How Does State Space Definition Influence the Measure of Chaotic Behavior?

Abstract

In the case of experimental data the largest Lyapunov exponent is a measure which is used to quantify the amount of chaos in a time series on the basis of a trajectory reconstructed in a phase (state) space. The authors’ goal was to analyze the influence of a state space definition on the measure of chaos. The time series which represent the joint angles of hip, knee and ankle joints were recorded using the motion capture technique in the CAREN Extended environment. Fourteen elderly subjects (‘65+’) participated in the experiments. Six state spaces based on univariate or multivariate time series describing a movement at individual joints were taken into consideration. The authors proposed a modified version of the False Nearest Neighbors algorithm adjusted for determining the embedding dimension in the case of a multivariate time series representing gait data (MultiFNN). The largest short-term Lyapunov exponent was computed in two variants for six scenarios of trials based on different assumptions regarding walking speed, platform inclination, and optional external perturbation. A statistical analysis confirmed a significant difference between values of the Lyapunov exponent for different state spaces. In addition, computation time was measured and averaged across the spaces.