Data Field-Based K-Means Clustering for Spatio-Temporal Seismicity Analysis and Hazard Assessment

emo e sensing A icle Da a Field-Based K-Means Clus e ing o Spa io-Tempo al Seismici y Analysis and Haza d Assessmen Xueyi Shang 1, Xibing Li 1,*, An onio Mo ales-Es eban 2, Gualbe o Asencio-Co és3ID and Zewei Wang 4 1School o Resou ces and Sa e y Enginee ing, Cen al Sou h Uni e si y, Changsha 410083, China; [email p o ec ed] 2 Depa men o Building S uc u es and Geo echnical Enginee ing, Uni e si y o Se ille, 41004 Se illa, Spain; [email p o ec ed] 3Depa men o Compu e Science, Pablo de Ola ide Uni e si y o Se ille, 41013 Se illa, Spain; [email p o ec ed] 4School o Ea hquake Sciences and Enginee ing, Sysu, Sun Ya -Sen Uni e si y, Guangzhou 510275, China; wangzw[email p o ec ed] *Co espondence: [email p o ec ed]; Tel.: +61-041-129-4835 Recei ed: 22 Janua y 2018; Accep ed: 14 Ma ch 2018; Published: 15 Ma ch 2018 Abs ac : Mic oseismic sensing aking ad an age o senso s can emo ely moni o seismic ac i i ies and e alua e seismic haza d. Compa ed wi h expe s’ seismic e en clus e s, clus e ing algo i hms a e mo e objec i e, and hey can handle many seismic e en s. Many me hods ha e been p oposed o seismic e en clus e ing and he K-means clus e ing echnique has become he mos amous one. Howe e , K-means can be a ec ed by noise e en s (la ge loca ion e o e en s) and ini ial clus e cen e s. In his pape , a da a ield-based K-means clus e ing me hodology is p oposed o seismici y analysis. The applica ion o syn he ic da a and eal seismic da a ha e shown i s e ec i eness in emo ing noise e en s as well as inding good ini ial clus e cen e s. Fu he mo e, we in oduced he ime pa ame e in o he K-means clus e ing p ocess and applied i o seismic e en s ob ained om he Chinese Yongshaba mine. The esul s show ha he ime-e en loca ion dis ance and da a ield-based K-means clus e ing can di ide seismic e en s by bo h space and ime, which p o ides a new insigh o seismici y analysis compa ed wi h e en loca ion dis ance and da a ield-based K-means clus e ing. The K zanowski-Lai (KL) index ob ains a maximum alue when he numbe o clus e s is i e: he ene gy index (EI) shows ha clus e s C1, C3 and C5 ha e e y c i ical pe iods. In conclusion, he ime-e en loca ion dis ance, and he da a ield-based K-means clus e ing can p o ide an e ec i e me hodology o seismici y analysis and haza d assessmen . In addi ion, u he s udy can be done by conside ing ime-e en loca ion-magni ude dis ances. Keywo ds: seismici y analysis; haza d assessmen ; spa io- empo al analysis; da a ield; K-means clus e ; ime-e en loca ion dis ance 1. In oduc ion Seismic e en clus e ing p o ides an e ec i e way o unde s and he unde lying in o ma ion (such as geological s uc u e in e p e a ion and seismic haza d assessmen ; a de ailed use o seismic e en clus e s is shown in Sec ion 2) o mic oseismic e en s [ 1 , 2 ]. This ask can be conduc ed by expe s, ollowing a wo-s ep clus e ing p ocess: selec ing he ime pe iod conce ned and di iding seismic e en s in o clus e s isually by e en loca ions. Howe e , expe s’ clus e s a e subjec i e, non-quan i a i e and ha e ouble handling a la ge numbe o seismic e en s [ 3 ]. Mo i a ed by his, a ious clus e ing algo i hms ha e been p oposed o seismici y analysis (e.g., K-means clus e , Remo e Sens. 2018,10, 461; doi:10.3390/ s10030461 www.mdpi.com/jou nal/ emo esensing Remo e Sens. 2018,10, 461 2 o 22 hie a chical clus e ing, and sel -o ganizing map (SOM)), among which K-means clus e ing echnique is he mos well-known one. Ye , K-means canno emo e noise e en s (which ha e ela i ely la ge e en loca ion e o s) and i uses andom ini ial clus e cen e s (which ha e a la ge in luence on clus e esul s). In addi ion, no K-means clus e ing-based me hodologies p e iously applied o his p oblem ha e conside ed ime pa ame e s. In ac , seismic e en s a e no only ela ed o he e en loca ion bu also o he e en ime [1]. In his wo k, a da a ield-based K-means clus e ing me hodology has been p oposed o spa io- empo al seismici y analysis. This me hod uses a da a ield-based h eshold alue o emo e noises and a new dis ance-based me hod o selec good ini ial clus e cen e s o K-means. Fu he mo e, we in oduced he ime-e en loca ion dis ance in o he K-means clus e ing p ocess, which p o ides a spa io- empo al insigh in o he seismici y analysis. The eal seismic da a applica ion shows i s e ec i eness in seismic haza d assessmen . In addi ion, we compa ed i s pe o mance wi h classical K-means clus e , hie a chical clus e ing and SOM clus e ing. The es o his pape is o ganized as ollows. In Sec ion 2, ela ed wo ks abou seismic e en clus e ing a e in oduced. Then, in Sec ion 3, he da a ield heo y is b ie ly illus a ed, and he da a ield-based algo i hm is p oposed o be e selec K-means ini ial clus e cen e s. In Sec ion 4, he p oposed da a ield-based K-means clus e ing was applied o 1210 mic oseismic e en s ob ained om he Chinese Yongshaba mine and he ene gy index (EI) is selec ed o assess seismic haza d o he clus e ing zones. The applica ion de ail e ec and compa isons wi h classical K-means clus e , hie a chical clus e ing and SOM clus e ing a e discussed in Sec ion 5. Finally, b ie conclusions o he wo k and u u e wo k a e shown in Sec ion 6. 