Bernoulli

  • Bernoulli
  • Volume 23, Number 4B (2017), 3166-3177.

Eigen structure of a new class of covariance and inverse covariance matrices

Heather Battey

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

There is a one to one mapping between a $p$ dimensional strictly positive definite covariance matrix $\Sigma$ and its matrix logarithm $L$. We exploit this relationship to study the structure induced on $\Sigma$ through a sparsity constraint on $L$. Consider $L$ as a random matrix generated through a basis expansion, with the support of the basis coefficients taken as a simple random sample of size $s=s^{*}$ from the index set $[p(p+1)/2]=\{1,\ldots,p(p+1)/2\}$. We find that the expected number of non-unit eigenvalues of $\Sigma$, denoted $\mathbb{E}[|\mathcal{A}|]$, is approximated with near perfect accuracy by the solution of the equation

\[\frac{4p+p(p-1)}{2(p+1)}[\log (\frac{p}{p-d})-\frac{d}{2p(p-d)}]-s^{*}=0.\] Furthermore, the corresponding eigenvectors are shown to possess only ${p-|\mathcal{A}^{c}|}$ non-zero entries. We use this result to elucidate the precise structure induced on $\Sigma$ and $\Sigma^{-1}$. We demonstrate that a positive definite symmetric matrix whose matrix logarithm is sparse is significantly less sparse in the original domain. This finding has important implications in high dimensional statistics where it is important to exploit structure in order to construct consistent estimators in non-trivial norms. An estimator exploiting the structure of the proposed class is presented.

Article information

Source
Bernoulli, Volume 23, Number 4B (2017), 3166-3177.

Dates
Received: December 2015
Revised: March 2016
First available in Project Euclid: 23 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.bj/1495505088

Digital Object Identifier
doi:10.3150/16-BEJ840

Mathematical Reviews number (MathSciNet)
MR3654802

Zentralblatt MATH identifier
06778282

Keywords
covariance matrix matrix logarithm precision matrix spectral theory

Citation

Battey, Heather. Eigen structure of a new class of covariance and inverse covariance matrices. Bernoulli 23 (2017), no. 4B, 3166--3177. doi:10.3150/16-BEJ840. https://projecteuclid.org/euclid.bj/1495505088


Export citation

References

  • [1] Anderson, T.W. (2003). An Introduction to Multivariate Statistical Analysis, 3rd ed. Wiley Series in Probability and Statistics. Hoboken, NJ: Wiley.
  • [2] Bickel, P.J. and Levina, E. (2008). Covariance regularization by thresholding. Ann. Statist. 36 2577–2604.
  • [3] Cribben, I., Haraldsdottir, R., Atlas, L., Wager, T. and Lindquist, M. (2012). Dynamic connectivity regression: Determining state-related changes in brain connectivity. NeuroImage 61 907–920.
  • [4] Fan, J. and Fan, Y. (2008). High-dimensional classification using features annealed independence rules. Ann. Statist. 36 2605–2637.
  • [5] Fan, J., Liao, Y. and Mincheva, M. (2013). Large covariance estimation by thresholding principal orthogonal complements. J. R. Stat. Soc. Ser. B. Stat. Methodol. 75 603–680.
  • [6] Johnstone, I.M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295–327.
  • [7] Körner, T.W. (2013). Vectors, Pure and Applied: A General Introduction to Linear Algebra. Cambridge: Cambridge Univ. Press.
  • [8] Mathew, B., Holand, A.M., Koistinen, P., Léon, J. and Sillanpää, M.J. (2016). Reparametrization-based estimation of genetic parameters in multi-trait animal model using integrated nested Laplace approximation. Theor. Appl. Genet. 129 215–225.
  • [9] Wu, W.B. and Pourahmadi, M. (2003). Nonparametric estimation of large covariance matrices of longitudinal data. Biometrika 90 831–844.
  • [10] Yuan, M. and Chen, J. (2016). Efficient Portfolio Selection in a Large Market. J. Financ. Econom. To appear.
  • [11] Zou, H. and Li, D. (2016). SURE information criteria for large covariance matrix estimation and their asymptotic properties. IEEE Trans. Inform. Theory 62 2153–2169.