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# predict

Class: CompactClassificationSVM

Predict labels for support vector machine classifiers

## Syntax

• label = predict(SVMModel,X) example
• [label,Score] = predict(SVMModel,X) example

## Description

example

label = predict(SVMModel,X) returns a vector of predicted class labels for predictor data X, based on the full or compact, trained SVM classifier SVMModel.

example

[label,Score] = predict(SVMModel,X) additionally returns class likelihood measures, i.e., either scores or posterior probabilities.

## Input Arguments

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### SVMModel — SVM classifierClassificationSVM classifier | CompactClassificationSVM classifier

SVM classifier, specified as a ClassificationSVM classifier or CompactClassificationSVM classifier returned by fitcsvm or compact, respectively.

### X — Predictor datanumeric matrix

Predictor data, specified as a numeric matrix.

Each row of X corresponds to one observation (also known as an instance or example), and each column corresponds to one variable (also known as a feature). The variables making up the columns of X should be the same as the variables that trained the SVMModel classifier.

The length of Y and the number of rows of X must be equal.

If you set 'Standardize',true in fitcsvm to train SVMModel, then the software standardizes the columns of X using the corresponding means in SVMModel.Mu and standard deviations in SVMModel.Sigma.

Data Types: double | single

## Output Arguments

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### label — Predicted class labelscategorical array | character array | logical vector | vector of numeric values | cell array of strings

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

label:

• Is the same data type as the observed class labels (Y) that trained SVMModel

• Has length equal to the number of rows of X

For one-class learning, the elements of label are the one class represented in the observed class labels.

### Score — Predicted class scores or posterior probabilitiesnumeric column vector | numeric matrix

Predicted class scores or posterior probabilities, returned as a numeric column vector or numeric matrix.

• For one-class learning, Score is a column vector with the same number of rows as the training observations (X). The elements are the positive class scores for the corresponding observations. You cannot obtain posterior probabilities for one-class learning.

• For two-class learning, Score is a two column matrix with the same number of rows as X.

• If you fit the optimal score-to-posterior probability transformation function using fitPosterior or fitSVMPosterior, then Score contains class posterior probabilities. That is, if the value of SVMModel.ScoreTransform is not none, then the elements of the first and second columns of Score are the negative class (SVMModel.ClassNames{1}) and positive class (SVMModel.ClassNames{2}) posterior probabilities for the corresponding observations, respectively.

• Otherwise, the elements of the first column are the negative class scores and the elements of the second column are the positive class scores for the corresponding observations.

Data Types: double | single

## Definitions

### Score

The SVM score for classifying observation x is the signed distance from x to the decision boundary ranging from -∞ to +∞. A positive score for a class indicates that x is predicted to be in that class, a negative score indicates otherwise.

The score is also the numerical, predicted response for x, $f\left(x\right)$, computed by the trained SVM classification function

$f\left(x\right)=\sum _{j=1}^{n}{\alpha }_{j}{y}_{j}G\left({x}_{j},x\right)+b,$

where $\left({\alpha }_{1},...,{\alpha }_{n},b\right)$ are the estimated SVM parameters, $G\left({x}_{j},x\right)$ is the dot product in the predictor space between x and the support vectors, and the sum includes the training set observations.

### Posterior Probability

The probability that an observation belongs in a particular class, given the data.

For SVM, the posterior probability is a function of the score, P(s), that observation j is in class k = {-1,1}.

• For separable classes, the posterior probability is the step function

$P\left({s}_{j}\right)=\left\{\begin{array}{l}\begin{array}{cc}0;& s<\underset{{y}_{k}=-1}{\mathrm{max}}{s}_{k}\end{array}\\ \begin{array}{cc}\pi ;& \underset{{y}_{k}=-1}{\mathrm{max}}{s}_{k}\le {s}_{j}\le \underset{{y}_{k}=+1}{\mathrm{min}}{s}_{k}\end{array}\\ \begin{array}{cc}1;& {s}_{j}>\underset{{y}_{k}=+1}{\mathrm{min}}{s}_{k}\end{array}\end{array},$

where:

• sj is the score of observation j.

