# predict

Classify observations using multiclass error-correcting output codes (ECOC) model

## Syntax

## Description

uses additional options specified by one or more name-value pair arguments. For example, you
can specify the posterior probability estimation method, decoding scheme, and verbosity
level.`label`

= predict(`Mdl`

,`X`

,`Name,Value`

)

`[`

uses any of the input argument combinations in the previous syntaxes and additionally
returns: `label`

,`NegLoss`

,`PBScore`

]
= predict(___)

An array of negated average binary losses (

`NegLoss`

). For each observation in`X`

,`predict`

assigns the label of the class yielding the largest negated average binary loss (or, equivalently, the smallest average binary loss).An array of positive-class scores (

`PBScore`

) for the observations classified by each binary learner.

## Examples

### Predict Test-Sample Labels of Training Data Using ECOC Model

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 an ECOC model using SVM binary classifiers. Specify a 30% holdout sample, standardize the predictors using an SVM template, and specify the class order.

t = templateSVM('Standardize',true); PMdl = fitcecoc(X,Y,'Holdout',0.30,'Learners',t,'ClassNames',classOrder); Mdl = PMdl.Trained{1}; % Extract trained, compact classifier

`PMdl`

is a `ClassificationPartitionedECOC`

model. It has the property `Trained`

, a 1-by-1 cell array containing the `CompactClassificationECOC`

model that the software trained using the training set.

Predict the test-sample labels. Print a random subset of true and predicted labels.

testInds = test(PMdl.Partition); % Extract the test indices XTest = X(testInds,:); YTest = Y(testInds,:); labels = predict(Mdl,XTest); idx = randsample(sum(testInds),10); table(YTest(idx),labels(idx),... 'VariableNames',{'TrueLabels','PredictedLabels'})

`ans=`*10×2 table*
TrueLabels PredictedLabels
__________ _______________
setosa setosa
versicolor virginica
setosa setosa
virginica virginica
versicolor versicolor
setosa setosa
virginica virginica
virginica virginica
setosa setosa
setosa setosa

`Mdl`

correctly labels all except one of the test-sample observations with indices `idx`

.

### Predict Test-Sample Labels of ECOC Model 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 rng(1); % For reproducibility

Train an ECOC model using SVM binary classifiers and specify a 30% holdout sample. Standardize the predictors using an SVM template, and specify the class order.

t = templateSVM('Standardize',true); PMdl = fitcecoc(X,Y,'Holdout',0.30,'Learners',t,'ClassNames',classOrder); Mdl = PMdl.Trained{1}; % Extract trained, compact classifier

`PMdl`

is a `ClassificationPartitionedECOC`

model. It has the property `Trained`

, a 1-by-1 cell array containing the `CompactClassificationECOC`

model that the software trained using the training set.

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

Map the coding design matrix (

*M*) and positive-class classification scores (*s*) for each learner to the binary loss for each observation.Use linear loss.

Aggregate 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. In this case, create a function handle (`customBL`

) to an anonymous binary loss function.

`customBL = @(M,s) median(1 - bsxfun(@times,M,s),2,'omitnan')/2;`

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

testInds = test(PMdl.Partition); % Extract the test indices XTest = X(testInds,:); YTest = Y(testInds,:); [label,NegLoss] = predict(Mdl,XTest,'BinaryLoss',customBL); idx = randsample(sum(testInds),10); classOrder

`classOrder = `*3x1 categorical*
setosa
versicolor
virginica

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

`ans=`*10×3 table*
TrueLabel PredictedLabel NegLoss
__________ ______________ _________________________________
setosa versicolor 0.18572 1.9878 -3.6735
versicolor virginica -1.3316 -0.12335 -0.045018
setosa versicolor 0.139 1.9261 -3.5651
virginica virginica -1.5133 -0.38263 0.39594
versicolor versicolor -0.8721 0.74774 -1.3756
setosa versicolor 0.48384 1.9973 -3.9811
virginica virginica -1.9364 -0.6751 1.1115
virginica virginica -1.579 -0.83337 0.91236
setosa versicolor 0.51003 2.1208 -4.1309
setosa versicolor 0.36121 2.0595 -3.9207

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 Posterior Probabilities Using ECOC Classifier

Train an ECOC classifier using SVM binary learners. First predict the training-sample labels and class posterior probabilities. Then predict the maximum class posterior probability at each point in a grid. Visualize the results.

