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Abstract Covariance matrices are fundamental to the analysis and forecast of economic, physical and biological systems. Although the eigenvalues $$\{\lambda _i\}$$ and eigenvectors $$\{\boldsymbol{u}_i\}$$ of a covariance matrix are central to such endeavours, in practice one must inevitably approximate the covariance matrix based on data with finite sample size $$n$$ to obtain empirical eigenvalues $$\{\tilde{\lambda }_i\}$$ and eigenvectors $$\{\tilde{\boldsymbol{u}}_i\}$$, and therefore understanding the error so introduced is of central importance. We analyse eigenvector error $$\|\boldsymbol{u}_i - \tilde{\boldsymbol{u}}_i \|^2$$ while leveraging the assumption that the true covariance matrix having size $$p$$ is drawn from a matrix ensemble with known spectral properties—particularly, we assume the distribution of population eigenvalues weakly converges as $$p\to \infty $$ to a spectral density $$\rho (\lambda )$$ and that the spacing between population eigenvalues is similar to that for the Gaussian orthogonal ensemble. Our approach complements previous analyses of eigenvector error that require the full set of eigenvalues to be known, which can be computationally infeasible when $$p$$ is large. To provide a scalable approach for uncertainty quantification of eigenvector error, we consider a fixed eigenvalue $$\lambda $$ and approximate the distribution of the expected square error $$r= \mathbb{E}\left [\| \boldsymbol{u}_i - \tilde{\boldsymbol{u}}_i \|^2\right ]$$ across the matrix ensemble for all $$\boldsymbol{u}_i$$ associated with $$\lambda _i=\lambda $$. We find, for example, that for sufficiently large matrix size $$p$$ and sample size $n> p$, the probability density of $$r$$ scales as $1/nr^2$. This power-law scaling implies that the eigenvector error is extremely heterogeneous—even if $$r$$ is very small for most eigenvectors, it can be large for others with non-negligible probability. We support this and further results with numerical experiments.
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Tracy–Widom distribution for the edge eigenvalues of elliptical model
Abstract In this paper, we study the largest eigenvalues of sample covariance matrices with elliptically distributed data. We consider the sample covariance matrix $$Q=YY^{*},$$ where the data matrix $$Y \in \mathbb{R}^{p \times n}$$ contains i.i.d. $$p$$-dimensional observations $$\textbf{y}_{i}=\xi _{i}T\textbf{u}_{i},\;i=1,\dots ,n.$$ Here $$\textbf{u}_{i}$$ is distributed on the unit sphere, $$\xi _{i} \sim \xi $$ is some random variable that is independent of $$\textbf{u}_{i}$$ and $$T^{*}T=\varSigma $$ is some deterministic positive definite matrix. Under some mild regularity assumptions on $$\varSigma ,$$ assuming $$\xi ^{2}$$ has bounded support and certain decay behaviour near its edge so that the limiting spectral distribution of $$Q$$ has a square root decay behaviour near the spectral edge, we prove that the Tracy–Widom law holds for the largest eigenvalues of $$Q$$ when $$p$$ and $$n$$ are comparably large. Based on our results, we further construct some useful statistics to detect the signals when they are corrupted by high dimensional elliptically distributed noise.
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- Award ID(s):
- 2306439
- PAR ID:
- 10583362
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- Information and Inference: A Journal of the IMA
- Volume:
- 14
- Issue:
- 2
- ISSN:
- 2049-8772
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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