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The fluctuations, under time reversal, of the natural time and the entropy distinguish similar looking electric signals of different dynamics

We show that the scale dependence of the fluctuations of the natural time itself under time reversal provides a useful tool for the discrimination of seismic electric signals (critical dynamics) from noises emitted from man made sources as well as for the determination of the scaling exponent. We present recent data of electric signals detected at the Earth's surface, which confirm that the value of the entropy in natural time as well as its value under time reversal are smaller than that of the entropy of a"uniform"distribution.


I. INTRODUCTION
In a time series comprising N events, the natural time χ k = k/N serves as an index 1,2,3 for the occurrence of the k-th event. In natural time analysis, the time evolution of the pair of the two quantities (χ k , Q k ) is considered, where Q k denotes in general a quantity proportional to the energy released during the k-th event.
In the case of dichotomous electric signals (e.g., seismic electric signal (SES) activities, i.e., low frequency ≤ 1Hz electric signals that precede earthquakes, e.g., see Refs.4,5,6,7,8,9,10,11,12,13) Q k stands for the duration of the k-th pulse (cf. The SES activities should not be confused with pulses of very short durations observed some minutes before earthquakes 14 ). It has been shown 15 that natural time domain is optimal for enhancing the signals' localization in the time-frequency space, thus conforming to the desire to reduce uncertainty and extract signal information as much as possible. The entropy S in natural time is defined 16 as the derivative with respect to q of the fluctuation function χ q − χ q at q = 1: where f (χ) = N k=1 p k f (χ k ) and p k = Q k / N n=1 Q n . It is dynamic entropy 17,18 and exhibits 19 concavity, positivity and Lesche 20,21 stability. Note that S should not be confused with Cov[χ, ln χ] ≡ χ ln χ − χ ln χ since in general ln χ = ln χ . The value of the entropy upon considering the time reversal T , i.e., T p k = p N −k+1 , is labelled by S − . The value of S − is 19,22,23 , in general, different from S, and thus S does satisfy the conditions to be "causal" in the following sense (see Ref. 19 and references therein): When studying a dynamical system evolving in time, the "causality" of an operator describing this evolution assures that the values assumed by the operator, at each time instant, depends solely on the past values of the system. Hence, a "causal" operator should be able to represent the evolution of the system according to the (true) time arrow, thus the operator can represent a real physical system evolving in time and reveal the differences arising upon time-reversal of the series.
The statistical properties of S and S − have been stud-ied in a variety of models 22,23 . In the case of a "uniform" distribution S = S − = S u (= ln 2/2 − 1/4 ≈ 0.0966).
The "uniform" distribution (defined in Refs.1,24) has been analytically studied in Ref. 17 and corresponds to the case when Q k are independent and identically distributed (IID) positive random variables of finite variance including the case of Markovian dichotomous electric signals studied in Ref. 16. The "uniform" distribution corresponds to p(χ) = 1, where p(χ) is a continuous probability density function (PDF) corresponding to the point probabilities p k used so far. When Q k of a "uniform" distribution are perturbed by a small linear trend, we find 22 (see also Eq.(4), below) that (S − S u )(S − − S u ) < 0 (cf. this simple example, which shows that S captures the effect of a linear trend, may be considered as clarifying the meaning of S, see Section V of Ref. 22). Another model studied is when the increments of Q k are positive IID, in this case we find 23 that S ≈ 0.048 and S − ≈ 0.088 which are both smaller than S u . The same holds, i.e., that both S and S − are smaller than S u , in the examples of an on-off intermittency model discussed in Ref. 22 as well as for a multiplicative cascades model 23 adjusted to describe turbulence data. A case of practical importance is that of the SES activities. SES activities (critical dynamics) exhibit infinitely ranged long-range temporal correlations 22,23,24 which are destroyed 23 after shuffling the durations Q k randomly. An interesting property emerged from the data analysis of several SES activities refers to the fact 22 that both S and S − values are smaller than the value of S u , i.e., in addition to the fact that for SES activities 1,2,3 the variance These findings -which do not hold 19 for "artificial" noises (AN) (i.e., electric signals emitted from manmade sources)-have been supported by numerical simulations in fractional Brownian motion (fBm) time series 22,23 that have an exponent α DF A , resulted from the Detrended Fluctuation Analysis (DFA) 25,26 , close to unity. This model have been applied since fBm (with a self-similarity index H ≈ 1) has been found 27 as an appropriate type of modeling process for the SES activities. These simulations resulted in values of S and S − that do obey relation (2) (see Fig.4 of Ref. 22) and κ 1 ≈ 0.070 (see Fig.3 of Ref. 23). It was then conjectured 22 that the validity of the relation (2) stems from infinitely ranged long-range temporal correlations (cf. H ≈ 1). On the other hand, for short-range temporal correlations (e.g. when modeling Q k by an autoregressive process Q k = aQ k−1 + g k + c, |a| < 1 and c stands for an appropriate constant to ensure positivity of Q k or Q k = |aQ k−1 + g k | where g k is Gaussian IID variables) the values of both S and S − approach (see Appendix A) that of S u and κ 1 → κ u , where κ u = 1/12 denotes the corresponding value of the"uniform" distribution 23 . The scope of this paper is twofold: First, in Section II, we point out the usefulness of the study of the fluctuations of the natural time itself under time reversal. In particular, it enables the determination of the scaling exponent, thus allowing the distinction of SES activities from similar looking AN. Second, in Section III, we provide the most recent experimental data that strengthen the validity of the relations (2) and (3) for SES activities. The earthquakes that followed the latter SES activities are described in Section IV. Section V, summarizes our conclusions.

