Joint Estimation of the Two-Level Gaussian Graphical Models Across Multiple Classes

Abstract

Lemma 1: Let { Â * , Γ(m) βkk * , Î * kk , P (m) kk * } be a local minimizer of Q1, then there exists a local minimizer { Â * * , Γ(m) βkk * * , Î * * kk , P (m) kk * * } of Q2, such that α * kk Γ(m) βkk * = α * * kk Γ(m) βkk * * and ιkk i,j * ρkk(m) i,j * = ιkk i,j * * ρkk(m) i,j * * . Similarly, let { Â * * , Γ(m) βkk * * , Î * * kk , P (m) kk * * } be a local minimizer of Q2, then there exists a local minimizer { Â * , Γ(m) βkk * , Î * kk , P (m) kk * } of Q1, such that α * kk Γ(m) βkk * = α * * kk Γ(m) βkk * * and ιkk i,j * ρkk(m) i,j * = ιkk i,j * * ρkk(m) i,j * * . Proof: Suppose { Â * , Γ(m) βkk * , Î * kk , P (m) kk * } is a local minimizer of Q1. We would like to prove { Â * * = * * = Ω(m) kk * * / η 12 M m=1 | Ω(m) kk * * | 1 ,

Gaussian graphical models with applications to omics analyses

“…An example of multi-level data in an omics context is genes and pathways; each gene belongs to one or more pathways, and one can estimate a two-level graphical model consisting of within-pathway edges between genes and between-pathway edges between pathways. 117,118 We close by noting that while GGMs are not a particularly new concept, their utility has been re-emphasized as technology has evolved over the last several decades. In the early 21st century, advances in omics and other big data outpaced existing statistical methods; the development of regularization-based approaches such as the graphical lasso was a transformative step forward.…”

Joint Gaussian Graphical Model Estimation: A Survey

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Multiple and multilevel graphical models

“…Shan, Qiao, Cheng, and Kim () proposed jointly estimating two‐level Gaussian graphical models by sharing the common two‐level conditional dependency structure during the estimation procedure. They further developed a joint estimation method for the multilevel Gaussian graphical models across multiple classes.…”

“…Here, “adjacent” means that two groups are placed next to each other in terms of the precision matrices' layout. By using the weighted GLASSO algorithm by Friedman et al (), Shan et al () then developed the joint estimation method for multilevel Gaussian graphical models across multiple classes.…”