Prediction in Projection: A new paradigm in delay-coordinate reconstruction

Abstract

Delay-coordinate embedding is a powerful, time-tested mathematical framework for reconstructing the dynamics of a system from a series of scalar observations. Most of the associated theory and heuristics are overly stringent for real-world data, however, and real-time use is out of the question due to the expert human intuition needed to use these heuristics correctly. The approach outlined in this thesis represents a paradigm shift away from that traditional approach. I argue that perfect reconstructions are not only unnecessary for the purposes of delay-coordinate based forecasting, but that they can often be less effective than reduced-order versions of those same models. I demonstrate this using a range of low- and high-dimensional dynamical systems, showing that forecast models that employ imperfect reconstructions of the dynamics---i.e., models that are not necessarily true embeddings---can produce surprisingly accurate predictions of the future state of these systems. I develop a theoretical framework for understanding why this is so. This framework, which combines information theory and computational topology, also allows one to quantify the amount of predictive structure in a given time series, and even to choose which forecast method will be the most effective for those data.