II. THE FLUCTUATIONS OF NATURAL TIME UNDER TIME REVERSAL
The way through which the entropy in natural time captures the influence of the effect of a small linear trend has been studied, as mentioned, in Ref. 22 on the basis of the parametric family of PDFs: p(χ; ǫ) = 1 + ǫ(χ − 1/2), where ǫ measures the extent of the linear trend. Such a family of PDFs shares the interesting property T p(χ; ǫ) = p(χ; −ǫ), i.e, the action of the time reversal is obtained by simply changing the sign of ǫ. It has been shown 22 that the entropy S(ǫ) ≡ S[p(χ; ǫ)], as well as that of the entropy under time reversal S − (ǫ) ≡ S[T p(χ; ǫ)], S − (ǫ) = S(−ǫ), depend non-linearly on the trend parameter ǫ: However, it would be extremely useful to obtain a linear measure of ǫ in natural time. Actually, this is simply the average of the natural time itself: If we consider the fluctuations of this simple measure upon time-reversal, we can obtain information on the long-range dependence of Q k . We shall show that a measure of the long-range dependence emerges in natural time if we study the dependence of its fluctuations under time-reversal ∆χ 2 l ≡ E[( χ − T χ ) 2 ] on the window length l that is used for the calculation. Since T p k = p l−k+1 , we have The sum of the resulting values over the number of the selected segments (different m 0 ) is assigned to E[. . .]. The full calculation refers to the case when m 0 takes all the L − l + 1(m 0 = 1, 2, . . . L − l + 1) possible values, whereas the Monte Carlo when m 0 is selected randomly.
By expanding the square in the last part of Eq.(6), we obtain The basic relation 17 that interrelates p k is l k=1 p k = 1 or equivalently p k = 1 − m =k p m . By subtracting from the last expression its value for k = l − k + 1, we obtain p k − p l−k+1 = − m =k (p m − p l−m+1 ), and thus By substituting Eq.(8) into Eq.(7), we obtain which simplifies to (10) The negative sign appears because (p k − p l−k+1 ) and (p m − p l−m+1 ) are in general anti-correlated due to Eq.(8).
Equation (10) implies that ∆χ 2 l measures the long-range correlations in Q k : If we assume that p k scales as 1/l, e.g. see 17 ), we have that so that where χ H is a scaling exponent.

