the traditionally trained model to `NumTrainingObservations`

.

**Data Types: **`double`

`Prior`

— Prior class probabilities

numeric vector | `"empirical"`

| `"uniform"`

This property is read-only.

Prior class probabilities, specified as `"empirical"`

,
`"uniform"`

, or a numeric vector. `incrementalClassificationECOC`

stores the `Prior`

value as a numeric vector.

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

`"empirical"` | Incremental learning functions infer prior class probabilities from the observed class relative frequencies in the response data during incremental training. |

`"uniform"` | For each class, the prior probability is 1/K, where
K is the number of classes. |

numeric vector | Custom, normalized prior probabilities. The order of the elements of
`Prior` corresponds to the elements of the
`ClassNames` property. |

The default `Prior`

value depends on how you create the model:

If you convert a traditionally trained model to create

`Mdl`

,`Prior`

is specified by the corresponding property of the traditionally trained model.Otherwise, the default value is

`"empirical"`

.

**Data Types: **`single`

| `double`

| `char`

| `string`

`ScoreTransform`

— Score transformation function to apply to predicted scores

`'none'`

This property is read-only.

Score transformation function to apply to the predicted scores, specified as
`'none'`

. An ECOC model does not support score transformation.

### Performance Metrics Parameters

`IsWarm`

— Flag indicating whether model tracks performance metrics

`false`

or `0`

| `true`

or `1`

Flag indicating whether the incremental model tracks performance metrics, specified as logical
`0`

(`false`

) or `1`

(`true`

).

The incremental model `Mdl`

is *warm*
(`IsWarm`

becomes `true`

) when incremental fitting
functions perform both of these actions:

Fit the incremental model to

`MetricsWarmupPeriod`

observations.Process

`MaxNumClasses`

classes or all class names specified by the`ClassNames`

name-value argument.

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

`true` or `1` | The incremental model `Mdl` is warm. Consequently, `updateMetrics` and `updateMetricsAndFit` track performance metrics in the `Metrics` property of `Mdl` . |

`false` or `0` | `updateMetrics` and `updateMetricsAndFit` do not track performance metrics. |

**Data Types: **`logical`

`Metrics`

— Model performance metrics

table

This property is read-only.

Model performance metrics updated during incremental learning by
`updateMetrics`

and `updateMetricsAndFit`

,
specified as a table with two columns and *m* rows, where
*m* is the number of metrics specified by the `Metrics`

name-value
argument.

The columns of `Metrics`

are labeled `Cumulative`

and `Window`

.

: Element`Cumulative`

is the model performance, as measured by metric`j`

, from the time the model became warm (`j`

`IsWarm`

is`1`

).: Element`Window`

is the model performance, as measured by metric`j`

, evaluated over all observations within the window specified by the`j`

`MetricsWindowSize`

property. The software updates`Window`

after it processes`MetricsWindowSize`

observations.

Rows are labeled by the specified metrics. For details, see the
`Metrics`

name-value argument of
`incrementalLearner`

or `incrementalClassificationECOC`

.

**Data Types: **`table`

`MetricsWarmupPeriod`

— Number of observations fit before tracking performance metrics

nonnegative integer

This property is read-only.

Number of observations the incremental model must be fit to before it tracks performance metrics in its `Metrics`

property, specified as a nonnegative integer.

The default `MetricsWarmupPeriod`

value depends on how you create
the model:

If you convert a traditionally trained model to create

`Mdl`

, the`MetricsWarmupPeriod`

name-value argument of the`incrementalLearner`

function sets this property. The default value of the argument is`0`

.Otherwise, the default value is

`1000`

.

For more details, see Performance Metrics.

**Data Types: **`single`

| `double`

`MetricsWindowSize`

— Number of observations to use to compute window performance metrics

positive integer

This property is read-only.

Number of observations to use to compute window performance metrics, specified as a positive integer.

The default `MetricsWindowSize`

value depends on how you create the model:

If you convert a traditionally trained model to create

`Mdl`

, the`MetricsWindowSize`

name-value argument of the`incrementalLearner`

function sets this property. The default value of the argument is`200`

.Otherwise, the default value is

`200`

.

For more details on performance metrics options, see Performance Metrics.

