CompactClassificationECOC

Compact multiclass model for support vector machines (SVMs) and other classifiers

Description

CompactClassificationECOC is a compact version of the multiclass error-correcting output codes (ECOC) model. The compact classifier does not include the data used for training the multiclass ECOC model. Therefore, you cannot perform certain tasks, such as cross-validation, using the compact classifier. Use a compact multiclass ECOC model for tasks such as classifying new data (predict).

Creation

You can create a CompactClassificationECOC model in two ways:

• Create a compact ECOC model from a trained ClassificationECOC model by using the compact object function.

• Create a compact ECOC model by using the fitcecoc function and specifying the 'Learners' name-value pair argument as 'linear', 'kernel', a templateLinear or templateKernel object, or a cell array of such objects.

Properties

After you create a CompactClassificationECOC model object, you can use dot notation to access its properties. For an example, see Train and Cross-Validate ECOC Classifier.

ECOC Properties

BinaryLearners — Trained binary learners

cell vector of model objects

Trained binary learners, specified as a cell vector of model objects. The number of binary learners depends on the number of classes in Y and the coding design.

The software trains BinaryLearner{j} according to the binary problem specified by CodingMatrix(:,j). For example, for multiclass learning using SVM learners, each element of BinaryLearners is a CompactClassificationSVM classifier.

Data Types: cell

BinaryLoss — Binary learner loss function

'binodeviance' | 'exponential' | 'hamming' | 'hinge' | 'linear' | 'logit' | 'quadratic'

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

The default BinaryLoss value depends on the score ranges returned by the binary learners. This table identifies what some default BinaryLoss values are when you use the default score transform (ScoreTransform property of the model is 'none').

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.

To potentially increase accuracy, specify a binary loss function other than the default during a prediction or loss computation by using the BinaryLoss name-value argument of predict or loss. For more information, see Binary Loss.

Data Types: char

CodingMatrix — Class assignment codes

numeric matrix

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.

Data Types: double | single | int8 | int16 | int32 | int64

LearnerWeights — Binary learner weights

numeric row vector

Binary learner weights, specified as a numeric row vector. The length of LearnerWeights is equal to the number of binary learners (length(Mdl.BinaryLearners)).

LearnerWeights(j) is the sum of the observation weights that binary learner j uses to train its classifier.

The software uses LearnerWeights to fit posterior probabilities by minimizing the Kullback-Leibler divergence. The software ignores LearnerWeights when it uses the quadratic programming method of estimating posterior probabilities.

Data Types: double | single

Other Classification Properties

CategoricalPredictors — Categorical predictor indices

vector of positive integers | []

Categorical predictor indices, specified as a vector of positive integers. CategoricalPredictors contains index values indicating that the corresponding predictors are categorical. The index values are between 1 and p, where p is the number of predictors used to train the model. If none of the predictors are categorical, then this property is empty ([]).

Data Types: single | double

ClassNames — Unique class labels

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

Unique class labels used in training, specified as a categorical or character array, logical or numeric vector, or cell array of character vectors. ClassNames has the same data type as the class labels Y. (The software treats string arrays as cell arrays of character vectors.) ClassNames also determines the class order.

Data Types: categorical | char | logical | single | double | cell

Cost — Misclassification costs

square numeric matrix

This property is read-only.

Misclassification costs, specified as a square numeric matrix. Cost has K rows and columns, where K is the number of classes.

Cost(i,j) is the cost of classifying a point into class j if its true class is i. The order of the rows and columns of Cost corresponds to the order of the classes in ClassNames.

Data Types: double

PredictorNames — Predictor names

cell array of character vectors

Predictor names in order of their appearance in the predictor data, specified as a cell array of character vectors. The length of PredictorNames is equal to the number of variables in the training data X or Tbl used as predictor variables.

Data Types: cell

ExpandedPredictorNames — Expanded predictor names

cell array of character vectors

Expanded predictor names, specified as a cell array of character vectors.

