# The Complexity Measures Associated with the Fluctuations of the Entropy in Natural Time before the Deadly México M8.2 Earthquake on 7 September 2017

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## Abstract

**:**

## 1. Introduction

## 2. Natural Time Analysis. The Entropy Defined in Natural Time and the Associated Complexity Measures

_{k}= k/N, which we term natural time χ. In this analysis [11,15], we ignore the time intervals between consecutive events, but preserve their order and energy Q

_{k}. We then study the pairs (χ

_{k}, Q

_{k}) or the pairs (χ

_{k}, p

_{k}) where

_{k}denote averages with respect to the distribution p

_{k}, i.e., <f(χ)> ≡ ∑f(χ

_{k})p

_{k}. The entropy S is a dynamic entropy that exhibits [18] concavity, positivity, and Lesche stability [19,20].

_{k}= p

_{N−k+}

_{1}, the value of S changes to a value S

_{−}:

_{−}in natural time under time reversal has been discussed in References. [11,21,22].

_{i}has been constructed [13] whose standard deviation is designated by δS

_{i}. The study of the effect of the change of scale i on δS

_{i}is made [14] by means of the complexity measure

_{100}is arbitrarily selected to stand for the δS value of a short scale, i.e., 100 events, while the numerator corresponds to a longer scale, e.g., i = 10

^{3}events. If instead of δS

_{i}the standard deviation σ(ΔS

_{i}) of the time series of ΔS

_{i}≡ S

_{i}− (S

_{−})

_{i}is used, we define [11,23] the complexity measure Λ

_{i},

_{100}) is arbitrarily selected to correspond to the standard deviation σ(ΔS

_{100}) of the time series of ΔS

_{i}of i = 100 events. In other words, this complexity measure quantifies how the statistics of ∆S

_{i}time series changes upon increasing the scale from 100 events to a longer scale e.g., i = 10

^{3}events.

## 3. Data and Analysis

_{k}should be [11,15] proportional to the energy emitted during the k-th earthquake of magnitude M

_{k}, we assumed ${Q}_{k}\propto {10}^{1.5{M}_{k}}$ [11]. To assure catalog completeness a magnitude threshold M ≥ 3.5 has been imposed. Furthermore, in order to investigate whether our results obey magnitude threshold invariance, we also repeat our calculations for M ≥ 4.0, as will be explained in the next Section.

_{−}, and therefrom their difference ∆S

_{i}, are calculated each time; we thus also form a new time series consisting of successive ∆S

_{i}values. The complexity measures λ

_{i}and Λ

_{i}are determined according to their definitions given in Equations (4) and (5), respectively.

## 4. Results

_{i}, (S

_{−})

_{i}, and ∆S

_{i}, respectively, versus the conventional time for all M ≥ 3.5 earthquakes in the Chiapas region during the period 1 January 2012 to the date of occurrence of the M8.2 earthquake for the scales 10

^{2}, 3 × 10

^{3}, and 4 × 10

^{3}events. The study of the first scale (10

^{2}events) is needed for the calculation of the denominator of Equations (4) and (5), while the selection of the other scales (3 × 10

^{3}events and longer) was made for the following reasons, as also explained in Ref. [7]. Since ∼11,500 earthquakes (M ≥ 3.5) occurred in this area from 1 January 2012 until the occurrence of the M8.2 earthquake on 7 September 2017, we find an average of around 170 earthquakes per month. We take into account that recent investigations by means of natural time analysis revealed that the fluctuations of the order parameter of seismicity exhibit [24] a minimum when a series of precursory low frequency (≤0.1 Hz) electric signals, termed Seismic Electric Signals (SES) activity (e.g., see [22]) (whose lead time is up to around 5.5 months [11]), is initiated. While this minimum of the order parameter of seismicity is observed during a period in which long-range correlations prevail between earthquake magnitudes, another stage appears before this minimum in which the temporal correlations between earthquake magnitudes exhibit a distinctly different behaviour, i.e., an evident anticorrelated behaviour [25]. The significant change between these two stages in the temporal correlations between earthquake magnitudes is likely to be captured by the time evolution of ΔS