**Data Types: **`single`

| `double`

## Object Functions

`fit` | Train ECOC classification model for incremental learning |

`updateMetricsAndFit` | Update performance metrics in ECOC incremental learning classification model given new data and train model |

`updateMetrics` | Update performance metrics in ECOC incremental learning classification model given new data |

`loss` | Loss of ECOC incremental learning classification model on batch of data |

`predict` | Predict responses for new observations from ECOC incremental learning classification model |

`perObservationLoss` | Per observation classification error of model for incremental learning |

`reset` | Reset incremental classification model |

## Examples

### Create Incremental Learner with Little Prior Information

To create an ECOC classification model for incremental learning, you must specify the maximum number of classes that you expect the model to process (`MaxNumClasses`

name-value argument). As you fit the model to incoming batches of data by using an incremental fitting function, the model collects new classes in its `ClassNames`

property. If the specified maximum number of classes is inaccurate, one of the following occurs:

Before an incremental fitting function processes the expected maximum number of classes, the model is cold. Consequently, the

`updateMetrics`

and`updateMetricsAndFit`

functions do not measure performance metrics.If the number of classes exceeds the maximum expected, the incremental fitting function issues an error.

This example shows how to create an ECOC model for incremental learning when the only information you specify is the expected maximum number of classes in the data. Also, the example illustrates the consequences when incremental fitting functions process all expected classes early and late in the sample.

For this example, consider training a device to predict whether a subject is sitting, standing, walking, running, or dancing based on biometric data measured on the subject. Therefore, the device has a maximum of 5 classes from which to choose.

**Process Expected Maximum Number of Classes Early in Sample**

Load the human activity data set. Randomly shuffle the data.

load humanactivity n = numel(actid); rng(1) % For reproducibility idx = randsample(n,n); X = feat(idx,:); Y = actid(idx);

For details on the data set, enter `Description`

at the command line.

Create an incremental ECOC model for multiclass learning. Specify a maximum of 5 classes in the data.

MdlEarly = incrementalClassificationECOC(MaxNumClasses=5)

MdlEarly = incrementalClassificationECOC IsWarm: 0 Metrics: [1x2 table] ClassNames: [1x0 double] ScoreTransform: 'none' BinaryLearners: {10x1 cell} CodingName: 'onevsone' Decoding: 'lossweighted' Properties, Methods

`MdlEarly`

is an `incrementalClassificationECOC`

model object. All its properties are read-only. `MdlEarly`

must be fit to data before you can use it to perform any other operations.

Display the coding design matrix.

MdlEarly.CodingMatrix

`ans = `*5×10*
1 1 1 1 0 0 0 0 0 0
-1 0 0 0 1 1 1 0 0 0
0 -1 0 0 -1 0 0 1 1 0
0 0 -1 0 0 -1 0 -1 0 1
0 0 0 -1 0 0 -1 0 -1 -1

Each row of the coding design matrix corresponds to a class, and each column corresponds to a binary learner. For example, the first binary learner is for classes 1 and 2, and the fourth binary learner is for classes 1 and 5, where both learners assume class 1 as a positive class.

Fit the incremental model to the training data by using the `updateMetricsAndFit`

function. Simulate a data stream by processing chunks of 50 observations at a time. At each iteration:

Process 50 observations.

Overwrite the previous incremental model with a new one fitted to the incoming observations.

Store the first model coefficient of the first binary learner $${\beta}_{11}$$, the cumulative metrics, and the window metrics to see how they evolve during incremental learning.

% Preallocation numObsPerChunk = 50; nchunk = floor(n/numObsPerChunk); mc = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]); beta11 = zeros(nchunk,1); % Incremental learning for j = 1:nchunk ibegin = min(n,numObsPerChunk*(j-1) + 1); iend = min(n,numObsPerChunk*j); idx = ibegin:iend; MdlEarly = updateMetricsAndFit(MdlEarly,X(idx,:),Y(idx)); mc{j,:} = MdlEarly.Metrics{"ClassificationError",:}; beta11(j) = MdlEarly.BinaryLearners{1}.Beta(1); end

`MdlEarly`

is an `incrementalClassificationECOC`

model object trained on all the data in the stream. During incremental learning and after the model is warmed up, `updateMetricsAndFit`

checks the performance of the model on the incoming observations, and then fits the model to those observations.

To see how the performance metrics and $${\beta}_{11}$$ evolve during training, plot them on separate tiles.

t = tiledlayout(2,1); nexttile plot(beta11) ylabel("\beta_{11}") xlim([0 nchunk]) nexttile plot(mc.Variables) xlim([0 nchunk]) ylabel("Classification Error") xline(MdlEarly.MetricsWarmupPeriod/numObsPerChunk,"--") legend(mc.Properties.VariableNames) xlabel(t,"Iteration")

The plots indicate that `updateMetricsAndFit`

performs the following actions:

Fit $${\beta}_{11}$$ during all incremental learning iterations.

