# kfoldPredict

Predict labels for observations not used for training

## Syntax

## Description

returns
class labels predicted by the cross-validated ECOC model composed
of linear classification models `Label`

= kfoldPredict(`CVMdl`

)`CVMdl`

. That is,
for every fold, `kfoldPredict`

predicts class labels
for observations that it holds out when it trains using all other
observations. `kfoldPredict`

applies the same data
used create `CVMdl`

(see `fitcecoc`

).

Also, `Label`

contains class labels for each
regularization strength in the linear classification models that compose `CVMdl`

.

returns
predicted class labels with additional options specified by one or
more `Label`

= kfoldPredict(`CVMdl`

,`Name,Value`

)`Name,Value`

pair arguments. For example,
specify the posterior probability estimation method, decoding scheme,
or verbosity level.

## Input Arguments

`CVMdl`

— Cross-validated, ECOC model composed of linear classification models

`ClassificationPartitionedLinearECOC`

model
object

Cross-validated, ECOC model composed of linear classification
models, specified as a `ClassificationPartitionedLinearECOC`

model
object. You can create a `ClassificationPartitionedLinearECOC`

model
using `fitcecoc`

and by:

Specifying any one of the cross-validation, name-value pair arguments, for example,

`CrossVal`

Setting the name-value pair argument

`Learners`

to`'linear'`

or a linear classification model template returned by`templateLinear`

To obtain estimates, kfoldPredict applies the same data used
to cross-validate the ECOC model (`X`

and `Y`

).

### 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.*

`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 contains names and descriptions of the built-in functions, where

*y*is the class label for a particular binary learner (in the set {-1,1,0}),_{j}*s*is the score for observation_{j}*j*, and*g*(*y*,_{j}*s*) is the binary loss formula._{j}Value Description Score Domain *g*(*y*,_{j}*s*)_{j}`'binodeviance'`

Binomial deviance (–∞,∞) log[1 + exp(–2 *y*)]/[2log(2)]_{j}s_{j}`'exponential'`

Exponential (–∞,∞) exp(– *y*)/2_{j}s_{j}`'hamming'`

Hamming [0,1] or (–∞,∞) [1 – sign( *y*)]/2_{j}s_{j}`'hinge'`

Hinge (–∞,∞) max(0,1 – *y*)/2_{j}s_{j}`'linear'`

Linear (–∞,∞) (1 – *y*)/2_{j}s_{j}`'logit'`

Logistic (–∞,∞) log[1 + exp(– *y*)]/[2log(2)]_{j}s_{j}`'quadratic'`

Quadratic [0,1] [1 – *y*(2_{j}*s*– 1)]_{j}^{2}/2The software normalizes the binary losses such that the loss is 0.5 when

*y*= 0. Also, the software calculates the mean binary loss for each class._{j}For a custom binary loss function, e.g.,

`customFunction`

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

.`customFunction`

should have this formwhere: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.

By default, if all binary learners are linear classification models using:

SVM, then

`BinaryLoss`

is`'hinge'`

Logistic regression, then

`BinaryLoss`

is`'quadratic'`

**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

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.

To use this option, you must:

Return the fourth output argument (

`Posterior`

).The linear classification models that compose the ECOC models must use logistic regression learners (that is,

`CVMdl.Trained{1}.BinaryLearners{1}.Learner`

must be`'logistic'`

).`PosteriorMethod`

must be`'kl'`

.

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'`

.

To use this option, you must return the fourth output argument (

`Posterior`

) and the linear classification models that compose the ECOC models must use logistic regression learners (that is,`CVMdl.Trained{1}.BinaryLearners{1}.Learner`

must be`'logistic'`

).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.

**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`

— Cross-validated, predicted class labels

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

Cross-validated, predicted class labels, returned as a categorical or character array, logical or numeric matrix, or cell array of character vectors.

In most cases, `Label`

is an *n*-by-*L*
array of the same data type as the observed class labels (`Y`

) used to create
`CVMdl`

. (The software treats string arrays as cell arrays of character
vectors.)
*n* is the number of observations in the predictor data
(`X`

) and *L*
is the number of regularization strengths in the linear classification
models that compose the cross-validated ECOC model. That is,
`Label(`

is the predicted class label for observation * i*,

*)*

`j`

*using the ECOC model of linear classification models that has regularization strength*

`i`

`CVMdl.Trained{1}.BinaryLearners{1}.Lambda(``j`

)

.If `Y`

is a character array and *L* >
1, then `Label`

is a cell array of class labels.

