Combining kernels for Gaussian Process Regression

How can I combine (add or multiply) kernels for GPR using fitgpr function?

Answer 1

[Ayush Anand]

Hi,

You can combine multiple predefined kernels for GPR using simple addition or multiplication. Gaussian kernels are inherently defined in a way such that when they are combined through addition or multiplication, the resulting kernel retains the characteristics of a valid Gaussian kernel.

Here's an example of how to combine kernels for GPR using the "fitrgp" function:

% Define XTrain, YTrain
% ...
% Define individual kernel functions 
kernel1 = @(x1,x2,theta) exp(-theta(1)*(pdist2(x1,x2).^2)); %Squared Exponential Kernel
kernel2 = @(x1,x2,theta) (1 + pdist2(x1,x2).^2/(2*theta(1)*theta(2))).^(-theta(2)); %Rational Quadratic Kernel

% Combine kernels by addition or multiplication. theta is a vector of hyperparameters for the combined kernel
combinedKernel = @(x1,x2,theta) ...
    (kernel1(x1,x2,theta(1:1)) + kernel2(x1,x2,theta(2:3)));

% Define initial values for the kernel parameters of combinedKernelAddition
initialTheta = [1, 1, 1]; % Set this as per initial conditions

% Train the GPR model using the combined kernel 
gprMdlAddition = fitrgp(XTrain, YTrain, 'KernelFunction', combinedKernel, ...
    'KernelParameters', initialTheta);

% Now you can use gprMdlAddition to make predictions
% ...

You can refer to the following link for reading more on the available kernels in MATLAB and how to use a custom kernel with "fitrgp" function:

https://www.mathworks.com/help/stats/fitrgp.html

Hope this helps!