Physica A 389 (2010) 1239–1252

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Physica A journal homepage: www.elsevier.com/locate/physa

Pattern synchrony in electrical signals related to earthquake activity R. Hernández-Pérez a,d,∗ , L. Guzmán-Vargas b , A. Ramírez-Rojas c , F. Angulo-Brown d a

Satélites Mexicanos, S.A. de C.V., Centro de Control Satelital Iztapalapa. Av. de las Telecomunicaciones S/N CONTEL Edif. SGA-II. México, D.F. 09310, Mexico

b

Unidad Profesional Interdisciplinaria en Ingeniería y Tecnologías Avanzadas, Instituto Politécnico Nacional, Av. IPN No. 2580, Col. Ticomán, México D.F. 07340, Mexico c Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana, Av. San Pablo 180, Col. Reynosa, México D.F., 02200, Mexico d

Departamento de Física, Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, Edif. 9 U.P. Zacatenco, México D.F. 07738, Mexico

article

info

Article history: Received 19 August 2009 Received in revised form 13 November 2009 Available online 4 December 2009 Keywords: Seismicity Time series analysis Complexity Statistical methods Pattern synchrony

abstract We apply the cross sample entropy method to geoelectrical time series collected from independent channels (North–South and East–West directions) monitored at two sites located in Mexico, to assess the presence of pattern synchrony between the signals, particularly in the proximity of earthquakes. To our best knowledge, this method has not been applied yet for the study of electrical signals related to earthquake activity. Moreover, we introduce the multiscale pattern synchrony analysis by extending the multiscale entropy technique to calculate the cross-entropy between two signals, which represents a novel approach to the study of pattern synchrony. The results obtained suggest that in the vicinity of an earthquake the geoelectrical signals exhibit pattern synchrony that persists for long sequences and through multiple scales, in addition to the presence of correlations in each channel. © 2009 Elsevier B.V. All rights reserved.

1. Introduction There have been many studies on developing a robust meaning of complexity, noticeably in physiological signals where complexity has been associated to healthy systems [1–5], and particularly in electrical signals related to earthquake activity [6–24]. With respect to the electroseismic signals, the study of electromagnetic phenomena that were possibly associated with earthquakes (EQs) was developed by Varotsos and Alexopoulos in 1984, through the introduction of the VAN method, which is based on the premise that large EQs are preceded by observable anomalous changes in the geoelectrical potential called seismic electric signals (SES) [25,26]. The geoelectrical signals are collected from two independent channels of the experimental setup, with one channel for each direction: North–South (NS) and East–West (EW). The studies on geoelectrical signals have been focused in searching for a possible relationship between changes in the statistical patterns, and other features of the signals, and the occurrence of EQs. Different approaches to analyze these signals have been followed, mainly: time-clustering behaviour [15–19], natural time [27,9,28], power spectrum [29,11,12], correlation profiles [21], multifractal analysis [14,30], information analysis [13,31] and multiscale entropy profiles [20]. These studies have suggested that in the vicinity of an EQ, the geoelectrical time-series exhibit complex behavior, mainly consisting of the appearance of long-range correlations. Moreover, these studies have been performed on a single channel and it has been found that each channel exhibits complex behavior in the vicinity of an EQ. In particular, several papers have been devoted to studying the self-potential signals collected at the Pacific Coast of Mexico using different tools such as spectral analysis, regularity, variability and correlations [11,12,20,21,29,32–34]. These works have reported changes in the correlation dynamics of each channel observed prior to the occurrence of EQs with



Corresponding author. Tel.: +52 5558047346. E-mail address: [email protected] (R. Hernández-Pérez).

