Greek Statistical InstituteProceedings of the 31st Panhellenic Statistics Conference (2018), pp. 278−291

Markovian Arrival Process modeling for the detectionof seismicity rate changes for the strong earthquakes

in Greece

P. Bountzis1, E. Papadimitriou1, G. Tsaklidis21Aristotle University of Thessaloniki, Geophysics Department, 54124, Thessaloniki,

Greecepmpountzp, ritsa@geo.auth.gr

2Aristotle University of Thessaloniki, Mathematics Department, 54124, Thessaloniki,Greece

tsaklidi@math.auth.gr

ABSTRACTThe Markovian Arrival Process (MAP) model is applied to a complete earthquake catalogwith magnitude M ≥ 6.5, in the broader area of Greece, for investigating the temporalvariations of seismicity. MAP is a stochastic process, that counts the regional earthquakeoccurrences and allows for transitions between the hidden states of a Markov Process,such that each state corresponds to a different level of seismicity rate. The inferenceregarding the choice of the optimal number of hidden states is made through Akaike andBayesian information criteria, and the estimation of the model parameters is accomplishedvia the EM algorithm. A local decoding procedure is implemented to reveal the timesof a change in the occurrence rate of earthquakes. The evaluated conditional intensityfunction demonstrates an alternation between higher and lower seismic activity levels,which is associated in some cases with major earthquake occurrences.

Keywords: hidden states, EM algorithm, forward-backward equations, seismicity rates.

1. INTRODUCTION

One of the most accepted feature of large earthquakes generation is the as-sumption of time independence (Michael, 2011; Daub et al., 2012; Parsons andGeist, 2012). Strong earthquakes occur according to a stationary Poisson process,meaning that they occur (arrive) according to a random process in time with astable occurrence rate, µ. However, brevity of the earthquake catalogs, as wellthe limited global number of large earthquake occurrences reduce the robust vali-dation of the statistical tests (Dimer de Oliveira, 2012; Lombardi and Marzocchi,2007).

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A possible non-stationarity of the time series of strong earthquakes (Bufe andPerkins, 2005; McCaffrey, 2008) is an interesting assumption. Transient pertur-bation caused by poroelastic relaxation probably increased the background rateduring the 1997-1998 Umbria-Marche, Central Italy (Lombardi et al. 2010). Eventhough global interactions between large earthquakes can not be assumed, thereis a significant increase of seismicity rate after major earthquakes in regional scalefar beyond the rupture zone (Papazachos et al., 2000; Parsons and Velasco, 2011;Stein and Toda, 2013).

In this work, we want to analyze the temporal behavior of strong earthquakesin the broader area of Greece. MAP model is applied in order to examine whetherthere are long-term fluctuations in the seismicity rate. It was introduced by Neuts(1979) and a more explicit definition was given later by Lucantoni et al., (1990).It concerns a two dimensional Markov process, {(Nt, Jt), t ≥ 0}, with the firstmarginal process being a counting one, Nt, in which the associated arrivals cor-respond to earthquake occurrence times, and the second one, Jt, being a Markovprocess with unobservable states that determine the arrival rate. MAPs provide ageneralization of the Poisson process and renewal models and the variable arrivalrate which is assumed, reveals the non-stationarity of the counting process. Theinterarrival times follow a phase type (PH) distribution and are conditionally butnot completely independent (He, 2014). Hence, correlations among intereventtimes can be captured. In addition any stochastic point process is the weak limitof a sequence of MAPs (Asmussen and Koole, 1993). A comprehensive review onMAPs and its applications, along with some important special cases are given inArtalejo et al., (2010) and Cordeiro and Kharoufeh, (2011).

Section 2 describes the dataset used in the current study and preliminaries ofMAP theory, along with the testing methodology. Section 3 presents the resultsof the applied method. Finally, in Section 4 the concluding remarks are drawnand some issues and recommendations are given.

2. DATA AND METHODOLOGY2.1 Data

Our aim is to reveal the temporal behavior of strong earthquake occurrenceswith magnitude M ≥ 6.5 in the broader area of Greece, and to examine whetherthere are seismicity rate changes related to major events. The focal mechanisms oflarge events have been thoroughly investigated since 1950 (Taymaz et al., 1991a,band Baker et al., 1997 among others), and a brief bibliography regarding eventswith M ≥ 6.5, is given in Papazachos and Papazachou (2003) and the referencestherein.

