Estimating the inverse of the covariance matrix, the precision matrices.
Two big data problems problems can arise here: large \(p\) (ambient dimension) and large \(n\) (sample size). Large \(p\) is a problem because the covariance matrix is a \(p \times p\) matrixand frequently we need to invert it to calculate some target estimand.
The obvious way
Estimate the covariance matrix then invert it. This is the baseline. 🏗
QUIC
Bayesian
🏗 Wishart priors?
Penalized
Structured
Bound distance from Covariance
covariance matrix or
References
Aragam, Bryon, Jiaying Gu, and Qing Zhou. 2017. “Learning Large-Scale Bayesian Networks with the Sparsebn Package.” arXiv:1703.04025 [Cs, Stat], March.
Avagyan, Vahe, and Xiaoling Mei. 2022. “Precision Matrix Estimation Under Data Contamination with an Application to Minimum Variance Portfolio Selection.” Communications in Statistics - Simulation and Computation 51 (4): 1381–1400.
Chen, Xiaohui, Mengyu Xu, and Wei Biao Wu. 2013. “Covariance and Precision Matrix Estimation for High-Dimensional Time Series.” The Annals of Statistics 41 (6).
Fan, Jianqing, Yuan Liao, and Han Liu. 2016. “An Overview of the Estimation of Large Covariance and Precision Matrices.” The Econometrics Journal 19 (1): C1–32.
Hsieh, Cho-Jui, Mátyás A. Sustik, Inderjit S. Dhillon, and Pradeep D. Ravikumar. 2014. “QUIC: Quadratic Approximation for Sparse Inverse Covariance Estimation.” Journal of Machine Learning Research 15 (1): 2911–47.
Hsieh, Cho-Jui, Mátyás A. Sustik, Inderjit S. Dhillon, Pradeep Ravikumar, and Russell A. Poldrack. 2013. “BIG & QUIC: Sparse Inverse Covariance Estimation for a Million Variables.” In Advances in Neural Information Processing Systems, 16. NIPS’13. Red Hook, NY, USA: Curran Associates Inc.
Janková, Jana, and Sara van de Geer. 2015. “Honest Confidence Regions and Optimality in High-Dimensional Precision Matrix Estimation.” arXiv:1507.02061 [Math, Stat], July.
Khoshgnauz, Ehsan. 2012. “Learning Markov Network Structure Using Brownian Distance Covariance.” arXiv:1206.6361 [Cs, Stat], June.
Kuismin, Markku O., and Mikko J. Sillanpää. 2017. “Estimation of Covariance and Precision Matrix, Network Structure, and a View Toward Systems Biology.” WIREs Computational Statistics 9 (6): e1415.
Lam, Clifford, and Jianqing Fan. 2009. “Sparsistency and Rates of Convergence in Large Covariance Matrix Estimation.” Annals of Statistics 37 (6B): 4254–78.
Le, Thien-Minh, and Ping-Shou Zhong. 2021. “High-Dimensional Precision Matrix Estimation with a Known Graphical Structure.” arXiv.
Meier, Alexander, Claudia Kirch, and Renate Meyer. 2020. “Bayesian Nonparametric Analysis of Multivariate Time Series: A Matrix Gamma Process Approach.” Journal of Multivariate Analysis 175 (January): 104560.
Mercer, A. McD. 2000. “Bounds for A–G, A–H, G–H, and a Family of Inequalities of Ky Fan’s Type, Using a General Method.” Journal of Mathematical Analysis and Applications 243 (1): 163–73.
Moscone, Francesco, Elisa Tosetti, and Veronica Vinciotti. 2017. “Sparse Estimation of Huge Networks with a Block-Wise Structure.” The Econometrics Journal 20 (3): S61–85.
Pleiss, Geoff, Jacob R. Gardner, Kilian Q. Weinberger, and Andrew Gordon Wilson. 2018. “Constant-Time Predictive Distributions for Gaussian Processes.” In. arXiv.
Pourahmadi, Mohsen. 2011. “Covariance Estimation: The GLM and Regularization Perspectives.” Statistical Science 26 (3): 369–87.
Saad, Yousef. 2003. Iterative Methods for Sparse Linear Systems: Second Edition. 2nd ed. SIAM.
Sharma, Rajesh. 2008. “Some More Inequalities for Arithmetic Mean, Harmonic Mean and Variance.” Journal of Mathematical Inequalities, no. 1: 109–14.
Ubaru, Shashanka, Jie Chen, and Yousef Saad. 2017. “Fast Estimation of \(tr(f(A))\) via Stochastic Lanczos Quadrature.” SIAM Journal on Matrix Analysis and Applications 38 (4): 1075–99.
Wang, Lingxiao, Xiang Ren, and Quanquan Gu. n.d. “Precision Matrix Estimation in High Dimensional Gaussian Graphical Models with Faster Rates,” 9.
Wu, Wei Biao, and Mohsen Pourahmadi. 2003. “Nonparametric Estimation of Large Covariance Matrices of Longitudinal Data.” Biometrika 90 (4): 831–44.
Yuan, Ming. 2010. “High Dimensional Inverse Covariance Matrix Estimation via Linear Programming.” The Journal of Machine Learning Research 11: 26.
Zhang, T., and H. Zou. 2014. “Sparse Precision Matrix Estimation via Lasso Penalized D-Trace Loss.” Biometrika 101 (1): 103–20.
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