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How can I map a Gaussian vector according to an equation that has a spatial autocorrelation function?

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Mohamad Al Hassan
Mohamad Al Hassan 2019-2-23
I have the following function:
I want to run some Monte Carlo simulations and at each simulation I have to generate new values of E at certain locations x and z. So basically I will have to generate a grid of points with coordinates and where I will evaluate E.
is a 0 mean Gaussian random field with a variance equal to 1. It has a spatial autocorrelation function where is the horizontal distance between two points and is the vertical distance between two points. The function R has the folllwing equation:
I know how to generate the vector Y if it has no correlation but I'm not sure how to do it with the autocorrelation function, I did the following but I'm not sure if this is correct:
Generated a grid of points (pts), code below:
x=-50:1:50;
z=0:1:20;
pts=zeros(length(x)*length(z),2);
for i=1:length(x)
for j=1:length(z)
pts((length(z)*(i-1)+1)+j-1,:)=[x(i),z(j)];
end
end
Generated a correlation matrix from the correlation function, code below:
cormat=zeros(length(pts),length(pts));
for i=1:length(pts)
for j=i:length(pts)
dx=pts(i,1)-pts(j,1);
dy=pts(i,2)-pts(j,2);
cormat(i,j)=exp(-abs(dx)/10.-abs(dy)/3.);
end
end
Generated a covariance matrix from the correlation matrix:
cormat=cormat'+cormat-eye(length(pts));
stdmat=eye(length(pts));
covmat=stdmat*cormat*stdmat;
Evaluated the function Y using MATLAB function mvnrnd:
y=mvnrnd(ones(1,length(pts)),covmat);

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