_{i}, hence we started the presentation of our study of ΔS

_{i}in Ref. [7] from the scale of i ∼ 10

^{3}events, i.e., around the maximum lead time of SES activities. This study led to the conclusion that at scales i = 3 × 10

^{3}or longer (e.g., 4 × 10

^{3}and 5 × 10

^{3}events), a pronounced minimum becomes evident at the date 14 June 2017 (when a M7 earthquake occurred, i.e., almost 3 months before the M8.2 earthquake that struck Mexico’s Chiapas state). Interestingly, this minimum of ΔS

_{i}was found to exhibit magnitude threshold invariance. Hence, in the following we will present our results for these scales and in addition focus our attention on what happens before the occurrence of the M7 earthquake on 14 June 2017 and before the M8.2 earthquake on 7 September 2017.

_{i}and Λ

_{i}on the following 6 dates: On 1 June 2017 (i.e., almost two weeks before the occurrence of the M7 event on 14 June), on 14 June 2017 (upon the occurrence of the last event before the M7 earthquake on 14 June 2017), on 1 July 2017, on 1 August 2017, on 1 September 2017 (6 days before the M8.2 earthquake), and on 7 September 2017 (upon the occurrence of the last small event before the M8.2 earthquake).

#### 4.1. Results for the Complexity Measure λ_{i}

_{i}, which is solely associated with the fluctuations δS

_{i}of the entropy in forward time. In Figure 2, we plot the λ

_{i}values versus the conventional time by considering all M ≥ 3.5 earthquakes from 1 January 2012 until the occurrence of the M8.2 earthquake. A close inspection of this figure in all scales investigated does not reveal any remarkable change before the occurrence of the M8.2 earthquake. The same conclusion is drawn when we plot in Figure 3 the λ

_{i}values versus the scale i of the number of M ≥ 3.5 events that occurred in the Chiapas region from 1 January 2012 until the dates mentioned above before the two earthquakes, i.e., the M7 earthquake on 14 June 2017 and the M8.2 earthquake on 7 September 2017. Remarkably, the resulting values for all six dates coincide for each i value without showing any precursory variation. In other words, even when considering the λ

_{i}value on 7 September 2017, upon the occurrence of a small event just before the M8.2 earthquake, the value at i = 5000 does not significantly differ from the other λ

_{i}values that correspond, for example, to that of 1 June 2017, i.e., more than 3 months before the M8.2 earthquake occurrence.

#### 4.2. Results for the Complexity Measure Λ_{i}

_{i}values versus the conventional time by considering all M ≥ 3.5 earthquakes in the Chiapas region from 1 January 2012 until the occurrence of the M8.2 earthquake. An inspection of this figure reveals that upon the occurrence of the M7 earthquake on 14 June 2017 an abrupt increase of Λ

_{i}is observed in all three scales.

_{i}values versus the scale i of the number of M ≥ 3.5 events in the Chiapas region from 1 January 2012 until the six dates mentioned above before the two earthquakes, i.e., the M7 earthquake on 14 June 2017 and the M8.2 earthquake on 7 September 2017. We observe that for the scales i = 3000 events and larger an evident abrupt increase of the Λ

_{i}value is observed upon the occurrence of the M7 earthquake on 14 June 2017. This date of the abrupt increase of Λ

_{i}remains invariant upon changing the magnitude threshold, for example, see the result also plotted in Figure 5 by increasing the magnitude threshold to M ≥ 4.0 instead of M ≥ 3.5 and considering that the number of earthquakes then decreases by a factor of around 4.

## 5. Discussion

_{i}does exhibit an evident precursory change almost three months before the M8.2 earthquake, the complexity measure λ

_{i}does not. This should not be considered as surprising in view of the following: Λ

_{i}is associated with the fluctuations of the entropy change under time reversal, and we know that ∆S

_{i}is a key quantity to determine the time of an impending dynamic phase transition [20] as, for example, is the case in sudden cardiac death risk. This is also the case with earthquakes because the observed earthquake scaling laws (e.g., [26]) indicate the existence of phenomena closely associated with the proximity of the system to a critical point. Thus, taking the view that a strong earthquake is a critical phenomenon (dynamic phase transition), it may not come as a surprise that the fluctuations of ∆S