2. Rela ed Wo ks Seismic e en clus e ing using clus e ing algo i hms has been s udied since he 1990s, and he K-means clus e ing algo i hm has become he mos amous one. K-means is a ha d-pa i ioning algo i hm p oposed by Ha igan and Wong [ 4 ]. I s a s wi h K andomly selec ed ini ial clus e cen e s and uses an i e a i e p ocess un il no da a poin changes he clus e assigna ions. Bu on e al. [ 5 ] and Wea he ill and Bu on [ 6 , 7 ] p oposed a line-sou ce K-means clus e ing echnique o be e iden i y ini ial clus e cen e s and hey applied i o cap u e he seismic spa ial a ia ion, in e p e he aul ype and analyze he p obabilis ic seismic haza d in he Ja a island and he Aegean egion. Ramdani e al. [ 8 ] ook ad an age o he inal K-Means clus e cen e s o ind subduc ion e idence benea h he Gib al a A c and he Andean egions. Rehman e al. [ 9 ] u ilizes he K-means algo i hm o iden i y he ea hquake spa ial di e ences and es ima e seismic haza d and isk in Pakis an. Mo ales-Es eban e al. [ 10 ] p oposed an e icien adap i e Mahalanobis-based K-means algo i hm and his has been applied o s udy he seismic ca alogues o C oa ia and he Ibe ian Peninsula. Shang e al. [ 11 ] p oposed a K zanowski–Lai and Silhoue e combined index o selec he op imum numbe o clus e s o K-means and hey in e p e ed he geological s uc u e in he Chinese Yongshaba mine. The K-means clus e is use ul o a la ge da ase clus e , and some s udies ha e been done o educe he e ec o ini ial clus e cen e s. Ne e heless, he abo e-men ioned me hods ha e no conside ed he ime pa ame e and he e ec o noise e en s. Hie a chical clus e ing is also widely used in seismici y pa i ioning. I is based on he co e idea ha an e en is mo e ela ed o a nea one han o a a e en , and i connec s e en s in o clus e s based on a p ese dis ance. Hie a chical clus e ing does no need o se a clus e numbe (jus selec a p ese dis ance) and i can clus e e en s in o di e en shapes. Wa dlaw e al. [ 12 ], F ohlich and Da is [ 13 ], Da is and F ohlich [ 14 ], Hudyma and Po in [ 15 ], and Hashemi and Mehdizadeh [ 16 ] used hie a chical clus e ing (single-link analysis) o s udy ea hquake ca alogues and seismic ac i i ies. No wi hs anding, hie a chical clus e ing has a endency o o m seismic e en s in o linea g oups [ 13 ] and he e may be many clus e s ha jus ha e ew e en s (see he discussion). Ano he commonly used seismic e en clus e me hod is he SOM [ 17 ]. The SOM con e s high dimension inpu space in o a low-dimension ( ypically wo-dimensional), which has a e y good isualizing image. Zamani and Hashemi [ 3 ] compa ed hie a chical clus e ing (Wa d’s me hod) and Remo e Sens. 2018,10, 461 3 o 22 SOM clus e ing wi h an applica ion o seismic zoning in I an. Mo eo e , Zamani e al. [ 18 ] applied he SOM o clus e ing ec onic zoning in I an. The same me hod was es ed in Zamani e al. [ 19 ], in which Wilk’s Lambda c i e ion was used o selec an op imum numbe o clus e s. Moja ab e al. [ 20 ] discussed he e ec o SOM inpu pa ame e s and p oposed a combined model o selec a clus e numbe o I an seismici y analysis. Besheli e al. [ 21 ] pe o med a K-means clus e ing along wi h a SOM on an I anian o eshock da abase and hey ound a o eshock zone which has a e y close ela ion wi h la ge ea hquakes. None heless, esea ch esul s [ 11 , 22 ] ha e shown ha he SOM clus e ing may only be alid o high seismic ac i i y a eas, while o low seismic ac i i y zones i may ha e bad clus e esul s. Fu he mo e, he SOM clus e ing may ha e discon inuous zones, which makes i ha d o in e p e clus e esul s [11,22]. Some o he clus e me hods a e p oposed o seismici y analysis. Ansa i e al. [ 23 ], Beni ez e al. [ 24 ] and Monem and Hashemy [ 25 ] used uzzy clus e s, including Ga h-Ge a clus e ing, p oposed Ga h-Ge a clus e ing, uzzy c-means clus e ing and Gus a son-Kessel (GK) clus e ing, o s udy seismic spa ial pa e n ecogni ion in I an, sou h-wes Colombia and he Ghaz in canal i iga ion ne wo k. The uzzy clus e s can be used o a la ge da ase clus e ; howe e , i s clus e esul is sensi i e o he ini ial clus e cen e s. Mukhopadhyay e al. [ 26 ] used a poin densi y clus e ing o gain insigh in o subduc ion kinema ics and seismic po en iali y. Geo goulas e al. [ 1 ] modi ied a densi y-based clus e ing (DBC) algo i hm and hen used a hie a chical agglome a i e scheme o seismic clus e ing. Nanda and Panda [ 27 ] p oposed wo DBC algo i hms and compa ed hem wi h he DBSCAN (densi y-based spa ial clus e ing o applica ions wi h noise). DBC can emo e noise e en s, ye when he densi y a ies much, i usually ob ains bad clus e esul s. Ma ínez-Ál a ez e al. [ 22 ] used a T iGen-based clus e ing algo i hm [ 28 , 29 ] o seismic zoning in he Ibe ian Peninsula. A compa ison o using di e en seismic clus e s