Multifractal test for nonlinear changes in time series

Abstract

The creativity and emergence of biological and psychological behavior are nonlinear. However, that does not necessarily mean only that the measurements of the behaviors are curvilinear. Furthermore, the linear model might fail to reduce these measurements to a sum of independent random factors, implying nonlinear changes over time. The present work reviews some of the concepts implicated in linear changes over time and details the mathematical steps involved. It introduces multifractality as a mathematical framework helpful in determining whether and to what degree the measured time series exhibits nonlinear changes over time. The 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 gives us the means to make those nonlinear excursions explicit and perhaps permit the development of theory that draws on both linear and nonlinear processes.