Load Fisher's iris data set. Specify the petal dimensions as the predictors and the species names as the response.

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

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

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

`t`

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

Train the ECOC classifier using the SVM template. Transform classification scores to class posterior probabilities (which are returned by `predict`

or `resubPredict`

) using the `'FitPosterior'`

name-value pair argument. Specify the class order using the `'ClassNames'`

name-value pair argument. Display diagnostic messages during training by using the `'Verbose'`

name-value pair argument.

Mdl = fitcecoc(X,Y,'Learners',t,'FitPosterior',true,... 'ClassNames',{'setosa','versicolor','virginica'},... 'Verbose',2);

Training binary learner 1 (SVM) out of 3 with 50 negative and 50 positive observations. Negative class indices: 2 Positive class indices: 1 Fitting posterior probabilities for learner 1 (SVM). Training binary learner 2 (SVM) out of 3 with 50 negative and 50 positive observations. Negative class indices: 3 Positive class indices: 1 Fitting posterior probabilities for learner 2 (SVM). Training binary learner 3 (SVM) out of 3 with 50 negative and 50 positive observations. Negative class indices: 3 Positive class indices: 2 Fitting posterior probabilities for learner 3 (SVM).

`Mdl`

is a `ClassificationECOC`

model. The same SVM template applies to each binary learner, but you can adjust options for each binary learner by passing in a cell vector of templates.

Predict the training-sample labels and class posterior probabilities. Display diagnostic messages during the computation of labels and class posterior probabilities by using the `'Verbose'`

name-value pair argument.

`[label,~,~,Posterior] = resubPredict(Mdl,'Verbose',1);`

Predictions from all learners have been computed. Loss for all observations has been computed. Computing posterior probabilities...

Mdl.BinaryLoss

ans = 'quadratic'

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

.

Display a random set of results.

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

`ans = `*3x1 cell*
{'setosa' }
{'versicolor'}
{'virginica' }

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

`ans=`*10×3 table*
TrueLabel PredLabel Posterior
______________ ______________ ______________________________________
{'virginica' } {'virginica' } 0.0039322 0.003987 0.99208
{'virginica' } {'virginica' } 0.017067 0.018263 0.96467
{'virginica' } {'virginica' } 0.014948 0.015856 0.9692
{'versicolor'} {'versicolor'} 2.2197e-14 0.87318 0.12682
{'setosa' } {'setosa' } 0.999 0.00025092 0.00074638
{'versicolor'} {'virginica' } 2.2195e-14 0.05943 0.94057
{'versicolor'} {'versicolor'} 2.2194e-14 0.97001 0.029985
{'setosa' } {'setosa' } 0.999 0.00024991 0.0007474
{'versicolor'} {'versicolor'} 0.0085642 0.98259 0.0088487
{'setosa' } {'setosa' } 0.999 0.00025013 0.00074717

The columns of `Posterior`

correspond to the class order of `Mdl.ClassNames`

.

Define a grid of values in the observed predictor space. Predict the posterior probabilities for each instance in the grid.

xMax = max(X); xMin = min(X); x1Pts = linspace(xMin(1),xMax(1)); x2Pts = linspace(xMin(2),xMax(2)); [x1Grid,x2Grid] = meshgrid(x1Pts,x2Pts); [~,~,~,PosteriorRegion] = predict(Mdl,[x1Grid(:),x2Grid(:)]);

For each coordinate on the grid, plot the maximum class posterior probability among all classes.

