On the Co-occurrence of Different Magnitude Earthquakes: Southwestern China Case

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1 Gaz Unversty Journal of Scence GU J Sc 29(3): (2016) On the Co-occurrence of Dfferent Magntude Earthquakes: Southwestern Chna Case Tanlu LI 1,, Salh ÇELEBİOĞLU 1, 1 Department of Statstcs, Faculty of Scence, Gaz Unversty, Ankara, Turkey ABSTRACT Receved:29/04/2016 Revsed:02/06/2016 Accepted: 06/06/2016 In ths study, we buld a Markov chan model for the earthquakes n Southwestern Chna by followng the maxmum entropy prncple whle the states of Markov chan are the co-occurrence of earthquakes wth dfferent magntudes. In ths case, an approxmaton s to focus on just the occurrences of the most serous magntude earthquakes and neglect the others. Our approxmaton to ths stuaton s to take all magntude earthquakes nto account f they occur at least once n any gven perod. In order to reveal the feature of ths Markov chan n respect of the frst passage tme dstrbutons, we run a long-term smulaton wth the occurrences of all the 3 categores of earthquakes. Fnally, we gve both the ftted dstrbutons and multnomal approxmatons for the dstrbuton of the frst passage tme for some states. Key words: Dscrete-tme Markov chan, Entropy, Frst passage tme, Multnomal dstrbuton approxmaton, Earthquakes n southwestern Chna 1. INTRODUCTION Consstng of Tbet Autonomous Regon, Schuan Provnce, Chongqng Muncpalty, Yunnan Provnce and Guzhou Provnce, southwestern Chna s the area that most suffered from earthquakes n Chna through 1970 to Whle the Schuan-Yunnan regon n southwest Chna lays on the boundary between Tbetan Plateau and South Chna platform, ths regon becomes an earthquake prone area n Southwestern Chna [1]. Moreover, Tbet Autonomous Regon takes a part of the Alpne-Hmalayan sesmc belt whch makes Southwestern Chna more vulnerable to earthquakes. Nowadays t s wdely accepted that t s mpossble to predct the exact tme, place and magntude of a comng earthquake. However, we can stll use statstcal tools n cooperaton wth the researchers n the feld of sesmology and geology to estmate sesmc actvty n a certan perod. In recent years there are lots of studes focusng on to estmate the probablty of earthquakes occurrence n certan perods. Usng Markov chan model for earthquake analyss becomes popular n earthquake forecastng studes recently. In 20 th century many researchers treated earthquake sequencng as a Markov process to explore the aftershocks [2] [3] [4]. In the begnnng of 21 st century Markov chan modellng started to be used as a tool n several studes n regonal earthquake forecastng [5] [6] [7]. Also there are some attempts to vew the global sesmc actvtes, Vasudevan and Cavers constructed a drected graph to show a Markov chan of global earthquake sequence [8] [9]. Correspondng author, e-mal: tanlu.l@gaz.edu.tr