can be ound in [11,30,31]. Only a ew o he abo e in es iga ions ha e conside ed he ime ac o . Ne e heless, he ime pa ame e can play an impo an ole in seismic e en clus e s. The commonly used clus e ing echniques conside ing a ime pa ame e a e he ime-e en loca ion and ime-e en loca ion-magni ude dis ance-based clus e s. F ohlich and Da is [ 13 ] and Da is and F ohlich [ 14 ] sugges ed equalizing he ime and space in e als using a scaling cons an and in oduced he ime-e en loca ion dis ance by lij =qdij2+ (a ij)2 , whe e d ij and ij a e he e en loca ion dis ance and ime in e al espec i ely. Then, hey used he hie a chical clus e o seismic e en clus e s. Baiesi and Paczuski [ 32 ] de ined he ime–e en loca ion-magni ude dis ance be ween e en iand jby nij =C ijlijd 10 −bmi , whe e d is he ac al dimension o epicen e s, bis he Gu enbe g–Rich e b- alue and mi is he e en magni ude. A single link me hod was used in his wo k o he seismic e en clus e . Zaliapin e al. [ 33 ] and Zaliapin and Ben-Zion [ 34 – 38 ] de ined he nij by magni ude-no malized ime componen Tij and space componen Rij ( Tij = ij 10 −qbmi and Rij =lijd 10 −pbmi , whe e q+p= 1). E en jis clus e ed o he main e en ii i sa is ies he ollowing condi ions: Tij <T0 , Rij <R0 and mj<mi . I is clea ha no ime pa ame e has been applied in K-means clus e ing. In his wo k, we jus in oduced he ime-e en loca ion dis ance in o a K-means clus e ing algo i hm. Fu he s udy can be done by using a ime-e en loca ion-magni ude dis ance. 3. P elimina y S udies and he P oposed Me hod In i s place, o p o e he limi a ions o he K-means echnique, a b ie s udy wi h syn he ic da a was pe o med in Sec ion 3.1. Nex , he da a ield heo y is in oduced in Sec ion 3.2. Then, he da a ield-based K-means clus e ing p ocedu e is p oposed in Sec ion 3.3. Finally, an applica ion es o da a ield-based K-means clus e ing is shown in Sec ion 3.4. Remo e Sens. 2018,10, 461 4 o 22 3.1. K-Means Clus e ing P elimina y S udy The classical K-means algo i hm is based on he ollowing s eps: ake a da ase o poin s ( x1 , x2 , . . . , xn ), andomly selec Kpoin s om he da a se as he ini ial clus e cen e s and alloca e each poin xi o he nea es cen e poin . Then, i calcula es he mean o each pa ame e espec i ely o each g oup o make up a se o new upda ed clus e cen e s. This p ocedu e is epea ed un il no poin s change hei clus e o he numbe o i e a ions eaches a p ese maximum. The mos widely used dis ance is he Euclidean unc ion and his was chosen o his wo k. The Euclidean dis ance be ween e en xi and xj is de ined in Equa ion (1). Then, he mean o he widely used clus e ing c i e ion-sum o Euclidean dis ance (SED) is used o e alua e he clus e pe o mance. We called his he MSED index. (The numbe o undenoised and denoised e en s is di e en , so we used he MSED ins ead o he SED o e alua e he clus e ing pe o mance). MSED is de ined in Equa ion (2), he smalle he MSED index o clus e s he be e he clus e ing esul s. dij = u u p ∑ l=1 (xi,l−xj,l)2(1) MSED =1 n K ∑ k=1 ∑ xi∈Ck u u p ∑ l=1 (xi,l−mk,l)2(2) whe e dij is he dis ance be ween e en xi and xj ,pis he dimension o he pa ame e s, C k is he k h clus e , Kis he clus e numbe , mkis he clus e cen e o Ck, and nis he numbe o clus e e en s. F om he p inciple o K-means clus e ing, we know ha he algo i hm may be a ec ed by he ini ial clus e cen e s. To p o e his issue, a syn he ic wo-dimension da ase wi h ou well-di e enced clus e s is used o es he e ec o K-means ini ial clus e cen e s (Figu e 1a). The algo i hm was es ed 15 imes by using andomly gene a ed ini ial clus e cen e s (using K= 4 o all he expe imen s) and hei MSED indexes a e shown in Figu e 1b. I is clea ha he K-means clus e ing can be hea ily a ec ed by he ini ial clus e cen e s: he MSED index a ies om 0.100 o 0.192. The K-means clus e ing esul s o six ypical MSED alues a e shown in Figu e 1c–h. I is clea ly seen ha o high MSED alues, qui e a lo o close e en s a e in di e en clus e s (Figu e 1c,h); o ela i ely high MSED alues, many close e en s a e in di e en clus e s (Figu e 1d,g); while o low MSED alues, he K-means clus e has good esul s (Figu e 1e, ). In conclusion, he lowe he MSED index, he be e he clus e ing esul . The e o e, he e is a high need o op imize he K-means ini ial clus e cen e s. Fu he mo e, he e a e some noise e en s placed in in e -clus e egions, which should be emo ed be o e clus e ing. Remo e Sens. 2018,10, 461 5 o 22 Remo e Sens. 2018, 10, x FOR PEER REVIEW 5 o 22 Figu e 1. K-means clus e ing esul s using di e en ini ial clus e cen e s o a ypical da ase . The colo ed ci cles a e he e en s and he same colo ed e en s co espond o a same clus e . (a) Da ase ; (b) MSED indexes o di e en andomly selec ed ini ial clus e cen e s (K = 4); (c–h) The clus e esul s o execu ions 2nd, 3 d, 4 h, 9 h, 11 h and 12 h o he K-means clus e ing p ocess (K = 4). Figu e 1. K-means clus e ing esul s using di e en ini ial clus e cen e s o a ypical da ase . The colo ed ci cles a e he e en s and he same colo ed e en s