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Classification edge for multiclass error-correcting output codes (ECOC) model

`e = edge(Mdl,tbl,ResponseVarName)`

`e = edge(Mdl,tbl,Y)`

`e = edge(Mdl,X,Y)`

`e = edge(___,Name,Value)`

returns the classification edge
(`e`

= edge(`Mdl`

,`tbl`

,`ResponseVarName`

)`e`

) for the trained multiclass error-correcting output codes (ECOC)
classifier `Mdl`

using the predictor data in table
`tbl`

and the class labels in
`tbl.ResponseVarName`

.

specifies options using one or more name-value pair arguments in addition to any of the
input argument combinations in previous syntaxes. For example, you can specify a decoding
scheme, binary learner loss function, and verbosity level.`e`

= edge(___,`Name,Value`

)

Compute the test-sample classification edge of an ECOC model with SVM binary classifiers.

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

, the response data `Y`

, and the order of the classes in `Y`

.

load fisheriris X = meas; Y = categorical(species); classOrder = unique(Y); % Class order rng(1); % For reproducibility

Train an ECOC model using SVM binary classifiers. Specify a 30% holdout sample for testing, standardize the predictors using an SVM template, and specify the class order.

t = templateSVM('Standardize',true); PMdl = fitcecoc(X,Y,'Holdout',0.30,'Learners',t,'ClassNames',classOrder); Mdl = PMdl.Trained{1}; % Extract trained, compact classifier

`PMdl`

is a `ClassificationPartitionedECOC`

model. It has the property `Trained`

, a 1-by-1 cell array containing the `CompactClassificationECOC`

model that the software trained using the training data.

Compute the test-sample edge.

```
testInds = test(PMdl.Partition); % Extract the test indices
XTest = X(testInds,:);
YTest = Y(testInds,:);
e = edge(Mdl,XTest,YTest)
```

e = 0.4573

The average of the test-sample margins is approximately 0.46.

Compute the mean of the test-sample weighted margins of an ECOC model.

Suppose that the observations in a data set are measured sequentially, and that the last 75 observations have better quality due to a technology upgrade. Incorporate this advancement by giving the better quality observations more weight than the other observations.

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

, the response data `Y`

, and the order of the classes in `Y`

.

load fisheriris X = meas; Y = categorical(species); classOrder = unique(Y); % Class order rng(1); % For reproducibility

Define a weight vector that assigns twice as much weight to the better quality observations.

n = size(X,1); weights = [ones(n-75,1);2*ones(75,1)];

Train an ECOC model using SVM binary classifiers. Specify a 30% holdout sample and the weighting scheme. Standardize the predictors using an SVM template, and specify the class order.

t = templateSVM('Standardize',true); PMdl = fitcecoc(X,Y,'Holdout',0.30,'Weights',weights,... 'Learners',t,'ClassNames',classOrder); Mdl = PMdl.Trained{1}; % Extract trained, compact classifier

`PMdl`

is a trained `ClassificationPartitionedECOC`

model. It has the property `Trained`

, a 1-by-1 cell array containing the `CompactClassificationECOC`

classifier that the software trained using the training data.

Compute the test-sample weighted edge using the weighting scheme.

testInds = test(PMdl.Partition); % Extract the test indices XTest = X(testInds,:); YTest = Y(testInds,:); wTest = weights(testInds,:); e = edge(Mdl,XTest,YTest,'Weights',wTest)

e = 0.4798

The average weighted margin of the test sample is approximately 0.48.

Perform feature selection by comparing test-sample edges from multiple models. Based solely on this comparison, the classifier with the greatest edge is the best classifier.

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

, the response data `Y`

, and the order of the classes in `Y`

.

load fisheriris X = meas; Y = categorical(species); classOrder = unique(Y); % Class order rng(1); % For reproducibility

Partition the data set into training and test sets. Specify a 30% holdout sample for testing.

