Testing time symmetry in time series using data compression dictionaries

Time symmetry, often called statistical time reversibility, in a dynamical process means that any segment of time-series output has the same probability of occurrence in the process as its time reversal. A technique, based on symbolic dynamics, is proposed to distinguish such symmetrical processes from asymmetrical ones, given a time-series observation of the otherwise unknown process. Because linear stochastic Gaussian processes, and static nonlinear transformations of them, are statistically reversible, but nonlinear dynamics such as dissipative chaos are usually statistically irreversible, a test will separate large classes of hypotheses for the data. A general-purpose and robust statistical test procedure requires adapting to arbitrary dynamics which may have significant time correlation of undetermined form. Given a symbolization of the observed time series, the technology behind adaptive dictionary data compression algorithms offers a suitable estimate of reversibility, as well as a statistical likelihood test. The data compression methods create approximately independent segments permitting a simple and direct null test without resampling or surrogate data. We demonstrate the results on various time-series-reversible and irreversible systems.