The software assigns the predicted label corresponding to the
class with the largest, negated, average binary loss (`NegLoss`

),
or, equivalently, the smallest average binary loss.

`NegLoss`

— Cross-validated, negated, average binary losses

numeric array

Cross-validated, negated, average binary losses, returned as
an *n*-by-*K*-by-*L* numeric
matrix or array. *K* is the number of distinct classes
in the training data and columns correspond to the classes in `CVMdl.ClassNames`

.
For *n* and *L*, see `Label`

. `NegLoss(`

is
the negated, average binary loss for classifying observation * i*,

*,*

`k`

*)*

`j`

*into class*

`i`

*using the linear classification model that has regularization strength*

`k`

`CVMdl.Trained{1}.BinaryLoss{1}.Lambda(``j`

)

.If

`Decoding`

is`'lossbased'`

, then`NegLoss(i,k,j)`

is the sum of the binary losses divided by the total number of binary learners.If

`Decoding`

is`'lossweighted'`

, then`NegLoss(i,k,j)`

is the sum of the binary losses divided by the number of binary learners for the*k*th class.

For more details, see Binary Loss.

`PBScore`

— Cross-validated, positive-class scores

numeric array

Cross-validated, positive-class scores, returned as an *n*-by-*B*-by-*L* numeric
array. *B* is the number of binary learners in the
cross-validated ECOC model and columns correspond to the binary learners
in `CVMdl.Trained{1}.BinaryLearners`

. For *n* and *L*,
see `Label`

. `PBScore(`

is
the positive-class score of binary learner * i*,

*,*

`b`

*)*

`j`

*b*for classifying observation

*into its positive class, using the linear classification model that has regularization strength*

`i`

`CVMdl.Trained{1}.BinaryLearners{1}.Lambda(``j`

)

.If the coding matrix varies across folds (that is, if the coding
scheme is `sparserandom`

or `denserandom`

),
then `PBScore`

is empty (`[]`

).

`Posterior`

— Cross-validated posterior class probabilities

numeric array

Cross-validated posterior class probabilities, returned as an *n*-by-*K*-by-*L* numeric
array. For dimension definitions, see `NegLoss`

. `Posterior(`

is
the posterior probability for classifying observation * i*,

*,*

`k`

*)*

`j`

*into class*

`i`

*using the linear classification model that has regularization strength*

`k`

`CVMdl.Trained{1}.BinaryLearners{1}.Lambda(``j`

)

.To return posterior probabilities, `CVMdl.Trained{1}.BinaryLearner{1}.Learner`

must
be `'logistic'`

.

## Examples

### Predict *k*-fold Cross-Validation Labels

Load the NLP data set.

`load nlpdata`

`X`

is a sparse matrix of predictor data, and `Y`

is a categorical vector of class labels.

Cross-validate an ECOC model of linear classification models.

rng(1); % For reproducibility CVMdl = fitcecoc(X,Y,'Learner','linear','CrossVal','on');

`CVMdl`

is a `ClassificationPartitionedLinearECOC`

model. By default, the software implements 10-fold cross validation.

Predict labels for the observations that `fitcecoc`

did not use in training the folds.

label = kfoldPredict(CVMdl);

Because there is one regularization strength in `CVMdl`

, `label`

is a column vector of predictions containing as many rows as observations in `X`

.

Construct a confusion matrix.

cm = confusionchart(Y,label);

### Specify Custom Binary Loss

Load the NLP data set. Transpose the predictor data.

```
load nlpdata
X = X';
```

For simplicity, use the label 'others' for all observations in `Y`

that are not `'simulink'`

, `'dsp'`

, or `'comm'`

.

