kfoldPredict

Classify observations in cross-validated ECOC model

Syntax

label = kfoldPredict(CVMdl)

label = kfoldPredict(CVMdl,Name,Value)

[label,NegLoss,PBScore] = kfoldPredict(___)

[label,NegLoss,PBScore,Posterior] = kfoldPredict(___)

Description

label = kfoldPredict(CVMdl) returns class labels predicted by the cross-validated ECOC model (ClassificationPartitionedECOC) CVMdl. For every fold, kfoldPredict predicts class labels for observations that it holds out during training. CVMdl.X contains both sets of observations.

The software predicts the classification of an observation by assigning the observation to the class yielding the largest negated average binary loss (or, equivalently, the smallest average binary loss).

label = kfoldPredict(CVMdl,Name,Value) returns predicted class labels with additional options specified by one or more name-value pair arguments. For example, specify the posterior probability estimation method, decoding scheme, or verbosity level.

[label,NegLoss,PBScore] = kfoldPredict(___) additionally returns negated values of the average binary loss per class (NegLoss) for validation-fold observations and positive-class scores (PBScore) for validation-fold observations classified by each binary learner, using any of the input argument combinations in the previous syntaxes.

If the coding matrix varies across folds (that is, the coding scheme is sparserandom or denserandom), then PBScore is empty ([]).

[label,NegLoss,PBScore,Posterior] = kfoldPredict(___) additionally returns posterior class probability estimates for validation-fold observations (Posterior).

To obtain posterior class probabilities, you must set 'FitPosterior',1 when training the cross-validated ECOC model using fitcecoc. Otherwise, kfoldPredict throws an error.

Examples

Predict k-Fold Cross-Validation Labels

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);
rng(1); % For reproducibility

Train and cross-validate an ECOC model using support vector machine (SVM) binary classifiers. Standardize the predictor data using an SVM template, and specify the class order.

t = templateSVM('Standardize',1);
CVMdl = fitcecoc(X,Y,'CrossVal','on','Learners',t,'ClassNames',classOrder);

CVMdl is a ClassificationPartitionedECOC model. By default, the software implements 10-fold cross-validation. You can specify a different number of folds using the 'KFold' name-value pair argument.

Predict the validation-fold labels. Print a random subset of true and predicted labels.

labels = kfoldPredict(CVMdl);
idx = randsample(numel(labels),10);
table(Y(idx),labels(idx),...
    'VariableNames',{'TrueLabels','PredictedLabels'})

ans=10×2 table
    TrueLabels    PredictedLabels
    __________    _______________

    setosa          setosa       
    versicolor      versicolor   
    setosa          setosa       
    virginica       virginica    
    versicolor      versicolor   
    setosa          setosa       
    virginica       virginica    
    virginica       virginica    
    setosa          setosa       
    setosa          setosa       

CVMdl correctly labels the validation-fold observations with indices idx.

Predict Cross-Validation Labels Using Custom Binary Loss Function

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
K = numel(classOrder);  % Number of classes
rng(1); % For reproducibility

Train and cross-validate an ECOC model using SVM binary classifiers. Standardize the predictor data using an SVM template, and specify the class order.

t = templateSVM('Standardize',1);
CVMdl = fitcecoc(X,Y,'CrossVal','on','Learners',t,'ClassNames',classOrder);

CVMdl is a ClassificationPartitionedECOC model. By default, the software implements 10-fold cross-validation. You can specify a different number of folds using the 'KFold' name-value pair argument.

SVM scores are signed distances from the observation to the decision boundary. Therefore, the domain is $$(-\infty ,\infty )$$. Create a custom binary loss function that:

• Maps the coding design matrix (M) and positive-class classification scores (s) for each learner to the binary loss for each observation

• Uses linear loss

• Aggregates the binary learner loss using the median

You can create a separate function for the binary loss function, and then save it on the MATLAB® path. Alternatively, you can specify an anonymous binary loss function. In this case, create a function handle (customBL) to an anonymous binary loss function.

customBL = @(M,s)nanmedian(1 - bsxfun(@times,M,s),2)/2;

Predict cross-validation labels and estimate the median binary loss per class. Print the median negative binary losses per class for a random set of 10 validation-fold observations.