2 636 GU J Sc, 29(3): (2016) / Tanlu LI, Salh ÇELEBİOĞLU, On the other hand, n order to buld a sutable Markov chan model, a reasonable tme nterval, whch we see t as a sngle perod, s needed to be found. By applyng the maxmum entropy prncple Ünal and Çeleboğlu obtaned a sutable tme nterval calculated for constructng a Markov chan wth a fnte state space [10]. In ths study we are gong to follow the maxmum entropy prncple as well. Another nterestng and meanngful topc that we focus on s the frst passage tme between two dfferent states, or for the same state that s tme of recurrence. Harrs has nvestgated the frst passage tme and recurrence dstrbutons n several specal cases lke random walks and large n-state fnte Markov chans [11]. Harrson and Knottenbelt also explored the characterstcs of passage tme dstrbutons n large n-state Markov chans, and appled ther mplementaton n substantal Markov chans wth over 1 mllon states and sem-markov chans [12]. Gül and Çeleboğlu have obtaned dstrbuton of the frst passage tme relatng to lumped states for an rreducble Markov chan wth fnte states [13]. And n ths study we fnd the approxmate dstrbutons of frst passage tmes and recurrence tmes of an rreducble Markov chan wth fnte states. In the ntroducton secton our research problem has been put up. Our data source and methodology are gven n Secton 2. In Secton 3 our modellng procedure of Markov chan applcaton on earthquakes occurrng n southwestern Chna s shown and meanwhle the obtaned results also take a part n Secton 3. Conclusons and further dscussons are gven n Secton DATA AND METHODOLOGY 2.1. Data Source In ths study our data source s obtaned from Chna Earthquakes Networks Center (CENC). Accordng to the data source durng Jan 1, 1970 and Dec 31, 2015, there were a total of earthquake occurrences n southwestern Chna. Consderng about the magntude of those earthquakes occurred n SE Chna, we can fgure out that there are 8674 out of tmes that the magntudes are no more than 4, and 2024 tmes that the magntudes of earthquakes, whle 4M 5 M 5 earthquakes whch could make a great damage to human bengs occurred 309 tmes durng that perod. Inspred by the dea of Ünal and Çeleboğlu, we classfed those earthquakes nto 3 dfferent types by ther magntudes as we mentoned before [10]. The applcaton of ths classfcaton s gven n Secton 3, and t s an mportant step of buldng our model Methodology Markov Chan In probablty theory, a stochastc process s a collecton of random varables consderng of the tme. The tme wll ether be a subset of natural numbers or a subset of [0, ), whch wll be nonnegatve real numbers [14]. Named after the Russan mathematcan Andrey A. Markov, Markov process s a stochastc process that satsfes the Markov property, whch can be presented as P follows: Let (,, ) be a probablty space wth a fltraton ( t,t T), for some (totally ordered) ndex set T ; and let ( S, ) be a measure space. Then an S- valued stochastc process X ( X, t T) adapted to the fltraton s sad to possess the Markov property wth respect to the { }, f for each A and each s, t T wth, P( X A ) P( X A X ). And for t t s t s t s t dscrete-tme case, S s a dscrete set wth the dscrete sgma algebra, T s the set of natural numbers and then we have: P( X j X, X x,..., X x ) P( X j X ). n n1 n2 n2 0 0 n n1 From ths we can easly see that the probablty of beng at the state j at n step, so here we can descrbe Markov chan wth the word "memoryless" [15]. th step only depends on the state at ( n 1) p Pr( X j X ),, j S, n N s n-step transton probablty from state to state ( n) Defnton ,0 n 0 j, and the process begns at t=0. When n=1, t becomes sngle-step probablty from state to state j. And when the transton probabltes are ndependent of the tme parameter n, whch means, j S, Pr( X j X ) Pr( X j X )... Pr( X j X ), n n n1 Any Markov chan whch satsfes ths property s called tme-homogeneous, or statonary Markov chan. The transton matrx P s the N N matrx whose elements are state to state j, and t s a stochastc matrx,.e.,, j S,0 p 1, p whch s the one-step transton probablty from S p. js and, 1 st

3 GU J Sc, 29(3): (2016) / Tanlu LI, Salh ÇELEBİOĞLU 637 Theorem For a Markov chan X, P{ X x,..., X x, X x X x } p p... p. 1 1 m1 m1 m m 0 0 x0x1 x1 x2 xm 1xm m, we have From Theorem we can obtan the followng corollary. Corollary For a Markov chan, we denote () 0 as the probablty of the ntal state, P{ X x, X x,..., X x, X x } ( x ) p p... p m1 m1 m m 0 0 x0x1 x1 x2 xm1 xm S. Hence we have Defnton For any two states and j, we denote ( n ) f, n to be the frst passage tme probablty that the chan spends n steps for passng to state j from state for the frst tme. From ths defnton we can easly get that ( n) f ( n 1) pb fbj n bs { j} Defnton p f f n1 ( n) n1 2,3,4,... s called the ever reachng probablty, whch s the probablty of reachng state j from state n fnte steps. Defnton A state j s sad to be accessble from ( n) a state (wrtten j), f n, p 0 [16]. Defnton A state s sad to communcate wth state j (wrtten j), f both j and j [16]. Defnton A set of states C s a communcatng class, f, j C, j [16]. Defnton A Markov chan s sad to be rreducble, f ts state space s a sngle communcatng class [16]. It s obvous that f t s possble to reach to any state from any state, that Markov chan s an rreducble Markov chan. Defnton A state s sad to be transent, f f 1, S. Defnton A state s sad to be recurrent, f t s not a transent state. Recurrent states whch have an nfnte mean recurrence tme are null recurrent. Recurrent states whch have a fnte mean recurrence tme are postve recurrent. Defnton A state has perod k f any return to state must occur n multples of k tme steps,.e., k gcd{ n 0: Pr( Xn X0 ) 0}. If k 1, then the state s sad to be aperodc, whch means the returns to state can occur at rregular tmes. When k2, k, the state s sad to be perodc wth perod k. A Markov chan s sad to be aperodc f every state s aperodc,.e., k 1. Theorem If a Markov chan s rreducble aperodc wth fnte states, and P s the transton matrx of the Markov chan, then the system of equatons ' P ' 1 S has a unque postve soluton. Ths soluton s called the lmt dstrbuton of Markov chan. ( ) Then we have lm p n j and n lm p 1 1 ( n) n ( n) nf n0 the mean return tme of state Entropy, where Entropy s a concept frst used n Physcs to descrbe the system complexty. In nformaton theory, entropy s the average nformaton contaned n each message. In 1948 Shannon brought a study wth the dea that entropy plays an mportant role n measurng the nformaton choce and uncertanty, and the source wth the maxmum entropy subject to the statstcal condtons s preferred to be retaned [17]. In 1956, Jaynes s study showed that the maxmum-entropy estmate s the least based estmate possble on the gven nformaton [18]. For the dscrete random varables entropy s defned as follows: Defnton Let X be a random varable havng the values 1 2 { x, x,... x } and p, 1,2,... n represents the probablty of n X x s respectvely. The entropy of X s defned as n H ( X ) c p log p, where c s an arbtrary 1 postve constant, and s taken as c 1 when the logarthm base s 2. In addton, log0 s regarded as 0 when we calculate. Moreover entropy has an applcaton n Markov chans [19].