_{i}, and hence Λ

_{i}, may serve for the estimation of the time of its occurrence. We note that in [12] the predictability of the OFC model was studied by using the entropy change under time reversal. It was found that the value of ΔS

_{i}exhibits a clear minimum (if we define ΔS = S − S

_{−}) [7] or maximum (if we define ΔS = S

_{−}− S) [12] before large avalanches in the OFC model, thus this minimum provides a decision variable for the prediction of a large avalanche. The prediction quality of this minimum was studied, following [27], by choosing the receiver operating characteristics (ROC) graph [28], see Figure 2 of [12]. The ROC graph was constructed for the OFC model, and it was found that the predictions made on the basis of ΔS are statistically significant, and, as concerns the origin of their predictive power, this should be attributed to the fact that ΔS is able to catch the ‘true’ time arrow.

_{i}possibly associated with strong earthquakes, we now investigate the following: At first glance, one may claim that the behavior of Λ

_{i}observed on 14 June 2017 seems to be very similar to what happens around the end of 2015, and both changes seem to be related to earthquakes around these times. In order to examine whether such a similarity is true, we now depict in Figure 6a the complexity measures Λ

_{i}versus the conventional time in an expanded time scale for M ≥ 3.5. A close inspection of this figure reveals that all the three complexity measures Λ

_{3000}, Λ

_{4000,}and Λ

_{5000}(which stand for the Λ

_{i}values at the scales i = 3000, 4000, and 5000, respectively) exhibit a strong and abrupt increase on 14 June 2017 (after the occurrence of a M7.0 earthquake). Such a strong and abrupt increase for all these three complexity measures is not observed at any other time. In particular, on 17 December 2015 after the occurrence of a M6.6 earthquake, only Λ

_{3000}and Λ

_{5000}exhibit a strong and abrupt increase, while the corresponding increase for Λ

_{4000}is much smaller and obviously does not scale with i. The behavior of the increase ΔΛ

_{i}(≡Λ

_{i}(t) − Λ

_{i}(t

_{EQ})) of the complexity measure Λ

_{i}upon the occurrence of a strong earthquake at t

_{EQ}is further studied in Figure 6b,c for the earthquakes of 14 June 2017 and 17 December 2015, respectively. An inspection of these two figures shows that while the abrupt increase on 14 June 2017 exhibits a scaling behavior of the form

_{i}three months before the M8.2 Chiapas earthquake as a precursor to strong earthquakes, we employ the most recent method of event coincidence analysis [32] as implemented by the CoinCalc package in R. More specifically, for every month we estimate the largest earthquake magnitude M

_{max}observed and compare it with a threshold ${M}_{thres}^{target}$, if ${M}_{\mathrm{max}}>{M}_{thres}^{target}$ a strong earthquake event occurred in that month. The precursor event time series is composed by the abrupt increase in events of Λ

_{i}(see Figure 4 and Figure 7a) and exhibits two events for i = 5000 and 3000 (one in December 2015 and one in June 2017), while for i = 4000 a single event (in June 2017). Figure 7 shows how the corresponding p-values vary upon increasing the threshold ${M}_{thres}^{target}$ to obtain the real situation by chance. We observe that only in three out of the nine results shown, these p-values exceed 10%, while for the largest threshold ${M}_{thres}^{target}$ the corresponding p-values are 4.8%, 2.8%, and 7.0% for i = 3000, 4000, and 5000, respectively, pointing to the statistical significance of the precursor under discussion.

_{i}

_{,}associated with the fluctuations of the entropy change of the seismicity in Japan (with M ≥ 3.5), was studied during the period from 2000 to 11 March 2011 and found to exhibit an abrupt increase upon the occurrence of a M7.8 earthquake on 22 December 2010. Remarkably, on the same date [25] the temporal correlations between earthquake magnitudes exhibited anticorrelated behavior with the lowest value (≈0.35) ever observed of the exponent in the detrended fluctuation analysis [33,34,35] we employed in that study. Details on the interconnection between the changes in the temporal correlations between earthquake magnitudes and the precursory ΔS minimum in Japan will be published shortly elsewhere.