Compute the performance metrics after the metrics warm-up period (dashed vertical line) only.

Compute the cumulative metrics during each iteration.

Compute the window metrics after processing 200 observations (4 iterations).

**Process Expected Maximum Number of Classes Late in Sample**

Rearrange the data set so that only the last 5000 samples contain the observations labeled with class 5.

Move all observations labeled with class 5 to the end of the sample.

idx5 = Y == 5; Xnew = [X(~idx5,:); X(idx5,:)]; Ynew = [Y(~idx5); Y(idx5)]; sum(idx5)

ans = 2653

Shuffle the last 5000 samples.

m = 5000; idx_shuffle = randsample(m,m); Xnew(end-m+1:end,:) = Xnew(end-m+idx_shuffle,:); Ynew(end-m+1:end) = Ynew(end-m+idx_shuffle);

An ECOC model trains a binary learner only when an incoming chunk contains observations for the classes that the binary learner treats as either positive or negative. Therefore, when the labels in incoming data are not well distributed for all expected classes, a good practice is to choose a coding design that does not have zeros in the coding matrix so that the software trains all binary learners for every chunk.

Create a new ECOC model for incremental learning. Specify the `onevsall`

coding design. In this design, one class is positive and the rest are negative for each binary learner.

`MdlLate = incrementalClassificationECOC(MaxNumClasses=5,Coding="onevsall")`

MdlLate = incrementalClassificationECOC IsWarm: 0 Metrics: [1x2 table] ClassNames: [1x0 double] ScoreTransform: 'none' BinaryLearners: {5x1 cell} CodingName: 'onevsall' Decoding: 'lossweighted' Properties, Methods

Display the coding design matrix.

MdlLate.CodingMatrix

`ans = `*5×5*
1 -1 -1 -1 -1
-1 1 -1 -1 -1
-1 -1 1 -1 -1
-1 -1 -1 1 -1
-1 -1 -1 -1 1

Fit the incremental model and plot the results. Store the first model coefficients of the first and fifth binary learners, $${\beta}_{11}$$ and $${\beta}_{51}$$.

mcnew = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]); beta11new = zeros(nchunk,1); beta51new = zeros(nchunk,1); for j = 1:nchunk ibegin = min(n,numObsPerChunk*(j-1) + 1); iend = min(n,numObsPerChunk*j); idx = ibegin:iend; MdlLate = updateMetricsAndFit(MdlLate,Xnew(idx,:),Ynew(idx)); mcnew{j,:} = MdlLate.Metrics{"ClassificationError",:}; beta11new(j) = MdlLate.BinaryLearners{1}.Beta(1); beta51new(j) = MdlLate.BinaryLearners{5}.Beta(1); end t = tiledlayout(3,1); nexttile plot(beta11new) xline(MdlLate.MetricsWarmupPeriod/numObsPerChunk,"--") xline((n-m)/numObsPerChunk,":") ylabel("\beta_{11}") xlim([0 nchunk]) nexttile plot(beta51new) xline(MdlLate.MetricsWarmupPeriod/numObsPerChunk,"--") xline((n-m)/numObsPerChunk,":") ylabel("\beta_{51}") xlim([0 nchunk]) nexttile plot(mcnew.Variables) xline(MdlLate.MetricsWarmupPeriod/numObsPerChunk,"--") xline((n-m)/numObsPerChunk,":") xlim([0 nchunk]) ylabel("Classification Error") legend(mcnew.Properties.VariableNames,Location="best") xlabel(t,"Iteration")

The `updateMetricsAndFit`

function trains the model throughout incremental learning. However, $${\beta}_{51}$$ does not change significantly until an incoming chunk contains observations with the fifth class (the dotted vertical line). Also, the function starts tracking performance metrics only after the model is fit to the expected number of classes.

### Specify All Class Names

Create an incremental ECOC model when you know all the class names in the data.

Consider training a device to predict whether a subject is sitting, standing, walking, running, or dancing based on biometric data measured on the subject. The class names map 1 through 5 to an activity.