4 638 GU J Sc, 29(3): (2016) / Tanlu LI, Salh ÇELEBİOĞLU, Defnton Suppose Markov chan s rreducble and aperodc and recurrent, and has a statonary dstrbuton. Let S be the state space of the Markov chan X wth the transton matrx P [ p ], and N N X, ts statonary dstrbuton. Then the entropy of Markov chan s gven by: H( X ) p log p., js 2 3. APPLICATION TO EARTHQUAKE DATA 3.1. Am and Content of the Applcaton In ths study, t s amed to estmate the sesmc rsk n southwestern Chna. In order to accomplsh ths task we make use of the applcaton of Markov chan model and entropy theory Applcaton Procedure As mentoned before, there were a total of earthquake occurrences n southwestern Chna n the years We classfed those earthquakes by ther magntudes M nto 3 classes: class I, ; and class III, M class II, 4 5 M 5. M 4 For the purpose of buldng a Markov chan, t should be decded the state space of the Markov chan. Here we use the number 1 to show that there s at least once the certan class earthquake occurrence n the gven perod, otherwse we record the earthquake stuaton as 0 n that perod. For example, f the class III earthquake ddn t occur n a certan perod, then we record the class III earthquake occurrence wth the number 0. Snce we categorzed earthquakes nto 3 classes, we have a seres of bnary numbers, whch s from 000 to 111, to express all the states n the gven perod. For example, state 101 means the class III and class I earthquake occurred at least once, and class II dd not occurred even once n that perod. Therefore state space of the Markov Chan s S {0,1,2,...7}, as n Table 1. ; Table 1. Lst of all states. State Class III Class II Class I And now our work becomes to fnd a sutable tme nterval t for obtanng realzaton steps of Markov chan. Fndng a sutable follows these 3 prncples: 1) should not be so small that there are too many transtons from state 0 to t t t state 0; 2) should not be too large that there are over transtons from state 7 to state 7; 3) should be wth a bg entropy of the Markov chan. At the same tme we also need to keep n mnd that as ncreases, the number of t t

5 GU J Sc, 29(3): (2016) / Tanlu LI, Salh ÇELEBİOĞLU 639 sampled transton gets dmnshng, whch makes the result less robust [6]. Fgure 1. Plot of entropy versus t. t 3 By those 3 prncples, we fnally found a days that fts to our Markov chan model, and accordng to ths tme nterval there are a total of 5600 perods through Wth ths we obtaned the matrx of transton frequences N, and the estmated transton matrx P of Markov chan, respectvely, as follows: N ; P Markov Chan Analyss For the elements of transton probablty matrx we can gve some comments whch reveal the co-occurrence of earthquakes. For example, for the state 5 we have p 55 0, whch means that an occurrence of state 5 s almost not followed by tself n the next perod, but the transton probablty t p ponts out that state 1 s the most probable among others after state 5. In respect of cases, the foregong comment s dentcal to say: cases III and I together s not followed by tself, but case I s the most probable case, whch s consstent wth the fact that the sesmcty s reduced after hgh magntude earthquakes. After obtanng the transton matrx, t s necessary to verfy whether ths process s a Markov chan. Besdes ths, we stll need to move on the frst passage tme dstrbuton problem.