• +1 and –1 denote the positive and negative classes, respectively.

• π is the prior probability that an observation is in the positive class.

• For inseparable classes, the posterior probability is the sigmoid function

$P\left({s}_{j}\right)=\frac{1}{1+\mathrm{exp}\left(A{s}_{j}+B\right)},$

where the parameters A and B are the slope and intercept parameters.

### Prior Probability

The prior probability is the believed relative frequency that observations from a class occur in the population for each class.

## Examples

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### Label Test Sample Observations of SVM Classifiers

load ionosphere
rng(1); % For reproducibility


Train an SVM classifier. Specify a 15% holdout sample for testing. It is good practice to specify the class order and standardize the data.

CVSVMModel = fitcsvm(X,Y,'Holdout',0.15,'ClassNames',{'b','g'},...
'Standardize',true);
CompactSVMModel = CVSVMModel.Trained{1}; % Extract trained, compact classifier
testInds = test(CVSVMModel.Partition);   % Extract the test indices
XTest = X(testInds,:);
YTest = Y(testInds,:);


CVSVMModel is a ClassificationPartitionedModel classifier. It contains the property Trained, which is a 1-by-1 cell array holding a CompactClassificationSVM classifier that the software trained using the training set.

Label the test sample observations. Display the results for the first 10 observations in the test sample.

[label,score] = predict(CompactSVMModel,XTest);
table(YTest(1:10),label(1:10),score(1:10,2),'VariableNames',...
{'TrueLabel','PredictedLabel','Score'})

ans =

TrueLabel    PredictedLabel     Score
_________    ______________    ________

'b'          'b'                -1.7178
'g'          'g'                 2.0003
'b'          'b'                -9.6847
'g'          'g'                 2.5619
'b'          'b'                -1.5481
'g'          'g'                 2.0984
'b'          'b'                -2.7017
'b'          'b'               -0.66307
'g'          'g'                 1.6047
'g'          'g'                 1.7731



### Predict Labels and Posterior Probabilities of SVM Classifiers

A goal of classification is to predict labels of new observations using a trained algorithm. Many applications train algorithms on large data sets, which can use resources that are better used elsewhere. This example shows how to efficiently label new observations using an SVM classifier.

Load the ionosphere data set. Suppose that the last 10 observations become available after training the SVM classifier.

load ionosphere

n = size(X,1);       % Training sample size
isInds = 1:(n-10);   % In-sample indices
oosInds = (n-9):n;   % Out-of-sample indices


Train an SVM classifier. It is good practice to standardize the predictors and specify the order of the classes. Conserve memory by reducing the size of the trained SVM classifier.

SVMModel = fitcsvm(X(isInds,:),Y(isInds),'Standardize',true,...
'ClassNames',{'b','g'});
CompactSVMModel = compact(SVMModel);
whos('SVMModel','CompactSVMModel')

  Name                 Size             Bytes  Class                                                 Attributes

CompactSVMModel      1x1              29576  classreg.learning.classif.CompactClassificationSVM
SVMModel             1x1             137545  ClassificationSVM



The positive class is 'g'. The CompactClassificationSVM classifier (CompactSVMModel) uses less space than the ClassificationSVM classifier (SVMModel) because the latter stores the data.

Estimate the optimal score-to-posterior-probability-transformation function.

CompactSVMModel = fitPosterior(CompactSVMModel,...
X(isInds,:),Y(isInds))

CompactSVMModel =

classreg.learning.classif.CompactClassificationSVM
PredictorNames: {1x34 cell}
ResponseName: 'Y'
ClassNames: {'b'  'g'}
ScoreTransform: '@(S)sigmoid(S,-1.968351e+00,3.122242e-01)'
Alpha: [88x1 double]
Bias: -0.2142
KernelParameters: [1x1 struct]
Mu: [1x34 double]
Sigma: [1x34 double]
SupportVectors: [88x34 double]
SupportVectorLabels: [88x1 double]



The optimal score transformation function (CompactSVMModel.ScoreTransform) is the sigmoid function because the classes are inseparable.

Predict the out-of-sample labels and positive class posterior probabilities. Since true labels are available, compare them with the predicted labels.

