Network exploration via the adaptive LASSO and SCAD penalties

TLDR

Graphical models are commonly used to explore networks by estimating sparse precision matrices, but positive‑definiteness constraints make the optimization challenging. The study introduces nonconcave and adaptive LASSO penalties to reduce bias in network estimation. The authors use a local linear approximation to recast precision matrix estimation as weighted L1 penalized likelihood problems solved by Friedman et al.’s algorithm, and validate the approach with real datasets, simulations, and asymptotic theory. Published in Biostatistics 9 (2008) 432–441.

Abstract

Graphical models are frequently used to explore networks, such as genetic networks, among a set of variables. This is usually carried out via exploring the sparsity of the precision matrix of the variables under consideration. Penalized likelihood methods are often used in such explorations. Yet, positive-definiteness constraints of precision matrices make the optimization problem challenging. We introduce nonconcave penalties and the adaptive LASSO penalty to attenuate the bias problem in the network estimation. Through the local linear approximation to the nonconcave penalty functions, the problem of precision matrix estimation is recast as a sequence of penalized likelihood problems with a weighted L1 penalty and solved using the efficient algorithm of Friedman et al. [Biostatistics 9 (2008) 432–441]. Our estimation schemes are applied to two real datasets. Simulation experiments and asymptotic theory are used to justify our proposed methods.