A Tutorial on Multifractality, Cascades, and Interactivity for Empirical Time Series in Ecological Science

Interactivity is a central theme of ecological psychology. According to Gibsonian views, behavior is the emergent property of interactions between organism and environment. Hence, an important challenge for ecological psychology has been to identify physical principles that provide an empirical window into interactivity. We suspect that multifractality, a concept from statistical physics, may be helpful in this regard, and we offer this article as a tutorial on multifractality with 2 main goals. First, we aim to describe multifractality with a series of simple, concrete, but progressively more elaborate examples that will incrementally elucidate the relationship between multifractality and interactivity. Second, we aim to describe a direct estimation method for computing the multifractal spectrum (e.g., Chhabra & Jensen, 1989 Chhabra, A. B. and Jensen, R. V. 1989. Direct determination of the f(α) singularity spectrum. Physics Review Letters, 62: 1327–1330. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), presenting it as an alternative that avoids the pitfalls of more popular methods and that may address more appropriately the measurements traditionally taken by ecological psychologists. In sum, this tutorial aims to unpack the theoretical background for an analytical method allowing rigorous test of interactivity in a variety of empirical settings.