The Annals of Applied Statistics

Penalized estimation in high-dimensional hidden Markov models with state-specific graphical models

Nicolas Städler and Sach Mukherjee

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Abstract

We consider penalized estimation in hidden Markov models (HMMs) with multivariate Normal observations. In the moderate-to-large dimensional setting, estimation for HMMs remains challenging in practice, due to several concerns arising from the hidden nature of the states. We address these concerns by $\ell_{1}$-penalization of state-specific inverse covariance matrices. Penalized estimation leads to sparse inverse covariance matrices which can be interpreted as state-specific conditional independence graphs. Penalization is nontrivial in this latent variable setting; we propose a penalty that automatically adapts to the number of states $K$ and the state-specific sample sizes and can cope with scaling issues arising from the unknown states. The methodology is adaptive and very general, applying in particular to both low- and high-dimensional settings without requiring hand tuning. Furthermore, our approach facilitates exploration of the number of states $K$ by coupling estimation for successive candidate values $K$. Empirical results on simulated examples demonstrate the effectiveness of the proposed approach. In a challenging real data example from genome biology, we demonstrate the ability of our approach to yield gains in predictive power and to deliver richer estimates than existing methods.

Article information

Source
Ann. Appl. Stat., Volume 7, Number 4 (2013), 2157-2179.

Dates
First available in Project Euclid: 23 December 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1387823314

Digital Object Identifier
doi:10.1214/13-AOAS662

Mathematical Reviews number (MathSciNet)
MR3161717

Zentralblatt MATH identifier
1283.62174

Keywords
HMM Graphical Lasso universal regularization model selection MMDL greedy backward pruning genome biology chromatin modeling

Citation

Städler, Nicolas; Mukherjee, Sach. Penalized estimation in high-dimensional hidden Markov models with state-specific graphical models. Ann. Appl. Stat. 7 (2013), no. 4, 2157--2179. doi:10.1214/13-AOAS662. https://projecteuclid.org/euclid.aoas/1387823314


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Supplemental materials

  • Supplementary material: Graphical Lasso with different penalty functions and supplementary figures. Optimization and performance of the Graphical Lasso with the penalty functions $\mathrm{Pen}_{\mathrm{invcov}}$, $\mathrm{Pen}_{\mathrm{parcor}}$ and $\mathrm{Pen}_{\mathrm{invcor}}$ introduced in Section 2.1. Additional Figures S2–S5 for Sections 3.1, 3.2 and 4.