kfoldPredict
Classify observations in cross-validated ECOC model
Syntax
Description
returns class labels predicted by the cross-validated ECOC model (label
= kfoldPredict(CVMdl
)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).
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
= kfoldPredict(CVMdl
,Name,Value
)
[
additionally returns negated values of the average binary loss per class
(label
,NegLoss
,PBScore
]
= kfoldPredict(___)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 ([]
).
[
additionally returns posterior class probability estimates for validation-fold
observations (label
,NegLoss
,PBScore
,Posterior
]
= kfoldPredict(___)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 . 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)median(1 - (M.*s),2,'omitnan')/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 0.37139 2.1298 -4.0012
versicolor versicolor -1.2169 0.3669 -0.65001
setosa versicolor 0.23932 2.0794 -3.8187
virginica virginica -1.9151 -0.19958 0.61472
versicolor versicolor -1.3746 0.45537 -0.58078
setosa versicolor 0.20061 2.2774 -3.978
virginica versicolor -1.4926 0.090735 -0.098156
virginica virginica -1.7666 -0.13461 0.4012
setosa versicolor 0.19994 1.9111 -3.611
setosa versicolor 0.16087 1.9681 -3.629
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 0.0086404 0.98243 0.0089302
versicolor virginica 2.2197e-14 0.12448 0.87552
setosa setosa 0.999 0.00022837 0.00076884
versicolor versicolor 2.2194e-14 0.98916 0.010845
virginica virginica 0.01232 0.012926 0.97475
virginica virginica 0.0015569 0.0015636 0.99688
virginica virginica 0.0042886 0.0043547 0.99136
setosa setosa 0.999 0.00028329 0.00071382
virginica virginica 0.0094727 0.0098238 0.9807
setosa setosa 0.999 0.00013558 0.00086196
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
) tocrossval
.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 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: 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 yj is the class label for a particular binary learner (in the set {–1,1,0}), sj is the score for observation j, and g(yj,sj) 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 yj = 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 inMdl.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:
| 'quadratic' |
All binary learners are SVMs. | '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'
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
Estimation options, specified as a structure array as returned by statset
.
To invoke parallel computing you need a Parallel Computing Toolbox™ license.
Example: Options=statset(UseParallel=true)
Data Types: struct
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 ofPosteriorMethod
.
Example: 'PosteriorMethod','qp'
Verbose
— Verbosity level
0
(default) | 1
Verbosity level, specified as 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)
).
NegLoss(i,k)
is the negated average binary loss for classifying observation
i into the kth class.
If
Decoding
is'lossbased'
, thenNegLoss(i,k)
is the negated sum of the binary losses divided by the total number of binary learners.If
Decoding
is'lossweighted'
, thenNegLoss(i,k)
is the negated sum of the binary losses divided by the number of binary learners for the kth class.
For more details, see Binary Loss.
PBScore
— Positive-class scores
numeric matrix
Positive-class scores for each binary learner, returned as a numeric
matrix. PBScore
is an
n-by-B matrix, where
n is the number of observations
(size(CVMdl.X,1)
) and B 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
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:
mkj 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.
sj is the score of binary learner j for an observation.
g is the binary loss function.
is the predicted class for the observation.
The software supports two decoding schemes:
Loss-based decoding [3] (
Decoding
is"lossbased"
) — The predicted class of an observation corresponds to the class that produces the minimum average of the binary losses over all binary learners.Loss-weighted decoding [4] (
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.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 yj is a class label for a particular binary learner (in the set {–1,1,0}), sj is the score for observation j, and g(yj,sj) 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 yj = 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 kfoldLoss
and
kfoldPredict
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:
mkj is the element (k,j) of the coding design matrix M.
I is the indicator function.
is the class posterior probability estimate for class k of an observation, k = 1,...,K.
rj is the positive-class posterior probability for binary learner j. That is, rj 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
where is the weight for binary learner j.
Sj is the set of observation indices on which binary learner j is trained.
is the weight of observation i.
The software minimizes the divergence iteratively. The first step is to choose initial values 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.is the solution of the system
where M01 is M with all mkj = –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
, wherec
is a natural number, then the software does the following to choose the set , 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 usingrand
, and then normalizes each vector to sum to 1.
At iteration t, the software completes these steps:
Compute
Estimate the next class posterior probability using
Normalize so that they sum to 1.
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:
Estimate the positive-class posterior probabilities, rj, for binary learners j = 1,...,L.
Using the relationship between rj and [Wu et al.], minimize
with respect to and the restrictions
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. “Separability of ternary codes for sparse designs of error-correcting output codes.” Pattern Recog. Lett. Vol. 30, Issue 3, 2009, pp. 285–297.
[4] 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.
[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
:
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™.
This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Version History
Introduced in R2014b
See Also
ClassificationPartitionedECOC
| ClassificationECOC
| edge
| fitcecoc
| statset
| predict
| quadprog
(Optimization Toolbox)
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