Partition = cvpartition(Y,'Holdout',0.30); testInds = test(Partition); % Indices for the test set XTest = X(testInds,:); YTest = Y(testInds,:);

`Partition`

defines the data set partition.

Define these two data sets:

`fullX`

contains all predictors.`partX`

contains the petal dimensions only.

fullX = X; partX = X(:,3:4);

Train an ECOC model using SVM binary classifiers for each predictor set. Specify the partition definition, standardize the predictors using an SVM template, and specify the class order.

t = templateSVM('Standardize',true); fullPMdl = fitcecoc(fullX,Y,'CVPartition',Partition,'Learners',t,... 'ClassNames',classOrder); partPMdl = fitcecoc(partX,Y,'CVPartition',Partition,'Learners',t,... 'ClassNames',classOrder); fullMdl = fullPMdl.Trained{1}; partMdl = partPMdl.Trained{1};

`fullPMdl`

and `partPMdl`

are `ClassificationPartitionedECOC`

models. Each model has the property `Trained`

, a 1-by-1 cell array containing the `CompactClassificationECOC`

model that the software trained using the corresponding training set.

Calculate the test-sample edge for each classifier.

fullEdge = edge(fullMdl,XTest,YTest)

fullEdge = 0.4573

partEdge = edge(partMdl,XTest(:,3:4),YTest)

partEdge = 0.4839

`partMdl`

yields an edge value comparable to the value for the more complex model `fullMdl`

.

`Mdl`

— Full or compact multiclass ECOC model`ClassificationECOC`

model object | `CompactClassificationECOC`

model
objectFull or compact multiclass ECOC model, specified as a
`ClassificationECOC`

or
`CompactClassificationECOC`

model
object.

To create a full or compact ECOC model, see `ClassificationECOC`

or `CompactClassificationECOC`

.

`tbl`

— Sample datatable

Sample data, specified as a table. Each row of `tbl`

corresponds to one
observation, and each column corresponds to one predictor variable. Optionally,
`tbl`

can contain additional columns for the response variable
and observation weights. `tbl`

must contain all the predictors used
to train `Mdl`

. Multicolumn variables and cell arrays other than cell
arrays of character vectors are not allowed.

If you train `Mdl`

using sample data contained in a
`table`

, then the input data for `edge`

must also be in a table.

If `Mdl.BinaryLearners`

contains linear or kernel classification
models (`ClassificationLinear`

or `ClassificationKernel`

model objects), then you cannot specify sample
data in a `table`

. Instead, pass a matrix (`X`

)
and class labels (`Y`

).

When training `Mdl`

, assume that you set
`'Standardize',true`

for a template object specified in the
`'Learners'`

name-value pair argument of `fitcecoc`

. In
this case, for the corresponding binary learner `j`

, the software standardizes
the columns of the new predictor data using the corresponding means in
`Mdl.BinaryLearner{j}.Mu`

and standard deviations in
`Mdl.BinaryLearner{j}.Sigma`

.

**Data Types: **`table`

`ResponseVarName`

— Response variable namename of variable in

`tbl`

Response variable name, specified as the name of a variable in `tbl`

. If
`tbl`

contains the response variable used to train
`Mdl`

, then you do not need to specify
`ResponseVarName`

.

If you specify `ResponseVarName`

, then you must do so as a character vector
or string scalar. For example, if the response variable is stored as
`tbl.y`

, then specify `ResponseVarName`

as
`'y'`

. Otherwise, the software treats all columns of
`tbl`

, including `tbl.y`

, as predictors.

The response variable must be a categorical, character, or string array, a logical or numeric vector, or a cell array of character vectors. If the response variable is a character array, then each element must correspond to one row of the array.

**Data Types: **`char`

| `string`

`X`

— Predictor datanumeric matrix

Predictor data, specified as a numeric matrix.