In our study a seismicity catalog of shallow events with depth D ≤ 45 kmand with magnitude M ≥ 6.5 is compiled, according to the most recent publish-ments. The catalog is assumed to be complete and homogeneous (Papazachos

279

and Papazachou, 2003) and includes 118 events, since 1845, within the bound-aries φ = 33o − 43o and λ = 19o − 30o (Figure 1). Paradisopoulou et al., (2010)provided evidence of interactions between strong earthquakes in Western Turkeyand North Anatolian Fault (NAF), due to stress changes in the area. Since ourgoal is to model temporal patterns of possible earthquake interactions, four eventsoutside the boundaries of our study area are included in the catalog. These areevents like the Duzce (Mw = 7.1) on 12 November, 1999 which occurred almostthree months after the earthquake of Izmit (Mw = 7.6) on 17 August, 1999.

Figure 1: Seismotectonic properties of Greece and its surrounding area along with theepicentral distribution of earthquakes with M ≥ 6.5 that occurred in the study area since1845.

The temporal distribution in conjunction with the cumulative number of events(Figure 2), illustrates the existence of active periods like in 1980-1983 which al-ternate with quiescent ones (1983-1995). Moreover, in some cases these changescould be associated with the occurrence of large earthquakes, like the AmorgosMw = 7.5 (Papadimitriou et al., 2005) event on 9 July, 1956 (red pentagon).

2.2 Methodolody

2.2.1 Markovian arrival process

As a first step, the definition of the Markovian arrival process (He, 2010;2014) is given. A counting (observed) process, {Nt}t≥0, determines the number ofarrivals (earthquake occurrences) in a time interval [0, t), driven by an irreducible

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Figure 2: Temporal distribution of events (left axis) and cumulative number of events(right axis) with M ≥ 6.5 in Greece and its surrounding area, since 1845. Red pentagonsare shown characteristic earthquakes of the catalog.

finite Markov process, {Jt}t≥0, where E = {1, . . . ,m},m ∈ N+ stands for the statespace. The evolution of the underlying (unobservable states) process Jt defines theseismic activity at each time t ≥ 0; that is, its coordinates indicate the rate of theearthquake occurrences. The bivariate stochastic process (N,J) := (Nt, Jt)t∈R+

is called Markovian Arrival Process (MAP), and it is a Markov process with aninfinitesimal generator of the structure,

Q =

D0 D1

D0 D1

. . .. . .

.

The elements of the m ×m matrices D0 = {qij(0)}i,j∈E and D1 = {qij(1)}i,j∈Ecompile the parameter space of the model, namely the transition rates from statei to j, i, j ∈ E, without or with an arrival, respectively; their elements are non-negative, where

qij(k) = limh→0P (Nt+h=k,Jt+h=j/Jt=i)

h

for all i, j ∈ E and k = 0, 1, except the diagonal elements of D0, which are definedas

qii(0) = −∑

j 6=i qij(0)−∑m

j=1 qij(1).

The latter is assumed to be non-singular and the transition times are finite withprobability equal to one. We assume a stationary version of the MAP, where theinitial state vector after an arbitrary arrival πarr is computed by the relation,

πTarr = −πTD0(πTD11m)−1,

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where superscript T stands for the transpose of a vector or matrix, 1m stands forthe unit column vector of dimension m and π is the stationary probability vectorof the Markov process Jt.

The model is efficient in capturing changes of the temporal behavior of theearthquake occurrence times, namely revealing the variations of the seismicityrate. In more detail, the MAP allows for transitions between the states, where toeach one of them corresponds an occurrence rate, qi. The rate is the i−th elementof D11m. The tractable structure of the process enables the physical interpreta-tion of the hidden states of the Markov process Jt, assuming a correspondencebetween occurrence rates and seismicity periods.

