On the principal components of sample covariance matrices

Abstract

We introduce a class of M × M sample covariance matrices Q which subsumes and generalizes several previous models. The associated population covariance matrix Σ = EQ is assumed to differ from the identity by a matrix of bounded rank. All quantities except the rank of Σ − IM may depend on M in an arbitrary fashion. We investigate the principal components, i.e. the top eigenvalues and eigenvectors, of Q. We derive precise large deviation estimates on the generalized components w , ξ i of the outlier and non-outlier eigenvectors ξ i . Our results also hold near the so-called BBP transition, where outliers are created or annihilated, and for degenerate or near-degenerate outliers. We believe the obtained rates of convergence to be optimal. In addition, we derive the asymptotic distribution of the generalized components of the non-outlier eigenvectors. A novel observation arising from our results is that, unlike the eigenvalues, the eigenvectors of the principal components contain information about the subcritical spikes of Σ. The proofs use several results on the eigenvalues and eigenvectors of the uncorrelated matrix Q, satisfying EQ = IM , as input: the isotropic local Marchenko-Pastur law established in [10], level repulsion, and quantum unique ergodicity of the eigenvectors. The latter is a special case of a new universality result for the joint eigenvalue-eigenvector distribution.

A memory-based method to select the number of relevant components in principal component analysis

“…Note that we also see from Fig. 1 that the best fit q and σ for E and G are quite similar, which matches the theoretical result of [57].…”

Eigenvalue Significance Testing for Genetic Association

“…However, the key is , for which no solution c exists in the range if is “too far” from the remaining eigenvalues (see Baik et al () for discussion of large eigenvalues and a distributional phase transition). The maximum bulk (non‐extreme) eigenvalue, appropriately rescaled, still approaches the Tracy–Widom distribution (Bloemendal et al, ), and so the essential nature of a test procedure based on MP/TW still follows when .…”

“…Despite the interest in eigenvalue testing following the theoretical results of Johnstone () for the regime where , there are surprisingly few available methods to put the theory into practice. However, we note that only in the past decade have Tracy–Widom results been extended to data such as genotypes that are non‐Gaussian and non‐symmetric (Bao et al, ), where true eigenvalues vary (Karoui, ; Tracy and Widom, ), and for the largest eigenvalue from a bulk when other eigenvalues are extreme (Bloemendal et al, ). To our knowledge, the relevant asymptotics have still not been established in full generality (Bao et al, ).…”

Spectral Measures of Spiked Random Matrices

“…Indeed, if s (S n ,v (n) 1 ) (z) − s µ α,Σ,θ (z) = O (ε n ) , then, the Helffer-Sjöstrand argument that we developed during the proof of Theorem 5 yields a convergence of the averaged-square projection onto the direction of the spike for averaging windows of size ε n , as long as ε > > n −1/2 . The limitation ε n > > n −1/2 corresponds to the optimal rate in the local laws of Knowles and Yin (10).…”

“…We give numerical simulations that agree with our predictions, see Subsections 2.2 and 3.2. In the covariance setting, Bloemendal, Knowles, Yau and Yin proved that individual square projections of non-outlier eigenvectors that are associated to eigenvalues in the vicinity of the edge (of b) converge towards a chi squared random variable with given variance (see [10,Theorem 2.20]). Although it requires an averaging step, our result completes the picture as it is concerned with eigenvectors associated to any fixed location of the bulk of the spectrum.…”