Create an incremental ECOC model for multiclass learning. Specify the class names.

classnames = 1:5; Mdl = incrementalClassificationECOC(ClassNames=classnames)

Mdl = incrementalClassificationECOC IsWarm: 0 Metrics: [1x2 table] ClassNames: [1 2 3 4 5] ScoreTransform: 'none' BinaryLearners: {10x1 cell} CodingName: 'onevsone' Decoding: 'lossweighted' Properties, Methods

`Mdl`

is an `incrementalClassificationECOC`

model object. All its properties are read-only.

`Mdl`

must be fit to data before you can use it to perform any other operations.

Load the human activity data set. Randomly shuffle the data.

load humanactivity n = numel(actid); rng(1) % For reproducibility idx = randsample(n,n); X = feat(idx,:); Y = actid(idx);

For details on the data set, enter `Description`

at the command line.

Fit the incremental model to the training data by using the `updateMetricsAndFit`

function. Simulate a data stream by processing chunks of 50 observations at a time. At each iteration:

Process 50 observations.

Overwrite the previous incremental model with a new one fitted to the incoming observations.

% Preallocation numObsPerChunk = 50; nchunk = floor(n/numObsPerChunk); % Incremental learning for j = 1:nchunk ibegin = min(n,numObsPerChunk*(j-1) + 1); iend = min(n,numObsPerChunk*j); idx = ibegin:iend; Mdl = updateMetricsAndFit(Mdl,X(idx,:),Y(idx)); end

### Configure Incremental Learning Options

In addition to specifying the maximum number of classes, prepare an incremental ECOC learner by specifying a metrics warm-up period and a metrics window size.

Load the human activity data set. Randomly shuffle the data. Orient the observations of the predictor data in columns.

load humanactivity n = numel(actid); rng(1) % For reproducibility idx = randsample(n,n); X = feat(idx,:)'; Y = actid(idx);

For details on the data set, enter `Description`

at the command line.

Create an incremental ECOC model for multiclass learning. Configure the model as follows:

Set the maximum number of classes to 5.

Specify a metrics warm-up period of 5000 observations.

Specify a metrics window size of 500 observations.

```
Mdl = incrementalClassificationECOC(MaxNumClasses=5, ...
MetricsWarmupPeriod=5000,MetricsWindowSize=500)
```

Mdl = incrementalClassificationECOC IsWarm: 0 Metrics: [1x2 table] ClassNames: [1x0 double] ScoreTransform: 'none' BinaryLearners: {10x1 cell} CodingName: 'onevsone' Decoding: 'lossweighted' Properties, Methods

`Mdl`

is an `incrementalClassificationECOC`

model object configured for incremental learning. By default, `incrementalClassificationECOC`

uses classification error loss to measure the performance of the model.

Fit the incremental model to the rest of the data by using the `updateMetricsAndFit`

function. At each iteration:

Simulate a data stream by processing a chunk of 50 observations.

Overwrite the previous incremental model with a new one fitted to the incoming observations. Specify that the observations are oriented in columns

Store the first model coefficient of the first binary learner $${\beta}_{11}$$, the cumulative metrics, and the window metrics to see how they evolve during incremental learning.

% Preallocation numObsPerChunk = 50; nchunk = floor(n/numObsPerChunk); ce = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]); beta11 = zeros(nchunk,1); % Incremental fitting for j = 1:nchunk ibegin = min(n,numObsPerChunk*(j-1) + 1); iend = min(n,numObsPerChunk*j); idx = ibegin:iend; Mdl = updateMetricsAndFit(Mdl,X(:,idx),Y(idx),ObservationsIn="columns"); ce{j,:} = Mdl.Metrics{"ClassificationError",:}; beta11(j) = Mdl.BinaryLearners{1}.Beta(1); end

`Mdl`

is an `incrementalClassificationECOC`

model object trained on all the data in the stream. During incremental learning and after the model is warmed up, `updateMetricsAndFit`

checks the performance of the model on the incoming observations, and then fits the model to those observations.

To see how the performance metrics and $${\beta}_{11}$$ evolve during training, plot them on separate tiles.

t = tiledlayout(2,1); nexttile plot(beta11) ylabel("\beta_{11}") xlim([0 nchunk]) xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"--") nexttile plot(ce.Variables) xlim([0 nchunk]) ylabel("Classification Error") xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"--") legend(ce.Properties.VariableNames) xlabel(t,"Iteration")

The plots indicate that `updateMetricsAndFit`

performs the following actions:

Fit $${\beta}_{11}$$ during all incremental learning iterations.

Compute the performance metrics after the metrics warm-up period (dashed vertical line) only.

