Simulating a 2D- Gaussian field. Results are not what I expected

Hi guys. So basically i want to do somthing simulate a 2D gaussian field from a RBF kernal where i can specify the hyper paramters and and area for which the field is generated over. Basically I want to do exactly this

Went into this thinking it should be simple but either I was wrong or im missing somthing. What im getting is slighty off what i want

the first image is for a low length scale and the second is for a length scale of 100. I dont understand why its stretching on the Y-axis like that

N=200;
x=linspace(0,10,N);
y=linspace(0,10,N);

[X,Y]=meshgrid(x,y);

%covarience matrix from RBF kernal
covm=Dkernal(1,X,Y,100);
%fix pos semi def error
covm = covm+.0001 * eye(N);

%generate gaussian process
R=chol(covm);
z = randn(N,N);
mu = zeros(N,1)';
x = mu + R'*z;

imagesc(x);

function [cov] = Dkernal(sigma,x,y,l)
    cov=sigma^2*exp(-(squareform(pdist(x.'))).^2/(2*l^2))+exp(-(squareform(pdist(y.'))).^2/(2*l^2));
end

Thats the code im using. Im pretty sure the error is somwhere in the kernal function but i cant be sure. Any advice on this would be appreciated

Answer 1

[Amish]

Hi Dave,

I understand that you’re trying to simulate a Gaussian field from an RBF kernel and want to figure out the issue with your existing code for the same.

Your intuition about the problem being in the Kernel’s function is correct as there is a slight modification needed in it.

It could be modified to use the ‘pdist2’ method to find the pairwise distances between the ‘x’ and ‘y’ points. Also, ensure proper cross multiplication for the calculation of ‘x’.

The modifications are as follows:

x = mu + z * R'; % Note the transpose
% modified kernel function
function [cov] = Dkernel(sigma, x, y, l)
cov = sigma^2 * exp(-pdist2(x, y).^2 / (2 * l^2));
end

You could also look at the documentation for reference:

https://www.mathworks.com/help/releases/R2021b/stats/pdist2.html

Hope this helps!