Multifractal test for nonlinearity of interactions across scales in time series

Abstract

The creativity and emergence of biological and psychological behavior tend to be nonlinear—biological and psychological measures contain degrees of irregularity. The linear model might fail to reduce these measurements to a sum of independent random factors (yielding a stable mean for the measurement), implying nonlinear changes over time. The present work reviews some of the concepts implicated in nonlinear changes over time and details the mathematical steps involved in their identification. It introduces multifractality as a mathematical framework helpful in determining whether and to what degree the measured series exhibits nonlinear changes over time. These mathematical steps include multifractal analysis and surrogate data production for resolving when multifractality entails nonlinear changes over time. Ultimately, when measurements fail to fit the structures of the traditional linear model, multifractal modeling allows making those nonlinear excursions explicit, that is, to come up with a quantitative estimate of how strongly events may interact across timescales. This estimate may serve some interests as merely a potentially statistically significant indicator, but we suspect that this estimate might serve more generally as a predictor of perceptuomotor or cognitive performance.

Multifractal Nonlinearity Moderates Feedforward and Feedback Responses to Suprapostural Perturbations

“…Below we outline Chhabra and Jensen’s (1989) multifractal analysis for non-negative series x(t) of length N. Greater detail and explanation of the conceptual background/interpretation are available in our review of specific types of interactivity responsible for multifractal fluctuations (Kelty-Stephen et al, 2013), our explanation of why multifractal modeling is more sensitive to nonlinearity than monofractal modeling (Kelty-Stephen & Wallot, 2017), and our explanation of how multifractal modeling is specifically relevant for modeling psychological measures to inform psychological theories (Kelty-Stephen et al, 2022).…”

“…The IAAFT procedure randomizes original values time-symmetrically around the autoregressive structure, thus generating surrogates that randomize phase ordering of the series’ spectral amplitudes while preserving only linear temporal correlations. To estimate how much observed multifractality reflects nonlinearity, Δα of each original series is compared to Δα of the surrogate series (Kelty-Stephen et al, 2013; 2022). Multifractality due to nonlinearity is quantified as the one-sample t -statistic (t MF ) comparing Δα of the original series to that of the surrogates (Figure 3).…”

Revisiting Research on Positive Psychology in Second and Foreign Language Education: Trends and Directions

“…Inspired by the "Complexity and Dynamic Systems Theory (CDST)", TSA aims to generate a thorough understanding of the dynamic and evolving essence of cognitive, emotional, or behavioral variables (Jin, 2022;Montgomery et al, 2008). TSA, according to Kelty-Stephen et al (2022), enables the retrodiction or prediction of dynamic and complex events in the past or future and, thus, can considerably contribute to the unraveling of the nuanced changes in teachers' and students' positive characteristics, behaviors, and emotions.…”