Y(~(ismember(Y,{'simulink','dsp','comm'}))) = 'others';

Create a linear classification model template that specifies optimizing the objective function using SpaRSA.

t = templateLinear('Solver','sparsa');

Cross-validate an ECOC model of linear classification models using 5-fold cross-validation. Specify that the predictor observations correspond to columns.

rng(1); % For reproducibility CVMdl = fitcecoc(X,Y,'Learners',t,'KFold',5,'ObservationsIn','columns'); CMdl1 = CVMdl.Trained{1}

CMdl1 = CompactClassificationECOC ResponseName: 'Y' ClassNames: [comm dsp simulink others] ScoreTransform: 'none' BinaryLearners: {6x1 cell} CodingMatrix: [4x6 double] Properties, Methods

`CVMdl`

is a `ClassificationPartitionedLinearECOC`

model. It contains the property `Trained`

, which is a 5-by-1 cell array holding a `CompactClassificationECOC`

models that the software trained using the training set of each fold.

By default, the linear classification models that compose the ECOC models use SVMs. 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 observationUses 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. Or, you can specify 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 out-of-fold observations.

[label,NegLoss] = kfoldPredict(CVMdl,'BinaryLoss',customBL); idx = randsample(numel(label),10); table(Y(idx),label(idx),NegLoss(idx,1),NegLoss(idx,2),NegLoss(idx,3),... NegLoss(idx,4),'VariableNames',[{'True'};{'Predicted'};... categories(CVMdl.ClassNames)])

`ans=`*10×6 table*
True Predicted comm dsp simulink others
________ _________ _________ ________ ________ _______
others others -1.2319 -1.0488 0.048758 1.6175
simulink simulink -16.407 -12.218 21.531 11.218
dsp dsp -0.7387 -0.11534 -0.88466 -0.2613
others others -0.1251 -0.8749 -0.99766 0.14517
dsp dsp 2.5867 6.4187 -3.5867 -4.4165
others others -0.025358 -1.2287 -0.97464 0.19747
others others -2.6725 -0.56708 -0.51092 2.7453
others others -1.1605 -0.88321 -0.11679 0.43504
others others -1.9511 -1.3175 0.24735 0.95111
simulink others -7.848 -5.8203 4.8203 6.8457

The software predicts the label based on the maximum negated loss.

### Estimate Posterior Class Probabilities

ECOC models composed of linear classification models return posterior probabilities for logistic regression learners only. This example requires the Parallel Computing Toolbox™ and the Optimization Toolbox™

Load the NLP data set and preprocess the data as in Specify Custom Binary Loss.

load nlpdata X = X'; Y(~(ismember(Y,{'simulink','dsp','comm'}))) = 'others';

Create a set of 5 logarithmically-spaced regularization strengths from through .

Lambda = logspace(-6,-0.5,5);

Create a linear classification model template that specifies optimizing the objective function using SpaRSA and to use logistic regression learners.

t = templateLinear('Solver','sparsa','Learner','logistic','Lambda',Lambda);

Cross-validate an ECOC model of linear classification models using 5-fold cross-validation. Specify that the predictor observations correspond to columns, and to use parallel computing.

rng(1); % For reproducibility Options = statset('UseParallel',true); CVMdl = fitcecoc(X,Y,'Learners',t,'KFold',5,'ObservationsIn','columns',... 'Options',Options);

Starting parallel pool (parpool) using the 'local' profile ... Connected to the parallel pool (number of workers: 6).

Predict the cross-validated posterior class probabilities. Specify to use parallel computing and to estimate posterior probabilities using quadratic programming.

[label,~,~,Posterior] = kfoldPredict(CVMdl,'Options',Options,... 'PosteriorMethod','qp'); size(label) label(3,4) size(Posterior) Posterior(3,:,4)

ans = 31572 5 ans = categorical others ans = 31572 4 5 ans = 0.0285 0.0373 0.1714 0.7627

Because there are five regularization strengths:

`label`

is a 31572-by-5 categorical array.`label(3,4)`

is the predicted, cross-validated label for observation 3 using the model trained with regularization strength`Lambda(4)`

.`Posterior`

is a 31572-by-4-by-5 matrix.`Posterior(3,:,4)`

is the vector of all estimated, posterior class probabilities for observation 3 using the model trained with regularization strength`Lambda(4)`

. The order of the second dimension corresponds to`CVMdl.ClassNames`

. Display a random set of 10 posterior class probabilities.