Compute the cumulative metrics during each iteration.

Compute the window metrics after processing 500 observations (10 iterations).

### Convert Traditionally Trained Model to Incremental Learner

Train an ECOC model for multiclass classification by using `fitcecoc`

. Then, convert the model to an incremental learner, track its performance, and fit the model to streaming data. Carry over training options from traditional to incremental learning.

**Load and Preprocess Data**

Load the human activity data set. Randomly shuffle the data.

load humanactivity rng(1) % For reproducibility n = numel(actid); idx = randsample(n,n); X = feat(idx,:); Y = actid(idx);

For details on the data set, enter `Description`

at the command line.

Suppose that the data collected when the subject was stationary (`Y`

<= 2) has double the quality than when the subject was moving. Create a weight variable that attributes 2 to observations collected from a stationary subject, and 1 to a moving subject.

W = ones(n,1) + (Y <= 2);

**Train ECOC Model**

Fit an ECOC model for multiclass classification to a random sample of half the data.

idxtt = randsample([true false],n,true); TTMdl = fitcecoc(X(idxtt,:),Y(idxtt),Weights=W(idxtt))

TTMdl = ClassificationECOC ResponseName: 'Y' CategoricalPredictors: [] ClassNames: [1 2 3 4 5] ScoreTransform: 'none' BinaryLearners: {10×1 cell} CodingName: 'onevsone' Properties, Methods

`TTMdl`

is a `ClassificationECOC`

model object representing a traditionally trained ECOC model.

**Convert Trained Model**

Convert the traditionally trained ECOC model to a model for incremental learning.

IncrementalMdl = incrementalLearner(TTMdl)

IncrementalMdl = incrementalClassificationECOC IsWarm: 1 Metrics: [1×2 table] ClassNames: [1 2 3 4 5] ScoreTransform: 'none' BinaryLearners: {10×1 cell} CodingName: 'onevsone' Decoding: 'lossweighted' Properties, Methods

`IncrementalMdl`

is an `incrementalClassificationECOC`

model object configured for incremental learning.

**Separately Track Performance Metrics and Fit Model**

Perform incremental learning on the rest of the data by using the `updateMetrics`

and `fit`

functions. Simulate a data stream by processing 50 observations at a time. At each iteration:

Call

`updateMetrics`

to update the cumulative and window classification error of the model given the incoming chunk of observations. Overwrite the previous incremental model to update the`Metrics`

property. Note that the function does not fit the model to the chunk of data—the chunk is "new" data for the model. Specify the observation weights.Call

`fit`

to fit the model to the incoming chunk of observations. Overwrite the previous incremental model to update the model parameters. Specify the observation weights.Store the classification error and first model coefficient of the first binary learner $${\beta}_{11}$$.

% Preallocation idxil = ~idxtt; nil = sum(idxil); numObsPerChunk = 50; nchunk = floor(nil/numObsPerChunk); ec = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]); beta11 = [IncrementalMdl.BinaryLearners{1}.Beta(1); zeros(nchunk,1)]; Xil = X(idxil,:); Yil = Y(idxil); Wil = W(idxil); % Incremental fitting for j = 1:nchunk ibegin = min(nil,numObsPerChunk*(j-1) + 1); iend = min(nil,numObsPerChunk*j); idx = ibegin:iend; IncrementalMdl = updateMetrics(IncrementalMdl,Xil(idx,:),Yil(idx), ... Weights=Wil(idx)); ec{j,:} = IncrementalMdl.Metrics{"ClassificationError",:}; IncrementalMdl = fit(IncrementalMdl,Xil(idx,:),Yil(idx),Weights=Wil(idx)); beta11(j+1) = IncrementalMdl.BinaryLearners{1}.Beta(1); end

`IncrementalMdl`

is an `incrementalClassificationECOC`

model object trained on all the data in the stream.

Alternatively, you can use `updateMetricsAndFit`

to update the performance metrics of the model given a new chunk of data, and then fit the model to the data.

Plot a trace plot of the performance metrics and estimated coefficient $${\beta}_{11}$$ on separate tiles.

t = tiledlayout(2,1); nexttile plot(ec.Variables) xlim([0 nchunk]) ylabel("Classification Error") legend(ec.Properties.VariableNames) nexttile plot(beta11) ylabel("\beta_{11}") xlim([0 nchunk]) xlabel(t,"Iteration")

The cumulative loss levels quickly and is stable, whereas the window loss jumps throughout the training.

$${\beta}_{11}$$ changes abruptly at first, then gradually levels off as `fit`

processes more chunks.