contourf(x1Grid,x2Grid,... reshape(max(PosteriorRegion,[],2),size(x1Grid,1),size(x1Grid,2))); h = colorbar; h.YLabel.String = 'Maximum posterior'; h.YLabel.FontSize = 15; hold on gh = gscatter(X(:,1),X(:,2),Y,'krk','*xd',8); gh(2).LineWidth = 2; gh(3).LineWidth = 2; title('Iris Petal Measurements and Maximum Posterior') xlabel('Petal length (cm)') ylabel('Petal width (cm)') axis tight legend(gh,'Location','NorthWest') hold off

### Estimate Test-Sample Posterior Probabilities Using Parallel Computing

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

Load the `arrhythmia`

data set. Examine the response data `Y`

, and determine the number of classes.

```
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%

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 its properties are empty (`[]`

). The software uses default values for all empty properties during training.

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

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

Train an ECOC model using parallel computing. Specify a 15% holdout sample, and fit posterior probabilities.

`pool = parpool; % Invokes workers`

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

options = statset('UseParallel',true); PMdl = fitcecoc(X,Y,'Learner',t,'Options',options,'Coding',Coding,... 'FitPosterior',true,'Holdout',0.15); Mdl = PMdl.Trained{1}; % Extract trained, compact classifier

`PMdl`

is a `ClassificationPartitionedECOC`

model. It has the property `Trained`

, a 1-by-1 cell array containing the `CompactClassificationECOC`

model that the software trained using the training set.

The pool invokes six workers, although the number of workers might vary among systems.

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 test-sample observations.

testInds = test(PMdl.Partition); % Extract the test indices XTest = X(testInds,:); YTest = Y(testInds,:); [~,~,~,posterior] = predict(Mdl,XTest,'Options',options); idx = randsample(sum(testInds),10); table(idx,YTest(idx),posterior(idx,1),... 'VariableNames',{'TestSampleIndex','TrueLabel','PosteriorNoArrhythmia'})

`ans=`*10×3 table*
TestSampleIndex TrueLabel PosteriorNoArrhythmia
_______________ _________ _____________________
11 6 0.60631
41 4 0.23674
51 2 0.13802
33 10 0.43831
12 1 0.94332
8 1 0.97278
37 1 0.62807
24 10 0.96876
56 16 0.29375
30 1 0.64512

## Input Arguments

`Mdl`

— Full or compact multiclass ECOC model

`ClassificationECOC`

model object | `CompactClassificationECOC`

model
object

Full or compact multiclass ECOC model, specified as a
`ClassificationECOC`

or
`CompactClassificationECOC`

model
object.

To create a full or compact ECOC model, see `ClassificationECOC`

or `CompactClassificationECOC`

.

`X`

— Predictor data to be classified

numeric matrix | table

Predictor data to be classified, specified as a numeric matrix or table.

By default, each row of `X`

corresponds to one observation, and
each column corresponds to one variable.

For a numeric matrix:

The variables that constitute the columns of

`X`

must have the same order as the predictor variables that train`Mdl`

.If you train

`Mdl`

using a table (for example,`Tbl`

), then`X`

can be a numeric matrix if`Tbl`

contains all numeric predictor variables. To treat numeric predictors in`Tbl`

as categorical during training, identify categorical predictors using the`CategoricalPredictors`

name-value pair argument of`fitcecoc`

. If`Tbl`

contains heterogeneous predictor variables (for example, numeric and categorical data types) and`X`

is a numeric matrix, then`predict`

throws an error.

For a table:

`predict`

does not support multicolumn variables or cell arrays other than cell arrays of character vectors.If you train

`Mdl`

using a table (for example,`Tbl`

), then all predictor variables in`X`

must have the same variable names and data types as the predictor variables that train`Mdl`

(stored in`Mdl.PredictorNames`

). However, the column order of`X`

does not need to correspond to the column order of`Tbl`

. Both`Tbl`

and`X`

can contain additional variables (response variables, observation weights, and so on), but`predict`

ignores them.If you train

`Mdl`

using a numeric matrix, then the predictor names in`Mdl.PredictorNames`

and the corresponding predictor variable names in`X`

must be the same. To specify predictor names during training, see the`PredictorNames`

name-value pair argument of`fitcecoc`

. All predictor variables in`X`

must be numeric vectors.`X`