co espond o a same clus e . ( a ) Da ase ; ( b )MSED indexes o di e en andomly selec ed ini ial clus e cen e s (K= 4); ( c – h ) The clus e esul s o execu ions 2nd, 3 d, 4 h, 9 h, 11 h and 12 h o he K-means clus e ing p ocess (K= 4). Remo e Sens. 2018,10, 461 6 o 22 3.2. Da a Field Theo y Mo i a ed by physical ield heo y, Wang e al. [ 39 ] p oposed a da a ield heo y o desc ibe he in e ac ion among a se o e en s. The po en ial alue (scala ield s eng h) o e en xi in he global da a ield is de ined as: ϕ(xi) = n ∑ j=1 mj×K(kxi−xjk σ)(3) whe e mj is he mass o e en xj ,K(x) is he uni po en ial unc ion, kxi−xjk ep esen s he dis ance be ween e en xiand e en xj, and σis he impac ac o . The mj is se o 1, o all he j alues, and he commonly used unc ion K(β) = e−β2 is selec ed. Then, he po en ial alue in he global da a ield o e en xi can be w i en as in Equa ion (4), whe e kxi−xjkwas de ined by he Euclidean unc ion. ϕ(xi) = n ∑ j=1 e−(kxi−xjk/σ)2= n ∑ j=1 e−( p ∑ l=1 (xi,l−xj,l)2/σ2)(4) The impac ac o con ols he in e ac ed dis ance be ween e en s and i s alue can ha e a hea y impac on po en ial alues. The ela ionship be ween he in e ac ed dis ance and he po en ial alue o Equa ion (4) is shown in Figu e 2a. I is easy o see ha he longe he dis ance be ween e en s he lowe he po en ial alue o a same impac ac o , and he highe he impac ac o he la ge he po en ial alue o a same in e ac ed dis ance. In o he wo ds, o a low impac ac o , he e will be many local maximum po en ial alues; while o a la ge impac ac o , he e will be ew local maximum po en ial alues. Some esea ch [ 39 , 40 ] has used po en ial en opy o ob ain an op imum impac ac o , and he minimum po en ial en opy co esponds o he op imum impac ac o . The po en ial en opy is de ined as in [39]: H=− n ∑ i=1 ϕ(xi) Zlogϕ(xi) Z(5) whe e Z=n ∑ i=1 ϕ(xi), 0 ≤H≤log(n). The po en ial en opy using di e en impac ac o s is shown in Figu e 2b. The po en ial en opy has a minimum alue when σ= 0.028. Then, he po en ial alue is calcula ed o each e en using Equa ion (4). The con ou map o po en ial alue is shown in Figu e 2c. I is easy o see ha he po en ial alue con ou has a good ag eemen wi h he da ase densi y. Po en ial alue may p o ide a good way o clus e ing (e.g., Figu e 2c): a po en ial h eshold alue can be used o di ide he da ase in o small da ase s. No wi hs anding, o an i egula da ase , i is di icul o p oduce adequa e clus e s based on a po en ial h eshold alue (e.g., Figu es 5a and 6). The e o e, ins ead o using a po en ial alue o clus e seismic e en s, we jus applied po en ial h eshold alues o emo e noise e en s and be e selec ed K-means ini ial clus e cen e s. Remo e Sens. 2018,10, 461 7 o 22 Remo e Sens. 2018, 10, x FOR PEER REVIEW 7 o 22 Figu e 2. Da a ield applica ion (a–c) and K-means clus e ing esul s based on he da a ield (d– ); (a) Po en ial alues based on di e en impac ac o s; (b) Rela ionship be ween po en ial en opy and he impac ac o ; (c) Con ou map o po en ial alues ( 0.028 σ =); (d) The MSED indexes based on di e en po en ial h eshold alues p o ϕ , whe e Min, O1, O2, …., O7 a e he minimum, i s , second, …, and se en h oc iles o po en ial alues, espec i ely; (e) The da a ield-based K-means clus e ing esul s using undenoised da a when he po en ial h eshold alue p o ϕ is equal o he second oc ile; ( ) The da a ield-based K-means clus e ing esul s wi h 7.5% da a emo ed and p o ϕ is equal o he second oc ile. Figu e 2. Da a ield applica ion ( a – c ) and K-means clus e ing esul s based on he da a ield ( d – ); ( a ) Po en ial alues based on di e en impac ac o s; ( b ) Rela ionship be ween po en ial en opy and he impac ac o ; ( c ) Con ou map o po en ial alues ( σ= 0.028); ( d ) The MSED indexes based on di e en po en ial h eshold alues ϕpo , whe e Min, O 1 ,O 2 , . . . ., O 7 a e he minimum, i s , second, . . . , and se en h oc iles o po en ial alues, espec i ely; ( e ) The da a ield-based K-means clus e ing esul s using undenoised da a when he po en ial h eshold alue ϕpo is equal o he second oc ile; ( ) The da a ield-based K-means clus e ing esul s wi h 7.5% da a emo ed and ϕpo is equal o he second oc ile. Remo e Sens. 2018,10, 461 8 o 22 3.3. Da a Field-Based K-Means Clus e ing P ocedu e The da a ield-based K-means clus e ing akes ad an age o he e en wi h high po en ial alue being mo e likely o be an ini ial clus e cen e and ha he dis ance be ween ini ial clus e cen e s should ha e a la ge dis ance. The e is a low e en densi y a ound he noise e en s and hey ha e la ge dis ances o mos e en s, which usually implies small po en ial alues. Then, a h eshold alue ϕ h can be se o emo e noise e en s. The low diag am o he da a ield-based K-means clus e ing p ocess is shown in Figu e 3con aining he ollowing s eps: S ep 1: Inpu da ase Uda ase (xi∈Uda ase ,i= 1, 2, . . . , nda ase ,nda ase =n). S ep 2: Selec he impac ac o o he da a ield by he impac ac o -po en ial en opy cu e calcula ed by Equa ion (5) and calcula e he po en ial alue ϕ (x i ) o each e en in Uda ase using