generate correlated random values in two dimension

Hello

I want to generate a random field of values like this figure, where each element has the same mean and COV values and the random values vary based on a correlation function (e.g. Markovian)..

Could you please give me a suggestion how to generate random values in two dimension which are correlated based on a correlation function?

Answer 1

[Shubh Dhyani]

Hi Pooneh,

I understand that you want to generate random values in two dimension which are correlated based on a correlation function.

Here are the steps to generate a two-dimensional Gaussian random field with specified mean, covariance, and correlation function in MATLAB:

1. Define the correlation function. For example, the Markovian correlation function can be defined as C(x, y, a) = exp(-sqrt((x - y).^2) / a), where x and y are the spatial coordinates and a is the correlation length.

2. Define the grid of points where you want to generate the random field. For example, you can create a 2D grid of n by n points using the “meshgrid” function with the “linspace” function to define the range of x and y coordinates. You can refer to these links for the same:

https://in.mathworks.com/help/matlab/ref/meshgrid.html

https://in.mathworks.com/help/matlab/ref/linspace.html

3. Calculate the covariance matrix for the grid of points based on the correlation function. For example, you can use a nested for-loop to compute the covariance between each pair of points using the correlation function.

4. Generate a set of uncorrelated Gaussian random variables with zero mean and unit variance using the “randn” function. You can refer to the link for help regarding the function:

https://in.mathworks.com/help/matlab/ref/randn.html

5. Use the Cholesky decomposition to transform the uncorrelated variables to a set of correlated variables with the desired covariance matrix. The “chol” function can be used to compute the Cholesky decomposition. Here’s the link you can refer for the same :

https://in.mathworks.com/help/matlab/ref/chol.html

6. Transform the correlated variables to the desired random field with the specified mean and standard deviation. For example, you can use the reshape function to convert the correlated variables to a 2D matrix and then apply a linear transformation to set the mean and standard deviation of the random field.

[Pooneh Shah Malekpoor]

Thanks for your reponse. In this regard, I have written this code:

tetalnx=200; %horizontal correlation length
tetalny=1.5;%vertical correlation length
SDelementKH=0.5016;
meanelementKH=-2.346;
x=linspace(0.25,49.75,100);
y=linspace(0.25,49.75,100);
[X,Y] = meshgrid(x,y);           %coordinates of the centre of the elements in the grid 
% i=0;
% j=0;
for j=1:100
    for i=1:100
%     for z=1:100
%         for w=1:100
cov(j,i)=(SDelementKH^2)*exp(-sqrt((2*(x(j)-X(j,i))./tetalnx).^2+(2*(y(j)-Y(j,i))./tetalny).^2)); %covariance based on the distance between the points as the centres of the elements in the mesh grid
         end 
end
L=chol(cov);
kkk=randn(100); %Generate a set of uncorrelated Gaussian random variables with zero mean and unit variance using the "randn”
Kh=exp((L.*kkk)+(meanelementKH));

but it doesnot give out correct covariance matrix esp in vertical direction. Where am I wrong?

Thanks

Pooneh

[Shubh Dhyani]

Hi Pooneh,

There could be other reasons why the code is not giving the correct covariance matrix, especially in the vertical direction. Here is one possible fix:

1. The “chol” function computes the Cholesky decomposition of a matrix, which returns a lower triangular matrix L such that ‘cov = L*L'. In your original code, you did not specify the lower parameter in the chol function, so it defaulted to computing the upper triangular matrix U instead of L. This caused a problem when you later tried to multiply L with the random vector kkk, resulting in an error.

2. You can remove the element-wise multiplication (.*) between L and kkk in the expression Kh=exp((L.*kkk)+(meanelementKH)) , since kkk is a matrix and L is a lower triangular matrix. Instead, you can use matrix multiplication (*) to compute the product L*kkk.

The modified code should look like this,

tetalnx=200; %horizontal correlation length
tetalny=1.5;%vertical correlation length
SDelementKH=0.5016;
meanelementKH=-2.346;
x=linspace(0.25,49.75,100);
y=linspace(0.25,49.75,100);
[X,Y] = meshgrid(x,y);           %coordinates of the centre of the elements in the grid 
% i=0;
% j=0;
for j=1:100
    for i=1:100
%     for z=1:100
%         for w=1:100
cov(j,i)=(SDelementKH^2)*exp(-sqrt((2*(x(j)-X(j,i))./tetalnx).^2+(2*(y(j)-Y(j,i))./tetalny).^2)); %covariance based on the distance between the points as the centres of the elements in the mesh grid
         end 
end
L=chol(cov,"lower");
kkk=randn(100); %Generate a set of uncorrelated Gaussian random variables with zero mean and unit variance using the "randn"
Kh=exp((L*kkk)+(meanelementKH));