If the model uses encoding for categorical variables, then ExpandedPredictorNames includes the names that describe the expanded variables. Otherwise, ExpandedPredictorNames is the same as PredictorNames.

Data Types: cell

Prior — Prior class probabilities

numeric vector

This property is read-only.

Prior class probabilities, specified as a numeric vector. Prior has as many elements as the number of classes in ClassNames, and the order of the elements corresponds to the order of the classes in ClassNames.

fitcecoc incorporates misclassification costs differently among different types of binary learners.

Data Types: double

ResponseName — Response variable name

character vector

Response variable name, specified as a character vector.

Data Types: char

ScoreTransform — Score transformation function to apply to predicted scores

'doublelogit' | 'invlogit' | 'ismax' | 'logit' | 'none' | function handle | ...

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

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

• For a built-in function, enter this code and replace function with a value in the table.

  Mdl.ScoreTransform = 'function';

  Value	Description
  "doublelogit"	1/(1 + e–2x)
  "invlogit"	log(x / (1 – x))
  "ismax"	Sets the score for the class with the largest score to 1, and sets the scores for all other classes to 0
  "logit"	1/(1 + e–x)
  "none" or "identity"	x (no transformation)
  "sign"	–1 for x < 0 0 for x = 0 1 for x > 0
  "symmetric"	2x – 1
  "symmetricismax"	Sets the score for the class with the largest score to 1, and sets the scores for all other classes to –1
  "symmetriclogit"	2/(1 + e–x) – 1

• For a MATLAB^® function or a function that you define, enter its function handle.

  Mdl.ScoreTransform = @function;

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

Data Types: char | function_handle

Object Functions

compareHoldout	Compare accuracies of two classification models using new data
discardSupportVectors	Discard support vectors of linear SVM binary learners in ECOC model
edge	Classification edge for multiclass error-correcting output codes (ECOC) model
gather	Gather properties of Statistics and Machine Learning Toolbox object from GPU
incrementalLearner	Convert multiclass error-correcting output codes (ECOC) model to incremental learner
lime	Local interpretable model-agnostic explanations (LIME)
loss	Classification loss for multiclass error-correcting output codes (ECOC) model
margin	Classification margins for multiclass error-correcting output codes (ECOC) model
partialDependence	Compute partial dependence
plotPartialDependence	Create partial dependence plot (PDP) and individual conditional expectation (ICE) plots
predict	Classify observations using multiclass error-correcting output codes (ECOC) model
shapley	Shapley values
selectModels	Choose subset of multiclass ECOC models composed of binary ClassificationLinear learners
update	Update model parameters for code generation

Examples

Reduce Size of Full ECOC Model

Reduce the size of a full ECOC model by removing the training data. Full ECOC models (ClassificationECOC models) hold the training data. To improve efficiency, use a smaller 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);

Train an ECOC model using SVM binary classifiers. Standardize the predictor data using an SVM template t, and specify the order of the classes. During training, the software uses default values for empty options in t.

t = templateSVM('Standardize',true);
Mdl = fitcecoc(X,Y,'Learners',t,'ClassNames',classOrder);

Mdl is a ClassificationECOC model.

Reduce the size of the ECOC model.

CompactMdl = compact(Mdl)

CompactMdl = 
  CompactClassificationECOC
             ResponseName: 'Y'
    CategoricalPredictors: []
               ClassNames: [setosa    versicolor    virginica]
           ScoreTransform: 'none'
           BinaryLearners: {3x1 cell}
             CodingMatrix: [3x3 double]


  Properties, Methods

CompactMdl is a CompactClassificationECOC model. CompactMdl does not store all of the properties that Mdl stores. In particular, it does not store the training data.