For a brief description of the process let us assume that the underlying pro-cess is in state i ∈ E, whereas Jt = i earthquakes occurrence follows a Poissonprocess with arrival rate qii(1), i = 1, ...,m. At the end of the sojourn time instate i, which is assumed to follow an exponential distribution, either a tran-sition to another state occurs without designating an arrival, with probabilityqij(0)/(−qii(0)), or a transition to another state designating an arrival, with prob-ability qij(1)/(−qii(0)), for i 6= j and j ∈ E.

In the estimation procedure the EM algorithm is used to obtain the maximumlikelihood estimates of the model parameters. It is an iterative algorithm whichin each step computes the loglikelihood function with the help of the forward-backward equations (McDonald and Zucchini, 1997) and reach a local maximumafter a predefined number of iterations or a converging criterion. The initialestimates of the parameters were chosen to match the mean value of the trace(interevent times). A special structure of the rate matrices is followed (Horvathand Okamura, 2013), where transitions among the hidden states are allowed onlywhen arrivals are taking place, implying that qij(0) = 0, for i 6= j.

2.2.2 Model selection

For the determination of the best number of hidden states visited in the timeinterval under study, we will apply Akaike Information Criterion (AIC) (Akaike,1974)

AIC = −2p+ 2logL, (1)

and the Bayesian Information Criterion (BIC) (Schwarz, 1978)

BIC = −p · logn+ 2logL, (2)

where p is the number of parameters, logL is the loglikelihood function and n thenumber of observations.

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2.2.3 Evaluation measures

First, the goodness-of-fit (gof) to assess the adjustment of the observed in-terevent time series to the proposed models is provided, through the Kolmogorov-Smirnov (K-S) (Chakravarty et al., 1967) and Anderson-Darling (A-D) (Stephens,1974) tests. We denote by τn the interarrival time between the (n− 1)th and thenth earthquake. We recall that each interarrival time, τn, is assumed to follow aPH distribution with representation (πarr,D0). Hence, the hitting time distribu-tion for an earthquake occurrence assuming that at time t = 0 an event occurred,is expressed by

F (x) = 1− πTarreD0x1m, x ≥ 0. (3)

In the sequel, the distance between the empirical distribution and the fittedones is evaluated, via the l2 distance and the Kullback-Leibler (KL) divergence(Kullback and Leibler, 1951). If we consider two continuous probability distri-butions P,Q with the same support S, then the two distance measures can bedefined as

dl2(P,Q) = ‖(P −Q)‖2 =

√∫S

(P −Q)2dτ (4)

dKL(P,Q) =

∫SP · log(

P

Q)dτ (5)

Another useful way to assess the goodness-of-fit of the MAP is through residualanalysis (Ogata, 1988). According to this technique, if we consider the transfor-mation τ = Λ(s) =

∫ s0 λ(t)dt, where λ(t) is the intensity function of a point

process, then the rescaled sample τi is a Poisson process with unit rate. Hence,when the fitted model achieves a good approximation of the data, (only) a smalldeclination from the bisector can be allowed.

2.2.4 Local decoding of the hidden states

The most likely path of the underlying process Jt that have generated thesequence of observations is estimated by the forward-backward equations, usedin the implementation of EM algorithm. According to McDonald and Zucchini(1997) these probabilities can be evaluated by

P (Jtk = i/HT ) =P (T1 = τ1, ..., Tn = τn, Jtk = i)

P (T1 = τ1, ..., Tn = τn)=

a[k]i · b[k + 1]i∑mj=1 a[k]j · b[k + 1]j

(6)

where HT = {τ1, ..., τn} is the complete history of the process, n is the number ofobservations and the forward-backward vectors can be expressed by

283

a[0] = πTarr, a[k] = a[k − 1]eD0τkD1

b[n] = 1m, b[k] = eD0τkD1b[k − 1].

They describe the evolution of the process from the first event up to the occurrencetime tk and from the latter to the last event, respectively. This procedure is oftencalled local decoding, and the estimated sequence of the hidden states is the localdecoding path.

The estimated intensity function can be easily obtained by

λtk =m∑i=1

qi ·a[k]i · b[k + 1]i∑mj=1 a[k]j · b[k + 1]j

, (7)

which denotes the expected occurrence rate at time tk, conditional on the completehistory of the observations, HT .

