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Register a new userStatistical inference for principal components of spiked covariance matrices
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Added by
Zhigang Bao
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
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In this paper, we study the asymptotic behavior of the extreme eigenvalues and eigenvectors of the high dimensional spiked sample covariance matrices, in the supercritical case when a reliable detection of spikes is possible. Especially, we derive the joint distribution of the extreme eigenvalues and the generalized components of the associated eigenvectors, i.e., the projections of the eigenvectors onto arbitrary given direction, assuming that the dimension and sample size are comparably large. In general, the joint distribution is given in terms of linear combinations of finitely many Gaussian and Chi-square variables, with parameters depending on the projection direction and the spikes. Our assumption on the spikes is fully general. First, the strengths of spikes are only required to be slightly above the critical threshold and no upper bound on the strengths is needed. Second, multiple spikes, i.e., spikes with the same strength, are allowed. Third, no structural assumption is imposed on the spikes. Thanks to the general setting, we can then apply the results to various high dimensional statistical hypothesis testing problems involving both the eigenvalues and eigenvectors. Specifically, we propose accurate and powerful statistics to conduct hypothesis testing on the principal components. These statistics are data-dependent and adaptive to the underlying true spikes. Numerical simulations also confirm the accuracy and powerfulness of our proposed statistics and illustrate significantly better performance compared to the existing methods in the literature. Especially, our methods are accurate and powerful even when either the spikes are small or the dimension is large.
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In this paper, we study the asymptotic behavior of the extreme eigenvalues and eigenvectors of the spiked covariance matrices, in the supercritical regime. Specifically, we derive the joint distribution of the extreme eigenvalues and the generalized components of their associated eigenvectors in this regime.
We study the asymptotic distributions of the spiked eigenvalues and the largest nonspiked eigenvalue of the sample covariance matrix under a general covariance matrix model with divergent spiked eigenvalues, while the other eigenvalues are bounded but otherwise arbitrary. The limiting normal distribution for the spiked sample eigenvalues is established. It has distinct features that the asymptotic mean relies on not only the population spikes but also the nonspikes and that the asymptotic variance in general depends on the population eigenvectors. In addition, the limiting Tracy-Widom law for the largest nonspiked sample eigenvalue is obtained. Estimation of the number of spikes and the convergence of the leading eigenvectors are also considered. The results hold even when the number of the spikes diverges. As a key technical tool, we develop a Central Limit Theorem for a type of random quadratic forms where the random vectors and random matrices involved are dependent. This result can be of independent interest.
We consider general high-dimensional spiked sample covariance models and show that their leading sample spiked eigenvalues and their linear spectral statistics are asymptotically independent when the sample size and dimension are proportional to each other. As a byproduct, we also establish the central limit theorem of the leading sample spiked eigenvalues by removing the block diagonal assumption on the population covariance matrix, which is commonly needed in the literature. Moreover, we propose consistent estimators of the $L_4$ norm of the spiked population eigenvectors. Based on these results, we develop a new statistic to test the equality of two spiked population covariance matrices. Numerical studies show that the new test procedure is more powerful than some existing methods.
The Riemannian geometry of covariance matrices has been essential to several successful applications, in computer vision, biomedical signal and image processing, and radar data processing. For these applications, an important ongoing challenge is to develop Riemannian-geometric tools which are adapted to structured covariance matrices. The present paper proposes to meet this challenge by introducing a new class of probability distributions, Gaussian distributions of structured covariance matrices. These are Riemannian analogs of Gaussian distributions, which only sample from covariance matrices having a preassigned structure, such as complex, Toeplitz, or block-Toeplitz. The usefulness of these distributions stems from three features: (1) they are completely tractable, analytically or numerically, when dealing with large covariance matrices, (2) they provide a statistical foundation to the concept of structured Riemannian barycentre (i.e. Frechet or geometric mean), (3) they lead to efficient statistical learning algorithms, which realise, among others, density estimation and classification of structured covariance matrices. The paper starts from the observation that several spaces of structured covariance matrices, considered from a geometric point of view, are Riemannian symmetric spaces. Accordingly, it develops an original theory of Gaussian distributions on Riemannian symmetric spaces, of their statistical inference, and of their relationship to the concept of Riemannian barycentre. Then, it uses this original theory to give a detailed description of Gaussian distributions of three kinds of structured covariance matrices, complex, Toeplitz, and block-Toeplitz. Finally, it describes algorithms for density estimation and classification of structured covariance matrices, based on Gaussian distribution mixture models.
Sample correlation matrices are employed ubiquitously in statistics. However, quite surprisingly, little is known about their asymptotic spectral properties for high-dimensional data, particularly beyond the case of null models for which the data is assumed independent. Here, considering the popular class of spiked models, we apply random matrix theory to derive asymptotic first-order and distributional results for both the leading eigenvalues and eigenvectors of sample correlation matrices. These results are obtained under high-dimensional settings for which the number of samples n and variables p approach infinity, with p/n tending to a constant. To first order, the spectral properties of sample correlation matrices are seen to coincide with those of sample covariance matrices; however their asymptotic distributions can differ significantly, with fluctuations of both the sample eigenvalues and eigenvectors often being remarkably smaller than those of their sample covariance counterparts.
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