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

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 dynamicsi.e., models that are not necessarily true embeddingscan 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.