3. MAIN RESULTS

3.1 Optimal model

We fitted models from two to five states, since we believe that there is no needfor more states in order to capture the temporal fluctuations of the earthquakeoccurrences with M ≥ 6.5. For the implementation of the EM algorithm, BuToolsprogram packages (Bodrog et al., 2014) were used in the MATLAB environment.We computed the AIC and BIC values through equations (1) and (2) and we chosethe models with the lowest values (Table 1.).

Table 1. The log likelihood, AIC and BIC values of the fitted models, wherethe number of states varies from 2 to 5.

# of S LogL AIC BIC

2 -156 325 3413 -155 334 3674 -137 315 3705 -133 326 409

AIC suggests the four state model (AIC=315), however BIC indicates thattwo states (BIC=341) are sufficient for describing the temporal behavior of thedata. We can see that BIC values are highly affected by the data shortage, sinceit takes into account the number of observations. In the sequel, we evaluate theperformance of both models, in order to be more accurate about the choice of theoptimal one.

3.2 Evaluation of the model

The goodness-of-fit of the proposed models to the empirical data is computed,and a comparison among the most popular theoretical distributions (11 in to-tal: Exponential, Extreme value, Gamma, Generalized Pareto, Inverse Gaussian,

284

Logistic, Lognormal, Nakagami (Nakagami, 1960), Normal, Rayleigh, Weibull) isderived. For all the tests a significance level a = 0.05 is assumed. As a firststep, in Table 2. we present the best four approximated distributions to the data(Exponential, Gamma, Weibull and Nakagami) according to the AIC, BIC andthe A-D p-values.

Table 2. A-D test statistics with the corresponding p-values, and the AIC andBIC values for the best four distributions.

DistributionA−D

cv = 2.492 p valueAIC BIC

Exponential 4.04 0.008 326 329Gamma 1.04 0.336 298 304Weibull 1.12 0.299 303 309

Nakagami 1.38 0.207 301 307

Both Weibull and Gamma distributions are well adapted, however the valuesof AIC (303, 298) and BIC (309, 304), indicate a better approximation to data forGamma compared to Weibull. In addition, high p−values (pgam = 0.336) of theformer are observed, strengthening the previous argument.

In the sequel, the hitting time distribution of the MAP with two and fourstates is evaluated according to equation (3), and the K-S gof test is applied. Wefurther computed the l2 and KL measures through equations (4) and (5) andthe results are presented in Table 3. In addition, Gamma distribution was alsoevaluated and compared with the proposed models.

Table 3. K-S test statistics with the corresponding p-values, and the l2 and KLvalues for the MAP with two and four states, as well for the Gamma

distribution.

ModelK − S

cv = 0.1135 p valuel2 KL

MAP (2) 0.0680 0.62 0.0006 0.1175MAP (4) 0.0061 0.75 0.0007 0.1271Gamma 0.0950 0.22 0.0959 0.1524

The gof tests for both models do not allow us to reject the null hypothesis(ksstatmap(2) = 0.0668 and ksstatmap(4) = 1.0996). Furthermore, we assume thatboth MAPs outperform Gamma distribution in terms of the K-S p-values, andaccording to the l2, KL distance measures.

Figure 3 shows the hitting time distribution for earthquake occurrences, un-der the condition that an earthquake occurred at t = 0, for both the empiricaldata and the proposed models. The cumulative distribution function (cdf) ofGamma distribution was also cited, in order to illustrate the outperformance ofthe models. It can be seen that for almost a time period of one year, MAP(2) andMAP(4) adapt better to data compared with Gamma distribution, especially for

285

Figure 3: Cumulative distributions of the data, fitted MAP(2), MAP(4) and Gammadistribution along with the 95% confidence zone.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

time (years)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Cum

ula

tive P

robabili

ty

ECDF

MAP(2)

MAP(4)

Gamma

95% upper conf. int.

95% lower conf. int.

t < 0.7 years. The difference between the cdfs of the two models is not significant,however, for the rest of the parer MAP(2) is considered as the optimal model.The study period is quite short compared to the duration of the physical processof earthquake generation, hence this makes us reluctant over the acceptance of amodel with many states.

For MAP(2) the transformed times τi are calculated, and the cumulative num-ber of the residuals versus the transformed times is plotted (Figure 4). We con-clude that the declination from the bisector y = x is acceptable.

