  Dimensionality reduction is ubiquitous in analysis of complex dynamics. The
conventional dimensionality reduction techniques, however, focus on reproducing
the underlying configuration space, rather than the dynamics itself. The
constructed low-dimensional space does not provide complete and accurate
description of the dynamics. Here I describe how to perform dimensionality
reduction while preserving the essential properties of the dynamics. The
approach is illustrated by analyzing the chess game - the archetype of complex
dynamics. A variable that provides complete and accurate description of chess
dynamics is constructed. Winning probability is predicted by describing the
game as a random walk on the free energy landscape associated with the
variable. The approach suggests a possible way of obtaining a simple yet
accurate description of many important complex phenomena. The analysis of the
chess game shows that the approach can quantitatively describe the dynamics of
processes where human decision-making plays a central role, e.g., financial and
social dynamics.
