Ergodic descriptors of nonergodic stochastic processes

TL;DR

This paper explores how to extract ergodic measures from nonergodic biological and psychological data, proposing fractal and multifractal indices as stable tools for analyzing far-from-equilibrium stochastic processes.

Contribution

It introduces the use of fractal and multifractal indices to achieve ergodic, stable statistical measures from nonergodic data in biological and psychological systems.

Findings

Linear statistics can be nonstationary and violate ergodicity.

Fractal and multifractal indices change in a time-independent way.

Adding fractal indices improves analysis of complex dynamics.

Abstract

The stochastic processes underlying the growth and stability of biological and psychological systems reveal themselves when far from equilibrium. Far from equilibrium, nonergodicity reigns. Nonergodicity implies that the average outcome for a group/ensemble (i.e., of representative organisms/minds) is not necessarily a reliable estimate of the average outcome for an individual over time. However, the scientific interest in causal inference suggests that we somehow aim at stable estimates of the cause that will generalize to new individuals in the long run. Therefore, the valid analysis must extract an ergodic stationary measure from fluctuating physiological data. So the challenge is to extract statistical estimates that may describe or quantify some of this nonergodicity (i.e., of the raw measured data) without themselves (i.e., the estimates) being nonergodic. We show that traditional linear statistics such as the standard deviation (SD), coefficient of variation (CV), and root mean square (RMS) can show nonstationarity, violating the ergodic assumption. Time series of statistics addressing sequential structure and its potential nonlinearity: fractality and multifractality, change in a time-independent way and fulfill the ergodic assumption. Complementing traditional linear indices with fractal and multifractal indices would empower the study of stochastic far-from-equilibrium biological and psychological dynamics.