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2016 | 4 | 1 | 189-201
Tytuł artykułu

Regularization for high-dimensional covariance matrix

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
4
Numer
1
Strony
189-201
Opis fizyczny
Daty
otrzymano
2015-10-09
zaakceptowano
2016-03-28
online
2016-04-12
Twórcy
  • School of Mathematics, University of Honghe, Yunnan, China
autor
  • School of Mathematics, University of Honghe, Yunnan, China
autor
  • School of Mathematics, University of Honghe, Yunnan, China
autor
  • School of Mathematics, University of Honghe, Yunnan, China
autor
  • School of Mathematics, University of Honghe, Yunnan, China
autor
  • School of Mathematics, University of Manchester, UK
Bibliografia
  • [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]
  • [2] P. Bickel and E. Levina. Covariance regularization by thresholding. Ann Stat, 36, (2008), 2577-2604. [WoS][Crossref]
  • [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]
  • [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]
  • [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]
  • [12] A. Rothman, E. Levina and J. Zhu. Generalized thresholding of large covariance matrices. J. Am. Stat. Assoc., 104, (2009), 177-186. [Crossref]
  • [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]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_spma-2016-0018
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