Data-Driven Dynamics: Estimation in the Presence of Noise

Dynamical systems theory is routinely applied to a mathematical model of a process rather than the process itself. In contrast, this dissertation advances a data-driven approach to dynamics in which conclusions are drawn directly from observations of the process of interest. The focus here is on laying mathematical foundations for this perspective; thus the data has been idealized as paths of stochastic differential equations (SDEs). An example that makes this summary concrete is the study of locomotion: A "routine'' approach is to analyze a model; for instance, is the runner's gait stable or unstable per the model? Instead, this dissertation is motivated by motion capture data---a time series reminiscent of a periodic process perturbed by noise. A prevailing theory is that organisms remain upright by using sensory information (akin to motion capture data) to subconsciously estimate the stability of their gait. Chapter 2 studies this estimation problem mathematically. It derives an inequality that quantifies the uncertainty intrinsic to Floquet multiplier estimates constructed from SDE paths. This inequality governs a sufficiently broad class of estimation strategies that the bound it establishes sheds light on the feasibility of the theory about how animals remain upright. The "data-first" perspective pursued in Chapter 2 is so underdeveloped in the context of dynamics that the other chapters of this dissertation arise naturally: Do certain types of noise yield better multiplier estimates? (Chapter 3.) And what should be done when observations of the process are costly? (Chapter 4.