Display the amount of memory each classifier uses.

whos('CompactMdl','Mdl')

  Name            Size            Bytes  Class                                                  Attributes

  CompactMdl      1x1             15116  classreg.learning.classif.CompactClassificationECOC              
  Mdl             1x1             28357  ClassificationECOC                                               

The full ECOC model (Mdl) is approximately double the size of the compact ECOC model (CompactMdl).

To label new observations efficiently, you can remove Mdl from the MATLAB® Workspace, and then pass CompactMdl and new predictor values to predict.

Train and Cross-Validate ECOC Classifier

Train and cross-validate an ECOC classifier using different binary learners and the one-versus-all coding design.

Load Fisher's iris data set. Specify the predictor data X and the response data Y. Determine the class names and the number of classes.

load fisheriris
X = meas;
Y = species;
classNames = unique(species(~strcmp(species,''))) % Remove empty classes 

classNames = 3x1 cell
    {'setosa'    }
    {'versicolor'}
    {'virginica' }

K = numel(classNames) % Number of classes

K = 3

You can use classNames to specify the order of the classes during training.

For a one-versus-all coding design, this example has K = 3 binary learners. Specify templates for the binary learners such that:

• Binary learner 1 and 2 are naive Bayes classifiers. By default, each predictor is conditionally, normally distributed given its label.

• Binary learner 3 is an SVM classifier. Specify to use the Gaussian kernel.

rng(1);  % For reproducibility
tNB = templateNaiveBayes();
tSVM = templateSVM('KernelFunction','gaussian');
tLearners = {tNB tNB tSVM};

tNB and tSVM are template objects for naive Bayes and SVM learning, respectively. The objects indicate which options to use during training. Most of their properties are empty, except those specified by name-value pair arguments. During training, the software fills in the empty properties with their default values.

Train and cross-validate an ECOC classifier using the binary learner templates and the one-versus-all coding design. Specify the order of the classes. By default, naive Bayes classifiers use posterior probabilities as scores, whereas SVM classifiers use distances from the decision boundary. Therefore, to aggregate the binary learners, you must specify to fit posterior probabilities.

CVMdl = fitcecoc(X,Y,'ClassNames',classNames,'CrossVal','on',...
    'Learners',tLearners,'FitPosterior',true);

CVMdl is a ClassificationPartitionedECOC cross-validated model. By default, the software implements 10-fold cross-validation. The scores across the binary learners have the same form (that is, they are posterior probabilities), so the software can aggregate the results of the binary classifications properly.

Inspect one of the trained folds using dot notation.

CVMdl.Trained{1}

ans = 
  CompactClassificationECOC
             ResponseName: 'Y'
    CategoricalPredictors: []
               ClassNames: {'setosa'  'versicolor'  'virginica'}
           ScoreTransform: 'none'
           BinaryLearners: {3x1 cell}
             CodingMatrix: [3x3 double]


  Properties, Methods

Each fold is a CompactClassificationECOC model trained on 90% of the data.

You can access the results of the binary learners using dot notation and cell indexing. Display the trained SVM classifier (the third binary learner) in the first fold.

CVMdl.Trained{1}.BinaryLearners{3}

ans = 
  CompactClassificationSVM
             ResponseName: 'Y'
    CategoricalPredictors: []
               ClassNames: [-1 1]
           ScoreTransform: '@(S)sigmoid(S,-4.016619e+00,-3.243499e-01)'
                    Alpha: [33x1 double]
                     Bias: -0.1345
         KernelParameters: [1x1 struct]
           SupportVectors: [33x4 double]
      SupportVectorLabels: [33x1 double]


  Properties, Methods

Estimate the generalization error.

genError = kfoldLoss(CVMdl)

genError = 0.0333

On average, the generalization error is approximately 3%.

More About

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

  $$\begin{array}{cccc}& \text{Learner 1}& \text{Learner 2}& \text{Learner 3}\\ \text{Class 1}& 1& 1& 0\\ \text{Class 2}& -1& 0& 1\\ \text{Class 3}& 0& -1& -1\end{array}$$

  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 call object functions, such as predict, loss, margin, edge, and so on.

