arXiv subjects

Nikita A. Sakhanenko

Publications and source records attributed to Nikita A. Sakhanenko.

7 recordsLinked to original sources

arXiv2017

Expansion of the Kullback-Leibler Divergence, and a new class of information metrics

Inferring and comparing complex, multivariable probability density functions is fundamental to problems in several fields, including probabilistic learning, network theory, and data analysis. Classification and prediction are the two faces of this class of problem. We take an approach here that simplifies many aspects of these problems by presenting a structured, series expansion of the Kullback-Leibler divergence - a function central to information theory - and devise a distance metric based on this divergence. Using the Möbius inversion duality between multivariable entropies and multivariable interaction information, we express the divergence as an additive series in the number of interacting variables, which provides a restricted and simplified set of distributions to use as approximation and with which to model data. Truncations of this series yield approximations based on the number of interacting variables. The first few terms of the expansion-truncation are illustrated and shown to lead naturally to familiar approximations, including the well-known Kirkwood superposition approximation. Truncation can also induce a simple relation between the multi-information and the interaction information. A measure of distance between distributions, based on Kullback-Leibler divergence, is then described and shown to be a true metric if properly restricted. The expansion is shown to generate a hierarchy of metrics and connects this work to information geometry formalisms. We give an example of the application of these metrics to a graph comparision problem that shows that the formalism can be applied to a wide range of network problems, provides a general approach for systematic approximations in numbers of interactions or connections, and a related quantitative metric.

cs.IT↗

arXiv2016

Multivariate information measures: a unification using Möbius operators on subset lattices

Information related measures are useful tools for multi variable data analysis, as measures of dependence among variables, and as descriptions of order in biological and physical systems. Information related measures, like marginal entropies, mutual / interaction / multi-information, have been used in a number of fields including descriptions of systems complexity and biological data analysis. The mathematical relationships among these measures are therefore of significant interest. Relations between common information measures include the duality relations based on Möbius inversion on lattices. These are the direct consequence of the symmetries of the lattices of the sets of variables (subsets ordered by inclusion). While the mathematical properties and relationships among these information-related measures are of significant interest, there has been, to our knowledge, no systematic examination of the full range of relationships and no unification of this diverse range of functions into a single formalism as we do here. In this paper we define operators on functions on these lattices based on the Möbius inversion idea that map the functions into one another (Möbius operators.) We show that these operators form a simple group isomorphic to the symmetric group S3. Relations among the set of functions on the lattice are transparently expressed in terms of the operator algebra, and, applied to the information measures, can be used to derive a wide range of relationships among measures. We describe a direct relation between sums of conditional log-likelihoods and previously defined dependency measures. The algebra is naturally generalized which yields more extensive relationships. This formalism provides a fundamental unification of information related measures, but isomorphism of all distributive lattices with the subset lattice implies broad potential application of these results.

cs.IT↗

arXiv2013

Describing the complexity of systems: multi-variable "set complexity" and the information basis of systems biology

Context dependence is central to the description of complexity. Keying on the pairwise definition of "set complexity" we use an information theory approach to formulate general measures of systems complexity. We examine the properties of multi-variable dependency starting with the concept of interaction information. We then present a new measure for unbiased detection of multi-variable dependency, "differential interaction information." This quantity for two variables reduces to the pairwise "set complexity" previously proposed as a context-dependent measure of information in biological systems. We generalize it here to an arbitrary number of variables. Critical limiting properties of the "differential interaction information" are key to the generalization. This measure extends previous ideas about biological information and provides a more sophisticated basis for study of complexity. The properties of "differential interaction information" also suggest new approaches to data analysis. Given a data set of system measurements differential interaction information can provide a measure of collective dependence, which can be represented in hypergraphs describing complex system interaction patterns. We investigate this kind of analysis using simulated data sets. The conjoining of a generalized set complexity measure, multi-variable dependency analysis, and hypergraphs is our central result. While our focus is on complex biological systems, our results are applicable to any complex system.

