Open Access
2016 Joint estimation of precision matrices in heterogeneous populations
Takumi Saegusa, Ali Shojaie
Electron. J. Statist. 10(1): 1341-1392 (2016). DOI: 10.1214/16-EJS1137

Abstract

We introduce a general framework for estimation of inverse covariance, or precision, matrices from heterogeneous populations. The proposed framework uses a Laplacian shrinkage penalty to encourage similarity among estimates from disparate, but related, subpopulations, while allowing for differences among matrices. We propose an efficient alternating direction method of multipliers (ADMM) algorithm for parameter estimation, as well as its extension for faster computation in high dimensions by thresholding the empirical covariance matrix to identify the joint block diagonal structure in the estimated precision matrices. We establish both variable selection and norm consistency of the proposed estimator for distributions with exponential or polynomial tails. Further, to extend the applicability of the method to the settings with unknown populations structure, we propose a Laplacian penalty based on hierarchical clustering, and discuss conditions under which this data-driven choice results in consistent estimation of precision matrices in heterogenous populations. Extensive numerical studies and applications to gene expression data from subtypes of cancer with distinct clinical outcomes indicate the potential advantages of the proposed method over existing approaches.

Citation

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Takumi Saegusa. Ali Shojaie. "Joint estimation of precision matrices in heterogeneous populations." Electron. J. Statist. 10 (1) 1341 - 1392, 2016. https://doi.org/10.1214/16-EJS1137

Information

Received: 1 January 2015; Published: 2016
First available in Project Euclid: 31 May 2016

zbMATH: 1341.62130
MathSciNet: MR3507368
Digital Object Identifier: 10.1214/16-EJS1137

Keywords: graph Laplacian , graphical modeling , heterogeneous populations , hierarchical clustering , high-dimensional estimation , precision matrix , Sparsity

Rights: Copyright © 2016 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.10 • No. 1 • 2016
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