  Understanding dynamics in complex systems is challenging because there are
many degrees of freedom, and those that are most important for describing
events of interest are often not obvious. The leading eigenfunctions of the
transition operator are useful for visualization, and they can provide an
efficient basis for computing statistics such as the likelihood and average
time of events (predictions). Here we develop inexact iterative linear algebra
methods for computing these eigenfunctions (spectral estimation) and making
predictions from a data set of short trajectories sampled at finite intervals.
We demonstrate the methods on a low-dimensional model that facilitates
visualization and a high-dimensional model of a biomolecular system.
Implications for the prediction problem in reinforcement learning are
discussed.
