Ergodic descriptors of non-ergodic stochastic processes

The stochastic processes underlying the growth and stability of biological and psychological systems reveal themselves when far-from-equilibrium. Far-from-equilibrium, non-ergodicity reigns. Non-ergodicity 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 non-ergodicity (i.e. of the raw measured data) without themselves (i.e. the estimates) being non-ergodic. We show that traditional linear statistics such as the standard deviation, coefficient of variation and root mean square can break ergodicity. Time series of statistics addressing sequential structure and its potential nonlinearity: fractality and multi-fractality, change in a time-independent way and fulfil the ergodic assumption. Complementing traditional linear indices with fractal and multi-fractal indices would empower the study of stochastic far-from-equilibrium biological and psychological dynamics.