[labels,PostProbs] = predict(CompactSVMModel,X(oosInds,:));
table(Y(oosInds),labels,PostProbs(:,2),'VariableNames',...
{'TrueLabels','PredictedLabels','PosClassPosterior'})

ans =

TrueLabels    PredictedLabels    PosClassPosterior
__________    _______________    _________________

'g'           'g'                0.98419
'g'           'g'                0.95545
'g'           'g'                0.67792
'g'           'g'                0.94447
'g'           'g'                0.98744
'g'           'g'                 0.9248
'g'           'g'                 0.9711
'g'           'g'                0.96986
'g'           'g'                0.97803
'g'           'g'                0.94361



PostProbs is a 10-by-2 matrix; its first column is the negative class posterior probabilities, and second column is the positive class posterior probabilities corresponding to the new observations.

### Plot Posterior Probability Regions for SVM Classifiers

Load Fisher's iris data set. Train the classifier using the petal lengths and widths, and remove the virginica species from the data.

load fisheriris
classKeep = ~strcmp(species,'virginica');
X = meas(classKeep,3:4);
y = species(classKeep);


Train an SVM classifier using the data. It is good practice to specify the order of the classes.

SVMModel = fitcsvm(X,y,'ClassNames',{'setosa','versicolor'});


Estimate the optimal score transformation function.

rng(1); % For reproducibility
[SVMModel,ScoreParameters] = fitPosterior(SVMModel);
ScoreParameters

Warning: Classes are perfectly separated. The optimal score-to-posterior
transformation is a step function.

ScoreParameters =

Type: 'step'
LowerBound: -0.8431
UpperBound: 0.6897
PositiveClassProbability: 0.5000



The optimal score transformation function is the step function because the classes are separable. The fields LowerBound and UpperBound of ScoreParameters indicate the lower and upper end points of the interval of scores corresponding to observations within the class-separating hyperplanes (the margin). No training observation falls within the margin. If a new score is in the interval, then the software assigns the corresonding observation a positive class posterior probability, i.e., the value in the PositiveClassProbability field of ScoreParameters.

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);
d = 0.01;
[x1Grid,x2Grid] = meshgrid(xMin(1):d:xMax(1),xMin(2):d:xMax(2));

[~,PosteriorRegion] = predict(SVMModel,[x1Grid(:),x2Grid(:)]);


Plot the positive class posterior probability region and the training data.

figure;
contourf(x1Grid,x2Grid,...
reshape(PosteriorRegion(:,2),size(x1Grid,1),size(x1Grid,2)));
h = colorbar;
h.Label.String = 'P({\it{versicolor}})';
h.YLabel.FontSize = 16;
caxis([0 1]);
colormap jet;

hold on
gscatter(X(:,1),X(:,2),y,'mc','.x',[15,10]);
sv = X(SVMModel.IsSupportVector,:);
plot(sv(:,1),sv(:,2),'yo','MarkerSize',15,'LineWidth',2);
axis tight
hold off


In two-class learning, if the classes are separable, then there are three regions: one where observations have positive class posterior probability 0, one where it is 1, and the other where it is the postiive class prior probability.

## Algorithms

• By default, the software computes optimal posterior probabilities using Platt's method [1]:

1. Performing 10-fold cross validation

2. Fitting the sigmoid function parameters to the scores returned from the cross validation

3. Estimating the posterior probabilities by entering the cross-validation scores into the fitted sigmoid function

• The software incorporates prior probabilities in the SVM objective function during training.

• For SVM, predict classifies observations into the class yielding the largest score (i.e., the largest posterior probability). The software accounts for misclassification costs by applying the average-cost correction before training the classifier. That is, given the class prior vector P, misclassification cost matrix C, and observation weight vector w, the software defines a new vector of observation weights (W) such that

${W}_{j}={w}_{j}{P}_{j}\sum _{k=1}^{K}{C}_{jk}.$

## References

[1] Platt, J. "Probabilistic outputs for support vector machines and comparisons to regularized likelihood methods." In Advances in Large Margin Classifiers. MIT Press, 1999, pages 61–74.