Compu a ional In elligence Techniques o
P edic ing Ea hquakes
F. Ma ínez-Ál a ez1,A.T oncoso
1, A. Mo ales-Es eban2,andJ.C.Riquelme
3
1Depa men o Compu e Science, Pablo de Ola ide Uni e si y o Se ille, Spain
{ ma al ,ali}@upo.es
2Depa men o Con inuum Mechanics, Uni e si y o Se ille, Spain
[email p o ec ed]
3Depa men o Compu e Science, Uni e si y o Se ille, Spain
[email p o ec ed]
Abs ac . Nowadays, much effo is being de o ed o de elop ech-
niques ha o ecas na u al disas e s in o de o ake p ecau iona y
measu es. In his pape , he ex ac ion o quan i a i e associa ion ules
and eg ession echniques a e used o disco e pa e ns which model he
beha io o seismic empo al da a o help in ea hquakes p edic ion.
Thus, a simple me hod based on he k–smalles and k–g ea es alues
is in oduced o mining ules ha a emp a explaining he condi ions
unde which an ea hquake may happen. On he o he hand pa e ns a e
disco e ed by using a ee-based piecewise linea model. Resul s om
seismic empo al da a p o ided by he Spanish’s Geog aphical Ins i u e
a e p esen ed and discussed, showing a ema kable pe o mance and he
significance o he ob ained esul s.
Keywo ds: ime se ies, quan i a i e associa ion ules, eg ession.
1 In oduc ion
A ime se ies is a sequence o alues obse ed o e ime and, he e o e, ch ono-
logically o de ed. Gi en his defini ion, i is usual o find da a ha can be ep-
esen ed as ime se ies in many esea ch fields.
The s udy o he pas beha io o a a iable may be ex emely aluable o
p edic i s u u e beha io . Assuming ha he na u e o he ea hquakes ime
se ies is s ochas ic, clus e ing echniques ha e shown ha hese ime se ies ex-
hibi some empo al pa e ns, making he modeling and subsequen p edic ion
possible [11].
This pape analyzes and o ecas s ea hquakes ime se ies by means o he
applica ion o wo classical echniques: Quan i a i e associa ion ules (QAR)
and eg ession.
A e ision o he la es published wo ks e eals ha he amoun o me a-
heu is ics and sea ch algo i hms ela ed o associa ion ules wi h con inuous
a ibu es is limi ed. Ne e heless, a classifie was p esen ed in [13] o ex ac
quan i a i e associa ion ules om unlabeled da a s eams. The main
no el y
F. Ma ínez-Ál a ez e al.
o his app oach lied on i s adap abili y o on-line ga he ed da a. Also, a me a-
heu is ic based on ough pa icle swa m echniques was p esen ed in [1]. In his
case, he special ea u e was he ob en ion o he alues de e mining he in e als
o he associa ion ules. They also e alua ed and es ed se e al new ope a o s in
syn he ic da a. A mul i-objec i e pa e o-based gene ic algo i hm was p esen ed
in [2]. The fi ness unc ion was o med by ou diffe en objec i es: suppo ,
confidence, comp ehensibili y o he ule (aimed a being maximized) and he
ampli ude o he in e als ha o ms he ule (in ended o be minimized). The
wo k published in [17] p esen ed a new app oach based on h ee no el algo-
i hms: Value-in e al clus e ing, in e al-in e al clus e ing and ma ix-in e al
clus e ing. Thei applica ion was ound especially use ul when mining complex
in o ma ion. Ano he gene ic algo i hm was used in [16] in o de o ob ain nu-
me ic associa ion ules. Howe e , he unique objec i e o be op imized in he
fi ness unc ion was he confidence. To ulfill his goal, he au ho s a oided he
specifica ion o he ac ual minimum suppo , which is he main con ibu ion o
his wo k. Finally, an ex ension o he well-known bina y-coded CHC algo i hm
is p esen ed in [10] o finding exis ing ela ions be ween a mosphe ic pollu ion
and clima ological condi ions.
Reg ession echniques ha e been widely used o o ecas ing ime se ies [5].
Thus, an empi ical s udy on sea wa e quali y p edic ion can be ound in [7].
