SVM-Classifier_Training(Gaussian-Kernel).m

%Train SVM Classifiers Using a Gaussian Kernel

%{
This example shows how to generate a nonlinear classifier with Gaussian kernel function. 
First, generate one class of points inside the unit disk in two dimensions, and another class of points 
in the annulus from radius 1 to radius 2. Then, generates a classifier based on the data 
with the Gaussian radial basis function kernel. The default linear classifier is obviously unsuitable 
for this problem, since the model is circularly symmetric. Set the box constraint parameter to Inf 
to make a strict classification, meaning no misclassified training points. Other kernel functions 
might not work with this strict box constraint, since they might be unable to provide a strict classification. 
Even though the rbf classifier can separate the classes, the result can be overtrained.

Generate 100 points uniformly distributed in the unit disk. To do so, generate a radius r as the square root 
of a uniform random variable, generate an angle t uniformly in (0,  ), and put the point at (r cos( t ), r sin( t )).
%}
rng(1); % For reproducibility
r = sqrt(rand(100,1)); % Radius
t = 2*pi*rand(100,1);  % Angle
data1 = [r.*cos(t), r.*sin(t)]; % Points

%{
Generate 100 points uniformly distributed in the annulus. The radius is again proportional to a square root, 
this time a square root of the uniform distribution from 1 through 4.
%}
r2 = sqrt(3*rand(100,1)+1); % Radius
t2 = 2*pi*rand(100,1);      % Angle
data2 = [r2.*cos(t2), r2.*sin(t2)]; % points

%Plot the points, and plot circles of radii 1 and 2 for comparison
figure;
plot(data1(:,1),data1(:,2),'r.','MarkerSize',15)
hold on
plot(data2(:,1),data2(:,2),'b.','MarkerSize',15)
ezpolar(@(x)1);ezpolar(@(x)2);
axis equal
hold off

%Put the data in one matrix, and make a vector of classifications
data3 = [data1;data2];
theclass = ones(200,1);
theclass(1:100) = -1;

%{
Train an SVM classifier with KernelFunction set to 'rbf' and BoxConstraint set to Inf. 
Plot the decision boundary and flag the support vectors
%}
%Train the SVM Classifier
cl = fitcsvm(data3,theclass,'KernelFunction','rbf',...
    'BoxConstraint',Inf,'ClassNames',[-1,1]);

% Predict scores over the grid
d = 0.02;
[x1Grid,x2Grid] = meshgrid(min(data3(:,1)):d:max(data3(:,1)),...
    min(data3(:,2)):d:max(data3(:,2)));
xGrid = [x1Grid(:),x2Grid(:)];
[~,scores] = predict(cl,xGrid);

% Plot the data and the decision boundary
figure;
h(1:2) = gscatter(data3(:,1),data3(:,2),theclass,'rb','.');
hold on
ezpolar(@(x)1);
h(3) = plot(data3(cl.IsSupportVector,1),data3(cl.IsSupportVector,2),'ko');
contour(x1Grid,x2Grid,reshape(scores(:,2),size(x1Grid)),[0 0],'k');
legend(h,{'-1','+1','Support Vectors'});
axis equal
hold off

%fitcsvm generates a classifier that is close to a circle of radius 1. The difference is due to the random training data.

%{
Training with the default parameters makes a more nearly circular classification boundary, 
but one that misclassifies some training data. Also, the default value of BoxConstraint is 1, 
and, therefore, there are more support vectors.
%}
cl2 = fitcsvm(data3,theclass,'KernelFunction','rbf');
[~,scores2] = predict(cl2,xGrid);

figure;
h(1:2) = gscatter(data3(:,1),data3(:,2),theclass,'rb','.');
hold on
ezpolar(@(x)1);
h(3) = plot(data3(cl2.IsSupportVector,1),data3(cl2.IsSupportVector,2),'ko');
contour(x1Grid,x2Grid,reshape(scores2(:,2),size(x1Grid)),[0 0],'k');
legend(h,{'-1','+1','Support Vectors'});
axis equal
hold off