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Three types of electric signals were analyzed: Ion current fluctuations in membrane channels (ICFMC), Seismic electric signals activities (SES), and "artificial" noises (AN). The wavelet transform, when applied to the conventional time domain, does not allow a classification of these signals, but does so in the "natural" time domain. A classification also becomes possible, if we study -(q) versus q, where chi stands for the "natural" time. For q values approximately between 1 and 2 the signals are classified and ICFMC lies between the other two types. For q=1, the "entropy" S identical with -ln of ICFMC almost equals that of a "uniform" distribution, while the AN and SES have larger and smaller S values, respectively. The recent [P. Varotsos, N. Sarlis, and E. Skordas, Phys. Rev. E 67, 021109 (2003)] finding that, in short time scales, both SES and AN (which are shown to be non-Markovian) result in comparable detrended fluctuation analysis exponents alpha in (1.0,1.5) is revisited. Even a Markovian dichotomous time series, in short time scales, leads to similar alpha exponents.

It has been shown that some dynamic features hidden in the time series of complex systems can be uncovered if we analyze them in a time domain called natural time χ . The order parameter of seismicity introduced in this time domain is the variance of χ weighted for normalized energy of each earthquake. Here, we analyze the Japan seismic catalog in natural time from January 1, 1984 to March 11, 2011, the day of the M9 Tohoku earthquake, by considering a sliding natural time window of fixed length comprised of the number of events that would occur in a few months. We find that the fluctuations of the order parameter of seismicity exhibit distinct minima a few months before all of the shallow earthquakes of magnitude 7.6 or larger that occurred during this 27-y period in the Japanese area. Among the minima, the minimum before the M9 Tohoku earthquake was the deepest. It appears that there are two kinds of minima, namely precursory and nonprecursory, to large earthquakes.criticality | seismic electric signals F or a time series comprised of N events, we define the natural time for the occurrence of the kth event by χ k = k=N (1), which means that we ignore the time intervals between consecutive events, but preserve their order. We also preserve their energy Q k . We then study the evolution of the pairðχ k ; p k Þ, whereQ n is the normalized energy. We postulated that the approach of a dynamical system to criticality can be identified by the variance κ 1 of natural time χ weighted for p k , namely,Earthquakes (EQs hereafter) exhibit complex correlations in time, space, and magnitude, and the opinion prevails (e.g., ref. 2 and references therein) that the EQs are critical phenomena. In natural time analysis of seismicity, the quantity κ 1 calculated from seismic catalogs serves as an order parameter (3, 4). Experiences have shown that the mainshock occurs in a few days to 1 wk after the κ 1 value in the candidate epicentral area approaches 0.070 (5). This was found useful in narrowing the lead time of EQ prediction. However, to trace the time evolution of κ 1 value, one needs to start the analysis of the seismic catalog at some time before the yet-to-occur mainshock. We chose, for the starting time for analysis, the initiation time of seismic electric signal (SES) activity. SESs are low-frequency (≤1 Hz) electric signals that precede EQs (6). The reason for this choice was based on the consideration that SESs are emitted when the focal zone enters the critical stage (7). In the case of the lack of SES data, as in the Tohoku EQ, we cannot adopt this approach. In this study, therefore, we instead examine the fluctuations of κ 1 near criticality, i.e., near the EQ occurrence. To compute the fluctuations, we apply the following procedure.First, take an excerpt comprised of W (≥100) successive EQs from the seismic catalog. We then form its subexcerpts consisting of the nth to (n + 5)th EQs, (n = 1, 2,. . .,W-5) and compute κ 1 for each of them. In so doing, we assign χ k = k=6 and the normalized energy p k = Q k = X 6 n = 1

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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.

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Nonextensive statistical mechanics, pioneered by Tsallis, has recently achieved a generalization of the Gutenberg-Richter law for earthquakes. This remarkable generalization is combined here with natural time analysis, which enables the distinction of two origins of self-similarity, i.e., the process' memory and the process' increments infinite variance. By using also detrended fluctuation analysis for the detection of long-range temporal correlations, we demonstrate the existence of both temporal and magnitude correlations in real seismic data of California and Japan. Natural time analysis reveals that the nonextensivity parameter q , in contrast to some published claims, cannot be considered as a measure of temporal organization, but the Tsallis formulation does achieve a satisfactory description of real seismic data for Japan for q=1.66 when supplemented by long-range temporal correlations.