3.3 Intensity function and sequence of hidden states

The decoded path of the underlying process Jt and the estimated intensityfunction are evaluated by equations (6) and (7), respectively and the results areshown in Figure 5. We observe that state 2 (q2 = 2.6170) (Figure 5, top plot)corresponds mainly to large shocks and in some cases to their immediate after-shocks or preshocks, since a declustering procedure has not been applied. This isthe case of the saros-marmara earthquake (Mw = 7.4) that occurred on 9 August,1912 (Abraseys and Finkel, 1987) and the Mw = 6.8 aftershock that followed justone month later. Another characteristic example is the Chirpan and Plovdiv 14and 18 April, 1928 events with magnitudes Mw = 6.5 and Mw = 6.9 (Karakostaset al., 2006), respectively. It is important to notice, that the model recognizes thispatterns and successfully makes a transition to the state of high occurrence rate.However, there are also some longer periods of relative higher seismicity ratelike the 09/07/1956-26/05/1957, which include the Amorgos Mw = 7.5 earth-

286

Figure 4: The residual process for the fitted MAP(2) model.

0 20 40 60 80 100

Transformed Time

0

20

40

60

80

100

Cum

ula

tive N

um

ber

of

Eve

nts

Figure 5: Top plot: The optimal path of the hidden states of the MAP(2) given theobserved sequence together with the magnitude time plot (with cyan color are shown eventsreferenced in the text). Bottom plot: The estimated intensity function of the MAP(2) andthe cumulative number of earthquakes.

1

2

Sta

tes

6.5

7

7.5

Magnitude

1855 1865 1875 1885 1895 1905 1915 1925 1935 1945 1955 1965 1975 1985 1995 2005 20150.5

1

1.5

2

2.5

Inte

nsity function lam

bda(t

)

0

20

40

60

80

100

120

Cum

ula

tive c

ounts

quake on 9 July, 1956, the Fethiye Mw = 7.2 event on 25 April, 1957 and theAbant Mw = 7.0 event on 26 May, 1957 (Abraseys, 1970). The model captures

287

this intense period with great success, and then makes a transition to state 1(q1 = 0.5095), which corresponds to a relative quiescent period with 2 earthquakeoccurrences in more than 7 years.

4. CONCLUSION

A Markovian arrival model was applied for investigating the temporal behaviorof strong earthquakes, M ≥ 6.5, in the broader area of Greece. The underlyingMarkov process Jt captures the seismicity evolution extremely well, and revealsseismicity rate changes associated in many cases with the occurrence of largeevents.

AIC and BIC demonstrated that two and four-state MAPs best capture theseismic activity of earthquakes with M ≥ 6.5 in the study area, however theformer was chosen, since we want to avoid an overparameterization problem. Thegoodness-of-fit tests verified the significance of the proposed model, which wasfurther compared to broadly accepted theoretical distributions. The evaluatedmeasures indicated the outperformance of the model.

The evaluation of the conditional intensity function and the optimal sequenceof hidden states, showed that temporal variations characterize the frequency ofstrong earthquakes, M ≥ 6.5, in the wider area of Greece. MAP with two statesseems adequate in capturing patterns of the real catalog, i.e., periods of seismicquiescence alternate with seismic active ones, hence fitted models with more statesdo not provide additional information.

Further investigation for the existence of spatial associations between theevents should be fulfilled. It is also important to apply a MAP on the declus-tered earthquake catalog, in order to examine the spatio-temporal behavior of themainshocks, in the broader area of Greece. Currently, the temporal variations oflarge earthquakes in a global scale, is under investigation.

