Has anybody programmed the mcKL expansion?

[confused_engineer]

TL;DR Summary

I have read the article in which this methodology is described, but I cannot understand what is going on. I need help to program it.

Hello everyone. I am trying to implement the mcKL expansion proposed in this article using Matlab and two vectors of correlated data of size 1000*50, meaning 50 realizations of two random processes measured 1000 times. As the article says, if two stochastic processes are correlated, one cannot just use two Karhunen-Loève expansions but must use one of the two special cases proposed in the article.

I am very confused by the mcKl. I can understand equation 26 of the paper; however, once I arrive at equations 27 to 30, I see K_km^ij and C_ij(s,t) defined as functions to each other. I have no idea how to calculate them analytically and much less using Matlab.

Has anybody programmed this before? What are the dimensions of Kkmij? And the ones of Cij(s,t)? What is the set of uncorrelated random variables eta? Can I use pca to obtain the elements of equation 26?

As you can see, I don’t understand this kind of special Karhunen-Loève expansion at all, so if someone could offer me sone guidance on how to perform this I would be extremely thankful.
Best regards.
Confused engineer.

 

[mmwave]

Dear Confused Engineer,

Thank you for reaching out about the mcKL expansion. I understand that it can be confusing to implement this technique, but I can offer some guidance to help you understand and use it in Matlab.

First, let's start with the dimensions of Kkmij and Cij(s,t). Kkmij is a matrix of size N x N, where N is the number of realizations (in your case, N=50). Cij(s,t) is a function that takes in two time points, s and t, and outputs a scalar value. So, the dimensions of Cij(s,t) are 1 x 1.

Next, to calculate Kkmij and Cij(s,t), you will need to use the equations provided in the paper. In particular, equations 27 to 30 describe how to calculate these quantities. You will need to use a loop to calculate Kkmij for each realization and then use this value to calculate Cij(s,t) for each time point.

As for the set of uncorrelated random variables eta, these can be obtained using a principal component analysis (PCA) on your data. This will give you a set of orthogonal basis functions that can be used to represent your correlated data. You can then use these basis functions to calculate Kkmij and Cij(s,t) as described in the paper.

I hope this helps clarify the dimensions and calculations needed for the mcKL expansion. Please feel free to reach out if you have any further questions or need additional assistance.