gCoda: Conditional Dependence Network Inference for Compositional Data

Abstract

The increasing quality and the reducing cost of high-throughput sequencing technologies for 16S rRNA gene profiling enable researchers to directly analyze microbe communities in natural environments. The direct interactions among microbial species of a given ecological system can help us understand the principles of community assembly and maintenance under various conditions. Compositionality and dimensionality of microbiome data are two main challenges for inferring the direct interaction network of microbes. In this article, we use the logistic normal distribution to model the background mechanism of microbiome data, which can appropriately deal with the compositional nature of the data. The direct interaction relationships are then modeled via the conditional dependence network under this logistic normal assumption. We then propose a novel penalized maximum likelihood method called gCoda to estimate the sparse structure of inverse covariance for latent normal variables to address the high dimensionality of the microbiome data. An effective Majorization-Minimization algorithm is proposed to solve the optimization problem in gCoda. Simulation studies show that gCoda outperforms existing methods (e.g., SPIEC-EASI) in edge recovery of inverse covariance for compositional data under a variety of scenarios. gCoda also performs better than SPIEC-EASI for inferring direct microbial interactions of mouse skin microbiome data.

Disentangling microbial associations from hidden environmental and technical factors via latent graphical models

“…In the microbiome context, covariance estimators are available both for compositional data [20,8] and for absolute microbial abundance data [54]. To account for transitive correlations, estimators of the inverse covariance (or precision) matrix are also available [30,16]. The inverse covariance matrix encodes the conditional dependency structure of the underlying variables and provides a parsimonious summary of (direct) taxon-taxon associations.…”

“…For our synthetic compositional covariance estimation experiments, we first created ten random replicates for cluster, band, and scale-free-type graphs each at node sizes of p ∈ [16,150]. For each graph, we set the number of edges to |E| := round(.04 × p(p−1)…”

“…Let L * = L 1 + L 2 + L 3 be a sum of three rank-1 matrices, L 1 = Ω −1 J, L 2 = JΩ −1 and L 3 = JΩ −1 J. We can compute the SVD by recursively applying (16).…”

A Framework to Incorporate D-trace Loss into Compositional Data Analysis

“…199 1. Band graph: for network recovery are investigated, including gCoda [10], SPIEC(MB) and 216 SPIEC(GL) [18]. We further consider an approximation method called aCDTr, which 217 approximates Σ with GΣ ln x G [18] and employs D-trace loss to estimate Θ = Σ −1 .…”

“…Since most species do not interact directly when the number of species p is large, we 125 further assume that the direct interaction network, or Θ, is sparse, which also helps to 126 solve the under-determined problem caused by compositionality and 127 dimensionality [9,10,29]. We employ the commonly used 1 penalty [25,29,30] to handle 128 the sparse assumption, and our sparse estimator of the precision matrix Θ is proposed as 129…”