can contain additional variables (response variables, observation weights, and so on), but`predict`

ignores them.

**Note**

If `Mdl.BinaryLearners`

contains linear classification models
(`ClassificationLinear`

), then you can orient
your predictor matrix so that observations correspond to columns and specify
`'ObservationsIn','columns'`

. However, you cannot specify
`'ObservationsIn','columns'`

for predictor data in a table.

When training `Mdl`

, assume that you set
`'Standardize',true`

for a template object specified in the
`'Learners'`

name-value pair argument of `fitcecoc`

. In
this case, for the corresponding binary learner `j`

, the software standardizes
the columns of the new predictor data using the corresponding means in
`Mdl.BinaryLearner{j}.Mu`

and standard deviations in
`Mdl.BinaryLearner{j}.Sigma`

.

**Data Types: **`table`

| `double`

| `single`

### 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: **`predict(Mdl,X,'BinaryLoss','quadratic','Decoding','lossbased')`

specifies a quadratic binary learner loss function and a loss-based decoding scheme for
aggregating the binary losses.

`BinaryLoss`

— Binary learner loss function

`'hamming'`

| `'linear'`

| `'logit'`

| `'exponential'`

| `'binodeviance'`

| `'hinge'`

| `'quadratic'`

| function handle

Binary learner loss function, specified as the comma-separated pair consisting of
`'BinaryLoss'`

and a built-in loss function name or function handle.

This table describes the built-in functions, where

*y*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 binary losses so 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, for example

`customFunction`

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

.`customFunction`

has this form:bLoss = customFunction(M,s)

`M`

is the*K*-by-*B*coding matrix stored in`Mdl.CodingMatrix`

.`s`

is the 1-by-*B*row vector of classification scores.`bLoss`

is the classification loss. This scalar aggregates the binary losses for every learner in a particular class. For example, you can use the mean binary loss to aggregate the loss over the learners for each class.*K*is the number of classes.*B*is the number of binary learners.

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

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 modelsLinear or kernel classification models of logistic regression learners Naive Bayes models
| `'quadratic'` |

All binary learners are SVMs or linear or kernel classification models of SVM learners. | `'hinge'` |

All binary learners are ensembles trained by
`AdaboostM1` or
`GentleBoost` . | `'exponential'` |

All binary learners are ensembles trained by
`LogitBoost` . | `'binodeviance'` |

You specify to predict class posterior probabilities by setting
`'FitPosterior',true` in `fitcecoc` . | `'quadratic'` |

Binary learners are heterogeneous and use different loss functions. | `'hamming'` |

To check the default value, use dot notation to display the `BinaryLoss`

property of the trained model at the command line.

**Example: **`'BinaryLoss','binodeviance'`

**Data Types: **`char`

| `string`

| `function_handle`

`Decoding`

— Decoding scheme

`'lossweighted'`

(default) | `'lossbased'`

Decoding scheme that aggregates the binary losses, specified as the comma-separated pair
consisting of `'Decoding'`

and `'lossweighted'`

or
`'lossbased'`

. For more information, see Binary Loss.

**Example: **`'Decoding','lossbased'`

`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`

`ObservationsIn`

— Predictor data observation dimension

`'rows'`

(default) | `'columns'`

Predictor data observation dimension, specified as the comma-separated pair consisting of
`'ObservationsIn'`

and `'columns'`

or
`'rows'`

. `Mdl.BinaryLearners`

must contain
`ClassificationLinear`

models.

**Note**

If you orient your predictor matrix so that
observations correspond to columns and specify
`'ObservationsIn','columns'`

, you
can experience a significant reduction in
execution time. You cannot specify
`'ObservationsIn','columns'`

for
predictor data in a table.

`Options`

— Estimation options

`[]`

(default) | structure array returned by `statset`

Estimation options, specified as the comma-separated pair consisting
of `'Options'`

and a structure array returned by `statset`

.

To invoke parallel computing:

You need a Parallel Computing Toolbox™ license.

Specify

`'Options',statset('UseParallel',true)`

.

`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 array | numeric array | cell array of character vectors

