Research of confidence intervals for precision matrix in high dimensional network data

doi:10.7523/j.ucas.2020.0038

Abstract

Abstract: With the development of the Internet, science, and technology, the surge of bigdata on an unprecedented scale has brought complex network data between different individuals. It is practical significance to uncover the network connection by studying the confidence intervals of the precision (inverse covariance) matrix in graphical models. One natural and important question is how to efficiently obtain confidence intervals of the precision matrix. This paper proposes the De-ISEE (De-innovated scalable efficient estimation) statistic, whose confidence intervals enjoy efficient computation while maintaining a desirable coverage rate. Both average coverage and computational advantages of the methods have been demonstrated by our numerical studies in network data. Moreover, this paper applies the De-ISEE method to riboflavin data and gene expression data, and finds that De-ISEE method could be an important tool for studying gene association.

Key words: network data, high-dimensional graphical models, confidence intervals, precision matrix, De-sparsified statistic

CLC Number:

O212.2

ZHENG Zemin, ZHOU Huiting. Research of confidence intervals for precision matrix in high dimensional network data[J]. Journal of University of Chinese Academy of Sciences, 2022, 39(3): 302-308.

References

[1] 王新栋,于华,江成. 社交网络关键节点检测的积极效应问题[J]. 中国科学院大学学报, 2019, 36(3):425-432. DOI:10.7523/j.issn.2095-6134.2019.03.017.
[2] Fan J, Liao Y, Liu H. An overview of the estimation of large covariance and precision matrices [J]. Economet J, 2016, 19(1): C1-C32. DOI:10.1111/ectj.12061.
[3] Chen X, Liu Y, Liu H, et al. Learning spatial-temporal varying graphs with applications to climate data analysis [ C]//Twenty-Fourth AAAI Conference on Artificial Intelligence: AAAI Press, 2010: 425-430.
[4] Schäfer J, Strimmer K. A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics [J]. Stat Appl Genet Mol Biol, 2005, 4(1): Article 32. DOI:10.2202/1544-6115.1175.
[5] Wainwright M J, Jordan M I. Graphical models, exponential families, and variational inference [J]. Found Trends Mach Learn, 2008, 1(1/2):1-305. DOI:10.1561/2200000001.
[6] Liu H, Lafferty J D, Wasserman L A. The nonparanormal: semiparametric estimation of high dimensional undirected graphs [J]. J Mach Learn Res, 2009,10(3): 2295-2328.
[7] Lauritzen S. Graphical models [M]. New York: Oxford Univ Press, 1996.
[8] Meinshausen N, Bühlmann P. High-dimensional graphs and variable selection with the lasso [J]. Ann Statist, 2006, 34(3): 1436-1462. DOI:10.1214/009053606000000281.
[9] Yuan M, Lin Y. Model selection and estimation in the Gaussian graphical model [J]. Biometrika, 2007, 94(1):19-35. DOI:10.1093/biomet/asm018.
[10] Bickel P, Levina E. Covariance regularization by thresholding [J]. Ann Statist, 2008, 36(6): 2577-2604. DOI:10.1214/08-aos600.
[11] Friedman J, Hastie T, Tibshirani R, et al. Sparse inverse covariance estimation with the graphical lasso [J]. Biostatistics, 2007, 9(3):432-441. DOI:10.1093/biostatistics/kxm045.
[12] Zhang T, Zou H. Sparse precision matrix estimation via lasso penalized D-trace loss[J]. Biometrika, 2014, 101:103-120. DOI:10.1093/biomet/ast059.
[13] Liu W D, Luo X. Fast and adaptive sparse precision matrix estimation in high dimensions[J]. J Multivariate Anal, 2015, 135: 153-162. DOI:10.1016/j.jmva.2014.11.005.
[14] Ren Z, Sun T N, Zhang C H, et al. Asymptotic normality and optimalities in estimation of large Gaussian graphical models [J]. Ann Statist, 2015, 43(3): 991-1026. DOI:10.1214/14-aos1286.
[15] Fan Y, Lyu J C. Innovated scalable efficient estimation in ultra-large Gaussian graphical models [J]. Ann Statist, 2016, 44(5): 2098-2126. DOI:10.1214/15-aos1416.
[16] 杨军,于丹. 修如旧模型中储存系统备件量的计算及其置信区间[J]. 中国科学院研究生院学报, 2004, 21(4): 441-446. DOI:10.7523/j.issn.2095-6134.2004.4.002.
[17] Cai T, Liu W D, Luo X. A constrained L₁ minimization approach to sparse precision matrix estimation [J]. J Amer Statist Assoc, 2011, 106(494): 594-607. DOI:10.1198/jasa.2011.tm10155.
[18] Nickl R, van de Geer S. Confidence sets in sparse regression [J]. Ann Statist, 2012, 41: 2852-2876. DOI:10.1214/13-aos1170.
[19] van de Geer S, Bühlmann P, Ritov Y, et al. On asymptotically optimal confidence regions and tests for high dimensional models [J]. Ann Statist, 2014, 42(3): 1166-1202. DOI:10.1214/14-aos1221.
[20] Meinshausen N. Assumption-free confidence intervals for groups of variables in sparse high-dimensional regression [J]. arXiv:1309.3489, 2013.
[21] Zhang C H, Zhang S S. Confidence intervals for low dimensional parameters in high dimensional liner models [J]. J Roy Statist Soc Ser B, 2014, 76(1): 217-242. DOI:10.1111/rssb.12026.
[22] Janková J, van de Geer S. Confidence intervals for high-dimensional inverse covariance estimation [J]. Electron J Stat, 2015, 9(1): 1205-1229. DOI:10.1214/15-ejs1031.
[23] Huang X, Li M. Confidence intervals for sparse precision matrix estimation via Lasso penalized D-trace loss [J]. Commun Stat Theor M, 2017, 46(24): 12299-12316. DOI:10.1080/03610926.2017.1295074.
[24] Janková J, van de Geer S. Inference in high-dimensional graphical models [J]. arXiv:1801.08512, 2018.
[25] Sun T N, Zhang C. Scaled sparse linear regression [J]. Biometrika, 2012, 99(4): 879-898. DOI:10.1093/biomet/ass043.