0378-4371/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2009.11.036

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M > 6: the signals exhibit long-range correlations, resembling 1/f -noise signals. In addition, another work was devoted to analyzing the mutual information index measured between the channels (EW and NS) [31], for which a changing behavior is observed when analyzing signals close to an EQ; which may suggest that the crust is anisotropic, therefore seismic stresses/waves propagate differently in each direction. Entropy, as it relates to dynamical systems, in this case represented by a time series, is the rate of information production and is quantified by the Kolmogorov–Sinai (KS) entropy [35]. In order to quantify the regularity and complexity of time series in the context of physiological signals, entropy-based algorithms are often used, such as the Approximate Entropy [2], the Sample Entropy [35] and the Multiscale Entropy [5]. Moreover, the cross-entropy versions of the Approximate and Sample entropy techniques were introduced as a measure of the similarity of two distinct, yet intertwined, time series [35,36], in particular in the analysis of biological signals, such as the study of hormone secretion [36–38], the relation between heart rate and blood pressure [39], functional connectivity in the brain [40], and renal sympathetic nerve activity in rats [41]. In particular, the Cross Sample Entropy is used to define the pattern synchrony between two signals, where synchrony refers to pattern similarity, not synchrony in time, in which patterns in one series appear (within a certain tolerance) in the other series. To our best knowledge, the Cross Sample Entropy technique has not been applied yet for the study of electrical signals related to earthquake activity, neither has the multiscale cross entropy been explored. Thus, in the present work we are interested in using the Cross Sample Entropy method to assess whether there is pattern synchrony between the signals from each channel, particularly in the proximity of an earthquake occurrence. Moreover, we are interested in studying this cross entropy in different time scales by extending the Multiscale Entropy method to include the cross entropy. The paper is organized as follows. In Section 2, we describe the entropy-based methods, in particular the Cross Sample Entropy. The data are described in Section 3. In Section 4 we present the results, and we discuss the findings in Section 5. Finally, concluding remarks are given in Section 6. 2. Methods 2.1. Entropy as measure of regularity Entropy, as it relates to dynamical systems, is the rate of information production, and it is quantified by the Kolmogorov–Sinai (KS) entropy. The KS entropy is useful to characterize the system dynamics: for instance, the KS entropy for deterministic systems is zero, since any state depends only on the initial conditions; while for uncorrelated random processes, the KS entropy reaches a maximum, since each new state is totally independent of the previous ones. However, the calculation of KS requires very long data sets. An alternative procedure to estimate the entropy of a signal was proposed: the K2 entropy, which is a lower bound of the KS entropy [42]. Later on, other approaches were developed based on K2 to estimate KS for short data sequences, such as the Approximate Entropy (AE ) that was introduced to quantify the regularity in time series [1]. 2.2. Approximate Entropy The AE was introduced to estimate KS for short data sequences and to quantify the regularity in time series [1,2]. The AE algorithm can be summarized as follows: for N data points, the statistics AE (m, r , N ) is approximately equal to the negative average natural logarithm of the conditional probability that two sequences that are similar for m points remain similar at the next point, within a tolerance r, where the tolerance is set as r × σ , being σ the standard deviation of the series [2,35]. Fig. 1 illustrates the procedure to find and count sequence matches within a time series. The procedure to calculate AE is as follows [35]: for a time series of N points u(j), we define a set of vectors xm (i) for i ∈ [1, N − m + 1], where xm (i) = {u(i + k)|0 ≤ k ≤ m − 1} is the vector of m samples from u(i) to u(i + m − 1). These vectors are also known as patterns: a sequence of data points from the time series. The distance between two such vectors, i.e., the maximum difference between their corresponding components is defined as: d[xm (i), xm (j)] = max{|u(i + k) − u(j + k)| : 0 ≤ k ≤ m − 1}. Let Bi be the number of vectors xm (j), with j ≤ N − m + 1, such that d[xm (i), xm (j)] ≤ r. Then, define the function Bi , (1) N −m+1 to quantify the probability of finding another vector within the distance r from the template vector xm (i). In calculating Cim (r ), the vector xm (i) is called the template, and an instance where a vector xm (j) is within r of it is called a template match. Now, the function Cim (r ) =

Φ m (r ) =

1 N −m+1

N −m+1

X i=1

ln Cim (r ),

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Fig. 1. A simulated time series u[1], . . . , u[N ] is shown to illustrate the procedure for finding sequence matches to calculate the Approximate (AE ) and Sample (SE ) entropies for the case m = 2 and a given tolerance r (a fraction of the standard deviation of the series). Dotted lines around points u[4], u[5] and u[6] represent u[4] ± r, u[5] ± r and u[6] ± r, respectively. We say that two data points match each other if the absolute difference between them is ≤r. Consider the two-component template sequence (u[4], u[5]) and the three-component template sequence (u[4], u[5], u[6]). A pattern is a sequence of data points from the time series. For the segment shown, there are two sequences (patterns), (u[21], u[22]) and (u[42], u[43]), that match the template sequence (u[4], u[5]), and only one sequence that matches the sequence (u[4], u[5], u[6]). Then, in this case there are two sequences matching the twocomponent template sequences and one sequence matching the three-component template sequence. These calculations are then repeated for the next two- and three-component template sequence, which are (u[5], u[6]) and (u[5], u[6], u[7]), respectively. The number of sequences that match each of the two- and three-component template sequences are again summed and added to the previous values. This counting provides the values for Bi (Eq. (1)) for AE ; or, the values for B0i (Eq. (2)) for SE , depending on whether the self-matches are counted or not.