### Specify Binary Learners

Customize binary learners of an `incrementalClassificationECOC`

model object by specifying the `Learners`

name-value argument.

First, configure binary learner properties by creating an `incrementalClassificationLinear`

object. Set the linear classification model type (`Learner`

) to logistic regression, and specify `Standardize`

as `true`

to standardize the predictor data.

binaryMdl = incrementalClassificationLinear(Learner="logistic", ... Standardize=true)

binaryMdl = incrementalClassificationLinear IsWarm: 0 Metrics: [1x2 table] ClassNames: [1x0 double] ScoreTransform: 'logit' Beta: [0x1 double] Bias: 0 Learner: 'logistic' Properties, Methods

Create an incremental ECOC model for multiclass learning. Specify the number of classes in the data as five, and set the binary learner template (`Learners`

) to `binaryMdl`

.

Mdl = incrementalClassificationECOC(MaxNumClasses=5,Learners=binaryMdl)

Warning: 'ScoreTransform' property for binary learner 1 is discarded because the binary learner is used in an ECOC model.

Mdl = incrementalClassificationECOC IsWarm: 0 Metrics: [1x2 table] ClassNames: [1x0 double] ScoreTransform: 'none' BinaryLearners: {10x1 cell} CodingName: 'onevsone' Decoding: 'lossweighted' Properties, Methods

The software displays a warning message that `incrementalClassificationECOC`

does not use the `ScoreTransform`

property in the binary learner. Because the software does not support score transformation for an ECOC model, `incrementalClassificationECOC`

does not use the property.

Display the `BinaryLearners`

property in `Mdl`

.

Mdl.BinaryLearners

`ans=`*10×1 cell array*
{1x1 incrementalClassificationLinear}
{1x1 incrementalClassificationLinear}
{1x1 incrementalClassificationLinear}
{1x1 incrementalClassificationLinear}
{1x1 incrementalClassificationLinear}
{1x1 incrementalClassificationLinear}
{1x1 incrementalClassificationLinear}
{1x1 incrementalClassificationLinear}
{1x1 incrementalClassificationLinear}
{1x1 incrementalClassificationLinear}

By default, `incrementalClassificationECOC`

uses the one-versus-one coding design, which requires 10 learners for five classes. Therefore, the `BinaryLearners`

property contains 10 binary learners of type `incrementalClassificationLinear`

.

## More About

### Incremental Learning

*Incremental learning*, or *online learning*, is a branch of machine learning concerned with processing incoming data from a data stream, possibly given little to no knowledge of the distribution of the predictor variables, aspects of the prediction or objective function (including tuning parameter values), or whether the observations are labeled. Incremental learning differs from traditional machine learning, where enough labeled data is available to fit to a model, perform cross-validation to tune hyperparameters, and infer the predictor distribution.

Given incoming observations, an incremental learning model processes data in any of the following ways, but usually in this order:

Predict labels.

Measure the predictive performance.

Check for structural breaks or drift in the model.

Fit the model to the incoming observations.

For more details, see Incremental Learning Overview.

### Adaptive Scale-Invariant Solver for Incremental Learning

The *adaptive scale-invariant solver for incremental
learning*, introduced in [5], is a gradient-descent-based objective solver for
training linear predictive models. The solver is hyperparameter free, insensitive to
differences in predictor variable scales, and does not require prior knowledge of the
distribution of the predictor variables. These characteristics make it well suited to
incremental learning.

The incremental fitting functions `fit`

and `updateMetricsAndFit`

use the more aggressive ScInOL2 version of the algorithm
to train binary learners. The functions always shuffles an incoming batch of data before
fitting the model.

### Error-Correcting Output Codes Model

An *error-correcting output codes (ECOC) model* reduces
the problem of classification with three or more classes to a set of binary classification
problems.

ECOC classification requires a coding design, which determines the classes that the binary learners train on, and a decoding scheme, which determines how the results (predictions) of the binary classifiers are aggregated.

Assume the following:

The classification problem has three classes.