Ha zikos e al. aced he p oblem o o ecas ing wa e quali y based on unde -
wa e senso s measu emen s, by means o a la ge a ie y o bo h linea and non-
linea me hods. Also, a new me hodology o build eg ession ees was in oduced
in [3]. The au ho s ans o med quan i a i e da a in o s a is ical momen s, and
cons uc ed a ee o es ima e he o ecas ing in e al o he a ge a iable.
Las , he p oblem o p edic ing he machine y deg ada ion and ending o aul
p opaga ion be o e eaching he ala m was s udied in [12]. In pa icula , he
au ho s p oposed an app oach based on eg ession ees o o ecas such ime
se ies.
The es o he pape is di ided as ollows. Sec ion 2 p o ides he me hodology
used in his wo k. The esul s o he app oach a e epo ed in Sec ion 3. Finally,
Sec ion 4 discusses he achie ed conclusions.
2 Me hodology
The me hods used o ex ac knowledge om ea hquakes ime se ies a e de-
sc ibed in his sec ion. The goal is o find pa e ns in da a ha p ecede he
appea ance o ea hquakes wi h a gi en magni ude.
2.1 Associa ion Rules Mining
Le F={F1, ..., Fn}be a se o ea u es wi h alues in Rdesc ibing an ea h-
quake. The desi ed ules a e defined by he ollowing equa ion:

i=1,...,n−1
Fi∈[li,u
i]⇒Fn∈[ln,u
n](1)
Compu a ional In elligence Techniques o P edic ing Ea hquakes
whe e liand ui ep esen s he lowe and uppe limi s o he in e al o Fi,
espec i ely and he limi s lnand una e gi en depending on he objec i e o
he p oblem o be sol ed. In he con ex o seismic ime se ies, Fn ep esen s he
ea hquake magni ude o be p edic ed and he limi s lnand undepend on he
equi ed size o he ea hquakes o be o ecas ed.
The p oposed me hod o ob ain QAR is desc ibed as ollows. Fi s , he da ase
is so ed by he ea u e Fn, ha is, by he consequen o he ule. Once he limi s
[ln,u
n]a e se , he ange o he emaining ea u es Fiis calcula ed as:
R(Fi)={Fisuch ha Fn∈[ln,u
n]}i=1, ..., n −1(2)
Le M
iand m
iwi h i=1, ..., n −1 wo unc ions defined by:
M
i:{1, ..., #(R(Fi))}−→R(Fi)
k−→ M
i(k)=kg ea es alue o R(Fi)(3)
m
i:{1, ..., #(R(Fi))}−→R(Fi)
k−→ M
i(k)=ksmalles alue o R(Fi)(4)
whe e #(R(Fi)) is he numbe o elemen s o he se R(Fi).
Le Sibe he se o pai o alues such ha he ampli ude o he in e al o
be sea ched o he ea u e Fiis sufficien ly small. Tha is,
Si={(k1,k
2)such ha M
i(k2)− m
i(k1)≤MAXi}(5)
whe e MAXiis he maximum allowed ampli ude o he ea u e Fiwhich is a
gi en pa ame e depending on he desi ed ules.
Thus, o any alue (ki
1,ki
2)∈Si, he ules buil by he k-g ea es and k-
smalles alues a e:

i=1,...,n−1
Fi∈[ m
i(ki
1), M
i(ki
2)] ⇒Fn∈[ln,u
n](6)
2.2 Reg ession: M5P Algo i hm
The second me hod used o ob ain pa e ns in seismic ime se ies is he M5P
algo i hm a ailable in WEKA [4]. The M5P app oach [15] ex ends o he M5 al-
go i hm by adding missing alues echniques and ans o ma ion o ea u es om
disc e e alues o bina y alues. The algo i hm M5 [14] p o ides a con en ional
decision- ee wi h linea eg ession unc ions a he nodes. The ee is ob ained
by a classical induc ion algo i hm bu he spli s a e ob ained by maximizing he
educ ion o he a iance and no maximizing he gain o in o ma ion.
