Multifractal test for nonlinearity of interactions across scales in time series

The creativity and emergence of biological and psychological behavior tend to be nonlinear, and correspondingly, 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 for 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 of independence failing to hold, but we suspect that this estimate might serve more generally as a predictor of perceptuomotor or cognitive performance.