Equa ion (4). S ep 3: Remo e noise e en s and ob ain a new da ase Uda ase -noise ( xj∈Uda ase -noise ,j= 1, 2, . . . ,n da ase -noise ,n da ase -noise ≤ n): I a po en ial alue is smalle han a p ede ined h eshold alue ϕ h , hen i s co esponding e en is ma ked as a noise e en and i is o be emo ed om he da ase Uda ase . I he e a e no noise e en s, hen he e is no need o do S ep 3 o a ϕ h ha is smalle han he minimum po en ial alue can be se o do S ep 3. ϕ h = Min-0.1 is used in his pape o no denoising, whe e Min-0.1 means he minimum po en ial alue minus 0.1. S ep 4: Selec K-means ini ial clus e cen e s based on he po en ial alue and a maximum dis ance-based algo i hm. (4-1) Selec possible ini ial clus e cen e s Upo ( xk∈Upo ,k= 1, 2, . . . ,n po ,n po ≤ n) by choosing e en s which ha e po en ial alues la ge han a p ede ined h eshold alue ϕpo; (4-2) Selec K-means ini ial clus e cen e s based on a maximum dis ance-based algo i hm. The maximum dis ance-based algo i hm con ains he ollowing s eps: (4-2-1) The e en which has a maximum po en ial alue in Upo is se o he i s ini ial clus e cen e m1; (4-2-2) Calcula e he dis ances be ween m1 and each poin in Upo - m1 , and he la ges dis ance co esponding e en is se o he second ini ial clus e cen e m2; (4-2-3) Calcula e he dis ances be ween m1 , m2 and each poin in Upo - m1 - m2 , hen selec he smalles dis ance o ob ain he dis ance se Vsd . Fo example, o e en xi ( xi∈Upo - m1 - m2 ), he e a e wo dis ances ( m1 , xi ) and ( m2 , xi ), and he Min(( m1 , xi ), ( m2 , xi )) is se o Vi sd . Calcula e he smalle dis ance o each e en in Upo - m1 - m2 , hen he smalles dis ances make up he dis ance se Vsd . The poin wi h he la ges alue in Vsd co esponds o he hi d ini ial clus e cen e m3; (4-2-4) Repea S ep (4-2-3) o ob ain m4,m5, . . . , mK. S ep 5: Pe o m he K-means clus e ing o da ase Uda ase -noise using he m1 , m2 , . . . , mK as he ini ial clus e cen e s. S ep 6: Selec he op imum numbe o clus e s using he K zanowski-Lai (KL) index (see Sec ion 4.3.2) and in e p e he clus e ing esul s. Remo e Sens. 2018,10, 461 9 o 22 Remo e Sens. 2018, 10, x FOR PEER REVIEW 9 o 22 Figu e 3. The low diag am o he da a ield-based K-means clus e ing. 3.4. Applica ion Tes The ela ionship be ween he MSED index and he h eshold alue p o ϕ o da a ield-based K- means clus e ing using undenoised da a is shown in Figu e 2d, whe e O1, O2, …., O7 a e he i s , second, …, and se en h oc iles o po en ial alues, espec i ely. I is clea ha he da a ield-based K- means clus e ing usually ob ains a smalle MSED index han he classical K-means algo i hm. This indica es ha he da a ield-based K-means clus e ing ob ains be e clus e s han hose p oduced by Figu e 3. The low diag am o he da a ield-based K-means clus e ing. 3.4. Applica ion Tes The ela ionship be ween he MSED index and he h eshold alue ϕpo o da a ield-based K-means clus e ing using undenoised da a is shown in Figu e 2d, whe e O 1 ,O 2 , . . . ., O 7 a e he i s , second, . . . , and se en h oc iles o po en ial alues, espec i ely. I is clea ha he da a ield-based Remo e Sens. 2018,10, 461 16 o 22 Remo e Sens. 2018, 10, x FOR PEER REVIEW 16 o 22 Figu e 9. LogEI o he i e clus e s in Figu e 8a. 4.3.3. Seismici y Analysis (1) Ene gy index The commonly used s a is ical pa ame e -ene gy index (EI) is selec ed o e alua e mic oseismic ac i i y. EI is a pa ame e ela ed o he concen a ion and accumula ion o s ess in a ock mass [45], and i is de ined as he a io be ween he adia ed ene gy and he a e age ene gy (expec ed ene gy) o a gi en seismic momen M. An EI > 1 signi ies ha mo e ene gy has been eleased han expec ed, while an EI < 1 signi ies ha less ene gy has been eleased han expec ed. Resea ch has shown ha when an EI > 1 he ock mass is accumula ing s ess, and when he ock mass s a s o ail, he EI d ops below one [46]. The EI is de ined as in [47]: log ()10 cd M EE EI EM + == (8) whe e 2 2 4c c J E R F πρ = is he seismic adia ion ene gy, ρ is he ock densi y, is he wa e p opaga ion speed, c J is he in eg al o he squa ed g ound speed, Fc is he adia ion pa e n Figu e 9. LogEI o he i e clus e s in Figu e 8a. 4.3.3. Seismici y Analysis (1) Ene gy index The commonly used s a is ical pa ame e -ene gy index (EI) is selec ed o e alua e mic oseismic ac i i y. EI is a pa ame e ela ed o he concen a ion and accumula ion o s ess in a ock mass [ 45 ], and i is de ined as he a io be ween he adia ed ene gy and he a e age ene gy (expec ed ene gy) o a gi en seismic momen M. An EI > 1 signi ies ha mo e ene gy has been eleased han expec ed, while an EI < 1 signi ies ha less ene gy has been eleased han expec ed. Resea ch has shown ha when an EI > 1 he ock mass is accumula ing s ess, and when he ock mass s a s o ail, he EI d ops below one [46]. The EI is de ined as in [47]: EI =E E(M)=E 10c+dlog M(8) whe e E=4πρ R2Jc Fc2is he seismic adia ion ene gy, ρis he ock densi y, is he wa e p opaga ion speed, Jc is he in eg al o he squa ed g ound speed, F c is he adia ion pa e n pa ame e : o a P Remo