Identifying multiplicative interactions between temporal scales of human movement variability

Conventional scaling analyses of human movement variability such as detrended fluctuation analyses assume that the movement variable can be decomposed into scale-dependent variation. However, these conventional scaling analyses are insensitive to multiplicative interactions within the movement variable. Multiplicative interactions refer to couplings between the scale-dependent variations across multiple scales that generate intermittent changes in the human movement variable. The mathematical concept for intermittent variability generated by multiplicative interactions is called multifractal variability. Multifractal variability is numerically defined by a spectrum of scaling exponents (i.e., a multifractal spectrum) that can be an important feature of coordinated movements and, consequently, relevant when aiming to identify movement disorders. In the current study, a new method is introduced based on detrended fluctuation analysis that can identify the multifractal spectrum from the temporal variation of local scaling exponents. The influence of multiplicative interactions on the local scaling exponents is tested by a Monte Carlo surrogate test. The methods are validated on multiplicative cascading processes with known multiplicative interactions. The application of the new methods is subsequently illustrated on an example of centre of pressure variations during quiet and relaxed standing. The results show that multiplicative interactions are present during periods with large movements of the center of gravity, where the movements of the centre of gravity and centre of pressure couple into coordinative structures. Further application and interpretation of the developed method for the study of human movement variability are discussed.