ΠΕΡΙΛΗΨΗ

Το μοντέλο Μαρκοβιανών Διαδικασιών Αφίξεων (ΜΔΑ) εφαρμόσθηκε σε πλήρη

σεισμικό κατάλογο, με σεισμούς μεγέθους M ≥ 6.5, στον ευρύτερο ελληνικό χώρο,με σκοπό τη μελέτη των χρονικών μεταβολών της σεισμικότητας και την εύρεση ση-

μείων αλλαγής του ρυθμού σεισμογένεσης. Η ΜΔΑ είναι μία στοχαστική διαδικασία,

που απαριθμεί το πλήθος σεισμών και μεταβαίνει μεταξύ των κρυφών καταστάσεων

μίας Μαρκοβιανής Διαδικασίας, έτσι ώστε κάθε κατάσταση να αντιστοιχεί σε ένα

διαφορετικό επίπεδο ρυθμού σεισμικότητας. Για τον καθορισμό του βέλτιστου αριθ-

μού κρυφών καταστάσεων χρησιμοποιήθηκαν τα κριτήρια πληροφορίας Akaike καιBayes, ενώ η εκτίμηση των παραμέτρων του μοντέλου πραγματοποιήθηκε μέσω τουαλγορίθμου ΕΜ. Η εκτίμηση των χρονικών στιγμών μετάβασης των καταστάσεων της

κρυφής Μαρκοβιανής Διαδικασίας έγινε με τη μπρος-πίσω διαδικασία. Η εκτιμώμενη

288

υπό συνθήκη συνάρτηση έντασης αναδεικνύει εναλλαγές μεταξύ υψηλών και χαμηλών

επιπέδων σεισμικής δραστηριότητας, σε μερικές περιπτώσεις σε συνδυασμό με τη

γένεση κύριων σεισμών.

Acknowledgements: The financial support by the European Union and Greece (Partner-ship Agreement for the Development Framework 2014-2020) under the Regional Oper-ational Programme Ionian Islands 2014-2020, for the project ” Telemachus - InnovativeOperational Seismic Risk Management System in the Region of Ionian Islands ” is grate-fully acknowledged. The software Generic Mapping Tools was used to plot the map ofthe study area (Wessel and Smith, 1998). Geophysics Department Contribution 914.

REFERENCES

Ambraseys, N. N. (1970). Some characteristic features of the Anatolian fault zone.Tectonophysics 9, 143-165.

Ambraseys, N. N., & Finkel, C. F. (1987). The Saros-Marmara earthquake of 9 August1912. Earthquake engineering & structural dynamics 15, 189-211.

Akaike, H. (1974). A new look at the statistical model identification. IEEE transactionson automatic control 19, 716-723.

Artalejo, J. R., & Gomez-Corral, A. (2010). Markovian arrivals in stochastic modelling:a survey and some new results (invited article with discussion: Rafael Perez-Ocon,Miklos Telek and Yiqiang Q. Zhao). SORT-Statistics and Operations ResearchTransactions 34, 101-156.

Asmussen, S., & Koole, G. (1993). Marked point processes as limits of Markovian arrivalstreams. Journal of Applied Probability 30, 365-372.

Baker, C., Hatzfeld, D. L., Lyon-Caen, H., Papadimitriou, E., & Rigo, A. (1997). Earth-quake mechanisms of the Adriatic Sea and Western Greece: implications for theoceanic subduction-continental collision transition. Geophysical Journal Interna-tional 131, 559-594.

Bodrog, L., Buchholz, P., Heindl, A., Horvath, A., Horvath, G., Kolossvary, I., MeszarosA., Nemeth Z., Papp J., Reinecke P., Telek, M. & Vecsei M. (2014). programpackages for computations with PH, ME distributions and MAP, RAP processes,User Manual; 2014.

Bufe, C. G., & Perkins, D. M. (2005). Evidence for a global seismic-moment releasesequence. Bulletin of the Seismological Society of America 95, 833-843.

Chakravarty, I. M., Roy, J. D., & Laha, R. G. (1967). Handbook of methods of appliedstatistics, New York: McGraw Hill

Cordeiro, J. D., & Kharoufeh, J. P. (2011). Batch markovian arrival processes (BMAP).Wiley Encyclopedia of Operations Research and Management Science. Hoboken,N.J.: John Wiley & Sons.

Daub, E. G., Ben Naim, E., Guyer, R. A., & Johnson, P. A. (2012). Are megaquakesclustered?. Geophysical Research Letters 39, L06308.

Dimer De Oliveira, F. (2012). Can we trust earthquake cluster detection tests?. Geo-physical Research Letters 39, L17305.

He, Q. M. (2010). Construction of continuous time markovian arrival processes. Journalof Systems Science and Systems Engineering 19, 351-366.

