  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.