is the average of the natural logarithms of the functions Cim (r ). Eckmann and Ruelle suggested the following approximation for the entropy of the underlying process: limr →0 limm→∞ limN →∞ (Φ m (r )−Φ m+1 (r )) [43]. However, this definition is not suited for the analysis of finite and noisy time series obtained in experiments because it requires infinite data sets. Therefore, Pincus defined the Approximate Entropy (AE ) as AE (m, r ) = lim

N →∞

 m  Φ (r ) − Φ m+1 (r ) ,

which for finite data sets is estimated by the statistics [2]: AE (m, r , N ) = Φ m (r ) − Φ m+1 (r ). 2.3. Sample Entropy The Approximate Entropy provides a measure of the degree of irregularity or randomness in a time series, and was introduced as a measure of system complexity: smaller values indicate greater regularity, and greater values convey more disorder and randomness; it is also useful to distinguish correlated stochastic processes and composite deterministic/stochastic models [2]. However, it has been found that AE is biased, leading to inconsistent results. This bias is produced since the AE counts self-matching sequences. The Sample Entropy (SE ) was introduced to reduce the bias by removing the counting of self-matched sequences in the AE algorithm [35]. Discounting the self-matches is justified since the entropy is conceived as a measure of the rate of information production [43]; then, self-matches do not add information. Therefore, the probability of finding another vector within the distance r from the template vector xm (i), without counting self-matches, is given by the following expression: Ci0m (r ) =

B0i N −m+1

,

(2)

where B0i is the number of vectors xm (j), with j ≤ N − m + 1 and j 6= i, such that d[xm (i), xm (j)] ≤ r. Then, the function U m provides the average of the terms Ci0m (r ) as follows: U m (r ) =

1 N −m+1

N −m+1

X i=1

Ci0m (r ).

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Finally, the Sample Entropy is defined as [35]:

  U m+1 (r ) SE (m, r ) = lim − ln , N →∞ U m (r ) which can be estimated by the statistics: SE (m, r , N ) = − ln

U m+1 (r ) U m (r )

.

It has been reported that SE agrees with theory much better than the AE statistics for different stochastic processes over a wide range of operating conditions and it improves the evaluation of time series regularity [35]. 2.4. Cross Sample Entropy Entropy can also be calculated between two signals, and this mutual entropy characterizes the probability of finding similar patterns within the signals. Therefore, the cross-entropy technique was introduced to measure the degree of asynchrony or dissimilarity of two time series [36,44]. When calculating the cross-entropies, the patterns that are compared are taken in pairs from the two different time series {u(i)} and {v(i)}, i = 1, . . . , N. The vectors are constructed as follows: xm (i) = [u(i), u(i + 1), u(i + 2), . . . , u(i + m − 1)] , ym (i) = [v(i), v(i + 1), v(i + 2), . . . , v(i + m − 1)] , with the vector distance defined as d[xm (i), ym (j)] = max{|u(i + k) − v(j + k)| : 0 ≤ k ≤ m − 1}. With this definition of distance, the SE algorithm can be applied to compare sequences from the template series to those of the target series to obtain the Cross Sample Entropy (CE ). It is usual that the two time series are first normalized by subtracting the mean value from each data series and then dividing it by the standard deviation. This normalization is valid since the main interest is to compare patterns. It is quite possible that no vectors in the target series can be found to be within a distance r of the template vector and then the value of CE is not defined. One important property of CE is that its value is independent of which signal is taken as a template. In particular, the Cross Sample Entropy is used to define the pattern synchrony between two signals, where synchrony refers to pattern similarity, not synchrony in time, wherein patterns in one series appear (within a certain tolerance) in the other series. Moreover, CE assigns a positive number to the similarity (synchronicity) of patterns in the two series, with larger values corresponding to greater common features in the pattern architecture and smaller values corresponding to large differences in the pattern architecture of the signals [40,37]. When no matches are found, a fixed negative value is assigned to CE to allow a better displaying of the results. The conceptual difference between pattern synchrony, as measured by the CE , and correlations, as measured by the crosscorrelation function, can be expressed as follows: let us suppose that we have two time series {x(k)} and {y(k)}. The CE deals with patterns: a sequence of data points of a certain length m is taken from the template time-series {x(k)} and this pattern is searched for in the target time-series {y(k)} within a tolerance r. However, the CE does not collect the time-stamp of the matching sequence in the time series {y(k)}, but counts the number of sequence matches of lengths m and m + 1. On the other hand, the objective of the cross-correlation function is to find the time lag τ for which the whole time series {x(k)} resembles {y(k)}, but the time series are not decomposed in sequences of points. Therefore, CE analysis is complementary to the cross-correlation and spectral analysis since it operates on different features of the signals (see the Appendix of Ref. [36]). 2.5. Multiscale Entropy Furthermore, Multiscale Entropy (ME ) was developed to measure the complexity of signals for different time scales [5]. This technique shows that long-range correlated noises are more complex than uncorrelated signals. In summary, the multiscale entropy method applied to a given time series x1 , . . . , xN , starts with a coarse-graining procedure, for a given scale factor τ , that consists of a moving average filtering given by: yτj =

