Sure Screening for Transelliptical Graphical Models

Consider the random vector $X=\left(X_{1},\ldots,X_{p}\right)^{T}$, and an undirected graph denoted by $G=\left(\mathcal{V},\mathcal{E}\right)$, where $\mathcal{V}=\left\{1,\dots,p\right\}$ is the set of nodes, and $\mathcal{E}$ is the set of edges describing the conditional dependence relationships among $X=\left(X_{1},\ldots,X_{p}\right)^{T}$. A pair $\left(j,j^{\prime}\right)$ is contained in the edge set $\mathcal{E}$ if and only if $X_{j}$ is conditionally dependent on $X_{j^{\prime}}$, given all remaining variables $X_{\mathcal{V}\backslash\{j,j^{\prime}\}}=\left\{X_{k};k\in\mathcal{V}\backslash\left\{j,j^{\prime}\right\}\right\}$.

In recent years, many methods have been developed to recover the structure of graphical models in the high dimensional setting. Many authors have studied the Gaussian graphical model, in which conditional dependence is encoded by the sparsity pattern of the inverse covariance matrix (Yuan and Lin,, 2007; Friedman et al.,, 2008; Rothman et al.,, 2008; Ravikumar et al.,, 2009, and references therein). Liu et al., (2009); Liu et al., 2012a introduced the nonparanormal distribution, which results from univariate monotonic transformations of the Gaussian distribution, and showed that the structural properties of the inverse covariance matrix of the Gaussian distribution carry over to the corresponding nonparanormal distribution. Ravikumar et al., (2010) and Anandkumar et al., (2012) considered recovering the structure of an Ising graphical model. Yang et al., (2012) studied a class of graphical models in which the node-wise conditional distributions arise from the exponential family. Moreover, Yang et al., (2014) and Chen et al., (2015) considered the problem of structure recovery for mixed graphical models.

Luo et al., (2015) proposed a computationally-efficient screening approach for Gaussian graphical models, which they called graphical sure screening (GRASS). They estimated an edge between the $j$th and $j^{\prime}$th nodes if the sample correlation between the $j$th and $j^{\prime}$th features exceeds some threshold $\gamma$. Then the $j$th node’s estimated neighborhood contains the true neighborhood with very high probability, under certain simple assumptions. GRASS requires only $\mathcal{O}(p^{2})$ operations, while most other existing methods for estimating the graph require $\mathcal{O}(p^{3})$ computations (Friedman et al.,, 2008).

However, GRASS requires that the data follows a multivariate normal distribution. In many settings, a reliance on exact normality is not desirable (Liu et al.,, 2009; Liu et al., 2012a, ; Liu et al., 2012b, ). In this paper, we propose transelliptical GRASS, an extension of the GRASS procedure to the transelliptical graphical model family, introduced by Liu et al., 2012b . We show that under a certain set of simple assumptions, the desirable statistical properties held by GRASS are also held by transelliptical GRASS. However, due to the relaxation of multivariate normality, the graph that we estimate represents the partial correlation, rather than conditional dependence, among variables.

The rest of this paper is organized as follows. In Section 2, we provide some background on transelliptical graphical models, and present a useful property of Kendall’s tau statistic. In Section 3, we establish the theoretical properties for transelliptical sure screening, which include the sure screening property, size control of the selected edge set, and the control of the expected false positive rate. Furthermore, we provide a choice of the threshold value that leads to the aforementioned desirable properties. Simulation studies are presented in Section 4, and an application to an equities data set is shown in Section 5. We close with a discussion in Section 6.

The transelliptical distribution (Liu et al., 2012b, ) is a generalization of the nonparanormal distribution (Liu et al.,, 2009; Liu et al., 2012a, ). The transelliptical distribution extends the elliptical distribution in much the same way that the nonparanormal extends the normal distribution. We first provide the definition of the elliptical distribution.

In Definition 1, $X\stackrel{{\scriptstyle d}}{{=}}Y$ indicates that $X$ and $Y$ have the same distribution. Many multivariate distributions, such as the multivariate normal and the multivariate t-distribution, belong to the elliptical distribution family.