A. Fractional Brownian motion and fractional Gaussian noise time series
In order to examine the validity of the above result Eq.(12) when Q k are coming from fBm or fractional Gaussian noise (fGn), we employed the following procedure: First, we generated fBm (or fGn) time-series X k (consisting of 2 × 10 4 points) for a given value of H using the Mandelbrot-Weierstrass function 28,29,30 as described in Ref. 22. Second, since Q k should be positive, we normalized the resulting X k time-series to zero mean and unit standard deviation and then added to the normalized time-series N k a constant factor c to ensure the positivity of Q k = N k + c (for the purpose of the present study we used c = 10). The resulting Q k timeseries were then analyzed and the fluctuations of ∆χ l versus the scale l are shown in Figs. 1(a) and 1(d) for fGn and fBm, respectively. The upper three panels of Fig.1 correspond to fGn while the lower three to fBm. We observe (see Fig.1(b)) that for fGn we have the interconnection: χ H ≈ H − 1 corresponding to descending curves(see Fig.1(a)), whereas for fBm the interconnection turns (see Fig.1(e)) to: χ H ≈ H corresponding to ascending curves(see Fig.1(d)).
In order to judge the merits or demerits of the procedure proposed here for the determination of the scaling exponent, we compare Figs.1(b) and 1(e) with Figs.1(c) and 1(f), respectively, that have been obtained by the well-established DFA method 25,26 . This comparison reveals that the results are more or less comparable for fGn, while for fBm the exponent χ H deviates less from the behavior of an ideal estimator of the true scaling exponent (drawn in dashed green) compared to α DF A , especially for the largest H values. The physical meaning of the present analysis was further investigated by performing the same procedure in the time-series of the durations of those signals analyzed in Ref. 19 that have enough number of pulses e.g.≈ 10 2 (cf. the signals depicted in Fig.3 could not be analyzed in view of the small number of pulses). The relevant results are shown in Fig.2. Their inspection interestingly indicates that all seven AN correspond to descending ∆χ l curves versus the scale l, while the three SES activities to ascending curves (in a similar fashion as in Figs.1(a) and 1(d), respectively) as expected from the fact that the latter exhibit 24 infinitely ranged long-ranged temporal correlations (having H close to unity), while the former do not. Hence, the method proposed here enables the detection of long-range correlations even for datasets of small size (≈ 10 2 ), thus allowing the distinction of SES activities from AN.

III. RECENT DATA OF SEISMIC ELECTRIC SIGNALS ACTIVITIES
First, Fig.3(a) depicts an electric signal, consisting of a number of pulses, that has been recorded on November 14, 2006 at a station labelled 31 PIR lying in western Greece (close to Pirgos city). This signal has been clearly collected at eleven measuring electric dipoles with electrodes installed at sites that are depicted in a map given in Ref. 31. The signal is presented (continuous line in red) in Fig. 3(a) in normalized units, i.e., by subtracting the mean value and dividing by the standard deviation. For the reader's convenience, the corresponding dichotomous representation is also drawn in Fig. 3(a) with a dotted (blue) line, while in Fig. 3(c) we show (in red crosses) how the signal is read in natural time. The computation of S and S − leads to the following values: S = 0.070 ± 0.012, S − = 0.051 ± 0.010. As for the variance κ 1 , the resulting value is κ 1 = 0.062 ± 0.010. These values more or less obey the conditions (2) and (3) that have been found to hold for other SES activities 22 . Note that the feature of this SES activity, it is similar to the one observed at the same station before the magnitude M ≈ 6.7 earthquake that occurred on Jan 8, 2006, see Ref. 32.
A closer inspection of Fig. 3(a) reveals the following experimental fact: An additional electric signal has been also detected (in the gray shaded area of Fig. 3(a)), which consists of pulses with markedly smaller amplitude than those of the SES activity discussed in the previous paragraph. This is reproduced (continuous line in red) in Fig. 3(b) in an expanded time scale and for the sake of the reader's convenience its dichotomous representation is also marked by the dotted (blue) line, which leads to the natural time representation shown (dotted blue) in Fig. 3(c). The computation of S and S − gives S = 0.077 ± 0.004, S − = 0.082 ± 0.004, while κ 1 is found to be κ 1 = 0.076 ± 0.005. Hence, these values also obey the conditions (2) and (3) for the classification of this signal as an SES activity.
The two aforementioned signals have been followed by two significant earthquakes as described in Section IV. This conforms to their classification as SES activities, which has been completed in an early version of this paper 33  An inspection of these values reveals that they obey the conditions (2) and (3) and hence both signals can be classified as SES activities. The procedure for the current study of the subsequent seismicity that occurred after these SES activities is described in the next Section.