1

jτ X

τ

i=(j−1)τ +1

xi ,

with 1 ≤ j ≤ N /τ . Note that the length of the coarse-grained time series is given by N /τ , and for scale one the original time series is recovered. Then, the SE is calculated for the coarse-grained time series {yτj }. This process is repeated for the desired scales τ to obtain the values of ME (τ ). It has been reported that ME reaches a very good agreement with theory for simulated signals: the entropy values for a random noise monotonically decrease whereas for long-range correlated noise, such as for the 1/f -noise, the entropy remains constant for several scales, indicating that these signals are structurally more complex

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Fig. 2. Location of the monitoring stations and the epicenters of the earthquakes occurring during the studied time period.

than uncorrelated signals [5,45]. When applied to biological signals, the ME provides consistent results indicating higher complexity for healthy dynamics showing long-range correlations than for certain pathologic conditions [45]. Moreover, our group has recently introduced the ME technique for the analysis of geoelectrical signals, finding that in the vicinity of an earthquake occurrence the signals on each channel exhibit complex behavior characterized by the presence of longrange correlations, and emphasizing the importance of ME as a complementary tool in the search for possible geoelectrical precursory phenomena of earthquakes [20,21]. Finally, to assess the multiscale pattern synchrony of the geoelectrical signals, we introduce an extension to the Multiscale Entropy (ME ) method [5] to include the calculation of the CE between the coarse-grained version of the two original time series. 3. Data We have used geoelectrical signals collected during the period from June 1994 to May 1996, by two monitoring stations located in the South Pacific coast of Mexico: Acapulco (16.85 N, 99.9 W) and Coyuca (17.1 N, 100.2 W) [11]. These time series are the electric self-potential fluctuations V between two electrodes buried 2 m into the ground and separated by a distance of 50 m, where each pair of electrodes was oriented in one direction: NS and EW, as indicated by the VAN methodology [25,26]; and the time series, one for each channel, were simultaneously recorded at each monitoring station, near the Middle American trench, which is the border between the Cocos and the American tectonic plates (see Fig. 2). In Fig. 3 we present representative time series for each channel for a short period of time collected at the Acapulco station. Due to technical adjustments, two different sampling rates were used: T = 2 s for Acapulco and T = 4 s for Coyuca [46]. During the period of study, two EQs with M > 6 occurred with epicenters within 250 km of the two monitoring stations. The first EQ occurred on September 14, 1995 with M = 7.4 and epicenter with coordinates (16.31 N, 98.88 W), with a focal depth of 22 km; the hypocenter was d = 112 km from Acapulco and d = 146.6 km from Coyuca. The second EQ occurred on February 24, 1996 with M = 7.0 and epicenter with coordinates (15.8 N, 98.25 W), with a focal depth of 3 km; the hypocenter was d = 220.02 km from Acapulco and d = 250.01 km from Coyuca. As can be seen from Fig. 2, the two earthquakes had epicenters located closer to the Acapulco station. The geoelectrical data are split into three time periods which showed a remarkably different behavior in the variability and correlation profiles of the signals [20,21]: region I from June 1994 to October 1994, region II from November 1994 to October 1995 (encompassing the occurrence of the EQ of September 14, 1995); and region III from November 1995 to May 1996 (encompassing the occurrence of the EQ of February 24, 1996). Finally, to have a reference as to what the CE and its multiscale version would look like for fractional Gaussian noises, we simulated signals with 215 points and a power spectral density given by 1/f β , 0 ≤ β ≤ 1, generated with the Fourier filtering method [47,48]. 4. Results In the following paragraphs we present the results for both single- and multi-scale versions of the CE , calculated on the simulated signals as well as on the geoelectrical signals from both monitoring stations.