Once he ee has been buil , he me hod compu es a linea model o each
node. La e he lea es o he ee a e p uned while he e o dec eases. Fo each
node, he e o is he mean o he absolu e alue o he diffe ence be ween he
F. Ma ínez-Ál a ez e al.
p edic ed and ac ual alues o each example eaching such node. This e o
is weigh ed depending on he numbe o examples which each ha node. The
p ocess is epea ed un il all examples a e co e ed o one o mo e ules.
Thus, M5P gene a es models ha a e compac and ela i ely comp ehensible.
3Resul s
This sec ion p esen s he esul s ob ained om he applica ion o he app oaches
in oduced in Sec ion 2. In pa icula , Sec ion 3.1 p o ides a desc ip ion o he
da a used. Sec ions 3.2 and 3.3 ga he all ele an esul s mined by means o
associa ion ules and decision- ee echniques, espec i ely.
3.1 Da a Desc ip ion
The da ase used in his wo k has been e ie ed om he ca alogue o Spanish’s
Geog aphical Ins i u e (SGI), which con ains he loca ion and magni ude o
Spanish ea hquakes.
Addi ionally, he b– alue pa ame e o he Gu enbe g–Rich e law has been
calcula ed, as i eflec s he ec onics and geophysical p ope ies o he ocks as
well as he fluid p essu e a ia ions in he cha ac e ized su ace [9].
Thus, each sample o ming he da ase is composed by ou a ibu es: Cu en
ea hquake magni ude, ime when he ea hquake occu ed, associa ed b- alue,
and magni ude o he p e iously occu ed ea hquake. No e ha ea hquakes
wi h magni ude lowe han 3.0 ha e been emo ed om he da ase , and bo h
a e shocks and o eshocks ha e been emo ed o a oid dependen da a, as ec-
ommended in [8].
Despi e he Ibe ian Peninsula is di ided in 27 seismogenic a eas acco ding o
SGI, only a eas 26 and 27 (Albo an Sea and Wes e n Azo es–Gib al a Faul ,
espec i ely) ha e been s udied, since hey a e he mos ac i e ones [11]. The
conside ed ea hquakes da e om 1981 o 2008, ha ing been analyzed a o al o
873 quakes.
3.2 Quan i a i e Associa ion Rules Ex ac ion
All mined associa ion ules o o ecas ea hquakes a e now in oduced and dis-
cussed. As he goal is o find pa e ns ha p ecede quake occu ences, he mag-
ni ude o he cu en ea hquake, Mc, has been o ced o be he only a ibu e
in he consequen .
The Mca ibu e has been di ided in h ee non-o e lapped in e als: [3.0,
3.5) o small ea hquakes, [3.5, 4.4) o medium ea hquakes, and [4.4, 6.2] o
la ge ea hquakes (no e ha he la ges e ie ed ea hquake magni ude is 6.2).
Tables 1, 2, and 3 show he ules ex ac ed o la ge, medium and small ea h-
quakes, espec i ely. No e ha Δb and Δ ep esen he inc emen o he b– alue
and he ime elapsed be ween he p e ious and cu en ea hquake, espec i ely.
Also, hemagni udeo heea hquake occu edp io hecu en one, Mp, has been
Compu a ional In elligence Techniques o P edic ing Ea hquakes
Table 1. Associa ion ules wi h consequen Mc∈[4.4,6.2]
Id An eceden Con . (%) Sup. (%) Li
#1 Δ ∈[0.02,0.08] ∧Δb ∈[−0.16,−0.10] ∧Mp∈[3.0,3.4] 75.0 5.7 12.4
#2 Δ ∈[0.00,0.07] ∧Δb ∈[−0.12,−0.05] ∧Mp∈[3.5,4.9] 87.5 13.2 14.4
#3 Δ ∈[0.00,0.33] ∧Δb ∈[−0.11,−0.01] ∧Mp∈[5.0,6.2] 80.0 7.6 13.2
di ided in non-o e lapped in e als, and he e o e, ansac ions o ming he da a
can be co e ed only by one ule. Finally, all ules ha e been assessed by means o
h ee well-known and widely used indices: Confidence, suppo , and li [6].
