Ergodicity breaking is a challenge for biological and psychological sciences.
Ergodicity is a necessary condition for linear causal modeling. Long-range
correlations and non-Gaussianity characterizing various biological and
psychological measurements break ergodicity routinely, threatening our capacity
for causal modeling. Long-range correlations (e.g., in fractional Gaussian
noise, a.k.a. "pink noise") break ergodicity--in raw Gaussian series, as well
as in some but not all standard descriptors of variability, i.e., in
coefficient of variation (CV) and root mean square (RMS) but not standard
deviation (SD) for longer series. The present work demonstrates that
progressive increases in non-Gaussianity conspire with long-range correlations
to break ergodicity in SD for all series lengths. Meanwhile, explicitly
encoding the cascade dynamics that can generate temporally correlated
non-Gaussian noise offers a way to restore ergodicity to our causal models.
Specifically, fractal and multifractal properties encode both scale-invariant
power-law correlations and their variety, respectively, both of which features
index the underlying cascade parameters. Fractal and multifractal descriptors
of long-range correlated non-Gaussian processes show no ergodicity breaking and
hence, provide a more stable explanation for the long-range correlated
non-Gaussian form of biological and psychological processes. Fractal and
multifractal descriptors offer a path to restoring ergodicity to causal
modeling in these fields.Comment: 33 pages, 11 figures. arXiv admin note: text overlap with
arXiv:2202.0109