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The CE is calculated between the NS and EW channels. The procedure starts by partitioning the data in non-overlapping calculation windows containing 5400 samples each. The values of the parameters for the calculation of the Cross Sample entropy were: a maximum sequence length of 15 samples and a tolerance r = 0.2σ , where σ is the standard deviation of the series. The usage of a tolerance of 20% is typical in virtually all clinical applications, since it provides an appropriate Approximate Entropy statistics for assessing irregularity in short data series that is in agreement with theory [35,37,38]. Since no previous studies on Cross Sample Entropy have been performed for geoelectrical signals, we adopted the value of 20% for the tolerance r. As a convention, we set the value of −1 for CE whenever either of the following conditions is met: (i) a single match is observed for a sequence with length m, whereas no matches are found for m + 1, therefore CE is infinite; or, (ii) if the number of matches for m and m + 1 are both zero, and then CE is not well defined. Moreover, we performed the same calculation for the shuffled version of the data, maintaining the CE parameters as well as the length of the calculation windows, in order to study the behavior of CE after correlations have been destroyed by the shuffling process. 4.1. Cross Sample Entropy results 4.1.1. Simulated signals Fig. 4 shows the CE profile for the simulated signals with power spectral density of the form f −β , with 0 ≤ β ≤ 1. For each value of the spectral exponent, ten independent realizations were performed and averaged to obtain the displayed results. As can be seen, CE stays well-defined for longer sequences when longer-range correlations are present in the signal (increasing β ). Specifically, we observe that for values of β close to the white noise fluctuations (β = 0), the pattern synchrony shows a high value and persists for a sequence length of around 8 samples, whereas for values of β close to one, the CE is slightly lower than for the uncorrelated case, but it persists for a larger sequence length such that for β = 1 it is around 12 samples. In addition, Fig. 4 shows the results of the CE between signals with different spectral exponents β1 and β2 , for different sequence lengths. As can be seen, the pattern synchrony between signals with correlations of longer range (for β → 1) persists for longer sequences. 4.1.2. Acapulco data The results of the CE calculation for the original and shuffled data from the Acapulco monitoring station are shown in Fig. 5. We observe regularity in the CE profile for region I. Moreover, the CE profile for the original data is not significantly different to the one obtained for the shuffled data, except that the CE reaches systematically higher values for the shuffled data. For the original data there is a period of time within June 1994 and towards the end of the region I for which the pattern synchrony remains for long sequences.

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Fig. 6. Acapulco station. Histograms of the maximum sequence length up to which CE is well-defined for the three epoch regions described in the text, corresponding to the original (top row) and to the shuffled data (bottom row).

Moreover, in order to assess the effect of the data shuffling on the CE calculation, we obtain the distribution of the maximum sequence length for which the CE is well-defined (for which there is pattern synchrony), in each calculation window. In other words, for each calculation window we obtain the value of the longest data-points sequence (pattern) for which there is pattern synchrony, such that for longer sequences the value of CE is not well-defined. In order to quantify the difference between the histograms obtained for the original and shuffled data sets, we use the kurtosis (denoted by k) and the skewness (denoted by s) of the distribution of the maximum sequence length. In Fig. 6 we see that for region I the histograms for the original and for the shuffled data are similar, with k = 7.00 and s = 2.32, and k = 6.94 and s = 2.32; respectively. In both cases, the majority of the calculation windows show the presence of pattern synchrony for sequences up to 7 data-points, although for the original data we observe that there are calculation windows for which the pattern synchrony is present for longer sequences. This suggests that the pattern synchrony between the channels in this region resembles the one exhibited by white noise-like signals. This result is connected to previous works [20,21], in which we have found that for region I the signals in each channel exhibit a variability and correlation profile similar to the one for white noise. On the other hand, we observe more variability in the CE profile for region II. In particular, notice the significant variation of the CE that occurred between January and April 1995. Also, notice that there is certain variability of the CE profile towards the end the region. Moreover, the histogram for the maximum sequence length for the data is remarkably different to the one for the shuffled data (see Fig. 6). For the original data the pattern synchrony is present for longer patterns than in the other two regions and there is a significant number of windows for which the pattern synchrony is present for sequences with the maximum length considered, and the histogram has k = 2.75 and s = 1.25. Once the data has been shuffled, the corresponding histogram, with k = 6.85 and s = 2.20, exhibits a shape similar to the histograms for the shuffled data of the other regions. From our previous studies on correlations and variability for the signals in separate channels [20,21], the geoelectrical signals for region II exhibit long-range correlation behavior; and the present results suggest that not only the channel signals individually exhibit long-range correlations, but also there is pattern synchrony between channels that persists longer than for the other regions. Finally, for region III we observe that the histogram for the original data is not as similar to that corresponding to the shuffled data as occurs for region I; since the histogram for the original data has k = 5.58 and s = 2.05; while for the shuffled data has k = 5.64 and s = 2.08. 4.1.3. Coyuca data The CE results for the geoelectrical signals collected by the Coyuca monitoring station are shown in Fig. 7, while the histograms for the distribution of the maximum sequence length is provided in Fig. 8.

