Entropy in the natural time domain

Abstract

A surrogate data analysis is presented, which is based on the fluctuations of the "entropy" S defined in the natural time domain [Phys. Rev. E 68, 031106 (2003)]]. This entropy is not a static one such as, for example, the Shannon entropy. The analysis is applied to three types of time series, i.e., seismic electric signals, "artificial" noises, and electrocardiograms, and it "recognizes" the non-Markovianity in all these signals. Furthermore, it differentiates the electrocardiograms of healthy humans from those of the sudden cardiac death ones. If deltaS and deltaSshuf denote the standard deviation when calculating the entropy by means of a time window sweeping through the original data and the "shuffled" (randomized) data, respectively, it seems that the ratio deltaSshuf /deltaS plays a key role. The physical meaning of deltaSshuf is investigated.

Entropy of seismic electric signals: Analysis in natural time under time reversal

“…[6]). We emphasize that the aforementioned points should not be misinterpreted as stating that the simple logistic map model treated here can capture the complex heart dynamics, but only can be seen in the following frame: Since sudden cardiac arrest (which may occur even if the electrocardiogram looks similar to that of H) may be considered as a dynamic phase transition [5,6], it is reasonable to expect that the entropy fluctuations significantly increase upon approaching the transition. …”

A plausible explanation of the b-value in the Gutenberg-Richter law from first Principles

“…11), 14) Static entropy solely depends on the probability distribution and hence remains unaltered when changing the order of the events, e.g., upon randomization ("shuffling"), while in a dynamic entropy the order of consecutive events plays an important role. The Shannon entropy is used, because our interest here is focused on the statistical properties, while when studying the dynamic evolution of a system, the "entropy" in the natural time S = <χlnχ > -<χ >ln<χ > should be preferred.…”

“…This was repeated for various b-values by keeping the total number (500,000) of events constant. The data for b~1 should be equivalent to the "shuffled" 14) data of an actual earthquake catalogue and their probability density functions (pdf) should be the same. (As it is explained further in the Discussion, we focus here on the self-similarity exponent that stems from the distribution of the process' increments only, not from the memory of the process.…”

“…Recall, that this figure may be regarded as depicting the probability distribution of the order parameter versus the value of the order parameter. Since the fluctuations of the order parameter become larger as we approach a critical point, 14) we now study the Shannon Information…”

“…Along these lines, the (random) "shuffling" of the original data provides a useful tool: If the self-similarity stems from the process memory only, then the exponents of the shuffled data are different from those in the original data and changed to H H = α DFA = 1/2 since the shuffling of the original data destroys the correlations and the resulting time series is without memory. 14) On the other hand, if the self-similarity stems only from the process' increments infinite variance, then the exponents of the original data do not change, i.e., they are equal to those in the shuffled data. In the general case, i.e., if the self-similarity stems from both origins, i.e., memory and increments' distribution, we observe a partial change of the exponents H H and α DFA .…”