Implementation of Correlated Gaussian Random Fields Model

[confused_engineer]

TL;DR

I have recently found an article with a useful algorithm. However, I don't understand it very well; thus, I would like to know if it is already implemented

Hello everyone. I have been recently working in an optimization model in the presence of uncertainty. I have read https://www.researchgate.net/publication/310742108_Efficient_Simulation_of_Stationary_Multivariate_Gaussian_Random_Fields_with_Given_Cross-Covariance in which, a methodology for generating correlated gaussian random fields. Unfortunately, I have no idea on how to implement it.

I was wondering if someone has seen an implementation of this methodology or knows where I can find one.

Best regards.
Confused engineer.

 

[jasonRF]

What exactly are you trying to do?

Here is a standard way that works as long as you don't have too many variables to fit in your computer. Let $N$ correlated random variables arranged in a vector $\mathbf{x}$, with mean $\mathbf{m} = E[\mathbf{x}]$ and covariance matrix $\mathbf{K} = E\left[ \left(\mathbf{x}-\mathbf{m}\right)\left(\mathbf{x}-\mathbf{m}\right)^T\right]$. One realization of this, $\mathbf{x}_i$, can be generated by starting with a vector $\mathbf{y}_i$ that contains independent, unit-variance, zero-mean Gaussian random numbers. Below is Matlab code to show how it can be done. Please do the math and verify that you believe the formula I am using is correct - doing such calculations is how you learn this stuff.

% code assumes you have already defined K as an NxN covariance matrix
% and m as an Nx1 vector
N = 100;
yi = randn(N,1); % unit-variance Gaussian-distributed random numbers
Ksqrt = sqrtm(K);  % matrix square-root
xi = Ksqrt*yi + m;