6 640 GU J Sc, 29(3): (2016) / Tanlu LI, Salh ÇELEBİOĞLU, Ch-Square Test For the purpose of testng whether our observed data can be accepted to be a Markov chan, here we use a ch-square test wth the hypothess: H0: Our observed data can be accepted as a Markov chan wth transton matrx P; H1: Our observed data cannot be accepted as a Markov chan wth transton matrx P Here we have the expected transton frequency matrx: The transton frequency matrx of our observed data has been shown n Secton 3.2. And now we have got a ch-square 2 2 test result cal , , and that means H0 s accepted Dstrbuton of Frst Passage Tme and Recurrence Tme We obtaned the lmtng dstrbuton of our Markov chan as follows: For states 4, 5, 6 and 7, ther probabltes n lmtng dstrbuton are less than whch can be regarded as rare states. And the numbers of recurrences for states 4, 5, 6 and 7 durng 5,600 perods are 66, 45, 49 and 76, respectvely. So n order to fgure out the features of ths Markov chan, we run a 1,000,000-perod smulaton. By the result of smulaton, the number of recurrences for states 4, 5, 6 and 7 durng 1,000,000 perods are 11968, 8246, 9091 and tmes, respectvely. Now we focus on state 4, whch there only has occurrences of class III earthquakes. We have 2 ways to fnd a ft dstrbuton of recurrence tme for state 4:1) Usng software lke EasyFt to ft a dstrbuton to our data; 2) Fgurng out a dstrbuton wth ts orgnal defnton and test t through hypotheses. For fttng dstrbuton by usng software, the result s shown as follows: The recurrence tme dstrbuton of state 4 for smulated data s sad to be a Dagum dstrbuton wth parameters p , , Ths result s accepted by Kolmogorov-Smrnov statstc value whch s whle the p-value s The expectaton of ths Dagum dstrbuton s 86.4, whch means for approxmately every days a recurrence for state 4 s expected n southwestern Chna.

7 GU J Sc, 29(3): (2016) / Tanlu LI, Salh ÇELEBİOĞLU 641 Fgure 2. The fttng curve of recurrence tme for state 4. From Table 2 n whch expected frequences of recurrence tme of state 4 for smulated data s presented, we should notce that the expected frequency s no less than 5 untl recurrence tme s over 286 perods. Furthermore, expected frequency s a decreasng seres as recurrence tme ncreases. Thus t s reasonable for us to combne those recurrence tmes whose expected frequences are less than 5 nto a sngle stuaton. By dong ths we are supposng to gve an approxmaton to the dstrbuton of recurrence tme for state 4. We come up wth a multnomal dstrbuton wth parameters as n1 (1) (2) ( n1) f f f f k1,,...,1 ( k ), and n=286 for state 4 case. Here we use a ch-square test to verfy the valdty of ths approxmaton to the dstrbuton of Table 2. Expected frequency of recurrence tme for state 4. recurrence tme for state 4, and the hypotheses are: H0: Approxmaton to the dstrbuton of recurrence tme of state 4 can be accepted as a multnomal dstrbuton 285 (k) (1) (2) (285) f44 f44 f44 f44 k 1 M(,,...,1 ) ; H1: Approxmaton to the dstrbuton of recurrence tme of state 4 cannot be accepted as a multnomal dstrbuton 285 (k) (1) (2) (285) f44 f44 f44 f44 k 1 M(,,...,1 ) the ch-square statstc values are 2 2 cal 284,0.05 accepted.. Snce , H0 s Recurrence Tme Expected Frequency Recurrence Tme Expected Frequency

8 642 GU J Sc, 29(3): (2016) / Tanlu LI, Salh ÇELEBİOĞLU, Although the sample space of recurrence tme for state 4 n the long term s the set of postve ntegers, we stll gve a ftted approxmaton for state 4. From ths approxmaton we can see that the probablty of sngle step recurrence for state 4 s f, whle (1) % the probablty of recurrence tme beng more than one perod s 92.65%. Moreover accordng to the lmtng dstrbuton, the expectaton of the recurrence tme for state 4 s 82.3 perods, whch means for approxmately every days a recurrence for state 4 s expected n southwestern Chna, meanwhle for the ftted multnomal dstrbuton the expectaton s 79.1 perods, whch s about days. Then we contnue to focus on the frst passage tme from state 4 to state 7, whch s the occurrence of class III earthquake only and the co-occurrence of all three category earthquakes. From 1,000,000-perod smulaton, we can see that there are a total of 5908 frst passage tmes from state 4 to state 7. In Table 3 t shows expected frequency of frst passage tme from state 4 to state 7 n 5908-tme frst passage case. Table 3. Expected frequency of frst passage tme from state 4 to state 7. Frst Passage Tme Expected Frequency Frst Passage Tme Expected Frequency Fgure 3. The fttng curve of frst passage tme from state 4 to state 7 As shown n Fgure 3, we found a 3-parameter Webull dstrbuton ftted for the dstrbuton of frst passage tme from state 4 to state 7. Ths Webull dstrbuton wth parameters , , 1 s accepted by

