In many applications, high-dimensional problem may occur often for various reasons, for example, when the number of variables under consideration is much bigger than the sample size, i.e., p >> n. For highdimensional data, the underlying structures of certain covariance matrix estimates are usually blurred due to substantial random noises, which is an obstacle to draw statistical inferences. In this paper, we propose a method to identify the underlying covariance structure by regularizing a given/estimated covariance matrix so that the noises can be filtered. By choosing an optimal structure from a class of candidate structures for the covariance matrix, the regularization is made in terms of minimizing Frobenius-norm discrepancy. The candidate class considered here includes the structures of order-1 moving average, compound symmetry, order-1 autoregressive and order-1 autoregressive moving average. Very intensive simulation studies are conducted to assess the performance of the proposed regularization method for very high-dimensional covariance problem. The simulation studies also show that the sample covariance matrix, although performs very badly in covariance estimation for high-dimensional data, can be used to correctly identify the underlying structure of the covariance matrix. The approach is also applied to real data analysis, which shows that the proposed regularization method works well in practice.
@article{bwmeta1.element.doi-10_1515_spma-2016-0018,
author = {Xiangzhao Cui and Chun Li and Jine Zhao and Li Zeng and Defei Zhang and Jianxin Pan},
title = {Regularization for high-dimensional covariance matrix},
journal = {Special Matrices},
volume = {4},
year = {2016},
pages = {189-201},
zbl = {1341.62156},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_spma-2016-0018}
}
Xiangzhao Cui; Chun Li; Jine Zhao; Li Zeng; Defei Zhang; Jianxin Pan. Regularization for high-dimensional covariance matrix. Special Matrices, Tome 4 (2016) pp. 189-201. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_spma-2016-0018/
[1] A. Belloni, V. Chernozhukov and L. Wang. Square-root lasso: Pivotal recovery of sparse signals via conic programming. Biometrika, 98, (2012), 791-806. [WoS] | Zbl 1228.62083
[2] P. Bickel and E. Levina. Covariance regularization by thresholding. Ann Stat, 36, (2008), 2577-2604. [WoS][Crossref] | Zbl 1196.62062
[3] M. Buscema. MetaNet: The Theory of Independent Judges, Substance Use & Misuse, 33, (1998), 439-461.
[4] T. Cai and W. Liu. Adaptive thresholding for sparse covariance estimation. J Am Stat Assoc, 106, (2011), 672-684. [Crossref][WoS] | Zbl 1232.62086
[5] X. Cui, C. Li, J. Zhao, L. Zeng, D. Zhang and J. Pan. Covariance structure regularization via Frobenius-norm discrepancy. Revised for Linear Algebra Appl. (2015).
[6] X. Deng and K. Tsui. Penalized covariance matrix estimation using a matrix-logarithm transformation. J Comput Stat Graph, 22, (2013), 494-512. [Crossref]
[7] D. Donoho. Aide-Memoire. High-dimensional data analysis: The curses and blessings of dimensionality. American Mathematical Society. Available at http://www.stat.stanford.edu/~donoho/Lectures/AMS2000/AMS2000.html, (2000).
[8] N. El Karoui. Operator norm consistent estimation of large dimensional sparse covariance matrices. Ann Stat, 36, (2008), 2712-2756. [WoS] | Zbl 1196.62064
[9] J. Fan, Y. Liao and M. Mincheva. Large covariance estimation by thresholding principal orthogonal complements. J Roy Stat Soc B, 75, (2013), 656-658.
[10] L. Lin, N. J. Higham and J. Pan. Covariance structure regularization via entropy loss function. Computational Statistics & Data Analysis, 72, (2014), 315-327. [Crossref][WoS]
[11] A. Rothman. Positive definite estimators of large covariance matrices. Biometrika, 99, (2012), 733-740. [WoS][Crossref] | Zbl 06085166
[12] A. Rothman, E. Levina and J. Zhu. Generalized thresholding of large covariance matrices. J. Am. Stat. Assoc., 104, (2009), 177-186. [Crossref] | Zbl 06448242
[13] L. Xue, S. Ma and H. Zou. Positive definite `1 penalized estimation of large covariance matrices. J. Am. Stat. Assoc., 107, (2012), 1480-1491. [WoS][Crossref] | Zbl 1258.62063