The coding design is one-versus-one. For three classes, this coding design is

$$\begin{array}{cccc}& \text{Learner1}& \text{Learner2}& \text{Learner3}\\ \text{Class1}& 1& 1& 0\\ \text{Class2}& -1& 0& 1\\ \text{Class3}& 0& -1& -1\end{array}$$

You can specify a different coding design by using the

`Coding`

name-value argument when you create a classification model.The model determines the predicted class by using the loss-weighted decoding scheme with the binary loss function

*g*. The software also supports the loss-based decoding scheme. You can specify the decoding scheme and binary loss function by using the`Decoding`

and`BinaryLoss`

name-value arguments, respectively, when you create a classification model or when you call the object functions`predict`

and`loss`

.

To build this classification model, the ECOC algorithm follows these steps.

Learner 1 trains on observations in Class 1 or Class 2, and treats Class 1 as the positive class and Class 2 as the negative class. The other learners are trained similarly.

Let

*M*be the coding design matrix with elements*m*, and_{kl}*s*be the predicted classification score for the positive class of learner_{l}*l*. The algorithm assigns a new observation to the class ($$\widehat{k}$$) that minimizes the aggregation of the losses for the*L*binary learners.$$\widehat{k}=\underset{k}{\text{argmin}}\frac{{\displaystyle \sum}_{l=1}^{B}\left|{m}_{kl}\right|g\left({m}_{kl},{s}_{l}\right)}{{\displaystyle \sum}_{l=1}^{B}\left|{m}_{kl}\right|}.$$

ECOC models can improve classification accuracy, compared to other multiclass models [4].

### Coding Design

The *coding design* is a matrix whose elements direct
which classes are trained by each binary learner, that is, how the multiclass problem is
reduced to a series of binary problems.

Each row of the coding design corresponds to a distinct class, and each column corresponds to a binary learner. In a ternary coding design, for a particular column (or binary learner):

A row containing 1 directs the binary learner to group all observations in the corresponding class into a positive class.

A row containing –1 directs the binary learner to group all observations in the corresponding class into a negative class.

A row containing 0 directs the binary learner to ignore all observations in the corresponding class.

Coding design matrices with large, minimal, pairwise row distances based on the Hamming measure are optimal. For details on the pairwise row distance, see Random Coding Design Matrices and [3].

This table describes popular coding designs.

Coding Design | Description | Number of Learners | Minimal Pairwise Row Distance |
---|---|---|---|

one-versus-all (OVA) | For each binary learner, one class is positive and the rest are negative. This design exhausts all combinations of positive class assignments. | K | 2 |

one-versus-one (OVO) | For each binary learner, one class is positive, one class is negative, and the rest are ignored. This design exhausts all combinations of class pair assignments. |
| 1 |

binary complete | This design partitions the classes into all binary
combinations, and does not ignore any classes. That is, all class
assignments are | 2^{K – 1} – 1 | 2^{K – 2} |

ternary complete | This design partitions the classes into all ternary
combinations. That is, all class assignments are
| (3 | 3^{K – 2} |

ordinal | For the first binary learner, the first class is negative and the rest are positive. For the second binary learner, the first two classes are negative and the rest are positive, and so on. | K – 1 | 1 |

dense random | For each binary learner, the software randomly assigns classes into positive or negative classes, with at least one of each type. For more details, see Random Coding Design Matrices. | Random, but approximately 10
log | Variable |

sparse random | For each binary learner, the software randomly assigns classes as positive or negative with probability 0.25 for each, and ignores classes with probability 0.5. For more details, see Random Coding Design Matrices. | Random, but approximately 15
log | Variable |

This plot compares the number of binary learners for the coding designs with
an increasing number of classes (*K*).

### 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*[3] (`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*[2] (`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 `loss`

and
`predict`

object functions), which measures how well an ECOC classifier
performs as a whole.

### 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.

## Algorithms

### Performance Metrics

The

`updateMetrics`

and`updateMetricsAndFit`

functions track model performance metrics (`Metrics`

) from new data only when the incremental model is*warm*(`IsWarm`

property is`true`

).If you create an incremental model by using

`incrementalLearner`

and`MetricsWarmupPeriod`

is 0 (default for`incrementalLearner`

), the model is warm at creation.Otherwise, an incremental model becomes warm after

`fit`

or`updateMetricsAndFit`

performs both of these actions:Fit the incremental model to

`MetricsWarmupPeriod`

observations, which is the*metrics warm-up period*.Fit the incremental model to all expected classes (see the

`MaxNumClasses`

and`ClassNames`

arguments of`incrementalClassificationECOC`

).

