edge

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

Syntax

e = edge(Mdl,tbl,ResponseVarName)

e = edge(Mdl,tbl,Y)

e = edge(Mdl,X,Y)

e = edge(___,Name,Value)

Description

e = edge(Mdl,tbl,ResponseVarName) returns the classification edge (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.

e = edge(Mdl,tbl,Y) returns the classification edge for the classifier Mdl using the predictor data in table tbl and the class labels in vector Y.

e = edge(Mdl,X,Y) returns the classification edge (e) for the classifier Mdl using the predictor data in matrix X and the class labels in vector Y.

e = edge(___,Name,Value) 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.

Examples

Test-Sample Edge of ECOC Model

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 = 0.6860

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

Mean of Test-Sample Weighted Margins of ECOC Model

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 = 0.7196

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

Select ECOC Model Features by Comparing Test-Sample Edges

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 = 0.6860

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

partEdge = 0.7259

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

Input Arguments

Mdl — Full or compact multiclass ECOC model

ClassificationECOC model object | CompactClassificationECOC model object

Full 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 data

table

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.

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 name

name 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 data

numeric 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 labels

categorical 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

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: 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 handle

Binary learner loss function, specified as the comma-separated pair consisting of 'BinaryLoss' and a built-in loss function name or function handle.

• This table describes the built-in functions, where y_j is the class label for a particular binary learner (in the set {–1,1,0}), s_j is the score for observation j, and g(y_j,s_j) is the binary loss formula.

  Value	Description	Score Domain	g(yj,sj)
  'binodeviance'	Binomial deviance	(–∞,∞)	log[1 + exp(–2yjsj)]/[2log(2)]
  'exponential'	Exponential	(–∞,∞)	exp(–yjsj)/2
  'hamming'	Hamming	[0,1] or (–∞,∞)	[1 – sign(yjsj)]/2
  'hinge'	Hinge	(–∞,∞)	max(0,1 – yjsj)/2
  'linear'	Linear	(–∞,∞)	(1 – yjsj)/2
  'logit'	Logistic	(–∞,∞)	log[1 + exp(–yjsj)]/[2log(2)]
  'quadratic'	Quadratic	[0,1]	[1 – yj(2sj – 1)]2/2

  The software normalizes binary losses so that the loss is 0.5 when y_j = 0. Also, the software calculates the mean binary loss for each class [1].

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

This table identifies the default BinaryLoss value, which depends on the score ranges returned by the binary learners.

Assumption	Default Value
All binary learners are any of the following: Classification decision trees Discriminant analysis models k-nearest neighbor models Linear or kernel classification models of logistic regression learners Naive Bayes models	'quadratic'
All binary learners are SVMs or 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'
You specify to predict class posterior probabilities by setting 'FitPosterior',true in fitcecoc.	'quadratic'
Binary learners are heterogeneous and use different loss functions.	'hamming'

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.

Note

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. You cannot specify 'ObservationsIn','columns' for predictor data in a table.

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

Output Arguments

e — Classification edge

numeric 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.

More About

Classification Edge

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.

Classification Margin

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.

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_kj is element (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_j is the score of binary learner j for an observation.

• g is the binary loss function.

• $$\widehat{k}$$ is the predicted class for the observation.

The software supports two decoding schemes:

• Loss-based decoding [2] (Decoding is 'lossbased') — The predicted class of an observation corresponds to the class that produces the minimum average of the binary losses over all binary learners.

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

• Loss-weighted decoding [3] (Decoding is 'lossweighted') — The predicted class of an observation corresponds to the class that produces the minimum average of the binary losses over the binary learners for the corresponding class.

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

  The denominator corresponds to the number of binary learners for class k. [1] suggests that loss-weighted decoding improves classification accuracy by keeping loss values for all classes in the same dynamic range.

The predict, resubPredict, and kfoldPredict functions return the negated value of the objective function of argmin as the second output argument (NegLoss) for each observation and class.

This table summarizes the supported binary loss functions, where y_j is a class label for a particular binary learner (in the set {–1,1,0}), s_j is the score for observation j, and g(y_j,s_j) is the binary loss function.

Value	Description	Score Domain	g(yj,sj)
"binodeviance"	Binomial deviance	(–∞,∞)	log[1 + exp(–2yjsj)]/[2log(2)]
"exponential"	Exponential	(–∞,∞)	exp(–yjsj)/2
"hamming"	Hamming	[0,1] or (–∞,∞)	[1 – sign(yjsj)]/2
"hinge"	Hinge	(–∞,∞)	max(0,1 – yjsj)/2
"linear"	Linear	(–∞,∞)	(1 – yjsj)/2
"logit"	Logistic	(–∞,∞)	log[1 + exp(–yjsj)]/[2log(2)]
"quadratic"	Quadratic	[0,1]	[1 – yj(2sj – 1)]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.

Tips

• 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.

References

[1] Allwein, E., R. Schapire, and Y. Singer. “Reducing multiclass to binary: A unifying approach for margin classifiers.” Journal of Machine Learning Research. Vol. 1, 2000, pp. 113–141.

[2] Escalera, S., O. Pujol, and P. Radeva. “Separability of ternary codes for sparse designs of error-correcting output codes.” Pattern Recog. Lett., Vol. 30, Issue 3, 2009, pp. 285–297.

[3] Escalera, S., O. Pujol, and P. Radeva. “On the decoding process in ternary error-correcting output codes.” IEEE Transactions on Pattern Analysis and Machine Intelligence. Vol. 32, Issue 7, 2010, pp. 120–134.

Extended Capabilities

Tall Arrays

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

Usage notes and limitations:

• edge does not support tall table data when Mdl contains kernel or linear binary learners.

For more information, see Tall Arrays.

Automatic Parallel Support

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

To run in parallel, specify the Options name-value argument in the call to this function and set the UseParallel field of the options structure to true using statset:

"Options",statset("UseParallel",true)

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

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

• The edge function does not support models trained using decision tree learners with surrogate splits.

• The edge function does not support models trained using SVM learners.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2014b