Evaluation of the variance of eigenvalues ​​of sample covariance matrices in Matlab

Question

I am trying to investigate the statistical variance of the eigenvalues ​​of sample covariance matrices using Matlab. To clarify, each covariance matrix sample is constructed from the number of finite vector images (struck by random white Gaussian noise). Then, compared with a large number of tests, a large number of such matrices are generated and created to evaluate statistical theoretical eigenvalues.

According to several sources (see, for example, [1, Eq.3] and [2, Eq.11]), the variance of each eigenvalue sample should be equal to the theoretical square of the eigenvalues ​​divided by the number of vector images used for each covariance matrix . However, the results I got from Matlab are not even close.

Is this a problem with my code? With matlab? (I have never had such problems with similar problems).

Here is a very simple example:

% Data vector length Lvec = 5; % Number of snapshots per sample covariance matrix N = 200; % Number of simulation trials Ntrials = 10000; % Noise variance sigma2 = 10; % Theoretical covariance matrix Rnn_th = sigma2*eye(Lvec); % Theoretical eigenvalues (should all be sigma2) lambda_th = sort(eig(Rnn_th),'descend'); lambda = zeros(Lvec,Ntrials); for trial = 1:Ntrials % Generate new (complex) white Gaussian noise data n = sqrt(sigma2/2)*(randn(Lvec,N) + 1j*randn(Lvec,N)); % Sample covariance matrix Rnn = n*n'/N; % Save sample eigenvalues lambda(:,trial) = sort(eig(Rnn),'descend'); end % Estimated eigenvalue covariance matrix b = lambda - lambda_th(:,ones(1,Ntrials)); Rbb = b*b'/Ntrials % Predicted (approximate) theoretical result Rbb_th_approx = diag(lambda_th.^2/N) 

Literature:

[1] Friedlander, B .; Weiss, AJ ;, “ On second-order statistics of eigenvectors of covariance sample matrices ,” Signal Processing, IEEE Transactions on, vol. 46, no.11, pp.3136-3139, November 1998. [2] Kaveh, M .; Barabell, A.;, “ MUSIC Statistical Performance and Minimum Norm Algorithms for Resolution of Plane Waves in Noise ,” Acoustics, Speech and Signal Processing, Transactions IEEE on, vol. 34, no.2, pp. 331-341, Apr 1986

Answer 1

According to the referral from your first reference:

“Formulas for statistics of the second order of eigenvectors were obtained in the statistical literature and are widely used. We point out the discrepancy between the statistics observed in numerical modeling and theoretical formulas due to the non-uniqueness of the definition from eigenvectors. We present two ways to resolve this discrepancy: the first includes a modification of theoretical formulas in accordance with the results of calculations, and the second is a simple change in calculations to match existing formulas. "

There seems to be a discrepancy, and it also sounds like two “solutions” are hacks, but without access to real paper, it’s hard to help.