Topological Information Data Analysis
Abstract
This paper presents methods that quantify the structure of statistical interactions within a given data set, and was first used in \cite{Tapia2018}. It establishes new results on the kmultivariate mutualinformations (I_k) inspired by the topological formulation of Information introduced in. In particular we show that the vanishing of all I_k for 2\leq k \leq 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 coordinates for binary variables, and that they are analytically independent on the probability simplex for any set of finite variables. The maximal positive I_k identifies the variables that covary the most in the population, whereas the minimal negative I_k identifies synergistic clusters and the variables that differentiatesegregate 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 kdependences following. We give an example of application of these methods to genetic expression and unsupervised celltype 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 higherorder 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 higherorder structure characteristic of biological systems.
 Publication:

Entropy
 Pub Date:
 September 2019
 DOI:
 10.3390/e21090869
 arXiv:
 arXiv:1907.04242
 Bibcode:
 2019Entrp..21..869B
 Keywords:

 Statistics  Other Statistics;
 Computer Science  Information Theory;
 Quantitative Biology  Neurons and Cognition
 EPrint:
 doi:10.3390/e21090869