Compression and diffusion: a joint approach to detect complexity

Abstract

The adoption of the Kolmogorov-Sinai (KS) entropy is becoming a popular research tool among physicists, especially when applied to a dynamical system fitting the conditions of validity of the Pesin theorem. The study of time series that are a manifestation of system dynamics whose rules are either unknown or too complex for a mathematical treatment, is still a challenge since the KS entropy is not computable, in general, in that case. Here we present a plan of action based on the joint action of two procedures, both related to the KS entropy, but compatible with computer implementation through fast and efficient programs. The former procedure, called Compression Algorithm Sensitive To Regularity (CASToRe), establishes the amount of order by the numerical evaluation of algorithmic compressibility. The latter, called Complex Analysis of Sequences via Scaling AND Randomness Assessment (CASSANDRA), establishes the complexity degree through the numerical evaluation of the strength of an anomalous effect. This is the departure, of the diffusion process generated by the observed fluctuations, from ordinary Brownian motion. The CASSANDRA algorithm shares with CASToRe a connection with the Kolmogorov complexity. This makes both algorithms especially suitable to study the transition from dynamics to thermodynamics, and 1 the case of non-stationary time series as well. The benefit of the joint action of these two methods is proven by the analysis of artificial sequences with the same main properties as the real time series to which the joint use of these two methods will be applied in future research work.

Scaling in non-stationary time series. (I)

“…If the scaling condition of Eq. (3) applies, it is convenient to measure the scaling parameter δ by the method of Diffusion Entropy (DE) [14,15,16,17,18,19,20] that, in principle, works independently of whether the second moment is finite or not. The DE method affords many advantages, including that of being totally independent of a constant bias.…”

Toward a dynamical model for prime numbers

“…Indeed in that range the invariant measure that is absolutely continuous with respect to the Lebesgue measure is infinite. For further properties we refer to [5], [13].…”

Stromatolites: why do we care?

“…This is the meaning of the question posed by the title of this paper. To answer this question, we have to draw the reader's attention to an earlier work [2], devoted to the problem of assessing the complexity of a process, as it is mirrored by a time series derived in some way from the process under study. The title of the paper of Ref.…”

“…The second method rests on converting first the time series to study into a diffusion process. Then, as a second step, we evaluate the entropy of this diffusion process [2], this being the reason why the method is called Diffusion Entropy (DE) method. We refer the reader to the paper of Ref.…”

“…We refer the reader to the paper of Ref. [2] for details on both method. Here we limit ourselves to observing that the connection between the two methods of analysis is explained as follows.…”