[label,NegLoss] = kfoldPredict(CVMdl,'BinaryLoss',customBL);

idx = randsample(numel(label),10);
classOrder

classOrder = 3x1 categorical
     setosa 
     versicolor 
     virginica 

table(Y(idx),label(idx),NegLoss(idx,:),'VariableNames',...
    {'TrueLabel','PredictedLabel','NegLoss'})

ans=10×3 table
    TrueLabel     PredictedLabel                 NegLoss             
    __________    ______________    _________________________________

    setosa          versicolor      0.37141       2.1296       -4.001
    versicolor      versicolor      -1.2166      0.36678     -0.65021
    setosa          versicolor      0.23932       2.0793      -3.8186
    virginica       virginica       -1.9151     -0.19953      0.61467
    versicolor      versicolor      -1.3745      0.45532     -0.58077
    setosa          versicolor      0.20061       2.2774       -3.978
    virginica       versicolor      -1.4926     0.090706    -0.098127
    virginica       virginica       -1.7667     -0.13466      0.40134
    setosa          versicolor      0.20011       1.9111      -3.6112
    setosa          versicolor      0.16118       1.9679      -3.6291

The order of the columns corresponds to the elements of classOrder. The software predicts the label based on the maximum negated loss. The results indicate that the median of the linear losses might not perform as well as other losses.

Estimate Cross-Validation Posterior Probabilities

Load Fisher's iris data set. Use the petal dimensions as the predictor data X. Specify the response data Y and the order of the classes in Y.

load fisheriris
X = meas(:,3:4);
Y = categorical(species);
classOrder = unique(Y);
rng(1); % For reproducibility

Create an SVM template. Standardize the predictors, and specify the Gaussian kernel.

t = templateSVM('Standardize',1,'KernelFunction','gaussian');

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

Train and cross-validate an ECOC classifier using the SVM template. Transform classification scores to class posterior probabilities (returned by kfoldPredict) using the 'FitPosterior' name-value pair argument. Specify the class order.

CVMdl = fitcecoc(X,Y,'Learners',t,'CrossVal','on','FitPosterior',true,...
    'ClassNames',classOrder);

CVMdl is a ClassificationPartitionedECOC model. By default, the software uses 10-fold cross-validation.

Predict the validation-fold class posterior probabilities. Use 10 random initial values for the Kullback-Leibler algorithm.

[label,~,~,Posterior] = kfoldPredict(CVMdl,'NumKLInitializations',10);

The software assigns an observation to the class that yields the smallest average binary loss. Because all the binary learners compute posterior probabilities, the binary loss function is quadratic.

Display a random set of results.

idx = randsample(size(X,1),10);
CVMdl.ClassNames

ans = 3x1 categorical
     setosa 
     versicolor 
     virginica 

table(Y(idx),label(idx),Posterior(idx,:),...
    'VariableNames',{'TrueLabel','PredLabel','Posterior'})

ans=10×3 table
    TrueLabel     PredLabel                   Posterior               
    __________    __________    ______________________________________

    versicolor    versicolor       0.00864       0.98243     0.0089298
    versicolor    virginica     2.2197e-14       0.12447       0.87553
    setosa        setosa             0.999    0.00022836    0.00076885
    versicolor    versicolor    2.2194e-14       0.98915      0.010848
    virginica     virginica       0.012318      0.012925       0.97476
    virginica     virginica      0.0015573     0.0015639       0.99688
    virginica     virginica      0.0042896     0.0043557       0.99135
    setosa        setosa             0.999    0.00028329    0.00071382
    virginica     virginica      0.0094654     0.0098159       0.98072
    setosa        setosa             0.999    0.00013559    0.00086195

The columns of Posterior correspond to the class order of CVMdl.ClassNames.

