Regularized rank-based estimation of high-dimensional nonparanormal graphical models

Abstract

A sparse precision matrix can be directly translated into a sparse Gaussian graphical model under the assumption that the data follow a joint normal distribution. This neat property makes high-dimensional precision matrix estimation very appealing in many applications. However, in practice we often face nonnormal data, and variable transformation is often used to achieve normality. In this paper we consider the nonparanormal model that assumes that the variables follow a joint normal distribution after a set of unknown monotone transformations. The nonparanormal model is much more flexible than the normal model while retaining the good interpretability of the latter in that each zero entry in the sparse precision matrix of the nonparanormal model corresponds to a pair of conditionally independent variables. In this paper we show that the nonparanormal graphical model can be efficiently estimated by using a rank-based estimation scheme which does not require estimating these unknown transformation functions. In particular, we study the rank-based graphical lasso, the rank-based neighborhood Dantzig selector and the rankbased CLIME. We establish their theoretical properties in the setting where the dimension is nearly exponentially large relative to the sample size. It is shown that the proposed rank-based estimators work as well as their oracle counterparts defined with the oracle data. Furthermore, the theory motivates us to consider the adaptive version of the rank-based neighborhood Dantzig selector and the rank-based CLIME that are shown to enjoy graphical model selection consistency without assuming the irrepresentable condition for the oracle and rank-based graphical lasso. Simulated and real data are used to demonstrate the finite performance of the rank-based estimators.

Graph estimation with joint additive models

“…However, as we will see, these two assumptions are essentially equivalent. Some recently proposed methods relax the multivariate Gaussian assumption using univariate transformations (Liu et al, 2009, 2012; Xue & Zou, 2012; Dobra & Lenkoski, 2011), restrictions on the graph structure (Liu et al, 2011), or flexible random forests (Fellinghauer et al, 2013). However, we will see that these methods may not capture realistic departures from multivariate Gaussianity.…”

Calibration and Multiple Robustness When Data Are Missing Not At Random

“…Rank-based correlation matrix estimation has been studied in a number of settings, including the nonparanormal graphical model [18,44,1], high dimensional structured covariance/precision matrix estimation [44,19,18], and sparse PCA model [11,27]. In the present paper, we only consider Kendall's tau-based estimator.…”

The cluster graphical lasso for improved estimation of Gaussian graphical models

“…As we have seen, this approach is quite natural due to a connection between the graphical lasso and single linkage clustering. However, in principle, one could perform clustering before estimating a graphical model using another technique, such as neighborhood selection [21], sparse partial correlation estimation [24], the nonparanormal [14, 35], or constrained ℓ 1 minimization [3]. We leave a full investigation of these approaches for future research.…”