e Sens. 2018,10, 461 17 o 22 wa e F c is 0.52, while o an S wa e F c is 0.63. E(M) is he a e age ene gy de i ed om he logE s. logM ela ion o a gi en momen M, and cand da e he eg ession cons an s. (2) Seismici y analysis o clus e s The leas squa es linea eg essions be ween log(ene gy) and log(momen ) o he i e clus e s o Figu e 8a a e shown in Figu e S2. Then, we calcula ed he EI h ough Equa ion (8) and he logEI o he i e clus e s a e shown in Figu e 9. Some esea ch [ 48 , 49 ] de ined he p edic i e pe iod and c i ical pe iod om he EI ime se ies: when he EI dec eases, his indica es ha pa s o he ock mass a e beginning o ha e an uns able s a us. This pe iod could be seen as a s ain-so ening s age. I is he beginning o po en ial damage and ega ded as a wa ning indica o . This pe iod is called a p edic i e pe iod. An inc easing EI means ha he ock mass is en e ing an uns able s a e, and his pe iod is called a c i ical pe iod. The quicke he EI dec eases, he la ge he isk o a big mic oseismic e en happening. So, when he EI dec eases sha ply, we can sugges he mine s o be ca e ul in he p edic i e pe iod and no wo k in he upcoming c i ical pe iod in he co esponding clus e a ea. Fo C1, he logEI has a sha p dec ease om 5 Ap il o 6 Ap il (a p edic i e pe iod)– he mine s should be ca e ul when mining in his a ea–and he logEI inc eases om 7 Ap il o 13 Ap il (a c i ical pe iod)– he mine s should no wo k in his a ea on hese days. Then, he e is a ela i ely small p edic i e pe iod ( om 13 Ap il o 14 Ap il) and a c i ical pe iod (15 Ap il). Fo C2, he logEI slowly dec eases om 23 Ap il o 27 Ap il (p edic i e pe iod), and he mine s should be ca e ul on 28 Ap il when mining. Fo C3, he logEI quickly dec eases on 10 Ap il (a p edic i e pe iod and special a en ion should be paid) and hen he logEI inc eases om 13 Ap il o 16 Ap il (a c i ical pe iod and he mine s should no wo k in his a ea). Fo C4, he logEI inc eases and dec eases al e na ely, which means he ock sys em eleases and abso bs ene gy s eadily and he e a e no p edic i e pe iods and c i ical pe iods. Fo C5, he e a e high logEI om 16 Ap il o 19 Ap il and he e is a sha p logEI dec easing om 25 Ap il o 28 Ap il (a p edic i e pe iod and he mine s should be e y ca e ul when mining), hen he logEI inc eases om 22 Ap il o 24 Ap il (a c i ical pe iod and he mine s should no wo k in his a ea). Though Figu e 8a shows ha C4 and C5 ha e a simila loca ion a ea, C5 has c i ical pe iods while C4 has no c i ical pe iods, which p o es ha he ime pa ame e can p o ide a new insigh o seismic clus e ing analysis. In conclusion, he e a e e y c i ical pe iods o C1, C3 and C5 when he mine s should no wo k. 5. Discussion This pape p oposed a da a ield-based K-means clus e ing me hodology o denoise noise e en s and selec ini ial clus e cen e s. Howe e , his can be a ec ed by he h eshold alue ϕpo , and i s pe o mance wi h espec o classical K-means clus e ing using denoised and undenoised e en s should be discussed. In addi ion, we also chose he commonly used hie a chical clus e ing and SOM clus e ing as compa isons. The MSED indexes o ime-e en loca ion dis ance and da a ield-based K-means clus e ing ( ϕpo = Min-0.1, Q1, Median and Q3, whe e Min, Q1, Median and Q3 co espond o he minimum, he i s qua ile, he second qua ile and he hi d qua ile o he po en ial alues, espec i ely), classical K-means clus e ing using denoised and undenoised e en s ( andomly selec ed ini ial clus e cen e s) and SOM clus e ing a e d awn in Figu e 10. Fo denoised e en clus e ing, his shows ha he MSED index o he da a ield-based K-means clus e ing is usually smalle han ha o he classical K-means clus e s. Fo a small clus e numbe (K= 2 and K= 3), he da a ield-based K-means ( ϕpo = Min-0.1, Q1 and Median) has a simila MSED index o he classical K-means. This is because he classical K-means clus e ing has a ela i ely good global sea ch o a small numbe o clus e s. The da a ield-based K-means clus e ing ( ϕpo =Q3) p oduces be e clus e esul s han he classical K-means algo i hm (K= 3 and K= 4). This is due o he highe po en ial alue e en being mo e likely o be a inal clus e cen e . Fo a ela i ely la ge clus e numbe (K= 5~7), he da a ield-based K-means clus e ing usually imp o es he classical K-means clus e ing, which p o es he e ec i eness o he da a ield-based algo i hm in selec ing ini ial clus e cen e s. Fo a la ge clus e Remo e Sens. 2018,10, 461 18 o 22 numbe (K= 8~10), he da a ield-based K-means clus e ing ( ϕpo =Q3) has a esul ha is nea ly as equal/bad as he classical K-means clus e ing. This is caused by he da a ield densi y-based algo i hm using a h eshold alue ϕpo o selec high po en ial alue e en s. Ye , he low po en ial alue e en s may ha e use ul ini ial clus e cen e s o a la ge clus e numbe , which esul s in he andomly selec ed ini ial clus e cen e s a imes ha ing a be e clus e esul . The da a ield-based K-means clus e ( ϕpo =Q1 and ϕpo = Median) has a good clus e esul in gene al and a h eshold alue ϕpo be ween Q1 and he