Each row of `X`

corresponds to one observation, and each column corresponds
to one variable. The variables in the columns of
`X`

must be the same as the
variables that trained the classifier
`Mdl`

.

The number of rows in `X`

must equal the number of rows in
`Y`

.

When training `Mdl`

, assume that you set
`'Standardize',true`

for a template object specified in the
`'Learners'`

name-value pair argument of `fitcecoc`

. In
this case, for the corresponding binary learner `j`

, the software standardizes
the columns of the new predictor data using the corresponding means in
`Mdl.BinaryLearner{j}.Mu`

and standard deviations in
`Mdl.BinaryLearner{j}.Sigma`

.

**Data Types: **`double`

| `single`

`Y`

— Class labelscategorical array | character array | string array | logical vector | numeric vector | cell array of character vectors

Class labels, specified as a categorical, character, or string array, a logical or numeric
vector, or a cell array of character vectors. `Y`

must have the same
data type as `Mdl.ClassNames`

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

The number of rows in `Y`

must equal the number of rows in
`tbl`

or `X`

.

**Data Types: **`categorical`

| `char`

| `string`

| `logical`

| `single`

| `double`

| `cell`

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

`edge(Mdl,X,Y,'BinaryLoss','exponential','Decoding','lossbased')`

specifies an exponential binary learner loss function and a loss-based decoding scheme for
aggregating the binary losses.`'BinaryLoss'`

— Binary learner loss function`'hamming'`

| `'linear'`

| `'logit'`

| `'exponential'`

| `'binodeviance'`

| `'hinge'`

| `'quadratic'`

| function handleBinary learner loss function, specified as the comma-separated pair consisting of
`'BinaryLoss'`

and a built-in loss function name or function handle.

This table describes the built-in functions, where

*y*is a 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._{j}For a custom binary loss function, for example

`customFunction`

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

.`customFunction`

has this form:where:bLoss = customFunction(M,s)

`M`

is the*K*-by-*L*coding matrix stored in`Mdl.CodingMatrix`

.`s`

is the 1-by-*L*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.*L*is the number of binary learners.

For an example of passing a custom binary loss function, see Predict Test-Sample Labels of ECOC Model Using Custom Binary Loss Function.

The default `BinaryLoss`

value depends on the score ranges returned
by the binary learners. This table describes some default
`BinaryLoss`

values based on the given assumptions.

Assumption | Default Value |
---|---|

All binary learners are SVMs or either linear or kernel classification models of SVM learners. | `'hinge'` |

All binary learners are ensembles trained by
`AdaboostM1` or
`GentleBoost` . | `'exponential'` |

All binary learners are ensembles trained by
`LogitBoost` . | `'binodeviance'` |

All binary learners are linear or kernel classification models of
logistic regression learners. Or, you specify to predict class
posterior probabilities by setting
`'FitPosterior',true` in `fitcecoc` . | `'quadratic'` |

To check the default value, use dot notation to display the
`BinaryLoss`

property of the trained model at the command
line.

**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'`

`'ObservationsIn'`

— Predictor data observation dimension`'rows'`

(default) | `'columns'`

Predictor data observation dimension, specified as the comma-separated pair consisting of
`'ObservationsIn'`

and `'columns'`

or
`'rows'`

. `Mdl.BinaryLearners`

must contain
`ClassificationLinear`

models.

If you orient your predictor matrix so that observations correspond
to columns and specify `'ObservationsIn','columns'`

,
you can experience a significant reduction in execution time.

`'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`

`'Weights'`

— Observation weights`ones(size(X,1),1)`

(default) | numeric vector | name of variable in `tbl`

Observation weights, specified as the comma-separated pair consisting of
`'Weights'`

and a numeric vector or the name of a variable in
`tbl`

. If you supply weights, `edge`

computes
the weighted classification edge.

If you specify `Weights`

as a numeric vector, then the size of
`Weights`

must be equal to the number of observations in
`X`

or `tbl`

. The software normalizes
`Weights`

to sum up to the value of the prior probability in the
respective class.

