2019/07/06 by Pierre Baudot, Monica Tapia, Daniel Bennequin +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Binary number #Fractal and DNA sequence analysis #Information theory #Random variable #Set (abstract data type) #Simple (philosophy) #Simplex #Statistical hypothesis testing #Statistical model #Topological and Geometric Data Analysis #Topological data analysis #advanced mathematical theories #cs.IT #math.IT #q-bio.NC #stat.OT
paper · pdf · doi:10.3390/e21090869
arxiv created 2019/07/06 · openalex created_date 2019/07/23 · openalex publication_date 2019/09/06 · arxiv updated 2019/10/02 · openalex updated_date 2026/08/05
This paper presents methods that quantify the structure of statistical interactions within a given data set, and was first used in \citeTapia2018. It establishes new results on the k-multivariate mutual-informations (Ik) inspired by the topological formulation of Information introduced in. In particular we show that the vanishing of all Ik for 2≤ k ≤ n of n random variables is equivalent to their statistical independence. Pursuing the work of Hu Kuo Ting and Te Sun Han, we show that information functions provide co-ordinates for binary variables, and that they are analytically independent on the probability simplex for any set of finite variables. The maximal positive Ik identifies the variables that co-vary the most in the population, whereas the minimal negative Ik identifies synergistic clusters and the variables that differentiate-segregate the most the population. Finite data size effects and estimation biases severely constrain the effective computation of the information topology on data, and we provide simple statistical tests for the undersampling bias and the k-dependences following. We give an example of application of these methods to genetic expression and unsupervised cell-type classification. The methods unravel biologically relevant subtypes, with a sample size of 41 genes and with few errors. It establishes generic basic methods to quantify the epigenetic information storage and a unified epigenetic unsupervised learning formalism. We propose that higher-order statistical interactions and non identically distributed variables are constitutive characteristics of biological systems that should be estimated in order to unravel their significant statistical structure and diversity. The topological information data analysis presented here allows to precisely estimate this higher-order structure characteristic of biological systems.