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

The `predict`

function 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). For an
observation with `NaN`

loss values, the function classifies the observation
into the majority class, which makes up the largest proportion of the training labels.

`label`

has the same data type as the class labels used to train
`Mdl`

and has the same number of rows as `X`

.
(The software treats string arrays as cell arrays of character
vectors.)

If `Mdl.BinaryLearners`

contains `ClassificationLinear`

models, then `label`

is an
*m*-by-*L* matrix, where *m* is the
number of observations in `X`

, and *L* is the number
of regularization strengths in the linear classification models
(`numel(Mdl.BinaryLearners{1}.Lambda)`

). The value
`label(i,j)`

is the predicted label of observation
`i`

for the model trained using regularization strength
`Mdl.BinaryLearners{1}.Lambda(j)`

.

Otherwise, `label`

is a column vector of length
*m*.

`NegLoss`

— Negated average binary losses

numeric matrix | numeric array

Negated average binary losses, returned as a numeric matrix or array.

If

`Mdl.BinaryLearners`

contains`ClassificationLinear`

models, then`NegLoss`

is an*m*-by-*K*-by-*L*array.*m*is the number of observations in`X`

.*K*is the number of distinct classes in the training data (`numel(Mdl.ClassNames)`

).*L*is the number of regularization strengths in the linear classification models (`numel(Mdl.BinaryLearners{1}.Lambda)`

).

`NegLoss(i,k,j)`

is the negated average binary loss for observation`i`

, corresponding to class`Mdl.ClassNames(k)`

, for the model trained using regularization strength`Mdl.BinaryLearners{1}.Lambda(j)`

.If

`Decoding`

is`'lossbased'`

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

is the sum of the binary losses divided by the 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.

Otherwise,

`NegLoss`

is an*m*-by-*K*matrix.

`PBScore`

— Positive-class scores

numeric matrix | numeric array

Positive-class scores for each binary learner, returned as a numeric matrix or array.

If

`Mdl.BinaryLearners`

contains`ClassificationLinear`

models, then`PBScore`

is an*m*-by-*B*-by-*L*array.*m*is the number of observations in`X`

.*B*is the number of binary learners (`numel(Mdl.BinaryLearners)`

).*L*is the number of regularization strengths in the linear classification models (`numel(Mdl.BinaryLearners{1}.Lambda)`

).

`PBScore(i,b,j)`

is the positive-class score for observation`i`

, using binary learner`b`

, for the model trained using regularization strength`Mdl.BinaryLearners{1}.Lambda(j)`

.Otherwise,

`PBScore`

is an*m*-by-*B*matrix.

`Posterior`

— Posterior class probabilities

numeric matrix | numeric array

Posterior class probabilities, returned as a numeric matrix or array.

If

`Mdl.BinaryLearners`

contains`ClassificationLinear`

models, then`Posterior`

is an*m*-by-*K*-by-*L*array. For dimension definitions, see`NegLoss`

.`Posterior(i,k,j)`

is the posterior probability that observation`i`

comes from class`Mdl.ClassNames(k)`

, for the model trained using regularization strength`Mdl.BinaryLearners{1}.Lambda(j)`

.Otherwise,

`Posterior`

is an*m*-by-*K*matrix.

## 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.

Suppose 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 decoding scheme of an ECOC model specifies how the software aggregates the binary losses and determines the predicted class for each 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.

Do not confuse the binary loss with the overall classification loss (specified by the
`LossFun`

name-value argument of the `loss`

and
`predict`

object functions), which measures how well an ECOC classifier
performs as a whole.

## 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.

[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

### Tall Arrays

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

Usage notes and limitations:

`predict`

does not support tall`table`

data when`Mdl`

contains kernel or linear binary learners.

For more information, see Tall Arrays.

### C/C++ Code Generation

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

Usage notes and limitations:

You can generate C/C++ code for both

`predict`

and`update`

by using a coder configurer. Or, generate code only for`predict`

by using`saveLearnerForCoder`

,`loadLearnerForCoder`

, and`codegen`

.Code generation for

`predict`

and`update`

— Create a coder configurer by using`learnerCoderConfigurer`

