MATLAB Examples

# Plot Posterior Probability Regions for SVM Classification Models

This example shows how to predict posterior probabilities of SVM models over a grid of observations, and then plot the posterior probabilities over the grid. Plotting posterior probabilities exposes decision boundaries.

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 = struct with fields: 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.