_{q}(e.g., see [37,38]). This has found application in the physics of earthquakes and especially in the description of the asperities [39,40] in the faults on which earthquakes occur through the entropic index q. Based on the earthquake magnitude distribution one can obtain [41,42] entropic index q and study its variation with time as we approach a strong EQ (for a recent review see [43]). Figure 8 depicts the q-value versus conventional time as it is estimated [42] for sliding windows of W = 1000 and W = 250 consecutive earthquakes in the Chiapas region for M ≥ 3.5 and M ≥ 4.0, shown as red and the blue lines, respectively. We observe that before the occurrence of the M8.2 Chiapas earthquake, the q-value starts to grow gradually during May 2017 and exhibits an abrupt increase upon the occurrence of the earthquake on 14 June 2017 for both magnitude thresholds. This behavior is reminiscent to that observed before the 1995 Kobe earthquake in Japan (e.g., see Figure 3 of [44]).

## 6. Conclusions

_{i}studied during the period 2012–2017, which is solely associated with the fluctuations of entropy in forward time, does not exhibit any obvious precursory change before the M8.2 earthquake on 7 September 2017.

_{i}associated with the fluctuations of entropy change under time reversal shows an evident increase on 14 June 2017, accompanied by an abrupt increase of the Tsallis entropic index q. Interestingly, on the same date reported by Sarlis et al. in [7], the entropy change under time reversal has been found to exhibit a minimum, as was found in the OFC model [12].