Estimate Cross-Validation Posterior Probabilities Using Parallel Computing

Train a multiclass ECOC model and estimate the posterior probabilities using parallel computing.

Load the arrhythmia data set. Examine the response data Y.

load arrhythmia
Y = categorical(Y);
tabulate(Y)

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

n = numel(Y);
K = numel(unique(Y));

Several classes are not represented in the data, and many of the other classes have low relative frequencies.

Specify an ensemble learning template that uses the GentleBoost method and 50 weak classification tree learners.

t = templateEnsemble('GentleBoost',50,'Tree');

t is a template object. Most of the options are empty ([]). The software uses default values for all empty options during training.

Because the response variable contains many classes, specify a sparse random coding design.

rng(1); % For reproducibility
Coding = designecoc(K,'sparserandom');

Train and cross-validate an ECOC model using parallel computing. Fit posterior probabilities (returned by kfoldPredict).

pool = parpool;                      % Invokes workers

Starting parallel pool (parpool) using the 'local' profile ...
connected to 6 workers.

options = statset('UseParallel',1);
CVMdl = fitcecoc(X,Y,'Learner',t,'Options',options,'Coding',Coding,...
    'FitPosterior',1,'CrossVal','on');

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

CVMdl is a ClassificationPartitionedECOC model. By default, the software implements 10-fold cross-validation. You can specify a different number of folds using the 'KFold' name-value pair argument.

The pool invokes six workers, although the number of workers might vary among systems. Because some classes have low relative frequency, one or more folds most likely do not contain observations from all classes.

Estimate posterior probabilities, and display the posterior probability of being classified as not having arrhythmia (class 1) given the data for a random set of validation-fold observations.

[~,~,~,posterior] = kfoldPredict(CVMdl,'Options',options);
idx = randsample(n,10);
table(idx,Y(idx),posterior(idx,1),...
    'VariableNames',{'OOFSampleIndex','TrueLabel','PosteriorNoArrhythmia'})

ans=10×3 table
    OOFSampleIndex    TrueLabel    PosteriorNoArrhythmia
    ______________    _________    _____________________

         171             1                0.33654       
         221             1                0.85135       
          72             16                0.9174       
           3             10              0.025649       
         202             1                 0.8438       
         243             1                 0.9435       
          18             1                0.81198       
          49             6               0.090154       
         234             1                0.61625       
         315             1                0.97187       

Input Arguments

CVMdl — Cross-validated ECOC model

ClassificationPartitionedECOC model

Cross-validated ECOC model, specified as a ClassificationPartitionedECOC model. You can create a ClassificationPartitionedECOC model in two ways:

• Pass a trained ECOC model (ClassificationECOC) to crossval.

• Train an ECOC model using fitcecoc and specify any one of these cross-validation name-value pair arguments: 'CrossVal', 'CVPartition', 'Holdout', 'KFold', or 'Leaveout'.

Name-Value Pair Arguments

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.

Example: kfoldPredict(CVMdl,'PosteriorMethod','qp') specifies to estimate multiclass posterior probabilities by solving a least-squares problem using quadratic programming.

'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 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 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.

• For a custom binary loss function, for example customFunction, specify its function handle 'BinaryLoss',@customFunction.

  customFunction has this form:

  bLoss = customFunction(M,s)

  where:

  • 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'

'NumKLInitializations' — Number of random initial values

0 (default) | nonnegative integer scalar

Number of random initial values for fitting posterior probabilities by Kullback-Leibler divergence minimization, specified as the comma-separated pair consisting of 'NumKLInitializations' and a nonnegative integer scalar.

If you do not request the fourth output argument (Posterior) and set 'PosteriorMethod','kl' (the default), then the software ignores the value of NumKLInitializations.

For more details, see Posterior Estimation Using Kullback-Leibler Divergence.

Example: 'NumKLInitializations',5

Data Types: single | double

'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).

'PosteriorMethod' — Posterior probability estimation method

'kl' (default) | 'qp'

Posterior probability estimation method, specified as the comma-separated pair consisting of 'PosteriorMethod' and 'kl' or 'qp'.