9 GU J Sc, 29(3): (2016) / Tanlu LI, Salh ÇELEBİOĞLU 643 Kolmogorov-Smrnov statstc whch s whle the p-value s The expectaton of ths 3- parameter Webull dstrbuton s 78.0 perods, n other words, for nearly every 234 days a frst passage from state 4 to state 7 s expected. Meanwhle we also gve a multnomal approxmaton to the dstrbuton of frst passage tme from state 4 to state 7. Ths tme we should notce that probablty of sngle-step transton from state 4 to state 7 s 0, thus our multnomal approxmaton s a dstrbuton wth parameters 211 ( k ) (2) (3) (211) f k 2 f f f,,..., for x=2,3, 212. Ths tme we also use a ch-square test to verfy the valdty of ths approxmaton to the dstrbuton of frst passage tme from state 4 to state 7, and the hypotheses are: H0: The approxmaton to the dstrbuton of frst passage tme from state 4 to state 7 can be accepted as a multnomal dstrbuton 211 ( k ) (2) (3) (211) f k 2 f f f M(,,...,1 ) , ; H1: The approxmaton to the dstrbuton of frst passage tme from state 4 to state 7 cannot be accepted as a multnomal dstrbuton 211 ( k ) (2) (3) (211) f k 2 f f f M(,,...,1 ) ch-square statstc values are 2 2 cal 209,0.05 whch means H0 s accepted.. The , From ths multnomal dstrbuton, t can be seen that the 5 th perod has the most possble probablty of frst passage tme from state 4 to state 7, and n ths case the probablty ncreases before the 5 th perod and decreases after the 5 th perod. Furthermore, 74.8 perods s the expectaton of the multnomal approxmaton for ths case, whch means for about every days a frst passage from state 4 to state 7 s expected. 4. CONCLUSION AND DISCUSSIONS In ths study we have bult a Markov chan model based on the earthquake data of southwestern Chna. It s verfed that the observed data follows a Markov chan. Moreover n order to fgure out features of ths Markov chan we have run a 1,000,000-perod smulaton. And smulated data are used to fnd a reasonable dstrbuton of recurrence tme and frst passage tme. By usng EasyFt software we fnd a ftted dstrbuton of recurrence tme for state 4, whch s a Dagum dstrbuton wth parameters p , , and we gve a multnomal approxmaton 285 (k) (1) (2) (285) f k 1 f f f M(,,...,1 ) to the dstrbuton of recurrence tme for state 4. Furthermore, we also explored the dstrbuton of frst passage tme from state 4 to state 7. A 3-parameter Webull dstrbuton wth parameters , , 1 s ftted by EasyFt software, and a multnomal dstrbuton 211 ( k ) (2) (3) (211) f k 2 f f f M(,,...,1 ) s gven as a multnomal approxmaton to the dstrbuton of frst passage tme from state 4 to state 7. The dfference found between the results ftted by software and gven by approxmaton s needed to be dscussed n a future study. Moreover some nterestng facts are needed to be dscussed such as the sngle-step transton probablty from state 4 to state 7 s 0 and the 5 th perod holds the largest probablty of frst passage tme from state 4 to state 7, from the sesmcty perspectve. In ths study we have nvestgated the dstrbutons of the recurrence tme of a rare state 4, the frst passage tme from a rare state 4 to a rare state 7. The recurrence tme of non-rare states, frst passage tme from a nonrare state to a rare state and the frst passage tme from a rare state to a non-rare state n ths southwestern Chna case wll be dscussed n our further study. CONFLICT OF INTEREST No conflct of nterest was declared by the authors. REFERENCES [1] Huang, J., Zhao, D. and Zheng, S., Lthospherc structure and ts relatonshp to sesmc and volcanc actvty n southwest Chna, Journal of Geophyscal Research: Sold Earth, 107.B10, (2002). [2] Vere-Jones, D. A Markov model for aftershock occurence, Pure and Appled Geophyscs, 64.1: pp , (1966). [3] Hagwara, Y., A stochastc model of earthquake occurrence and the accompanyng horzontal land deformaton, Tectonophyscs, 26.1: pp , (1975). [4] Ogata, Y., Statstcal models for earthquake occurrences and resdual analyss for pont processes, Journal of the Amercan Statstcal Assocaton, : pp. 9-27, (1988). [5] Tsapanos, T.M. and Papadopoulou, A.A., A dscrete Markov model for earthquake occurrences n Southern Alaska and Aleutan Islands, Journal of the Balkan Geophyscal Socety, 2.3: pp , (1999).

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