Eigenvalues of large sample covariance matrices of spiked population models

Abstract

We consider a spiked population model, proposed by Johnstone, in which all the population eigenvalues are one except for a few fixed eigenvalues. The question is to determine how the sample eigenvalues depend on the non-unit population ones when both sample size and population size become large. This paper completely determines the almost sure limits of the sample eigenvalues in a spiked model for a general class of samples.

Performance of Eigenvalue-Based Signal Detectors with Known and Unknown Noise Level

“…A similar expression holds for t GLRT (α), with s α replaced by s ′ α,K,N , found by inverting (21) and, in general, also having a weak dependence on N, K. Inserting these expressions into (27) gives that, for N ≫ (1/Kρ) 2 , the difference is roughly…”

“…2) Detection probability: Under H 1 , the asymptotic distribution of λ 1 in the joint limit N, K → ∞ is characterized by a phase transition phenomenon [21]. In the case of a single signal, the critical detection threshold for N, K → ∞ can be expressed directly in terms of the SNR as [22] …”

“…If the SNR ρ is lower than the critical value, the limiting distribution of λ 1 under H 1 is the same as that of the largest eigenvalue under H 0 , thus nullifying the statistical power of a largest eigenvalue test. If ρ > ρ crit , on the contrary, the distribution of T RLRT is asymptotically Gaussian [17], [21], with…”

Exact and Asymptotic Analysis of Largest Eigenvalue Based Spectrum Sensing

“…The results in (Baik & Silverstein, 2005) prove that the largest eigenvalue of a correlated central Wishart matrix converges to a Gaussian distribution N (µ 2 , σ 2 2 ) with mean µ 2 = Nξ 1 1 + K/N ξ 1 −1 ,a n dv a ria n c eσ 2 2 = Nξ 2 1 1 − K/N (ξ 1 −1) 2 . The convergence occurs when…”

SOS, lost in a high dimensional space

“…There are large differences between the population eigenvalues. The Karoui correction models the eigenvalues similar to the spiked population model: almost all eigenvalues are equal except for a few considerably larger ones (see [31] and [32]). It also sets a few population eigenvalues to zero.…”