IV. THE SEISMIC ACTIVITY THAT FOLLOWED THE SES ACTIVITIES
We discriminate that during the last decade SES activities are publicized only when their amplitude indicates that the impending earthquake has an expected 13,23 magnitude comparable to 6.0 unit or larger. . (Note that preseismic information based on SES activities is issued only when the magnitude of the strongest EQ of the impending EQ activity is estimated -by means of the SES amplitude 34 -to be comparable to 6.0 units or larger 13 .) Here, we show that the occurrence times of the aforementioned two EQs can be estimated by following the procedure described in Refs.1,22,23,35 and using the order parameter of seismicity proposed in Ref. 35, i.e, the normalized power spectrum in natural time Π(φ) as φ → 0 (see also below). We study how the seismicity evolved after the recording of the SES activities on November 14, 2006, at PIR station (which were classified as SES activities in the initially submitted version of the present paper on November 16, 2006 .4), by considering three magnitude thresholds M thres = 3.2, 3.4 and 3.6 (hence six combinations were studied in total). If we set the natural time for seismicity zero at the initiation of the SES activity at 17:19 UT on November 14, 2006, we form time series of seismic events 36 in natural time for various time windows as the number N of consecutive (small) EQs increases. We then compute the normalized power spectrum 1, 22,23,35 which holds when the system enters the critical stage (ω = 2πφ, where φ stands for the natural frequency 1,2,3,13 ). The date and the time of the occurrence of each small earthquake (with M ≥ 3.4) that occurred in the area B, is written in green in each panel (see also Table I). An inspection of Fig.5(a) reveals that the red line approaches the blue line as N increases and a coincidence occurs at the small event of magnitude 3.7 that occurred at 03:22 UT on January 17, 2007, i.e., roughly two days before the 5.2-units EQ at 22:25 UT on January 18,2007. A similar behavior is observed in Fig.5(b) in which we see that a coincidence occurs at the small event of magnitude 3.6 at 18:40 UT on January 31, 2007, i.e., roughly three days before the strong EQ of magnitude 5.8-units that occurred at 13:43 UT on February 3, 2007. To ensure that these two coincidences in Figs.5(a) and (b) are true ones 1,3,13,31,35 (see also below) we also calculate the evolution of the quantities κ 1 ,S and S − and the results are depicted in Fig.6 for the three magnitude thresholds for each of the aforementioned two areas A and B.
The conditions for a coincidence to be considered as true are the following (e.g., see Ref.1, see also 3,13,31,35 ): First, the 'average' distance D between the empirical and the theoretical Π(φ)(i.e., the red and the blue line, respectively, in Fig.5) should be 1,2,13,31,35 smaller than or equal to 10 −2 . See Fig.7, where we plot D versus the conventional time during the whole period after the recording of the SES activities on November 14, 2006, for both areas, i.e., the large one (area A) and the small (area B) and the three magnitude thresholds. For the sake of the readers convenience, the mean value of the results obtained for the three thresholds is also shown in black. Second, in the examples observed to date 1,3,13,31,34,35 , a few events before the coincidence leading to the strong EQ, the evolving Π(φ) has been found to approach that of Eq.(1), i.e., the blue one in Fig.5 , from below (cf. this reflects that during this approach the κ 1 -value decreases as the number of events increases). In addition, both values S and S − should be smaller than S u at the coincidence. Finally, since the process concerned is selfsimilar (critical dynamics), the time of the occurrence of the (true) coincidence should not change, in principle, upon changing either the (surrounding) area or the magnitude threshold used in the calculation. Note that in Fig.7, at the last small events ,i.e., the rightmost in Figs.5(a) and 5(b), respectively (i.e., the magnitude 3.7 event on January 17, 2007 and the second event of magnitude 3.6 on January 31, 2007) just before the occurrences of the 5.2-units and 5.8-units EQs, in both areas A and B, the mean value (see the black thick lines in Fig.7) of D obtained from the three magnitude thresholds become smaller than or equal to 10 −2 . Hence, these two coincidences can be considered as true.
In summary, the SES activities recorded on November 14, 2006, at PIR station (presented in Figs.3(a),(b)) have been followed by two EQs with magnitudes 5.2-units and 5.8-units that occurred on January 18 and February 3, 2007. The time of the occurrences of these two EQs are determined within a narrow range of a few days upon analyzing, in natural time, the seismicity subsequent to the SES activities. In order to determine the occurrence time of the impending EQs, we currently apply the procedure explained in the previous subsection by studying the seismicity in the areas A, B, C (see Fig.9). Since the result should exhibit spatial scale invariance, the epicenter(s) will lie either in the area B or in C depending on whether the areas A and B or A and C show true coincidence.

V. CONCLUSIONS
First, the scale dependence of the fluctuations of the natural time under time reversal distinguish similar looking electric signals emitted from systems of different dynamics providing a useful tool for the determination of the scaling exponent. In particular, SES activities (critical dynamics) are distinguished from noises emitted from man-made electrical sources.
Second, recent data of SES activities are presented which confirm that the value of the entropy in natural time as well as its value under time reversal are smaller than that of a "uniform" distribution.

