A combinatorial view of stochastic processes: White noise

TL;DR

This paper introduces a combinatorial approach to white noise using ordinal pattern analysis, providing new insights into its structure and enabling simple statistical analysis of time series data.

Contribution

It develops a novel combinatorial framework for white noise based on ordinal patterns and permutation classes, with exact probability distributions and practical applications.

Findings

Exact p.m.f. of a permutation-based functional for white noise

Approximation of the p.m.f. by a Gaussian distribution

Application to nanoparticle diffusion data

Abstract

White noise is a fundamental and fairly well understood stochastic process that conforms the conceptual basis for many other processes, as well as for the modeling of time series. Here we push a fresh perspective toward white noise that, grounded on combinatorial considerations, contributes to give new interesting insights both for modelling and theoretical purposes. To this aim, we incorporate the ordinal pattern analysis approach which allows us to abstract a time series as a sequence of patterns and their associated permutations, and introduce a simple functional over permutations that partitions them into classes encoding their level of asymmetry. We compute the exact probability mass function (p.m.f.) of this functional over the symmetric group of degree $n$, thus providing the description for the case of an infinite white noise realization. This p.m.f. can be conveniently approximated by a continuous probability density from an exponential family, the Gaussian, hence providing natural sufficient statistics that render a convenient and simple statistical analysis through ordinal patterns. Such analysis is exemplified on experimental data for the spatial increments from tracks of gold nanoparticles in 3D diffusion.