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Fig. 7. Cross Sample Entropy analysis for geoelectrical time series from the Coyuca station: Original (top) and shuffled data (bottom).

We observe regularity in the CE profile for region I. As can be seen in Fig. 8, the histogram for the original data is wider than for the shuffled data, indicating that the pattern synchrony for the original data is defined for longer sequences. The histogram for the original data has k = 4.22 and s = 1.58; while for the shuffled data has k = 7.75 and s = 2.40. Notice that the original data from Coyuca in region I exhibits pattern synchrony for longer sequences than for the Acapulco station, where the CE profile for the original data resembles the one obtained for the shuffled data. On the other hand, for region II we notice the significant variation of the CE that occurs mainly between April and June 1995. Moreover, the variability of CE continues for the remaining part of the region. Comparing to the results for the Acapulco station (see Fig. 5), it can be seen that this signature occurred later for the Coyuca station, which was farther away from the EQ epicenter than Acapulco. Moreover, from the histogram for the original data in region II we observe that the CE is welldefined for long sequences for most calculation windows, with a significant number of cases for sequences with 15 samples; showing that the pattern synchrony between the channels in this region persists for long sequences, which is consistent with the observed behavior for the Acapulco data. Moreover, the histogram for the shuffled data is narrower than for the original data. The histogram for the original data has k = 3.58 and s = 1.36; and the one for the shuffled data has k = 6.79 and s = 2.18. Finally, for region III we see that the CE profile at the beginning of the region shows pattern synchrony for long sequences, with some gaps towards the middle and the end of the region, on which the CE is defined for shorter sequences. Again, we observe that the CE profile for the shuffled data still shows pattern synchrony on a non-negligible number of cases for relatively long sequences, with a more even distribution. The variability in the CE profile in this region is captured by the histograms shown in Fig. 8, on which we observe that after the shuffling, most of the calculation windows give positive values of CE for sequences of 7 and 8 samples. Comparing to the results from Acapulco, for the Coyuca station we observe that the effect of the shuffling reduced the persistence of the pattern synchrony less than for the Acapulco station. Again, this difference could be due to the different local properties of the crust [29]. Also, notice the significant difference between the histograms for the original (k = 3.18 and s = 0.92) and the shuffled data (k = 3.89 and s = 1.57).

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4.2. Multiscale Cross Entropy results 4.2.1. Simulated signals We generate signals with 215 samples with spectral exponents of β = 0, 0.5 and 1.0. The Multiscale Cross Entropy (MCE ) was calculated for sequence lengths up to 12 samples, with tolerance r = 0.2 and scale factors up to 40. Fig. 9 shows the average for ten independent realizations for each value of β . As can be seen for the three signals, the MCE is well-defined even for large scales, but for increasing sequence lengths as β approaches to one. This suggests that signals with long-range correlations exhibit pattern synchrony for longer sequences and in different scales. Notice that for large scales, the MCE for the three signals shown in Fig. 9 seems to reach a stable value. For the case of the white noise, this result may seem counter-intuitive. One thing that should be taken into consideration is that the calculation of MCE comprises a moving-average filter operation, therefore, for large scales more points are considered in the filtering window; which results in having a smoother version of the signal (reduced variability), and therefore the probability to find similar sequences, within the tolerance r, does not vanish but reaches a stable value. This implies that the statistics is poorer as larger scales are considered, since the subseries for the coarse-graining procedure have less data points [5,45]. 4.2.2. Geoelectrical signals The Multiscale Cross Entropy was calculated for the geoelectrical signals for sequence lengths up to 15 samples, with tolerance r = 0.2 and for 40 scales. Since a typical file for an observation period of one month has around 1 million records for the Acapulco station, and around half of this for Coyuca (due to the difference in the sampling rate), we split the data files in non-overlapping calculation windows of 42 300 samples for Acapulco and 21 600 samples for Coyuca, corresponding to a 24 h period for each station. For each calculation window a surface is obtained whose height gives the value of the MCE , while the x and y axes represent the scale factors and the sequence length, respectively. However, since there are several windows for each month of data, displaying the whole set of results is a difficult task for the present work. Therefore, we select to show the results for some selected scales: 5, 10, 20 and 30; for both the original and shuffled data from the two stations. Fig. 10 shows the MCE analysis results for the selected scales of the geoelectrical signals, and its shuffled version, from Acapulco station. Notice that the MCE profile for region II shows pattern synchrony for longer sequences and for larger scales than in the other regions, where MCE profiles are similar to the results obtained for simulated signals with short-range correlations. Moreover, the results for the shuffled data show that the shuffling process has almost completely destroyed the pattern synchrony for different scales.