The ECOC algorithm follows these steps.

1. 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.

2. Let M be the coding design matrix with elements m_kl, and s_l be the predicted classification score for the positive class of learner l. The algorithm assigns a new observation to the class ($$\widehat{k}$$) that minimizes the aggregation of the losses for the B 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 [1].

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 [2].

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.	K(K – 1)/2	1
binary complete	This design partitions the classes into all binary combinations, and does not ignore any classes. That is, all class assignments are –1 and 1 with at least one positive class and one negative class in the assignment for each binary learner.	2K – 1 – 1	2K – 2
ternary complete	This design partitions the classes into all ternary combinations. That is, all class assignments are 0, –1, and 1 with at least one positive class and one negative class in the assignment for each binary learner.	(3K – 2K + 1 + 1)/2	3K – 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 log2K	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 log2K	Variable

This plot compares the number of binary learners for the coding designs with increasing K.

Algorithms

Random Coding Design Matrices

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

1. The software generates one of these matrices:

   1. Dense random — The software assigns 1 or –1 with equal probability to each element of the K-by-L_d coding design matrix, where $$L_d\approx \lceil 10{\log }_{2}K\rceil$$.

   2. Sparse random — The software assigns 1 to each element of the K-by-L_s coding design matrix with probability 0.25, –1 with probability 0.25, and 0 with probability 0.5, where $$L_s\approx \lceil 15{\log }_{2}K\rceil$$.

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

3. 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 ([2]) 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.

Support Vector Storage

By default and for efficiency, fitcecoc empties the Alpha, SupportVectorLabels, and SupportVectors properties for all linear SVM binary learners. fitcecoc lists Beta, rather than Alpha, in the model display.

To store Alpha, SupportVectorLabels, and SupportVectors, pass a linear SVM template that specifies storing support vectors to fitcecoc. For example, enter:

t = templateSVM('SaveSupportVectors',true)
Mdl = fitcecoc(X,Y,'Learners',t);

You can remove the support vectors and related values by passing the resulting ClassificationECOC model to discardSupportVectors.

References

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

[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.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• The predict and update functions support code generation.

• When you train an ECOC model by using fitcecoc, the following restrictions apply.

  • All binary learners must be either SVM classifiers or linear classification models. For the Learners name-value argument, you can specify:

    • 'svm' or 'linear'

    • An SVM template object or a cell array of such objects (see templateSVM)

    • A linear classification model template object or a cell array of such objects (see templateLinear)

  • When you generate code using a coder configurer for predict and update, the following additional restrictions apply for binary learners.

    • If you use a cell array of SVM template objects, the value of Standardize for SVM learners must be consistent. For example, if you specify 'Standardize',true for one SVM learner, you must specify the same value for all SVM learners.

    • If you use a cell array of SVM template objects, and you use one SVM learner with a linear kernel ('KernelFunction','linear') and another with a different type of kernel function, then you must specify 'SaveSupportVectors',true for the learner with a linear kernel.

    For details, see ClassificationECOCCoderConfigurer. For information on name-value arguments that you cannot modify when you retrain a model, see Tips.

  • Code generation limitations for SVM classifiers and linear classification models also apply to ECOC classifiers, depending on the choice of binary learners. For more details, see Code Generation of the CompactClassificationSVM class and Code Generation of the ClassificationLinear class.

For more information, see Introduction to Code Generation.

GPU Arrays

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

Usage notes and limitations:

• The following object functions fully support GPU arrays:

  • discardSupportVectors

  • gather

• The following object functions offer limited support for GPU arrays:

  • compareHoldout

  • edge

  • loss

  • margin

  • partialDependence

  • plotPartialDependence

  • predict

• The object functions execute on a GPU if either of the following apply:

  • The model was fitted with GPU arrays.

  • The predictor data that you pass to the object function is a GPU array.

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

Version History

Introduced in R2014b