[Paper Review] Large random matrix approach for testing independence of a large number of Gaussian time series

Philippe Loubaton, Alexis Rosuel|arXiv (Cornell University)|Jul 17, 2020

Random Matrices and Applications34 references4 citations

TL;DR

This paper develops a large random matrix theory framework to analyze linear spectral statistics (LSS) of frequency-smoothed periodogram estimators for high-dimensional Gaussian time series. Under asymptotic regimes where sample size N→∞ and dimension M, along with smoothing span B, grow proportionally such that M/N→0, the empirical eigenvalue distribution of the estimated spectral coherency matrix converges to the Marcenko-Pastur law, enabling precise deterministic approximation of LSS with error bounds of order O(1/M + √M/N + (M/N)^3).

ABSTRACT

The asymptotic behaviour of Linear Spectral Statistics (LSS) of the smoothed periodogram estimator of the spectral coherency matrix of a complex Gaussian high-dimensional time series $(\y_n)_{n \in \mathbb{Z}}$ with independent components is studied under the asymptotic regime where the sample size $N$ converges towards $+\infty$ while the dimension $M$ of $\y$ and the smoothing span of the estimator grow to infinity at the same rate in such a way that $\frac{M}{N} ightarrow 0$. It is established that, at each frequency, the estimated spectral coherency matrix is close from the sample covariance matrix of an independent identically $\mathcal{N}_{\mathbb{C}}(0,\I_M)$ distributed sequence, and that its empirical eigenvalue distribution converges towards the Marcenko-Pastur distribution. This allows to conclude that each LSS has a deterministic behaviour that can be evaluated explicitly. Using concentration inequalities, it is shown that the order of magnitude of the supremum over the frequencies of the deviation of each LSS from its deterministic approximation is of the order of $\frac{1}{M} + \frac{\sqrt{M}}{N}+ (\frac{M}{N})^{3}$ where $N$ is the sample size. Numerical simulations supports our results.

Motivation & Objective

• To study the asymptotic behavior of linear spectral statistics (LSS) of the frequency-smoothed periodogram estimator of the spectral coherency matrix in high-dimensional Gaussian time series.
• To establish that under high-dimensional asymptotics (N→∞, M,N→∞ with M/N→0), the estimated coherency matrix behaves like a sample covariance matrix of i.i.d. complex Gaussian vectors.
• To derive deterministic approximations for LSS of the eigenvalues of the estimated coherency matrix at each frequency.
• To quantify the deviation of LSS from their deterministic limits using concentration inequalities.
• To support the development of a statistical test for mutual independence among a large number of time series components using spectral properties.

Proposed method

• Uses the frequency-smoothed periodogram estimator to construct a consistent estimate of the spectral coherency matrix, defined as $\hat{\mathbf{C}}(\nu) = \mathrm{diag}(\hat{\mathbf{S}}(\nu))^{-1/2} \hat{\mathbf{S}}(\nu) \mathrm{diag}(\hat{\mathbf{S}}(\nu))^{-1/2}$.
• Applies large random matrix theory, particularly the Marcenko-Pastur law, to show that the empirical eigenvalue distribution of $\hat{\mathbf{C}}(\nu)$ converges to the Marcenko-Pastur distribution with parameter $c = \lim M/N$.
• Employs stochastic domination and concentration inequalities to bound the deviation of LSS from their deterministic limits, with the supremum deviation over frequency $\nu \in [0,1]$ controlled by $u_N = \frac{1}{B} + \frac{\sqrt{B}}{N} + \left(\frac{B}{N}\right)^3$.
• Derives a stochastic representation of the estimated coherency matrix $\hat{\mathbf{C}}(\nu)$ via decomposition into a reference Wishart-type matrix and a perturbation term.
• Uses the Helffer-Sjöstrand formula and resolvent techniques to analyze the trace of functions of random matrices.
• Applies the Hanson-Wright inequality and integration-by-parts formulas to control moments and cumulants of quadratic forms in Gaussian vectors.

Experimental results

Research questions

• RQ1 How does the empirical eigenvalue distribution of the frequency-smoothed estimated spectral coherency matrix behave in high-dimensional asymptotic regimes?
• RQ2 Can linear spectral statistics (LSS) of the estimated coherency matrix be approximated deterministically under high-dimensional asymptotics?
• RQ3 What is the rate of convergence of LSS to their deterministic limits, and how can the deviation be quantified?
• RQ4 How does the smoothing span B and sample size N jointly affect the accuracy of spectral statistics in high-dimensional time series?
• RQ5 Can the theoretical framework support a formal test for mutual independence among components of a high-dimensional Gaussian time series?

Key findings

• Under the asymptotic regime $N \to \infty$, $M(N) = \mathcal{O}(N^\alpha)$ with $\alpha \in (1/2,1)$, and $c_N = M/N \to c \in (0,1)$, the empirical eigenvalue distribution of $\hat{\mathbf{C}}(\nu)$ converges weakly to the Marcenko-Pastur distribution with parameter $c$.
• The linear spectral statistics $\frac{1}{M}\mathrm{Tr}\left(f(\hat{\mathbf{C}}(\nu))\right)$ converge to a deterministic limit that can be explicitly computed via the Stieltjes transform of the Marcenko-Pastur law.
• The supremum over $\nu \in [0,1]$ of the deviation of LSS from its deterministic approximation is of order $\mathcal{O}\left(\frac{1}{M} + \frac{\sqrt{M}}{N} + \left(\frac{M}{N}\right)^3\right)$ with high probability.
• The perturbation between the estimated coherency matrix and the reference Wishart matrix is shown to be stochastically dominated with error scaling $\mathcal{O}\left(\left(\frac{B}{N}\right)^k\right)$ for any $k \geq 1$.
• Numerical simulations confirm the theoretical error bounds and validate the deterministic approximation of LSS.
• The framework enables the construction of a test for mutual uncorrelatedness (i.e., independence) of components in high-dimensional time series based on spectral statistics.