Median, and a clus e numbe smalle han 9 is sugges ed in he da a ield-based K-means clus e ing algo i hm. Remo e Sens. 2018, 10, x FOR PEER REVIEW 18 o 22 numbe (K = 8~10), he da a ield-based K-means clus e ing ( p o ϕ = Q3) has a esul ha is nea ly as equal/bad as he classical K-means clus e ing. This is caused by he da a ield densi y-based algo i hm using a h eshold alue p o ϕ o selec high po en ial alue e en s. Ye , he low po en ial alue e en s may ha e use ul ini ial clus e cen e s o a la ge clus e numbe , which esul s in he andomly selec ed ini ial clus e cen e s a imes ha ing a be e clus e esul . The da a ield-based K-means clus e ( p o ϕ = Q1 and p o ϕ = Median) has a good clus e esul in gene al and a h eshold alue p o ϕ be ween Q1 and he Median, and a clus e numbe smalle han 9 is sugges ed in he da a ield-based K-means clus e ing algo i hm. The classical K-means clus e ing using denoised e en s usually has a smalle MSED index han ha o classical K-means clus e ing using undenoised e en s. This is caused by he noise e en s being ela i ely a away om mos e en s (Figu e 11a). The SOM clus e ing has a e y la ge MSED index compa ed wi h K-means-based clus e s (Figu e 10). This is due o he SOM clus e s some imes mixing oge he (Figu e 11b). Fu he mo e, he mixed clus e s make i ha d o in e p e he clus e esul s. Fo he hie a chical clus e ing, we ied di e en clus e numbe s: o a small clus e numbe , 99% o seismic e en s a e in one clus e ; o 100 clus e s, 36% and 50% o e en s a e in wo clus e s; o 200 clus e s, 42% and 28% o e en s a e in wo clus e s; and o 400 clus e s, 25% and 12%, 5%, 7% o e en s a e in ou clus e s. Fo hese numbe s o clus e s, e e y o he clus e makes up less han 1%. This is b ough abou by he hie a chical clus e ing connec ing close e en s oge he . Al hough he e a e some a away e en s in he seismic da a, e e y e en will usually be di ided in o one clus e o se e al close e en s connec ed in o one clus e . The e o e, i is ha d o ob ain a ce ain numbe o clus e s which con ain a ela i ely la ge numbe o e en s, and he e a e many e en s ha will be ea ed as noises ( he clus e s wi h ew e en s). In conclusion, he da a ield-based K-means clus e ing can ha e a good clus e esul as well as a good in e p e a ion compa ed wi h he abo e discussed clus e s. Figu e 10. The MSED indexes o ime-e en loca ion dis ance and da a ield-based K-means clus e ing, classical K-means clus e ing using denoised and undenoised e en s and SOM clus e ing using denoised e en s. The Min, Q1, Median and Q3 co espond o he minimum, i s qua ile, second qua ile and hi d qua ile o po en ial alues, espec i ely. Figu e 10. The MSED indexes o ime-e en loca ion dis ance and da a ield-based K-means clus e ing, classical K-means clus e ing using denoised and undenoised e en s and SOM clus e ing using denoised e en s. The Min, Q1, Median and Q3 co espond o he minimum, i s qua ile, second qua ile and hi d qua ile o po en ial alues, espec i ely. The classical K-means clus e ing using denoised e en s usually has a smalle MSED index han ha o classical K-means clus e ing using undenoised e en s. This is caused by he noise e en s being ela i ely a away om mos e en s (Figu e 11a). The SOM clus e ing has a e y la ge MSED index compa ed wi h K-means-based clus e s (Figu e 10). This is due o he SOM clus e s some imes mixing oge he (Figu e 11b). Fu he mo e, he mixed clus e s make i ha d o in e p e he clus e esul s. Fo he hie a chical clus e ing, we ied di e en clus e numbe s: o a small clus e numbe , 99% o seismic e en s a e in one clus e ; o 100 clus e s, 36% and 50% o e en s a e in wo clus e s; o 200 clus e s, 42% and 28% o e en s a e in wo clus e s; and o 400 clus e s, 25% and 12%, 5%, 7% o e en s a e in ou clus e s. Fo hese numbe s o clus e s, e e y o he clus e makes up less han 1%. This is b ough abou by he hie a chical clus e ing connec ing close e en s oge he . Al hough he e a e some a away e en s in he seismic da a, e e y e en will usually be di ided in o one clus e o se e al close e en s connec ed in o one clus e . The e o e, i is ha d o ob ain a ce ain numbe o clus e s which con ain a ela i ely la ge numbe o e en s, and he e a e many e en s ha will be ea ed as noises ( he clus e s wi h ew e en s). In conclusion, he da a ield-based K-means clus e ing can ha e a good clus e esul as well as a good in e p e a ion compa ed wi h he abo e discussed clus e s. Remo e Sens. 2018,10, 461 19 o 22 Remo e Sens. 2018, 10, x FOR PEER REVIEW 19 o 22 Figu e 11. The ime-e en loca ion dis ance based clus e esul s (K = 5) o classical K-means using undenoised e en s and SOM clus e ing using denoised e en s. (a) Classical K-means using undenoised e en s; (b) SOM clus e ing using denoised e en s. 6. Conclusions In his pape , a da a ield-based K-means clus e ing me hod has been p oposed o spa io- empo al seismici y analysis. This me hod akes ad an age o a da a ield-based h eshold alue o emo e noises as well as a new dis ance-based algo i hm o ob ain good ini