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On the other hand, Fig. 11 shows the results of the MCE analysis of the data from Coyuca station. As can be seen, there is a higher variability in the MCE profiles than for Acapulco data; showing the presence of multiscale pattern synchrony for the three regions, but more significantly for region II. Moreover, the results for the shuffled data show that the shuffling process has significantly destroyed the pattern synchrony for different scales. 5. Discussion Regarding the variation in the CE profile for region II for the Acapulco data that occurred between January and April 1995 (see Fig. 5), it could have been related to the occurrence of the M = 7.4 EQ on September 14, 1995. On the other hand, the pattern synchrony profile for region II for Coyuca data shows a noticeable variability starting on April 1995 and that extends to the remaining part of the region and to region III as well (see Fig. 7). One thing to notice is that this signature did not occur immediately before the EQ, in fact, it was observed several months earlier. A similar result was reported in Ref. [29], when performing spectral analysis on the signals from the same monitoring stations. According to this study, the electric activity observed by each monitoring station is neither the same nor simultaneous, but the bursts propagate through the crust with an intensity that depends on the distance from the EQ epicenter to the detector, on the electrical properties of the soil at the monitoring station as well as on the properties of the medium between the EQ epicenter and the detector [29]. Moreover, independent studies [49,50] have shown that the SES amplitude recorded at a given measuring station depends on the inhomogeneities around the station as well as on the geoelectrical structure, in general, of the medium between the EQ epicenter and the station. Moreover, it can be seen that this signature occurred later for the Coyuca station, which was farther away from the EQ epicenter than Acapulco. This result is consistent with that reported in Ref. [29], in which it was suggested that the electric pulses are generated by local effects in the sites of the stations, perhaps correlated to a kind of stress pulses traveling away from the epicentral region; and therefore these signals appear earlier at the closest monitoring station. In fact, in Ref. [29] it was estimated that for the geological zone here considered, the traveling stress pulse has an apparent speed of 1–2 km/day. However, in Ref. [29] true electric anomalies were considered which were evaluated through the integral of the spectral density function for each channel. Nevertheless, in the present work we are calculating the pattern synchrony between orthogonal channels, which provides information about the pattern similarity between the signals from each channel. From our previous studies on correlations and variability for the signals in separate channels [21], the geoelectrical signals for region II for both stations exhibit long-range correlations behavior; and the present results suggest that not only the channel signals individually exhibit long-range correlations, but there is also pattern synchrony between channels that persists longer than for the other regions. In other words, when the orthogonal electrodes measure only very local random

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fluctuations of the electric field (the background noise), the pattern synchrony between the channels is not sustained for long sequences, resembling the results for the white noise. However, when the source of the electric field is relatively far away from the electrodes, perhaps associated with a regional zone affected by a certain excitation caused by a seismic front traveling from the focal zone, such as was reported in Ref. [29], the pattern synchrony between the channels persists for longer sequences. Therefore, the pattern synchrony is a possible expression of a dominant regional field mounted over the local background noise. Moreover, the results from the multiscale cross entropy analysis (see Figs. 10 and 11) for data in region II from both stations indicate that the pattern synchrony persists for long sequences and large scales. In a previous work [20], we found that for the data in region II, each channel exhibited a correlated behavior when applying the multiscale entropy analysis. Our present results indicate that, in addition to the appearance of correlated dynamics in each channel, it is observed that pattern synchrony between the channels persists for other scales. As can be seen, the results of the multiscale pattern synchrony for both stations differ between each other, for the reasons discussed previously for the single scale pattern synchrony. The present results are consistent with those deduced by analyzing the time-series in a new time domain termed natural time [6,7]: when analyzing the original time series of SES activities, the entropy S defined in natural time [51] and the entropy S− upon time reversal are obtained; then [27] both values (S and S− ) are found to be smaller than the entropy calculated for the shuffled data. This comes from the fact that, since SES activities are characterized by critical dynamics, they exhibit infinitely ranged temporal correlations [9], which are however destroyed upon randomly shuffling the original data. Moreover the fact that the entropy of the shuffled data is found to be equal to that of a uniform distribution (white noise), reveals that the self-similarity of SES activities stems solely from long-range temporal correlations, i.e., from the process memory only (and not from the process’ increments infinite variance, see Ref. [28] for details). 6. Conclusions The contribution of this work is the introduction of the Cross Sample Entropy (CE ) analysis to the study of electrical signals related to earthquake activity, which to our best knowledge has not been applied to these kind of signals. The CE analysis gives information about the pattern synchrony between two signals. We calculate the CE profile between the NS and EW channels for data collected from two monitoring stations located in the Pacific Coast of Mexico. Our main result suggests that at a certain point before the occurrence of an earthquake, the pattern synchrony between the channels is

