Context trees and dynamics
A.I. Mees · AIP conference proceedings · 2000
Given a real-world system with behavior which appears complex, it is difficult to separate the effects of chaos, high dimensionality and noise except in the rare cases where a high-quality model is available. Although great progress has been made in modeling such systems, there is little that is rigorous and most algorithms are slow. To gain understanding it seems necessary to idealize in some way, though not, of course, in the traditional way, which is by linearization. In this paper we simplify the problem by assuming that the system outputs symbols from a finite alphabet, rather than outputting a real number. With this simplification and a reasonable assumption which is the discrete analogue of the standard embedding theorem, it is possible to use known results in data compression theory to produce very fast reconstruction algorithms with guaranteed asymptotic optimality. The models that result can be used to simulate and to predict as well as to calculate all the usual dynamically interesting quantities such as topological entropy.