If you specify `Weights`

as the name of a variable in
`tbl`

, you must do so as a character vector or string scalar. For
example, if the weights are stored as `tbl.w`

, then specify
`Weights`

as `'w'`

. Otherwise, the software
treats all columns of `tbl`

, including `tbl.w`

, as
predictors.

**Data Types: **`single`

| `double`

| `char`

| `string`

`e`

— Classification edgenumeric scalar | numeric vector

Classification edge, returned
as a numeric scalar or vector. `e`

represents the weighted mean of
the classification margins.

If `Mdl.BinaryLearners`

contains `ClassificationLinear`

models, then `e`

is a
1-by-*L* vector, where *L* is the number of
regularization strengths in the linear classification models
(`numel(Mdl.BinaryLearners{1}.Lambda)`

). The value
`e(j)`

is the edge for the model trained using regularization
strength `Mdl.BinaryLearners{1}.Lambda(j)`

.

Otherwise, `e`

is a scalar value.

The *classification edge* is
the weighted mean of the *classification margins*.

One way to choose among multiple classifiers, for example to perform feature selection, is to choose the classifier that yields the greatest edge.

The *classification margin* is, for each observation,
the difference between the negative loss for the true class and the maximal negative loss
among the false classes. If the margins are on the same scale, then they serve as a
classification confidence measure. Among multiple classifiers, those that yield greater
margins are better.

A *binary loss* is a function
of the class and classification score that determines how well a binary
learner classifies an observation into the class.

Suppose the following:

*m*is element (_{kj}*k*,*j*) of the coding design matrix*M*(that is, the code corresponding to class*k*of binary learner*j*).*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.

In *loss-based decoding*
[Escalera et al.], the class producing the minimum sum of the binary losses over
binary learners determines the predicted class of an observation, that is,

$$\widehat{k}=\underset{k}{\text{argmin}}{\displaystyle \sum _{j=1}^{L}\left|{m}_{kj}\right|g}({m}_{kj},{s}_{j}).$$

In *loss-weighted decoding*
[Escalera et al.], the class producing the minimum average of the binary losses
over binary learners determines the predicted class of an observation, that is,

$$\widehat{k}=\underset{k}{\text{argmin}}\frac{{\displaystyle \sum _{j=1}^{L}\left|{m}_{kj}\right|g}({m}_{kj},{s}_{j})}{{\displaystyle \sum}_{j=1}^{L}\left|{m}_{kj}\right|}.$$

Allwein et al. suggest that loss-weighted decoding improves classification accuracy by keeping loss values for all classes in the same dynamic range.

This table summarizes the supported loss functions, where
*y _{j}* is a class label for a particular binary
learner (in the set {–1,1,0}),

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 such that the loss is 0.5 when
*y _{j}* = 0, and aggregates using the average
of the binary learners [Allwein et al.].

Do not confuse the binary loss with the overall classification loss (specified by the
`'LossFun'`

name-value pair argument of the `loss`

and
`predict`

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

To compare the margins or edges of several ECOC classifiers, use template objects to specify a common score transform function among the classifiers during training.

[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
Recogn*. Vol. 30, Issue 3, 2009, pp. 285–297.

Calculate with arrays that have more rows than fit in memory.

This function fully supports tall arrays. For more information, see Tall Arrays (MATLAB).

Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.

To run in parallel, set the `'UseParallel'`

option to `true`

.

Set the `'UseParallel'`

field of the options structure to `true`

using `statset`

and specify the `'Options'`

name-value pair argument in the call to this function.

For example: `'Options',statset('UseParallel',true)`

For more information, see the `'Options'`

name-value pair argument.

For more general information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).

`ClassificationECOC`

| `CompactClassificationECOC`

| `fitcecoc`

| `loss`

| `margin`

| `predict`

| `resubEdge`

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