ial clus e cen e s o K- means. The me hod has been es ed by i s applica ion o mic oseismic e en s ob ained om he Chinese Yongshaba mine, which conside s bo h a ime pa ame e and loca ion pa ame e s. The KL index shows ha he me hod has he bes clus e ing esul when K = 5 and he EI shows ha C1, C3 and C5 ha e e y c i ical pe iods. C4 and C5 ha e simila clus e zones. Ne e heless, C4 has no c i ical pe iods, which p o es ha he ime-e en loca ion dis ance-based K-means clus e ing can p o ide a new way o seismici y analysis compa ed wi h e en loca ion dis ance-based K-means clus e ing. The da a ield-based K-means clus e ing usually achie es be e clus e esul s compa ed wi h classical K-means clus e ing when he h eshold alue po ϕ is be ween Q1 and he Median o he po en ial alues and he clus e numbe is smalle han 9. Compa isons wi h he classical K-means clus e ing, hie a chical clus e ing and SOM clus e ing show he e ec i eness o he p oposed clus e ing me hod. The po en ial alue-based denoising is e y applicable o o he da ase s, whe e e en s need o be di ided in o clus e ed and non-clus e ed e en s. Also, he po en ial alue can be used o show e en densi y. The p oposed maximum dis ance-based algo i hm can also p o ide good ini ial clus e cen e s o some o he clus e s, such as he FCM clus e . The e ec i eness o in oducing spa io- empo al dis ance in o K-means shows ha i may also be use ul in o he clus e algo i hms. Mo eo e , u he s udy can be done by in oducing ime-e en loca ion-magni ude dis ance in o he da a ield-based K-means clus e ing o seismici y analysis. Supplemen a y Ma e ials: The ollowing a e a ailable online a www.mdpi.com/link. Figu e S1: Seismic e en loca ions wi h 5%, 7.5%, 12.5% and 15% o da a emo ed. (a) Seismic e en loca ions wi h 5% o da a emo ed; Figu e 11. The ime-e en loca ion dis ance based clus e esul s (K= 5) o classical K-means using undenoised e en s and SOM clus e ing using denoised e en s. ( a ) Classical K-means using undenoised e en s; (b) SOM clus e ing using denoised e en s. 6. Conclusions In his pape , a da a ield-based K-means clus e ing me hod has been p oposed o spa io- empo al seismici y analysis. This me hod akes ad an age o a da a ield-based h eshold alue o emo e noises as well as a new dis ance-based algo i hm o ob ain good ini ial clus e cen e s o K-means. The me hod has been es ed by i s applica ion o mic oseismic e en s ob ained om he Chinese Yongshaba mine, which conside s bo h a ime pa ame e and loca ion pa ame e s. The KL index shows ha he me hod has he bes clus e ing esul when K= 5 and he EI shows ha C1, C3 and C5 ha e e y c i ical pe iods. C4 and C5 ha e simila clus e zones. Ne e heless, C4 has no c i ical pe iods, which p o es ha he ime-e en loca ion dis ance-based K-means clus e ing can p o ide a new way o seismici y analysis compa ed wi h e en loca ion dis ance-based K-means clus e ing. The da a ield-based K-means clus e ing usually achie es be e clus e esul s compa ed wi h classical K-means clus e ing when he h eshold alue ϕpo is be ween Q1 and he Median o he po en ial alues and he clus e numbe is smalle han 9. Compa isons wi h he classical K-means clus e ing, hie a chical clus e ing and SOM clus e ing show he e ec i eness o he p oposed clus e ing me hod. The po en ial alue-based denoising is e y applicable o o he da ase s, whe e e en s need o be di ided in o clus e ed and non-clus e ed e en s. Also, he po en ial alue can be used o show e en densi y. The p oposed maximum dis ance-based algo i hm can also p o ide good ini ial clus e cen e s o some o he clus e s, such as he FCM clus e . The e ec i eness o in oducing spa io- empo al dis ance in o K-means shows ha i may also be use ul in o he clus e algo i hms. Mo eo e , u he s udy can be done by in oducing ime-e en loca ion-magni ude dis ance in o he da a ield-based K-means clus e ing o seismici y analysis. Remo e Sens. 2018,10, 461 20 o 22 Supplemen a y Ma e ials: The ollowing a e a ailable online a www.mdpi.com/2072-4292/10/3/461/s1. Figu e S1: Seismic e en loca ions wi h 5%, 7.5%, 12.5% and 15% o da a emo ed. (a) Seismic e en loca ions wi h 5% o da a emo ed; (b) Seismic e en loca ions wi h 7.5% o da a emo ed; (c) Seismic e en loca ions wi h 12.5% o da a emo ed; (d) Seismic e en loca ions wi h 15% o da a emo ed. Figu e S2: Leas squa es linea eg ession be ween log(ene gy) and log(momen ) o he i e clus e s shown in Figu e 8a. Acknowledgmen s: The au ho s g a e ully acknowledge he inancial suppo o he Na ional Key Resea ch and De elopmen P og am o China (2016YFC0600706). The au ho s would also like o hank Dong Liu, Yongyong Zhou and Jing Yang o hei help. Au ho Con ibu ions: Xueyi Shang w o e he pape ; Xibing Li p o ided he o iginal idea and seismic da a; A. Mo ales-Es eban and Gualbe o Asencio-Co és modi ied he pape and ga e some use ul sugges ions; Zewei Wang w o e pa o he p og am code. Con lic s o In e es : The au ho s decla e no con lic o in e es . Re e ences 1. 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