how to construct a gaussian process kernel in 2D

Say a gaussian process regression take with predictor X in 2D, i.e.

X = [x1, x2]

I am wondering how to construct a kernel function in 2D for

fitrgp(X, y, 'KernelFunction', kfcn)

In 1D input case, kernel could be in shape as:

kfcn = @(XN,XM,theta) (exp(theta(2))^2)*exp(-(pdist2(XN,XM).^2)/(2*exp(theta(1))^2));

But in 2D, I am confused how to define the function with indivial magnitude and length scale parameters for x1 and x2.

Perhaps in form as following?

% additive kernel
kfcn = @(XN,XM,theta) (exp(theta(2))^2)*exp(-(pdist2(XN(:,1),XM(:,1)).^2)/(2*exp(theta(1))^2))...
    + (exp(theta(4))^2)*exp(-(pdist2(XN(:,2),XM(:,2)).^2)/(2*exp(theta(3))^2));

or

% multiplication kernel
kfcn = @(XN,XM,theta) (exp(theta(2))^2)*exp(-(pdist2(XN(:,1),XM(:,1)).^2)/(2*exp(theta(1))^2)) ...
    * (exp(theta(4))^2)*exp(-(pdist2(XN(:,2),XM(:,2)).^2)/(2*exp(theta(3))^2));

Answer 1

[Yash]

Hi Mono,

For a 2D input case, you can define a kernel function that takes two inputs and returns a scalar value. One possible way to define a kernel function is to use the squared exponential kernel, which is a popular choice for Gaussian process regression. The squared exponential kernel can be defined as follows:

% squared exponential kernel
kfcn = @(XN,XM,theta) (exp(theta(1))^2)*exp(-0.5*(pdist2(XN,XM).^2)/exp(theta(2))^2);

Here, XN and XM are the input matrices, where each row represents a 2D input point, and theta is a vector of hyperparameters that control the magnitude and length scale of the kernel. The first element of theta controls the magnitude of the kernel, and the second element controls the length scale.

The above kernel function assumes that the same length scale applies to both dimensions of the input. If you want to use different length scales for each dimension, you can modify the kernel function as follows:

% modified squared exponential kernel
kfcn = @(XN,XM,theta) (exp(theta(1))^2)*exp(-0.5*(pdist2(XN(:,1),XM(:,1)).^2)/exp(theta(2))^2) ...
    .* exp(-0.5*(pdist2(XN(:,2),XM(:,2)).^2)/exp(theta(3))^2);

Here, the first element of theta controls the magnitude of the kernel, the second element controls the length scale for the first dimension, and the third element controls the length scale for the second dimension.

Hope this helps.

[mono]

It works, thank you.