• If PosteriorMethod is 'kl', then the software estimates multiclass posterior probabilities by minimizing the Kullback-Leibler divergence between the predicted and expected posterior probabilities returned by binary learners. For details, see Posterior Estimation Using Kullback-Leibler Divergence.

• If PosteriorMethod is 'qp', then the software estimates multiclass posterior probabilities by solving a least-squares problem using quadratic programming. You need an Optimization Toolbox™ license to use this option. For details, see Posterior Estimation Using Quadratic Programming.

• If you do not request the fourth output argument (Posterior), then the software ignores the value of PosteriorMethod.

Example: 'PosteriorMethod','qp'

'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

Output Arguments

label — Predicted class labels

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

Predicted class labels, returned as a categorical or character array, logical or numeric vector, or cell array of character vectors.

label has the same data type and number of rows as CVMdl.Y.

The software predicts the classification of an observation by assigning the observation to the class yielding the largest negated average binary loss (or, equivalently, the smallest average binary loss).

NegLoss — Negated average binary losses

numeric matrix

Negated average binary losses, returned as a numeric matrix. NegLoss is an n-by-K matrix, where n is the number of observations (size(CVMdl.X,1)) and K is the number of unique classes (size(CVMdl.ClassNames,1)).

PBScore — Positive-class scores

numeric matrix

Positive-class scores for each binary learner, returned as a numeric matrix. PBScore is an n-by-L matrix, where n is the number of observations (size(CVMdl.X,1)) and L is the number of binary learners (size(CVMdl.CodingMatrix,2)).

If the coding matrix varies across folds (that is, the coding scheme is sparserandom or denserandom), then PBScore is empty ([]).

Posterior — Posterior class probabilities

numeric matrix

Posterior class probabilities, returned as a numeric matrix. Posterior is an n-by-K matrix, where n is the number of observations (size(CVMdl.X,1)) and K is the number of unique classes (size(CVMdl.ClassNames,1)).

You must set 'FitPosterior',1 when training the cross-validated ECOC model using fitcecoc in order to request Posterior. Otherwise, the software throws an error.

More About

Binary Loss

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_kj is element (k,j) of the coding design matrix M (that is, the code corresponding to class k of binary learner j).

• 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.

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}), s_j is the score for observation j, and g(y_j,s_j).

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 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.

Algorithms

The software can estimate class posterior probabilities by minimizing the Kullback-Leibler divergence or by using quadratic programming. For the following descriptions of the posterior estimation algorithms, assume that:

• m_kj is the element (k,j) of the coding design matrix M.

• I is the indicator function.

• $${\widehat{p}}_k$$ is the class posterior probability estimate for class k of an observation, k = 1,...,K.

• r_j is the positive-class posterior probability for binary learner j. That is, r_j is the probability that binary learner j classifies an observation into the positive class, given the training data.

Posterior Estimation Using Kullback-Leibler Divergence

By default, the software minimizes the Kullback-Leibler divergence to estimate class posterior probabilities. The Kullback-Leibler divergence between the expected and observed positive-class posterior probabilities is

$$\Delta (r,\widehat{r})={\displaystyle \sum _{j=1}^L{w}_j}\left[r_j\log \frac{r_j}{{\widehat{r}}_j}+\left(1-r_j\right)\log \frac{1-r_j}{1-{\widehat{r}}_j}\right],$$

where $$w_j={\displaystyle \sum _{S_j}{w}_i^{\ast }}$$ is the weight for binary learner j.

• S_j is the set of observation indices on which binary learner j is trained.

• $$w_i^{\ast }$$ is the weight of observation i.

The software minimizes the divergence iteratively. The first step is to choose initial values $${\widehat{p}}_k^{(0)};k=1,\mathrm{...},K$$ for the class posterior probabilities.

• If you do not specify 'NumKLIterations', then the software tries both sets of deterministic initial values described next, and selects the set that minimizes Δ.