## Author Contributions

## Conflicts of Interest

## References

- Mega, E.R. Deadly Mexico earthquake had unusual cause. Nature
**2017**, 549. [Google Scholar] [CrossRef] - Wade, L. Unusual quake rattles Mexico. Science
**2017**, 357, 1084. [Google Scholar] [CrossRef] [PubMed] - Witze, A. Pair of deadly Mexico quakes puzzles scientists. Nature
**2017**, 549. [Google Scholar] [CrossRef] [PubMed] - Pérez-Campos, X.; Kim, Y.; Husker, A.; Davis, P.M.; Clayton, R.W.; Iglesias, A.; Pacheco, J.F.; Singh, S.K.; Manea, V.C.; Gurnis, M. Horizontal subduction and truncation of the Cocos Plate beneath central Mexico. Geophys. Res. Lett.
**2008**, 35, l18303. [Google Scholar] [CrossRef] - Manea, V.; Manea, M.; Ferrari, L.; Orozco-Esquivel, T.; Valenzuela, R.; Husker, A.; Kostoglodov, V. A review of the geodynamic evolution of flat slab subduction in Mexico, Peru, and Chile. Tectonophysics
**2016**, 695 (Suppl. C), 27–52. [Google Scholar] [CrossRef] - Witze, A. Deadly Mexico quakes not linked. Nature
**2017**, 549, 442. [Google Scholar] [CrossRef] [PubMed] - Sarlis, N.V.; Skordas, E.S.; Varotsos, P.A.; Ramírez-Rojas, A.; Flores-Márquez, E.L. Natural time analysis: On the deadly Mexico M8.2 earthquake on 7 September 2017. Phys. A
**2018**, 506, 625–634. [Google Scholar] [CrossRef] - Olami, Z.; Feder, H.J.S.; Christensen, K. Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes. Phys. Rev. Lett.
**1992**, 68, 1244–1247. [Google Scholar] [CrossRef] [PubMed] - Ramos, O.; Altshuler, E.; Måløy, K.J. Quasiperiodic events in an earthquake model. Phys. Rev. Lett.
**2006**, 96, 098501. [Google Scholar] [CrossRef] [PubMed] - Caruso, F.; Kantz, H. Prediction of extreme events in the OFC model on a small world network. Eur. Phys. J.
**2011**, 79, 7–11. [Google Scholar] [CrossRef] - Varotsos, P.A.; Sarlis, N.V.; Skordas, E.S. Natural Time Analysis: The new view of time. In Precursory Seismic Electric Signals, Earthquakes and Other Complex Time-Series; Springer: Berlin/Heidelberg, Germany, 2011. [Google Scholar]
- Sarlis, N.; Skordas, E.; Varotsos, P. The change of the entropy in natural time under time-reversal in the Olami-Feder-Christensen earthquake model. Tectonophysics
**2011**, 513, 49–53. [Google Scholar] [CrossRef] - Varotsos, P.A.; Sarlis, N.V.; Skordas, E.S.; Lazaridou, M.S. Entropy in natural time domain. Phys. Rev. E
**2004**, 70, 011106. [Google Scholar] [CrossRef] [PubMed] - Varotsos, P.A.; Sarlis, N.V.; Skordas, E.S.; Lazaridou, M.S. Natural entropy fluctuations discriminate similar-looking electric signals emitted from systems of different dynamics. Phys. Rev. E
**2005**, 71, 011110. [Google Scholar] [CrossRef] [PubMed] - Varotsos, P.; Sarlis, N.V.; Skordas, E.S.; Uyeda, S.; Kamogawa, M. Natural time analysis of critical phenomena. Proc. Natl. Acad. Sci. USA
**2011**, 108, 11361–11364. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Varotsos, P.A.; Sarlis, N.V.; Skordas, E.S. Spatio-temporal complexity aspects on the interrelation between seismic electric signals and seismicity. Pract. Athens Acad.
**2001**, 76, 294–321. [Google Scholar] - Varotsos, P.A.; Sarlis, N.V.; Skordas, E.S. Attempt to distinguish electric signals of a dichotomous nature. Phys. Rev. E
**2003**, 68, 031106. [Google Scholar] [CrossRef] [PubMed] - Varotsos, P.A.; Sarlis, N.V.; Tanaka, H.K.; Skordas, E.S. Some properties of the entropy in the natural time. Phys. Rev. E
**2005**, 71, 032102. [Google Scholar] [CrossRef] [PubMed] - Lesche, B.J. Instabilities of Rényi entropies. Stat. Phys.
**1982**, 27, 419–422. [Google Scholar] [CrossRef] - Lesche, B. Rényi entropies and observables. Phys. Rev. E
**2004**, 70, 017102. [Google Scholar] [CrossRef] [PubMed] - Varotsos, P.A.; Sarlis, N.V.; Skordas, E.S.; Lazaridou, M.S. Identifying sudden cardiac death risk and specifying its occurrence time by analysing electrocardiograms in natural time. Appl. Phys. Lett.
**2007**, 91, 064106. [Google Scholar] [CrossRef] - Varotsos, P.A.; Sarlis, N.V.; Skordas, E.S.; Lazaridou, M.S. Fluctuations, under time reversal, of the natural time and the entropy distinguish similar looking electric signals of different dynamics. J. Appl. Phys.
