Time-Reversibility, Causality and Compression-Complexity

Time-irreversibility test for random-length time series: The matching-time approach applied to DNA

In this work, we implement the so-called matching-time estimators for estimating the entropy rate as well as the entropy production rate for symbolic sequences. These estimators are based on recurrence properties of the system, which have been shown to be appropriate for testing irreversibility, especially when the sequences have large correlations or memory. Based on limit theorems for matching times, we derive a maximum likelihood estimator for the entropy rate by assuming that we have a set of moderately short symbolic time series of finite random duration. We show that the proposed estimator has several properties that make it adequate for estimating the entropy rate and entropy production rate (or for testing the irreversibility) when the sample sequences have different lengths, such as the coding sequences of DNA. We test our approach with controlled examples of Markov chains, non-linear chaotic maps, and linear and non-linear autoregressive processes. We also implement our estimators for genomic sequences to show that the degree of irreversibility of coding sequences in human DNA is significantly larger than that for the corresponding non-coding sequences.

Causality in Reversed Time Series: Reversed or Conserved?

The inference of causal relations between observable phenomena is paramount across scientific disciplines; however, the means for such enterprise without experimental manipulation are limited. A commonly applied principle is that of the cause preceding and predicting the effect, taking into account other circumstances. Intuitively, when the temporal order of events is reverted, one would expect the cause and effect to apparently switch roles. This was previously demonstrated in bivariate linear systems and used in design of improved causal inference scores, while such behaviour in linear systems has been put in contrast with nonlinear chaotic systems where the inferred causal direction appears unchanged under time reversal. The presented work explores the conditions under which the causal reversal happens—either perfectly, approximately, or not at all—using theoretical analysis, low-dimensional examples, and network simulations, focusing on the simplified yet illustrative linear vector autoregressive process of order one. We start with a theoretical analysis that demonstrates that a perfect coupling reversal under time reversal occurs only under very specific conditions, followed up by constructing low-dimensional examples where indeed the dominant causal direction is even conserved rather than reversed. Finally, simulations of random as well as realistically motivated network coupling patterns from brain and climate show that level of coupling reversal and conservation can be well predicted by asymmetry and anormality indices introduced based on the theoretical analysis of the problem. The consequences for causal inference are discussed.