and then generate code by using`generateCode`

. Then you can update model parameters in the generated code without having to regenerate the code.Code generation for

`predict`

— Save a trained model by using`saveLearnerForCoder`

. Define an entry-point function that loads the saved model by using`loadLearnerForCoder`

and calls the`predict`

function. Then use`codegen`

(MATLAB Coder) to generate code for the entry-point function.

To generate single-precision C/C++ code for

`predict`

, specify the name-value argument`"DataType","single"`

when you call the`loadLearnerForCoder`

function.This table contains notes about the arguments of

`predict`

. Arguments not included in this table are fully supported.Argument Notes and Limitations `Mdl`

If you use

`saveLearnerForCoder`

to save a model that is equipped to predict posterior probabilities, and use`loadLearnerForCoder`

to load the model, then`loadLearnerForCoder`

cannot restore the`ScoreTransform`

property into the MATLAB^{®}Workspace. However,`loadLearnerForCoder`

can load the model, including the`ScoreTransform`

property, within an entry-point function at compile time for code generation.For the usage notes and limitations of the model object, see Code Generation of the

`CompactClassificationECOC`

object.

`X`

For general code generation,

`X`

must be a single-precision or double-precision matrix or a table containing numeric variables, categorical variables, or both.In the coder configurer workflow,

`X`

must be a single-precision or double-precision matrix.The number of observations in

`X`

can be a variable size, but the number of variables in`X`

must be fixed.If you want to specify

`X`

as a table, then your model must be trained using a table, and your entry-point function for prediction must do the following:Accept data as arrays.

Create a table from the data input arguments and specify the variable names in the table.

Pass the table to

`predict`

.

For an example of this table workflow, see Generate Code to Classify Data in Table. For more information on using tables in code generation, see Code Generation for Tables (MATLAB Coder) and Table Limitations for Code Generation (MATLAB Coder).

`Posterior`

This output argument is not supported. Name-value pair arguments Names in name-value arguments must be compile-time constants. `BinaryLoss`

The value for the

`'BinaryLoss'`

name-value pair argument must be a compile-time constant. For example, to use the`'BinaryLoss','logit'`

name-value pair argument in the generated code, include`{coder.Constant('BinaryLoss'),coder.Constant('logit')}`

in the`-args`

value of`codegen`

(MATLAB Coder).To set the

`'BinaryLoss'`

name-value pair argument to a custom binary loss function in the generated code, define a custom function on the MATLAB search path, and specify the name of the custom function instead of its function handle. The custom function name must be a compile-time constant. For example, if you define a custom function named`customFunction`

, then include`{coder.Constant('BinaryLoss'),coder.Constant('customFunction')}`

in the`-args`

value of`codegen`

(MATLAB Coder).

`NumKLInitializations`

This name-value pair argument is not supported. `ObservationsIn`

The value for the

`ObservationsIn`

name-value argument must be a compile-time constant. For example, to use the`"ObservationsIn","columns"`

name-value argument in the generated code, include`{coder.Constant("ObservationsIn"),coder.Constant("columns")}`

in the`-args`

value of`codegen`

(MATLAB Coder).`Options`

This name-value pair argument is not supported. `PosteriorMethod`

This name-value pair argument is not supported. `Verbose`

If you plan to generate a MEX file without using a coder configurer, then you can specify `Verbose`

. Otherwise,`codegen`

does not support`Verbose`

.

For more information, see Introduction to Code Generation.

### 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).

### GPU Arrays

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

Usage notes and limitations:

The

`predict`

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

`predict`

function does not support models trained using SVM learners.

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

## Version History

**Introduced in R2014b**

## See Also

`CompactClassificationECOC`

| `ClassificationECOC`

| `fitcecoc`

| `statset`

| `resubPredict`

| `quadprog`

(Optimization Toolbox) | `loss`

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