cs.IT↗

arXiv2013

On Lattices and the Dualities of Information Measures

Measures of dependence among variables, and measures of information content and shared information have become valuable tools of multi-variable data analysis. Information measures, like marginal entropies, mutual and multi-information, have a number of significant advantages over more standard statistical methods, like their reduced sensitivity to sampling limitations than statistical estimates of probability densities. There are also interesting applications of these measures to the theory of complexity and to statistical mechanics. Their mathematical properties and relationships are therefore of interest at several levels. Of the interesting relationships between common information measures, perhaps none are more intriguing and as elegant as the duality relationships based on Mobius inversions. These inversions are directly related to the lattices (posets) that describe these sets of variables and their multi-variable measures. In this paper we describe extensions of the duality previously noted by Bell to a range of measures, and show how the structure of the lattice determines fundamental relationships of these functions. Our major result is a set of interlinked duality relations among marginal entropies, interaction information, and conditional interaction information. The implications of these results include a flexible range of alternative formulations of information-based measures, and a new set of sum rules that arise from path-independent sums on the lattice. Our motivation is to advance the fundamental integration of this set of ideas and relations, and to show explicitly the ways in which all these measures are interrelated through lattice properties. These ideas can be useful in constructing theories of complexity, descriptions of large scale stochastic processes and systems, and in devising algorithms and approximations for computations in multi-variable data analysis.

cs.IT↗

arXiv2010

Markov Logic Networks in the Analysis of Genetic Data

Complex, non-additive genetic interactions are common and can be critical in determining phenotypes. Genome-wide association studies (GWAS) and similar statistical studies of linkage data, however, assume additive models of gene interactions in looking for genotype-phenotype associations. These statistical methods view the compound effects of multiple genes on a phenotype as a sum of partial influences of each individual gene and can often miss a substantial part of the heritable effect. Such methods do not use any biological knowledge about underlying genotype-phenotype mechanisms. Modeling approaches from the AI field that incorporate deterministic knowledge into models to perform statistical analysis can be applied to include prior knowledge in genetic analysis. We chose to use the most general such approach, Markov Logic Networks (MLNs), as a framework for combining deterministic knowledge with statistical analysis. Using simple, logistic regression-type MLNs we have been able to replicate the results of traditional statistical methods. Moreover, we show that even with simple models we are able to go beyond finding independent markers linked to a phenotype by using joint inference that avoids an independence assumption. The method is applied to genetic data on yeast sporulation, a phenotype governed by non-linear gene interactions. In addition to detecting all of the previously identified loci associated with sporulation, our method is able to identify four loci with small effects. Since their effect on sporulation is small, these four loci were not detected with methods that do not account for dependence between markers due to gene interactions. We show how gene interactions can be detected using more complex models, which can be used as a general framework for incorporating systems biology with genetics.

q-bio.GN↗

arXiv2007

Analysis of Proton Radiography Images of Shock Melted/Damaged Tin

Tin coupons were shock damaged/melted under identical conditions with a diverging high explosive shock wave. Proton Radiography images and velocimetry data from experiments with seven different tin coupons of varying thickness are analyzed. Comparing experiments with identical samples allowed us to distinguish between repeatable and random features. Shapes and velocities of the main fragments are deterministic functions of the coupon thickness; random differences exist only at a small scale. Velocities of the leading layer and of the main fragment differ by the same value independently of coupon thicknesses, which is likely related to the separation energy of metal layers.

physics.data-an↗

arXiv2006

Application of Support Vector Regression to Interpolation of Sparse Shock Physics Data Sets

Shock physics experiments are often complicated and expensive. As a result, researchers are unable to conduct as many experiments as they would like - leading to sparse data sets. In this paper, Support Vector Machines for regression are applied to velocimetry data sets for shock damaged and melted tin metal. Some success at interpolating between data sets is achieved. Implications for future work are discussed.

cs.AI↗