He, Q. M. (2014). Fundamentals of matrix-analytic methods, New York: Springer.

289

Horvath, G., & Okamura, H. (2013). A fast EM algorithm for fitting marked Marko-vian arrival processes with a new special structure. In European Workshop onPerformance Engineering, Berlin, Heidelberg: Springer.

Karakostas, V., Papadimitriou, E., Gospodinov, D., & Ranguelov, B. (2006). Slip dis-tribution of the 1928 Chirpan and Plovdiv main shocks and earthquake triggering.In VIth International Conference of SGEM, 119-127.

Kullback, S., & Leibler, R. A. (1951). On information and sufficiency. The annals ofmathematical statistics 22, 79-86.

Lombardi, A. M., & Marzocchi, W. (2007). Evidence of clustering and nonstationarityin the time distribution of large worldwide earthquakes. Journal of GeophysicalResearch: Solid Earth 112, B02303.

Lombardi, A. M., Cocco, M., & Marzocchi, W. (2010). On the increase of backgroundseismicity rate during the 1997-1998 Umbria-Marche, Central Italy, sequence: ap-parent variation or fluid-driven triggering?. Bulletin of the Seismological Society ofAmerica 100, 1138-1152.

Lucantoni, D. M., Meier-Hellstern, K. S., & Neuts, M. F. (1990). A single-server queuewith server vacations and a class of non-renewal arrival processes. Advances inApplied Probability 22, 676-705.

McCaffrey, R. (2008). Global frequency of magnitude 9 earthquakes. Geology 36, 263-266.

MacDonald, I. L., & Zucchini, W. (1997). Hidden Markov and other models for discrete-valued time series, London, U.K.: Chapman & Hall.

Michael, A. J. (2011). Random variability explains apparent global clustering of largeearthquakes. Geophysical Research Letters 38, L21301.

Nakagami, M. (1960). The m-Distribution, a general formula of intensity of rapid fading.Statistical Methods in Radio Wave Propagation 47, 3-36.

Neuts, M. F. (1979). A versatile Markovian point process. Journal of Applied Probability16, 764-779.

Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysisfor point processes. Journal of the American Statistical association 83, 9-27.

Papadimitriou, E., Sourlas, G., & Karakostas, V. (2005). Seismicity variations in theSouthern Aegean, Greece, before and after the large (M7. 7) 1956 Amorgos earth-quake due to evolving stress. Pure and applied geophysics 162, 783-804.

Papazachos, B. C., Karakaisis, G. F., Papazachos, C. B., & Scordilis, E. M. (2000).Earthquake triggering in the north and east Aegean plate boundaries due to theAnatolia westward motion. Geophysical research letters 27, 3957-3960.

Paradisopoulou, P. M., Papadimitriou, E. E., Karakostas, V. G., Taymaz, T., Kilias,A., & Yolsal, S. (2010). Seismic hazard evaluation in western Turkey as revealedby stress transfer and time-dependent probability calculations. Pure and appliedgeophysics 167, 1013-1048.

Papazachos, B.C. and Papazachou C.B. (2003). The earthquakes of Greece, Thessaloniki:Ziti Publications.

Parsons, T., & Geist, E. L. (2012). Were global M ≥ 8.3 earthquake time intervalsrandom between 1900 and 2011?. Bulletin of the Seismological Society of America102, 1583-1592.

Parsons, T., & Velasco, A. A. (2011). Absence of remotely triggered large earthquakesbeyond the mainshock region. Nature Geoscience 4, 312-316.

290

Schwarz, G. (1978). Estimating the dimension of a model. The annals of statistics 6,461-464.

Stephens, M. A. (1974). EDF statistics for goodness of fit and some comparisons. Jour-nal of the American statistical Association 69, 730-737.

Stein, R. S., & Toda, S. (2013). Megacity megaquakes?Two near misses. Science 341,850-852.

Taymaz, T., Eyidogan, H., & Jackson, J. (1991a). Source parameters of large earth-quakes in the East Anatolian Fault Zone (Turkey). Geophysical Journal Interna-tional 106, 537-550.

Taymaz, T., Jackson, J., & McKenzie, D. (1991b). Active tectonics of the north andcentral Aegean Sea. Geophysical Journal International 106, 433-490.

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