Testing time symmetry in time series using data compression dictionaries

Abstract

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.

Equal heartbeat intervals and their effects on the nonlinearity of permutation-based time irreversibility in heart rate

“…L. Lacasa et al [22] estimate the degree of irreversibility using the Kullback-CLeibler divergence between the in and out degree distributions based on horizontal visibility graph. And some symbolic methods, like the 'false flipped symbols' proposed by C. Daw [23], a data compression method introduced by M. Kennel [24], the ternary coding symbolic approach provided by C. Cammarota [25] and so forth [26,27], are proposed and show promising nonlinearity detection. J. Martinez et al [28] detect time reversibility by measuring the Jensen-Shannon divergence of time forward as well as its time-reversed counterpart by means of permutation, and M. Zanin et al [29] adopt the KL divergence to compare the probability distributions of symmetric order patterns.…”

Visibility Algorithms: A Short Review

“…This is the case of chaotic maps in which entropy production via instabilities in the forward time direction is quantitatively different to the amount of past information lost. In other words, those whose positive Lyapunov exponents, which characterize chaos in the forward process, differ in magnitude with negative ones, which characterize chaos in the backward process (Kennel, 2004). Several chaotic maps have been analyzed and the degree of reversibility of their associated time series has been estimated using using KLD, showing that for dissipative chaotic series it is positive while it vanishes for an example of conservative chaos.…”

“…Several methods to measure time irreversibility have been proposed (Andrieux et al, 2007;Cammarota & Rogora, 2007;Costa et al, 2005;Daw et al, 2000;Diks et al, 1995;Gaspard, 2004;Kennel, 2004;Wang et al, 2005;Yang et al, 2003). The majority of them perform a time series symbolization, typically making an empirical partition of the data range (Daw et al, 2000) (note that such a transformation does not alter the reversible character of the output series (Kennel, 2004)) and subsequently analyze the symbolized series, through statistical comparison of symbol strings occurrence in the forward and backwards series or using compression algorithms (Cover & Thomas, 2006;Kennel, 2004;Roldan & Parrondo, 2011). The first step requires an extra amount of ad hoc information (such as range partitioning or size of the symbol alphabet) and therefore the output of these methods eventually depend on these extra parameters.…”

Compression-based data mining of sequential data

“…For example, in the early days of computational biology, lossless compression was routinely used to classify and analyze DNA sequences. We refer to, e.g., Allison et al (2000), Baronchelli et al (2005), Farach et al (1995, Frank et al (2000), Gatlin (1972), Kennel (2004), Kit (1998), Loewenstern et al (1995), Yianilos (1999), Mahnoney (2003, Unpublished), Needham and Dowe (2001), Segen (1990), and Teahan et al (2000), and references therein for a sampler of the rich literature existing on this subject.…”