The

`Metrics`

property of the incremental model stores two forms of each performance metric as variables (columns) of a table,`Cumulative`

and`Window`

, with individual metrics in rows. When the incremental model is warm,`updateMetrics`

and`updateMetricsAndFit`

update the metrics at the following frequencies:`Cumulative`

— The functions compute cumulative metrics since the start of model performance tracking. The functions update metrics every time you call the functions and base the calculation on the entire supplied data set.`Window`

— The functions compute metrics based on all observations within a window determined by`MetricsWindowSize`

, which also determines the frequency at which the software updates`Window`

metrics. For example, if`MetricsWindowSize`

is 20, the functions compute metrics based on the last 20 observations in the supplied data (`X((end – 20 + 1):end,:)`

and`Y((end – 20 + 1):end)`

).Incremental functions that track performance metrics within a window use the following process:

Store a buffer of length

`MetricsWindowSize`

for each specified metric, and store a buffer of observation weights.Populate elements of the metrics buffer with the model performance based on batches of incoming observations, and store corresponding observation weights in the weights buffer.

When the buffer is full, overwrite the

`Window`

field of the`Metrics`

property with the weighted average performance in the metrics window. If the buffer overfills when the function processes a batch of observations, the latest incoming`MetricsWindowSize`

observations enter the buffer, and the earliest observations are removed from the buffer. For example, suppose`MetricsWindowSize`

is 20, the metrics buffer has 10 values from a previously processed batch, and 15 values are incoming. To compose the length 20 window, the functions use the measurements from the 15 incoming observations and the latest 5 measurements from the previous batch.

The software omits an observation with a

`NaN`

score when computing the`Cumulative`

and`Window`

performance metric values.

### Custom Coding Design Matrices

Custom coding matrices must have a certain form. The software validates a custom coding matrix by ensuring:

Every element is –1, 0, or 1.

Every column contains as least one –1 and one 1.

For all distinct column vectors

*u*and*v*,*u*≠*v*and*u*≠ –*v*.All row vectors are unique.

The matrix can separate any two classes. That is, you can move from any row to any other row following these rules:

Move vertically from 1 to –1 or –1 to 1.

Move horizontally from a nonzero element to another nonzero element.

Use a column of the matrix for a vertical move only once.

If it is not possible to move from row

*i*to row*j*using these rules, then classes*i*and*j*cannot be separated by the design. For example, in the coding design$$\left[\begin{array}{cc}1& 0\\ -1& 0\\ 0& 1\\ 0& -1\end{array}\right]$$

classes 1 and 2 cannot be separated from classes 3 and 4 (that is, you cannot move horizontally from –1 in row 2 to column 2 because that position contains a 0). Therefore, the software rejects this coding design.

### Random Coding Design Matrices

For a given number of classes *K*, the software generates random coding
design matrices as follows.

The software generates one of these matrices:

Dense random — The software assigns 1 or –1 with equal probability to each element of the

*K*-by-*L*coding design matrix, where $${L}_{d}\approx \lceil 10{\mathrm{log}}_{2}K\rceil $$._{d}Sparse random — The software assigns 1 to each element of the

*K*-by-*L*coding design matrix with probability 0.25, –1 with probability 0.25, and 0 with probability 0.5, where $${L}_{s}\approx \lceil 15{\mathrm{log}}_{2}K\rceil $$._{s}

If a column does not contain at least one 1 and one –1, then the software removes that column.

For distinct columns

*u*and*v*, if*u*=*v*or*u*= –*v*, then the software removes*v*from the coding design matrix.

The software randomly generates 10,000 matrices by default, and retains the matrix with the largest, minimal, pairwise row distance based on the Hamming measure ([3]) given by

$$\Delta ({k}_{1},{k}_{2})=0.5{\displaystyle \sum}_{l=1}^{L}\left|{m}_{{k}_{1}l}\right|\left|{m}_{{k}_{2}l}\right|\left|{m}_{{k}_{1}l}-{m}_{{k}_{2}l}\right|,$$

where
*m _{kjl}* is an element of
coding design matrix

*j*.

## 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. “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.

[3] 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.

[4] Fürnkranz, Johannes. “Round Robin
Classification.” *J. Mach. Learn. Res.*, Vol. 2, 2002, pp.
721–747.

[5] Kempka, Michał, Wojciech Kotłowski, and Manfred K. Warmuth. "Adaptive Scale-Invariant Online Algorithms for Learning Linear Models." Preprint, submitted February 10, 2019. https://arxiv.org/abs/1902.07528.

## Version History

**Introduced in R2022a**

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