  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.
