MULTIVARIATE MULTISCALE ANALYSIS OF NATURAL AND ENGINEERED DYNAMICAL SYSTEMS

Abstract

Most natural and engineered systems possess both spatial and temporal multiscale characteristics. In some complex systems (e.g., molecular dynamics and muscle fatigue dynamics), fast-time scales may dominate and/or obscure slow-time processes. In many practical situations, fast-time data is used to infer slow-time dynamics. The objective of this dissertation is to investigate the application of the newly developed multivariate analysis method of smooth orthogonal decomposition (SOD) to a general class of hierarchical dynamical systems. In particular, the SOD methodology was applied to a muscle fatigue accumulation study and protein dynamics. The concept of phase space warping (PSW) refers to the deformations of fasttime phase space trajectories due to slow-time parameter drift. By characterizing these deformations caused by the slow-time processes from only the fast-time dynamics, feature vectors can be developed which contain the slow-time information. The idea of SOD is to extract smooth deterministic trends from multivariate data by considering the temporal and spatial characteristics of the data set. By doing so, SOD identies time functions (SOCs) that are smoothest in time and have maximal variance. Therefore, it is our hope that the slow-time dynamics captured in the PSW feature space is in a one-to-one relationship with the smooth deterministic trends identied by SOD. We discuss several advances to the already existing technology of PSW and SOD which have led to considerable improvements in the results. The three main modications to the PSW/SOD methodology are: (1) a new weighting function is used for the PSW feature estimation, (2) SOD is applied to the nonlinear (i.e., polynomial) expansion of the original SOD coordinates, and (3) ad-hoc F test is used to statistically determine the number of SOD coordinates needed to reconstruct muscle fatigue. For the muscle fatigue accumulation, we hypothesize that: (1) SOD analysis of measured fast-time motion kinematics can be used to reconstruct the slow-time dynamics of muscle fatigue to establish a mapping between the kinematics and fatigue, and (2) a nonlinear extension of SOD can identify more optimal fatigue coordinates to provide a lower-dimensional reconstruction of the fatigue dynamics. These hypotheses were tested using movement kinematics recorded in ten subjects performing a high and low sawing motion and three load carrying walking Army soldiers. Independent local and global fatigue markers indicated by traditional surface electromyography (EMG) (recording of the electrical signals generated by individual muscles during a task) and breath-by-breath oxygen consumption _V O2, respectively, were used for validation. SOD