Calculation of Fourier coefficients using SAMBA methodology

[confused_engineer]

TL;DR Summary

I am working with the SAMBA article. I have the random variables and the reconstructions. However, I don't know how to calculate the Fourier coefficients.

Hello everyone.

I have 4 samples of 50 elements from 4 unknown random variables obtained from a Karhunen-Loève decomposition using Matlab's pca (each one is a column of size 50 from the coefficient matrix). I am following the article SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos.

I have managed to calculate the reconstruction of the original data, its mean and standard deviation as per equation 42. The 4 unknown random variables are the coeffs from Matlab.

To arrive to these results, I have built 4 matrices J (see eq 21) and obtained the corresponding eigenvalues (weights) and eigenvectors (nodes), I have obtained the moments of each coeff and go on from there. Since I am not using sparse grids, this leaves me with 2⁴=16 combinations of nodes and their associated weights. The reconstruction of the original data using Matlab's scores multiplied by the value of these nodes leaves me with 16 reconstructions (each one is the sum of the score multiplied by the node, also, I am calculating 4 moments for each random variable). If I calculate the mean and std using the weights, I get results similar to the original data.

Unfortunately, I am lost trying to calculate the Fourier coefficients (or their approximation, α with a hat) between equations 28 and 30. I don't understand very well what are Y and ψ. I think that they might be the reconstruction measured at the same points that the original data is and the reconstruction generated using each random variable respectively. Can someone please tell me if this is the right idea?

I apologize if I am not making my post easy to understand, but I also don't understand very well what I am doing. Any help is appreciated. I will answer any asked question the best I can.

Best regards.
Confused engineer.

 

[mmwave]

Hello Confused engineer,

Thank you for sharing your progress and questions with us. It sounds like you are on the right track with your calculations and are making good progress in understanding the SAMBA method. To answer your question about the Fourier coefficients, Y and ψ represent the original data and the reconstruction generated using each random variable, respectively. The Fourier coefficient, α, is a measure of the correlation between the original data and the reconstruction at a specific frequency or wavelength. It is calculated by taking the dot product of Y and ψ and dividing by the norm of ψ.

I would suggest reviewing the equations and definitions in the SAMBA article again, as well as consulting with a colleague or mentor who is familiar with the method. It can also be helpful to break down the equations and calculations step by step to better understand each component and its purpose.

I hope this helps clear up some confusion and I wish you the best of luck in your research. Don't hesitate to ask any further questions or provide updates on your progress. Science is all about collaboration and learning from each other. Keep up the good work!
Scientist