Question about the PCA function

[confused_engineer]

TL;DR Summary

The results obtained from appying the pca command dont' match with the theoretical results. The obtained random variables should be standard gaussian. However, I don't obtain that.

Greetings everyone.

I have generated a gaussian random process composed of 500 realizations and 501 observations. The used random variables are gaussian normal.
I have then applied the pca analysis to that process (Mathwork's help). However, if I plot the histograms of the coeffs I don't find gaussian random variables, the variables are gaussian but not standard as specified in page 10 of the Followed article, just before section 6.

Is there something wrong with my code (see below) am I misunderstanding the article or the pca function from MATLAB?

I need help since I can't see a solution to this problem.
Thanks.

close all
clear
clc

 
[X,Y] = meshgrid(0:0.002:1,0:0.002:1);
Z=exp((-1)*abs(X-Y));

tam=size(X, 1);

number_realizations=500;realizacion_mat=zeros(tam, number_realizations);
cov_mat=cov(Z);
[evec_mal, evalM_mal]=eig(cov_mat);
eval_mal=eig(evalM_mal);
num_eval=size(eval_mal,1);

for i=1:num_eval
    eval(i)=eval_mal(num_eval-i+1);
    evec(:,i)=evec_mal(:,num_eval-i+1);
end
figure
hold on
for j=1:number_realizations
  
    realizacion=zeros(tam, 1);
    for i=1:tam
        v_a = normrnd(0,1);
        realizacion=realizacion+sqrt(eval(i))*evec(:,i)*v_a;
    end
    realizacion_mat(:,j)=realizacion;
    plot(realizacion)
    clear('realizacion')
end

[coeff,score,latent,tsquared,explained,mu] = pca(realizacion_mat,'Centered',false);

realizaciones=size (realizacion_mat, 2);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
number_figures=6;
number_bins=10;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%for i=1:number_figures
    a= num2str(i);
    figure
    histogram(coeff(i,:), number_bins)
    xlabel(['\omega'])
    ylabel(['\xi_' ,num2str(i), '(\omega)'])
end

 

[mmwave]

Thank you for sharing your work and seeking help with your problem. I am a scientist with experience in PCA analysis and I would be happy to provide some insights and suggestions.

Firstly, I would like to commend you for your thorough approach and the clear presentation of your code. This makes it easier for others to understand and help with your problem.

From your description, it seems that you have correctly applied the PCA analysis to your gaussian random process. However, the issue you are facing is with the resulting histograms of the coefficients not being gaussian. Based on the information you have provided, I believe there could be a few reasons for this.

1. The number of realizations may not be enough to accurately represent the process:

In your code, you have generated 500 realizations of the process, which may not be enough to accurately represent the underlying distribution. The number of realizations required for accurate representation may depend on the complexity of your process and the number of observations. I would suggest trying with a larger number of realizations and see if there is any improvement in the gaussianity of the coefficients.

2. The underlying process may not be purely gaussian:

It is possible that the underlying process itself is not purely gaussian, which could result in non-gaussian coefficients. In such cases, the PCA analysis may not be the best approach. I would suggest checking the distribution of your initial data and if it is not purely gaussian, you may need to explore other statistical methods that are better suited for non-gaussian data.

3. The use of the "Centered" parameter in the PCA function:

In your code, you have specified the 'Centered' parameter as false, which means that the data is not being centered before performing the PCA analysis. This could also affect the resulting coefficients and histograms. I would suggest trying with the 'Centered' parameter set to true and see if there is any improvement in the gaussianity of the coefficients.

Overall, I would suggest exploring these possibilities and making appropriate changes to your code to see if there is any improvement in the results. I hope this helps and wish you all the best in finding a solution to your problem.