  • $${\widehat{p}}_k^{(0)}=1/K;k=1,\mathrm{...},K.$$

  • $${\widehat{p}}_k^{(0)};k=1,\mathrm{...},K$$ is the solution of the system

    $$M_{01}{\widehat{p}}^{(0)}=r,$$

    where M₀₁ is M with all m_kj = –1 replaced with 0, and r is a vector of positive-class posterior probabilities returned by the L binary learners [Dietterich et al.]. The software uses lsqnonneg to solve the system.

• If you specify 'NumKLIterations',c, where c is a natural number, then the software does the following to choose the set $${\widehat{p}}_k^{(0)};k=1,\mathrm{...},K$$, and selects the set that minimizes Δ.

  • The software tries both sets of deterministic initial values as described previously.

  • The software randomly generates c vectors of length K using rand, and then normalizes each vector to sum to 1.

At iteration t, the software completes these steps:

1. Compute

   $${\widehat{r}}_j^{(t)}=\frac{{\displaystyle \sum _{k=1}^K{\widehat{p}}_k^{(t)}}I(m_{kj}=+1)}{{\displaystyle \sum _{k=1}^K{\widehat{p}}_k^{(t)}}I(m_{kj}=+1\cup m_{kj}=-1)}.$$

2. Estimate the next class posterior probability using

   $${\widehat{p}}_k^{(t+1)}={\widehat{p}}_k^{(t)}\frac{{\displaystyle \sum _{j=1}^L{w}_j}\left[r_{j}I\left(m_{kj}=+1\right)+\left(1-r_j\right)I\left(m_{kj}=-1\right)\right]}{{\displaystyle \sum _{j=1}^L{w}_j}\left[{\widehat{r}}_j^{(t)}I\left(m_{kj}=+1\right)+\left(1-{\widehat{r}}_j^{(t)}\right)I\left(m_{kj}=-1\right)\right]}.$$

3. Normalize $${\widehat{p}}_k^{(t+1)};k=1,\mathrm{...},K$$ so that they sum to 1.

4. Check for convergence.

For more details, see [Hastie et al.] and [Zadrozny].

Posterior Estimation Using Quadratic Programming

Posterior probability estimation using quadratic programming requires an Optimization Toolbox license. To estimate posterior probabilities for an observation using this method, the software completes these steps:

1. Estimate the positive-class posterior probabilities, r_j, for binary learners j = 1,...,L.

2. Using the relationship between r_j and $${\widehat{p}}_k$$ [Wu et al.], minimize

   $${\displaystyle \sum }_{j=1}^L{\left[-r_j{\displaystyle \sum _{k=1}^K{\widehat{p}}_k}I\left(m_{kj}=-1\right)+\left(1-r_j\right){\displaystyle \sum _{k=1}^K{\widehat{p}}_k}I\left(m_{kj}=+1\right)\right]}^2$$

   with respect to $${\widehat{p}}_k$$ and the restrictions

   $$\begin{array}{l}0\le {\widehat{p}}_k\le 1\\ {\displaystyle \sum _k{\widehat{p}}_k}=1.\end{array}$$

   The software performs minimization using quadprog (Optimization Toolbox).

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] Dietterich, T., and G. Bakiri. “Solving Multiclass Learning Problems Via Error-Correcting Output Codes.” Journal of Artificial Intelligence Research. Vol. 2, 1995, pp. 263–286.

[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.

[4] 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.

[5] Hastie, T., and R. Tibshirani. “Classification by Pairwise Coupling.” Annals of Statistics. Vol. 26, Issue 2, 1998, pp. 451–471.

[6] Wu, T. F., C. J. Lin, and R. Weng. “Probability Estimates for Multi-Class Classification by Pairwise Coupling.” Journal of Machine Learning Research. Vol. 5, 2004, pp. 975–1005.

[7] Zadrozny, B. “Reducing Multiclass to Binary by Coupling Probability Estimates.” NIPS 2001: Proceedings of Advances in Neural Information Processing Systems 14, 2001, pp. 1041–1048.

Extended Capabilities

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.

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

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