Display a random sample of cross-validated labels and posterior probabilities for the model trained using `Lambda(4)`

.

idx = randsample(size(label,1),10); table(Y(idx),label(idx,4),Posterior(idx,1,4),Posterior(idx,2,4),... Posterior(idx,3,4),Posterior(idx,4,4),... 'VariableNames',[{'True'};{'Predicted'};categories(CVMdl.ClassNames)])

ans = 10×6 table True Predicted comm dsp simulink others ________ _________ __________ __________ ________ _________ others others 0.030275 0.022142 0.10416 0.84342 simulink simulink 3.4954e-05 4.2982e-05 0.99832 0.0016016 dsp others 0.15787 0.25718 0.18848 0.39647 others others 0.094177 0.062712 0.12921 0.71391 dsp dsp 0.0057979 0.89703 0.015098 0.082072 others others 0.086084 0.054836 0.086165 0.77292 others others 0.0062338 0.0060492 0.023816 0.9639 others others 0.06543 0.075097 0.17136 0.68812 others others 0.051843 0.025566 0.13299 0.7896 simulink simulink 0.00044059 0.00049753 0.70958 0.28948

## 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:

*m*is element (_{kj}*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*is the score of binary learner_{j}*j*for an observation.*g*is the binary loss function.$$\widehat{k}$$ is the predicted class for the observation.

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

*Loss-weighted decoding*[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.$$\widehat{k}=\underset{k}{\text{argmin}}\frac{{\displaystyle \sum _{j=1}^{B}\left|{m}_{kj}\right|g}({m}_{kj},{s}_{j})}{{\displaystyle \sum}_{j=1}^{B}\left|{m}_{kj}\right|}.$$

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*is the score for observation

_{j}*j*, and

*g*(

*y*,

_{j}*s*) is the binary loss function.

_{j}Value | Description | Score Domain | g(y,_{j}s)_{j} |
---|---|---|---|

`"binodeviance"` | Binomial deviance | (–∞,∞) | log[1 +
exp(–2y)]/[2log(2)]_{j}s_{j} |

`"exponential"` | Exponential | (–∞,∞) | exp(–y)/2_{j}s_{j} |

`"hamming"` | Hamming | [0,1] or (–∞,∞) | [1 – sign(y)]/2_{j}s_{j} |

`"hinge"` | Hinge | (–∞,∞) | max(0,1 – y)/2_{j}s_{j} |

`"linear"` | Linear | (–∞,∞) | (1 – y)/2_{j}s_{j} |

`"logit"` | Logistic | (–∞,∞) | log[1 +
exp(–y)]/[2log(2)]_{j}s_{j} |

`"quadratic"` | Quadratic | [0,1] | [1 – y(2_{j}s –
1)]_{j}^{2}/2 |

The software normalizes binary losses 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 `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:

*m*is the element (_{kj}*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*is the positive-class posterior probability for binary learner_{j}*j*. That is,*r*is the probability that binary learner_{j}*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

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

where $${w}_{j}={\displaystyle \sum _{{S}_{j}}{w}_{i}^{\ast}}$$ is the weight for binary learner *j*.

*S*is the set of observation indices on which binary learner_{j}*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 $${\widehat{p}}_{k}^{(0)};\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=1,\mathrm{...},K$$ 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.$${\widehat{p}}_{k}^{(0)}=1/K;\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=1,\mathrm{...},K.$$

$${\widehat{p}}_{k}^{(0)};\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=1,\mathrm{...},K$$ is the solution of the system

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

where

*M*_{01}is*M*with all*m*= –1 replaced with 0, and_{kj}*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 $${\widehat{p}}_{k}^{(0)};\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=1,\mathrm{...},K$$, 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:

Compute

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

Estimate the next class posterior probability using

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

Normalize $${\widehat{p}}_{k}^{(t+1)};\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=1,\mathrm{...},K$$ 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,

*r*, for binary learners_{j}*j*= 1,...,*L*.Using the relationship between

*r*and $${\widehat{p}}_{k}$$ [Wu et al.], minimize_{j}$$\sum}_{j=1}^{L}{\left[-{r}_{j}{\displaystyle \sum _{k=1}^{K}{\widehat{p}}_{k}}I\left({m}_{kj}=-1\right)+\left(1-{r}_{j}\right){\displaystyle \sum _{k=1}^{K}{\widehat{p}}_{k}}I\left({m}_{kj}=+1\right)\right]}^{2$$

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.

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## 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).

## Version History

**Introduced in R2016a**

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