tion
We first introduce the notation of matrix norms used in the rest of the paper. For a vector a = (a1, . . . , ap)T ∈ R, define |a|1 =
Pp
i=1|ai| and |a|2 = pPp
i=1a2i. For
a matrix A = (aij) ∈ Rp×q, we define the entrywise `1 norm |A|1 = Pp
i=1 Pq
j=1|aij|
and the entrywise `∞ norm |A|∞ = max1≤i≤p,q≤j≤q|aij|. We further define the ma-
trix `1 norm by kAkL1 = max1≤j≤q
Pp
i=1|aij|, the matrix `∞ norm by kAkL∞ =
max1≤i≤p
Pq
j=1|aij|, the spectral normkAk2 = max|x|2≤1|Ax|2and the Frobenius norm
bykAkF =
qP
4.2.1
Covariate Adjusted Gaussian Graphical Model
We consider the following model
y= Γ0x+z, (4.2.1)
wherey= (y1, . . . , yp)T is a random vector denoting the expression levels forpgenes,
x= (x1, . . . , xq)T is a random vector describe the coding for q genetic markers, Γ0 is
ap×qunknown coefficient matrix,zis ap×1 normal random vector with mean zero, covariance matrix Σ0 and precision matrix Ω0 = (ωij0) = Σ−01. Since the segregating
population in the genetical genomics experiments can be viewed as random due to the random recombination process, x can be treated as a random variable. We further assume xand z are independent. Assume that we have n i.i.d observations (xk,yk)
(k = 1, . . . , n) for the model.
Model (4.2.1) is similar to the seemingly unrelated regression (sur) model in Zeller (1962), which aimed to improve the estimation efficiency of the effects of ge- netic variants on gene expressions by considering the residual correlations of the gene expressions of many genes. However, the model is viewed differently here with a focus on improving the estimation accuracy of the conditional dependency structure of y
by adjusting for the covariates x.
In eQTL studies, each row of Γ0 is assumed to be sparse since one gene is expected
to have only a few genetic regulators. The precision matrix Ω0 is also expected to be
conditional dependency and can be use to construct a conditional dependency graph. To be specific, let G = (V, E) be a graph representing conditional independence relations between components of y. The vertex set V has p components y1, . . . , yp
and the edge setE consists of pairs (i, j), where (i, j)∈E if there is an edge between
yi and yj. The edge between yi and yj is excluded from E if and only if zi and
zj are independent given other zk (k 6= i, j). Since z follow a multivariate normal
distribution, the conditional independence of zi and zj leads to ωij = 0. We are
interested in detecting the non-zero entries of Ω0so that we can construct a conditional
independency graph foryafter the effects of the covariatesxonyare adjusted. Such a graphical model is called the covariate-adjusted Gaussian graphical model.
In this paper, we consider the setting where bothpandq can be much larger than
n. The assumption fits the real application of analysis of genetical genomics data where there are usually thousands of genes and genetic markers, but relatively small sample sizes.
4.2.2
Estimation of
Γ
0When q = 1, many novel methods have been developed for estimation of Γ0,
including the methods based on the `1 minimization such as thelasso in Tibshirani (1996) and the Dantzig selector in Cand`es & Tao (2007). We propose to develop method for estimating Γ0 usingl1 minimization that can be treated as a multivariate
Let ¯y=n−1Pn
k=1yk, ¯x=n−1
Pn
k=1xk, ¯z=n−1
Pn
k=1zk. Then it follows that
yk−y¯ = Γ0(xk−x¯) +zk−z¯. (4.2.2) Set Sxy =n−1 Pn k=1(yk−y¯)(xk−x¯)T and Sxx= n−1 Pn k=1(xk−x¯)(xk−x¯)T. We
propose to estimate Γ by solving the following the optimization problem: min|Γ|1 subj ¯ ¯ ¯Sxy−ΓSxx ¯ ¯ ¯ ∞ ≤λn, 1≤i≤p, 1≤j ≤q, (4.2.3)
where λn is a tuning parameter that will be specified later. Note that (4.2.3) is
equivalent to the following p optimization problems: for 1≤i≤p, min|γi|1 subj ¯ ¯ ¯Sxy,i−γTi Sxx ¯ ¯ ¯ ∞ ≤λn, 1≤j ≤q, (4.2.4) where Γ =: (γ1, . . . ,γp)T and S
xy =: (Sxy,1, . . . , Sxy,p)T. This is exactly the Dantzig
selector formulation for simple regression analysis for the ith gene and its solution can therefore be obtained by solving the corresponding linear programming problem (Cand`es & Romberg, 2005) or by some alternative methods (Becker & Grant, 2010; Lu, 2009; ?). This simple observation is useful for implementation and technical analysis. In this paper and the R package capmewe develop, we implement a linear programming optimization using the primal dual and interior point algorithm.
4.2.3
Estimation of
Ω
0Suppose now we have an estimation ˆΓ by (4.2.3). After plugging ˆΓ into the equations (4.2.2), Ω0 can be estimated by the method of constrained`1-minimization
proposed in Caiet al. (In press). Let
Syy = 1 n n X k=1 (yk−Γˆxk)(yk−Γˆxk)T.
We estimate Ω0 by solving the optimization problem:
min|Ω|1 subj |Ip×p−SyyΩ|∞≤τn, (4.2.5)
where τn is a tuning parameter. Let ˆΩ1 = (ˆωij1) be any solution of (4.2.5). This
constrainedl1 minimization approach is the same as the climeproposed in Caiet al.
(In press), except that Syy depend on the estimated regression coefficient matrix
ˆ
Γ. Note that we do not impose the symmetry condition on ˆΩ1 and as a result the
solution is not symmetric in general. The final capme estimator of Ω0, denoted by
ˆ
Ω = (ˆωij), is obtained by symmetrizing the estimator as follows.
ˆ
Ω = (ˆωij), where ˆωij = ˆωji = ˆωij1I{|ωˆ1ij| ≤ |ωˆji1|}+ ˆωji1I{|ωˆij1|>|ωˆji1|}. (4.2.6)
As in (4.2.4), the problem (4.2.6) can be decomposed into p optimization problems. For 1≤i≤p, let ˆωi be the solution of the following convex optimization problem
where ωi is a vector in Rp, ei is a standard unit vector in Rp with 1 in the i-th
coordinate and 0 in all other coordinates.