We study the dependence of the spectral density of the covariance matrix ensemble on the power spectrum of the underlying multivariate signal. The white noise signal leads to the celebrated Marchenko-Pastur formula. We demonstrate results for some colored noise signals.
Deep Dive into Spectral Density of Sample Covariance Matrices of Colored Noise.
We study the dependence of the spectral density of the covariance matrix ensemble on the power spectrum of the underlying multivariate signal. The white noise signal leads to the celebrated Marchenko-Pastur formula. We demonstrate results for some colored noise signals.
The covariance matrix is a fundamental object in the multivariate statistics and probability theory. A sample covariance matrix use only part of the data and is determined by the number of samples. But it has the same population size as the covariance matrix. When the population size is not large and the number of sampling points is sufficient the sample covariance matrix is a good approximate of the covariance matrix. Unfortunately, we usually investigate data with a sampling rate that is not sufficient and select the number of samples to be comparable with the population size. In this case the sample covariance matrix is no longer a good approximation to the covariance matrix.
Marchenko and Pastur [5] were discussing a limiting case when the ratio p between the population size m and the number of samples n remains constant and n grows without bounds. They studied the sample covariance matrix c defined by the formula
where x i k stands for the normalized (i. e. with zero-mean) independent and identically distributed random data. The upper and lower indices denote the population and sample index respectively.
The spectral density of c depends in the limit only on the variance σ 2 of x and on the population-to-sample ratio p
where
For p > 1, there is an additional Dirac measure at λ = 0 of mass 1 -1 p .
The formula (2) describes the spectral density of the sample covariance matrices of a white noise signal. So the power spectrum of the signal vector x i k is constant. In many situations however the signal is not accessible directly. What is actually measured is its filtered image. For instance if we deal with the EEG signal we do not measure directly the cerebral signal but only its image filtered through the tissues in the skull. The natural question is of course to what degree the spectral density of the sample covariance matrix depends on such signal filtering. We show that spectral density (2) is universal in certain circumstances and that it represents a special case of the general probability distribution which depends on the power spectrum of the signal.
The measuring device has a finite sampling rate that leads to a discrete set of the measured values. For that reason we will use a discrete Fourier transformation (DFT) for the frequency analysis. In our notation DFT is defined as
where f is the sampling rate. The Fourier transform X of a real vector x is complex and fulfills
For real x it is therefore useful to use another transformation
where [a] means the integer part of a. The remaining two elements are defined separately. Since X 1 is real and equal to the sum of x i , we define X 1 = X 1 .
For even n we take X n
is real. For odd n we use
The transformed vector X is real and contains the full information on the frequency properties of the original vector x. The definition of the covariance matrix (1) can be easily rewritten using the discrete Fourier transformation and the transformation (5):
where the rows of the matrices X and X are the transformed rows of the matrix x.
Colored noise is a random signal with a non-flat power spectrum. We are interested in the question how the profile of the power spectrum influence the spectral density of the sample covariance matrix. In what follows we assume that the data matrix x has independent rows with identical power spectra and zero mean. Then the elements of the matrix X are also of mean zero -see the definition (6). Moreover the elements in the rows of the matrix x are independent. The transform X leads therefore also to a matrix with independent rows. Since the signal phase is random we get
and hence
where the angle brackets denote the sample mean. To find the spectral density of the covariance matrix ensemble we use now the a theorem of Girko [4].
Theorem 1: Let A be a m × [cm] random matrix with independent entries of a zero-mean that satisfy the condition
for some bound B < ∞. Moreover, let for each m be v m a function v m : [0, 1] × [0, c] → R defined by:
and suppose that v m converges uniformly to a limiting bounded function v for m → ∞. Then the limiting eigenvalue distribution ρ(λ) of the covariation matrix AA T exists and for every τ ≥ 0 satisfies:
with u(µ, τ ) solving the equation
The solution of the equation ( 12) exists and is unique in the class of functions u(µ, τ ) ≥ 0, analytical on τ and continuous on µ
Let us use this theorem taking A = X. We immediately see that c = n m .
Since all the rows of the matrix X have identical power spectra, the function v(µ, ν) will not depend on µ. The equation (12) shows that the function u(µ, τ ) is also µ independent. Inserting v(ν) and u(τ ) into the equations ( 11) and (12), we get
and
The spectral density is determined by the function v(ν) (that itself is a function of the power spectrum). However, to solve the equations ( 13) and ( 14) for a general power spectrum profile is extremely difficult. So in next chapters we will try to get an exact formula for the spectral den
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