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sustained for relatively large patterns. This behavior was observed for simulated signals with long-range correlations. In addition, we observed that there are periods for which the CE profile for the data is very similar to the one obtained for the shuffled data; which in turn resembles the one obtained for simulated signals with short-range correlations. Moreover, when the orthogonal electrodes measure only very local random fluctuations of the electric field (the local background noise), the pattern synchrony between the channels is not sustained for long sequences, resembling the results of white noise. However, when the traveling stress front arrives at the local region of each station, the electric field relatively far away from the electrodes, the pattern synchrony between the channels persists for longer sequences. Therefore, the pattern synchrony is a possible expression of a dominant relatively far field mounted over the very local background noise. On the other hand, we have extended the Multiscale Entropy method to include the calculation of the cross entropy between the signals in both channels, which represents a novel approach to the study of the pattern synchrony between two signals. Moreover, the Multiscale Cross Entropy (MCE ) profile observed for the data close to the occurrence of a M = 7.4 EQ shows that the pattern synchrony persists for long sequences and different scales. In addition, we notice that the behavior of the MCE profile for data from epoch regions far from the EQ occurrence is similar to the results obtained for simulated signals with short-range correlations. Nevertheless, more work should be performed to improve the interpretation and applicability of this tool for the study of multiscale pattern synchrony for time series in different research fields, such as for physiological signals. Results from previous works based on the study of correlations and variability concluded that in the vicinity of an EQ the geoelectrical signals individually exhibit a complex behavior [11,20,21]. Our results suggest that, in addition to this, the signals exhibit pattern synchrony between channels for long sequences and through multiple scales. Acknowledgements This work was partially supported by CONACYT (Grant 49128-F-26020), COFAA-IPN, EDI-IPN, México. The authors wish to thank to the referees whose suggestions allowed the improvement of this manuscript. References [1] S.M. Pincus, Approximate entropy as a measure of system complexity, Proc. Natl. Acad. Sci. 88 (1991) 2297–2301. [2] S.M. Pincus, Approximate entropy (ApEn) as a complexity measure, Chaos 5 (1995) 110–117. [3] A. Eke, P. Hermán, L. Kocsis, L.R. Kozak, Fractal characterization of complexity in temporal physiological signals, Physiol. Meas. 23 (2002) R1–R38.

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Predicting Synchrony in Heterogeneous Pulse Coupled ...
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Physica A Strong anticipation: Sensitivity to long-range ...
Available online 17 May 2008 ... Participants were instructed to synchronize, to the best of their ... Explanations of anticipation in cognitive science and neuroscience ... from time delays in appropriately coupled ''master'' and ''slave'' ..... eac

Synchrony in Dyadic Social Interaction / Fabian ...
Positive and negative affect scale (PANAS). Relationship quality: Post-‐interaction/Rapport questionnaire (IRQ). Future interaction questionnaire (FIQ). Inclusion of other in the self scale (IOS). Nonverbal synchrony: Visual images from the video r

Hierarchical synchrony of phase oscillators in modular ...
Jan 18, 2012 - ... Mathematics, University of Colorado at Boulder, Boulder, Colorado 80309, USA ...... T. M. Antonsen, R. T. Faghih, M. Girvan, E. Ott, and J. H..

Cluster synchrony in systems of coupled phase ...
Sep 16, 2011 - Cluster synchrony in systems of coupled phase oscillators with higher-order coupling. Per Sebastian Skardal,1,* Edward Ott,2 and Juan G. Restrepo1. 1Department of Applied Mathematics, University of Colorado at Boulder, Colorado 80309,

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Signals of adaptation in genomes
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REGULARITY IN MAPPINGS BETWEEN SIGNALS ...
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Visual search for a target changing in synchrony with ...
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Pattern formation in spatial games - Semantic Scholar
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Design Patterns in Ruby: State Pattern - GitHub
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