Partial Information Decomposition and the Information Delta: A Geometric Unification Disentangling Non-Pairwise Information.

Information theory provides robust measures of multivariable interdependence, but classically does little to characterize the multivariable relationships it detects. The Partial Information Decomposition (PID) characterizes the mutual information between variables by decomposing it into unique, redundant, and synergistic components. This has been usefully applied, particularly in neuroscience, but there is currently no generally accepted method for its computation. Independently, the Information Delta framework characterizes non-pairwise dependencies in genetic datasets. This framework has developed an intuitive geometric interpretation for how discrete functions encode information, but lacks some important generalizations. This paper shows that the PID and Delta frameworks are largely equivalent. We equate their key expressions, allowing for results in one framework to apply towards open questions in the other. For example, we find that the approach of Bertschinger et al. is useful for the open Information Delta question of how to deal with linkage disequilibrium. We also show how PID solutions can be mapped onto the space of delta measures. Using Bertschinger et al. as an example solution, we identify a specific plane in delta-space on which this approach's optimization is constrained, and compute it for all possible three-variable discrete functions of a three-letter alphabet. This yields a clear geometric picture of how a given solution decomposes information.

CINOEDV: a co-information based method for detecting and visualizing n-order epistatic interactions.

BACKGROUND: Detecting and visualizing nonlinear interaction effects of single nucleotide polymorphisms (SNPs) or epistatic interactions are important topics in bioinformatics since they play an important role in unraveling the mystery of "missing heritability". However, related studies are almost limited to pairwise epistatic interactions due to their methodological and computational challenges. RESULTS: We develop CINOEDV (Co-Information based N-Order Epistasis Detector and Visualizer) for the detection and visualization of epistatic interactions of their orders from 1 to n (n ≥ 2). CINOEDV is composed of two stages, namely, detecting stage and visualizing stage. In detecting stage, co-information based measures are employed to quantify association effects of n-order SNP combinations to the phenotype, and two types of search strategies are introduced to identify n-order epistatic interactions: an exhaustive search and a particle swarm optimization based search. In visualizing stage, all detected n-order epistatic interactions are used to construct a hypergraph, where a real vertex represents the main effect of a SNP and a virtual vertex denotes the interaction effect of an n-order epistatic interaction. By deeply analyzing the constructed hypergraph, some hidden clues for better understanding the underlying genetic architecture of complex diseases could be revealed. CONCLUSIONS: Experiments of CINOEDV and its comparison with existing state-of-the-art methods are performed on both simulation data sets and a real data set of age-related macular degeneration. Results demonstrate that CINOEDV is promising in detecting and visualizing n-order epistatic interactions. CINOEDV is implemented in R and is freely available from R CRAN: http://cran.r-project.org and https://sourceforge.net/projects/cinoedv/files/ .

HC-HDSD: A method of hypergraph construction and high-density subgraph detection for inferring high-order epistatic interactions.

Detecting epistatic interactions, or nonlinear interactive effects of Single Nucleotide Polymorphisms (SNPs), has gained increasing attention in explaining the "missing heritability" of complex diseases. Though much work has been done in mapping SNPs underlying diseases, most of them constrain to 2-order epistatic interactions. In this paper, a method of hypergraph construction and high-density subgraph detection, named HC-HDSD, is proposed for detecting high-order epistatic interactions. The hypergraph is constructed by low-order epistatic interactions that identified using the normalized co-information measure and the exhaustive search. The hypergraph consists of two types of vertices: real ones representing main effects of SNPs and virtual ones denoting interactive effects of epistatic interactions. Then, both maximal clique centrality algorithm and near-clique mining algorithm are employed to detect high-density subgraphs from the constructed hypergraph. These high-density subgraphs are inferred as high-order epistatic interactions in the HC-HDSD. Experiments are performed on several simulation data sets, results of which show that HC-HDSD is promising in inferring high-order epistatic interactions while substantially reducing the computation cost. In addition, the application of HC-HDSD on a real Age-related Macular Degeneration (AMD) data set provides several new clues for the exploration of causative factors of AMD.