**2008**, 103, 014906. [Google Scholar] [CrossRef] [Green Version] - Nicholas, V.; Sarlis, S.; Christopoulos, R.G.; Bemplidaki, M.M. Change ΔS of the entropy in natural time under time reversal: Complexity measure upon change of scale. Eur. Lett.
**2015**, 109, 18002. [Google Scholar] [CrossRef] - Varotsos, P.A.; Sarlis, N.V.; Skordas, E.S.; Lazaridou, M.S. Seismic Electric Signals: An additional fact showing their physical interconnection with seismicity. Tectonophysics
**2013**, 589, 116–125. [Google Scholar] [CrossRef] - Varotsos, P.A.; Sarlis, N.V.; Skordas, E.S. Study of the temporal correlations in the magnitude time series before major earthquakes in Japan. J. Geophys. Res.
**2014**, 119, 9192–9206. [Google Scholar] [CrossRef] [Green Version] - Turcotte, D.L. Fractals and Chaos in Geology and Geophysics, 2nd ed.; Cambridge University Press: Cambridge, UK, 1997. [Google Scholar]
- Garber, A.; Hallerberg, S.; Kantz, H. Predicting extreme avalanches in self-organized critical Sandpiles. Phys. Rev. E
**2009**, 80, 026124. [Google Scholar] [CrossRef] [PubMed] - Fawcett, T. An Introduction to ROC Analysis. Pattern Recognit. Lett.
**2006**, 27, 861–874. [Google Scholar] [CrossRef] - Lifshitz, I.M.; Slyozov, V.V. The kinetics of precipitation from supersaturated solid solutions. J. Phys. Chem. Solids
**1961**, 19, 35–50. [Google Scholar] [CrossRef] - Wagner, C. Theorie der Alterung von NiederschlagendurchUmlosen (Ostwald-Reifung). Z. Elektrochem.
**1981**, 65, 581–591. [Google Scholar] - Bray, A.J. Theory of phase-ordering kinetics. Adv. Phys.
**1994**, 43, 357–459. [Google Scholar] [CrossRef] [Green Version] - Siegmund, J.F.; Siegmund, N.; Donner, R.V. CoinCalc—A new R package for quantifying simultaneities of event series. Comput. Geosci.
**2017**, 98, 64–72. [Google Scholar] [CrossRef] - Peng, C.K.; Buldyrev, S.V.; Havlin, S.; Simons, M.; Stanley, H.E.; Goldberger, A.L. Mosaic organization of DNA nucleotides. Phys. Rev. E
**1944**, 49, 1685–1689. [Google Scholar] [CrossRef] - Peng, C.K.; Buldyrev, S.V.; Goldberger, A.L.; Havlin, S.; Mantegna, R.N.; Simons, M.; Stanley, H.E. Statisticalproperties of DNA sequences. Phys. A
**1995**, 221, 180–192. [Google Scholar] [CrossRef] - Peng, C.K.; Havlin, S.; Stanley, H.E.; Goldberger, A.L. Quantification of scaling exponents and crossoverphenomena in nonstationary heartbeat time series. Chaos
**1995**, 5, 82–87. [Google Scholar] [CrossRef] [PubMed] - Tsallis, C. Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys.
**1988**, 52, 479–487. [Google Scholar] [CrossRef] - Tsallis, C. Introduction to Nonextensive Statistical Mechanics; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Tsallis, C. The Nonadditive Entropy Sq and Its Applications in Physics and Elsewhere: Some Remarks. Entropy
**2011**, 13, 1765–1804. [Google Scholar] [CrossRef] [Green Version] - Sotolongo-Costa, O.; Posadas, A. Fragment-Asperity Interaction Model for Earthquakes. Phys. Rev. Lett.
**2004**, 92, 048501. [Google Scholar] [CrossRef] [PubMed] - Silva, R.; França, G.S.; Vilar, C.S.; Alcaniz, J.S. Nonextensive models for earthquakes. Phys. Rev. E
**2006**, 73, 026102. [Google Scholar] [CrossRef] [PubMed] - Sarlis, N.V.; Skordas, E.S.; Varotsos, P.A. Nonextensivity and natural time: The case of seismicity. Phys. Rev. E
**2010**, 82, 021110. [Google Scholar] [CrossRef] [PubMed] - Telesca, L. Maximum Likelihood Estimation of the Nonextensive Parameters of the Earthquake Cumulative 375 Magnitude Distribution. Bull. Seismol. Soc. Am.
**2012**, 102, 886–891. [Google Scholar] [CrossRef] - Vallianatos, F.; Michas, G.; Papadakis, G. Nonextensive Statistical Seismology: An Overview. In Complexity of Seismic Time Series; Chelidze, T., Vallianatos, F., Telesca, L., Eds.; Elsevier: New York, NY, USA, 2018; pp. 25–59. [Google Scholar]
- Papadakis, G.; Vallianatos, F.; Sammonds, P. A Nonextensive Statistical Physics Analysis of the 1995 Kobe, 379 Japan Earthquake. Pure Appl. Geophys.
**2015**, 172, 1923–1931. [Google Scholar] [CrossRef]