From now on, we assume that $A$ in (2.1) is a full rank $p\times p$ matrix so that $\Sigma$ is positive definite. In addition, assume that $\Sigma$ has unit diagonal elements. We let $\rho_{jj^{\prime}}$ denote the $\left(j,j^{\prime}\right)$th entry of $\Sigma$. We also assume that the scalar random variable $r$ in (2.1) has density $g(\cdot)$ and ${\mathbb{E}}(r^{2})<\infty$.

Given a transelliptical distribution $\mbox{TE}_{p}\left(\Sigma,r;f_{1},\dots,f_{p}\right)$, we can define an undirected graph $G=\left(\mathcal{V},\mathcal{E}\right)$, where $\mathcal{V}=\left\{1,\ldots,p\right\}$, and $\left(j,j^{\prime}\right)\in\mathcal{E}$ if and only if $\left(\Sigma^{-1}\right)_{jj^{\prime}}\neq 0$ — that is, if and only if the latent variables $Z_{j}=f_{j}(X_{j})$ and $Z_{j^{\prime}}=f_{j^{\prime}}(X_{j^{\prime}})$ are partially correlated (Liu et al., 2012b, ). In the special case of a nonparanormal distribution, a zero entry of the precision matrix $\Sigma^{-1}$ further implies conditional independence between the corresponding pair of random variables.

Next, we present the definition and some theoretical properties of the Kendall’s tau statistic.

Now suppose that we have $n$ independent draws from $X=(X_{1},\ldots,X_{p})^{T}\sim\mbox{TE}_{p}(\Sigma,r;f_{1},\dots,f_{p})$, such that $X_{ij}$ is the value of the $j$th variable in the $i$th observation. Then there is a simple connection between the population-level Kendall’s tau statistic $\tau_{jj^{\prime}}$ and $\rho_{jj^{\prime}}$.

Lemma 1 motivated Liu et al., 2012b to estimate $\rho_{jj^{\prime}}$ using

We examined the Yahoo! Finance stock price data, which is described in Liu et al., 2012b , and available in the huge package in R on CRAN. The data consists of 1258 daily closing prices for 452 stocks in the S$\&$P 500 index between January 1, 2003 and January 1, 2008. The stocks are categorized into 10 Global Industry Classification Standard (GICS) sectors. Let $S_{ij}$ denote the closing price of the $j$th stock on the $i$th day. We construct a $1257\times 452$ data matrix $X$ such that $X_{ij}=\log\left(S_{(i+1)j}/S_{ij}\right)$ for $i=1,\dots,1257$ and $j=1,\dots,452$. We standardize each stock to have mean zero and standard deviation one, as in Tan et al., (2015).

We applied transelliptical GRASS with $\gamma_{n,jj^{\prime}}$=0.5, 0.6, 0.7, and GL($\widehat{S}^{\tau}$) with $\lambda$=0.5, 0.6, 0.7. Figure 3 indicates that in all estimated graphs, stocks from the same GICS sector tend to be highly-connected, indicating that both methods provide informative graph estimates. Furthermore, plots $(c)$, $(d)$, $(e)$ and $(f)$ of Figure 3 also show that the graph estimates from transelliptical grass and GL($\widehat{S}^{\tau}$) are quite similar provided that $\gamma_{n,jj^{\prime}}=\lambda$. In fact, the arguments in Witten et al., (2011) can be used to establish the fact that when $\gamma_{n,jj^{\prime}}=\lambda$, the connected components of GRASS and GL($\widehat{S}^{\tau}$) are identical.

In this paper, we have proposed transelliptical GRASS, a simple and efficient procedure for recovering the structure of a high-dimensional transelliptical graphical model. Transelliptical GRASS is a natural extension of the GRASS proposal of Luo et al., (2015) to the non-normal setting. Transelliptical GRASS shares the attractive theoretical and computational properties of the original GRASS proposal. We have established that it performs almost as well as methods that assume Gaussianity when the data are Gaussian, and much better than methods that assume Gaussianity in the case of non-Gaussian data. Therefore, in general, there is little cost to applying transelliptical GRASS instead of the original GRASS proposal.