APPENDIX A: THE CASE OF SIGNALS THAT EXHIBIT SHORT-RANGE TEMPORAL CORRELATIONS
Here, we present results of modeling Q k by shortranged temporal correlated time-series. Two examples were treated by numerical simulation: (i) A stationary autoregressive process Q k = aQ k−1 + g k + c, |a| < 1, where g k are Gaussian IID random variables, and c stands for an appropriate constant to ensure positivity of Q k . (ii) Q k = |aQ k−1 + g k |. Figure 10(a) depicts the results for S,S − and κ 1 for the first example versus the number of Q k , whereas Fig.10(b) refers to the second example. In both cases S and S − converge to S u whereas κ 1 to the value κ u = 1/12 corresponding to the "uniform" distribution. For the reader's convenience, the values of       in area A (see Fig.9) after the initiation of the SES activity on July 2, 2007 ( Fig.8(a)) until 03:27 UT of September 25, 2007 is shown in Fig.11(a). The evolution of the corresponding parameters κ 1 , S, S − and D calculated for three magnitude thresholds, i.e.,M thres = 3.0, 3.1 and 3.2 are shown in Figs.11(b), (c) and (d) respectively. To investigate the spatial invariance, the computation was repeated for several smaller areas, three of which are shown in Figs.12,13 and 14 (which are different from the areas B and C of Fig.9) along with the evolution of the corresponding parameters. The same was repeated for an area (see Fig.15) somewhat larger than A. An inspection of all these figures, i.e., Figs.11 to15, suggests that presumably a true coincidence has just been approached, thus being very close to the critical point.

APPENDIX C: THE SEISMIC ACTIVITY THAT FOLLOWED THE MOST RECENT SES ACTIVITIES AT PAT AND PIR
Here, we report the update results of the seismic activity that followed the SES activity at PAT on October 9, 2008 38 and the SES activity at PIR on December 12, 2008 39 by following the procedure described by Sarlis et al. 40 . The subsequent seismicity of the former SES activity was studied in the area N 38.6 37.5 E 23.3 19.8 while that of the latter in the selectivity map of PIR depicted in Fig. 16.  Fig.16.
We now present the results, when considering the seismicity until early in the morning of February 19, 2009. Figure 19, depicts the results for M thres =3.2 in the se-lectivity map of PAT (i.e., in the area N 38.6 37.5 E 23.3 19.8 ), which show that the aforementioned probability Prob(κ 1 ) versus κ 1 maximizes at κ 1 ≈ 0.070 upon the occurrence of a M L =3.2 event at 00:35 UT on February 19, 2009 with epicenter at 37.8 o N21.2 o E. As for the results in the selectivity map of PIR (i.e., the shaded area in Fig.16), they are shown in Fig.20 for M thres =3.4 and reveal the following: After the aforementioned strong earthquake on February 16, 2009, the bimodal curve Prob(κ 1 ) versus κ 1 shows a clear local maximum at κ 1 ≈ 0.070 upon the oc- Updated note on March 13, 2009: Several hours before the occurrence of the aforementioned magnitude 6.0 earthquake on February 16, 2009, an electrical anomaly of significant amplitude, see Fig.21, appeared at PIR. It basically consists of three bay like transient changes, thus having a form essentially different than the SES activities that preceded the major earthquakes on February 14, 2008 andJune 8, 2008. This electrical anomaly could be in principle attributed to the 6.0 earthquake on February 16, 2009 that occurred almost 11 hours later, but this earthquake was also preceded by the SES activity at PIR on December 12, 2008. Alternatively, this anomaly might be related to a new impending earthquake. For this reason, a natural time analysis of the seismicity subsequent to the electrical anomaly of February 16, 2009 was carried out. In this calculation we considered the events that occurred in the PIR selectivity map shown in Fig.22 (cf. this, which is slightly different than that depicted in Fig.16, has been drawn by taking into account the whole region to the west of the Hellenides). The results of the analysis shown in Fig.23 indicate that for M thres =3.6 and 3.5 (see a and b, respectively) the probability Prob(κ 1 ) maximizes at κ 1 ≈ 0.070 upon the occurrence of a M L =3.6 event at 02:48 UT on March 9, 2009, while for M thres =3.3 the maximization occurs upon the occurrence of a M L =3.3 event on March 10, 2009. Interestingly, a similar analysis (subsequent to the SES activity at PAT) in the selectivity map of PAT, i.e., in the area N 38.6 37.5 E 23.3 19.8 , reveals that a maximization of Prob(κ 1 ) at κ 1 ≈ 0.070 also occurs on March 10, 2009 (see Fig.24). We are currently investigating the spatial and magnitude threshold invariance of the aforementioned maxima to examine whether they actually point to an approach to the critical point. 5.0 proposed by the Global Seismological Services were used. Then, the normalized power spectrum is given[1-4,9] by