This example shows how to use a custom kernel function, such as the sigmoid kernel, to train SVM classifiers, and adjust custom kernel function parameters.
Generate a random set of points within the unit circle. Label points in the first and third quadrants as belonging to the positive class, and those in the second and fourth quadrants in the negative class.
rng(1); % For reproducibility n = 100; % Number of points per quadrant r1 = sqrt(rand(2*n,1)); % Random radii t1 = [pi/2*rand(n,1); (pi/2*rand(n,1)+pi)]; % Random angles for Q1 and Q3 X1 = [r1.*cos(t1) r1.*sin(t1)]; % Polar-to-Cartesian conversion r2 = sqrt(rand(2*n,1)); t2 = [pi/2*rand(n,1)+pi/2; (pi/2*rand(n,1)-pi/2)]; % Random angles for Q2 and Q4 X2 = [r2.*cos(t2) r2.*sin(t2)]; X = [X1; X2]; % Predictors Y = ones(4*n,1); Y(2*n + 1:end) = -1; % Labels
Plot the data.
figure;
gscatter(X(:,1),X(:,2),Y);
title('Scatter Diagram of Simulated Data')
Write a function that accepts two matrices in the feature space as inputs, and transforms them into a Gram matrix using the sigmoid kernel.
function G = mysigmoid(U,V) % Sigmoid kernel function with slope gamma and intercept c gamma = 1; c = -1; G = tanh(gamma*U*V' + c); end
Save this code as a file named mysigmoid on your MATLAB® path.
Train an SVM classifier using the sigmoid kernel function. It is good practice to standardize the data.
Mdl1 = fitcsvm(X,Y,'KernelFunction','mysigmoid','Standardize',true);
Mdl1 is a ClassificationSVM classifier containing the estimated parameters.
Plot the data, and identify the support vectors and the decision boundary.
% Compute the scores over a grid
d = 0.02; % Step size of the grid
[x1Grid,x2Grid] = meshgrid(min(X(:,1)):d:max(X(:,1)),...
min(X(:,2)):d:max(X(:,2)));
xGrid = [x1Grid(:),x2Grid(:)]; % The grid
[~,scores1] = predict(Mdl1,xGrid); % The scores
figure;
h(1:2) = gscatter(X(:,1),X(:,2),Y);
hold on
h(3) = plot(X(Mdl1.IsSupportVector,1),...
X(Mdl1.IsSupportVector,2),'ko','MarkerSize',10);
% Support vectors
contour(x1Grid,x2Grid,reshape(scores1(:,2),size(x1Grid)),[0 0],'k');
% Decision boundary
title('Scatter Diagram with the Decision Boundary')
legend({'-1','1','Support Vectors'},'Location','Best');
hold off
You can adjust the kernel parameters in an attempt to improve the shape of the decision boundary. This might also decrease the within-sample misclassification rate, but, you should first determine the out-of-sample misclassification rate.
Determine the out-of-sample misclassification rate by using 10-fold cross validation.
CVMdl1 = crossval(Mdl1); misclass1 = kfoldLoss(CVMdl1); misclass1
misclass1 =
0.1350
The out-of-sample misclassification rate is 13.5%.
Write another sigmoid function, but Set gamma = 0.5;.
function G = mysigmoid2(U,V) % Sigmoid kernel function with slope gamma and intercept c gamma = 0.5; c = -1; G = tanh(gamma*U*V' + c); end
Save this code as a file named mysigmoid2 on your MATLAB® path.
Train another SVM classifier using the adjusted sigmoid kernel. Plot the data and the decision region, and determine the out-of-sample misclassification rate.
Mdl2 = fitcsvm(X,Y,'KernelFunction','mysigmoid2','Standardize',true);
[~,scores2] = predict(Mdl2,xGrid);
figure;
h(1:2) = gscatter(X(:,1),X(:,2),Y);
hold on
h(3) = plot(X(Mdl2.IsSupportVector,1),...
X(Mdl2.IsSupportVector,2),'ko','MarkerSize',10);
title('Scatter Diagram with the Decision Boundary')
contour(x1Grid,x2Grid,reshape(scores2(:,2),size(x1Grid)),[0 0],'k');
legend({'-1','1','Support Vectors'},'Location','Best');
hold off
CVMdl2 = crossval(Mdl2);
misclass2 = kfoldLoss(CVMdl2);
misclass2
misclass2 =
0.0450
After the sigmoid slope adjustment, the new decision boundary seems to provide a better within-sample fit, and the cross-validation rate contracts by more than 66%.
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