# 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 fast-time 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 identifies 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 identified 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 modifications 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˙O*_{2}, respectively, were used for validation. ^ SOD analysis indicates that the linear SOCs are contaminated with more high-frequency content than the corresponding nonlinear coordinates for the same modal index across all subjects and experiments. This resulted in considerably fewer nonlinear SOCs needed to reconstruct the muscle fatigue trend compared to the linear SOCs. Between subject variability *R*^{ 2} values, which are used to determine the quality of fits, indicate roughly a 20 dimensional subspace is needed to reconstruct the EMG trends using linear SOD analysis. However, utilizing nonlinear SOD, it was shown that in certain cases, where fatigue was actually present, a three to five dimensional subspace of cubic SOCs was sufficient. Similar results were seen in traditional *V˙O*_{2} trends. Thus, nonlinear SOD extracts more optimal coordinates than linear SOD, and in doing so, reduces the required dimensionality by roughly one order of magnitude. In addition, roughly 20 linear SOD modes were needed to capture a majority of the energy in the movement kinematics, which confers with the *R*^{ 2} values. ^ For the protein dynamics we hypothesize that: (1) SOD analysis of a full scale short time simulation can identify slow-time modes, and (2) slow coordinate subspaces will contain the peptide conformational geometry and information about the amount of time the peptide stays in each conformation. These were demonstrated by simulating a five residue cyclic pentapeptide-cyclo(d-Pro1-Ala2-Ala3-Ala4-Ala5) which was previously assumed to oscillate between two conformational states. SOD analysis of Cartesian position and velocity components of the atoms in the backbone chain show some manifestation of rigid body modes. SOD analysis of transformed generalized internal coordinates extracts five dominant slow-time modes whose corresponding coordinates extract the conformational transitions of the peptide. Furthermore, the slow-time subspace identifies 33 conformational states instead of two and the total time spent in each conformation is found to range from 0.107% to 11.478% of the total time. ^

#### Subject Area

Engineering, Mechanical

#### Recommended Citation

David B Segala,
"Multivariate multiscale analysis of natural and engineered dynamical systems"
(2011).
*Dissertations and Master's Theses (Campus Access).*
Paper AAI3449439.

http://digitalcommons.uri.edu/dissertations/AAI3449439