**Figure 1.**Plot of the entropies S

_{i}(

**a**) and (S

_{−})

_{i}(

**b**) as well as the entropy change ∆S

_{i}under time reversal (

**c**) versus the conventional time for the three scales i = 10

^{2}(red), 3 × 10

^{3}(green), and 4 × 10

^{3}(blue) events when analysing all earthquakes with M ≥ 3.5. The vertical lines ending at circles depict the earthquake magnitudes, which are read in the right scale.

**Figure 2.**Plot of the values of the complexity measure λ

_{i}versus the conventional time that correspond to the scales i = 3 × 10

^{3}(green), 4 × 10

^{3}(blue), and 5 × 10

^{3}(cyan) events when considering all earthquakes in the Chiapas region with M ≥ 3.5 since 2012.

**Figure 3.**Plot of the complexity measure λ

_{i}versus the scale i (number of events) for all M ≥ 3.5 earthquakes in the Chiapas region since 1 January 2012. The λ

_{i}values are calculated for each i value at the following dates: 1 June 2017 (yellow solid circles), 14 June 2017 (cyan squares), 1 July 2017 (red plus), 1 August 2017 (blue star), 1 September 2017 (green cross), and 7 September 2017 (red circle, until the last event before the M8.2 earthquake on 7 September 2017).

**Figure 4.**Plot of the values of the complexity measure Λ

_{i}versus the conventional time that correspond to the scales i = 3 × 10

^{3}(green), 4 × 10

^{3}(blue), and 5 × 10

^{3}(cyan) events when considering all earthquakes in the Chiapas region with M ≥ 3.5 since 2012.

**Figure 5.**Plot of the complexity measure Λ

_{i}versus the scale i (number of events) for all M ≥ 3.5 earthquakes (

**a**) as well as for all M ≥ 4.0 earthquakes (

**b**) in the Chiapas region since 1 January 2012. The Λ

_{i}values are calculated for each i value at the following dates: 1 June 2017 (yellow solid circles), 14 June 2017 (cyan squares), 1 July 2017 (red plus), 1 August 2017 (blue star), 1 September 2017 (green cross) and 7 September 2017 (red circle, until the last event before the M8.2 earthquake on 7 September 2017).

**Figure 6.**(

**a**) The same as in Figure 4 but in an expanded timescale; (

**b**,

**c**) plot of the change ΔΛ

_{i}of the complexity measure Λ

_{i}after the occurrence of the M7.0 earthquake on 14 June 2017 (

**b**) and the M6.6 earthquake on 17 December 2015 (

**c**) versus the time elapsed (t − t

_{0}) in days. In both cases, the value of t

_{0}has been selected approximately 30 min after earthquake occurrence.

**Figure 7.**Results from the event coincidence analysis between the earthquake events of magnitude $M>{M}_{thres}^{target}$ and the events of abrupt increase of Λ

_{i}(see Figure 4 and Figure 6a). The probability to obtain by chance (p-value) the observed results when considering the events of abrupt increase of Λ

_{i}as precursors to strong earthquakes versus the threshold ${M}_{thres}^{target}$ for various scales i. The p-values have been obtained by using the compute code eca.es of the CoinCalc package [32] of R.

**Figure 8.**The entropic index q versus conventional time for sliding windows W = 1000 (red) and W = 250 (blue) consecutive earthquakes in the Chiapas region since 2012 for M ≥ 3.5 and M ≥ 4.0, respectively.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ramírez-Rojas, A.; Flores-Márquez, E.L.; Sarlis, N.V.; Varotsos, P.A.
The Complexity Measures Associated with the Fluctuations of the Entropy in Natural Time before the Deadly México M8.2 Earthquake on 7 September 2017. *Entropy* **2018**, *20*, 477.
https://doi.org/10.3390/e20060477

**AMA Style**

Ramírez-Rojas A, Flores-Márquez EL, Sarlis NV, Varotsos PA.
The Complexity Measures Associated with the Fluctuations of the Entropy in Natural Time before the Deadly México M8.2 Earthquake on 7 September 2017. *Entropy*. 2018; 20(6):477.
https://doi.org/10.3390/e20060477

**Chicago/Turabian Style**

Ramírez-Rojas, Alejandro, Elsa Leticia Flores-Márquez, Nicholas V. Sarlis, and Panayiotis A. Varotsos.
2018. "The Complexity Measures Associated with the Fluctuations of the Entropy in Natural Time before the Deadly México M8.2 Earthquake on 7 September 2017" *Entropy* 20, no. 6: 477.
https://doi.org/10.3390/e20060477