The bes ules mined o la ge ea hquakes (Mc∈[4.4,6.2])a eshownin
Table 1. These ules sha e a common ea u e, which is ha hey all p esen
ema kable and nega i e Δb.Mo eo e ,Δ is small in all ules, excep o ule
#3, which allows ime in e als up o 0.33. F om he 53 ea hquakes ha sa is y
ha Mc∈[4.4,6.2], 14 a e co e ed by ules #1, #2 and #3, which ep esen s a
suppo o 26.4%. On he o he hand, i is no iceable he high confidence eached
by all o hem: 80.8% on a e age. Finally, he in e es ingness o he ules (o li )
is 13.3 on a e age. Assuming ha a li g ea e han 1 leads o conside he ule
as in e es ing [6], he ob ained alues indica e ha he ex ac ed ules p o ide
meaning ul knowledge.
Table 2 shows he QAR ob ained o medium ea hquakes, ha is, wi h Mc∈
[3.5,4.4). The mos significa i e ea u e ha sha e all he ules is ha he b–
alue does no a y much (i s alue anges om Δb =−0.07 o Δb =0.02).
Also ema kable is ha he occu ence o hese ea hquakes akes place a e
mode a ely sho ime pe iods ( he ime elapsed be ween ea hquakes a ies
om Δ =0.00 o Δ =0.20). As o he quali y o he esul s, 86 ea hquakes
ou o 344 we e co e ed by ules #4, #5 and #6, which means a suppo o
25.0%. The confidence was o 76.0% on a e age which can be conside ed high.
Las , he li measu e also confi ms ha he ules a e high quali y, since i has
alues g ea e han 1, in pa icula , 1.9 on a e age.
Table 3 ep esen s he bes QAR disco e ed o small ea hquakes (Mc∈
[3.0,3.5)). The b– alue is now cha ac e ized by mode a e and posi i e inc emen s
(Δb anges om 0.01 o 0.04). Mo eo e , in con as o wha happens wi h
medium and la ge ea hquakes, he ime elapsed is high, a ying om Δ =0.10
o Δ =0.32. A o al o 476 small ea hquakes we e e ie ed, om which 46
ha e been co e ed by ules #7, #8 and #9, which imply a suppo o 9.7%.
Especially no iceable is he confidence eached by hese ules which is 85.7% on
a e age. Again, he li measu e is g ea e han 1 o all ules, in pa icula , 1.7
on a e age.
Table 2. Associa ion ules wi h consequen Mc∈[3.5,4.4)
Id An eceden Con . (%) Sup. (%) Li
#4 Δ ∈[0.04,0.20] ∧Δb ∈[−0.07,−0.01] ∧Mp∈[3.0,3.5] 79.0 8.7 2.0
#5 Δ ∈[0.00,0.02] ∧Δb ∈[−0.01,0.00] ∧Mp∈[3.6,4.5] 78.6 12.8 2.0
#6 Δ ∈[0.00,0.05] ∧Δb ∈[−0.02,0.02] ∧Mp∈[4.6,5.9] 70.6 3.6 1.8

F. Ma ínez-Ál a ez e al.
Table 3. Associa ion ules wi h consequen Mc∈[3.0,3.5)
Id An eceden Con . (%) Sup. (%) Li
#7 Δ ∈[0.13,0.32] ∧Δb ∈[0.01,0.04] ∧Mp∈[3.0,3.2] 100 2.5 1.8
#8 Δ ∈[0.10,0.19] ∧Δb ∈[0.01,0.03] ∧Mp∈[3.3,3.4] 88.0 4.6 1.6
#9 Δ ∈[0.11,0.32] ∧Δb ∈[0.00,0.03] ∧Mp∈[3.5,5.7] 85.7 2.5 1.6
3.3 M5P Resul s
This sec ion p o ides he esul ob ained om he applica ion o he M5P e-
g esso . Fig. 1 illus a es he ee buil by his algo i hm. Thus, M5P ound ou
linea models (LM), whose equa ions a e lis ed below:
LM 1: Mc=−0.0160Δ −11.2781Δb +0.3237Mp+2.3766 (7)
LM 2: Mc=−0.0795Δ −0.4022Δb +0.1889Mp+2.7826 (8)
LM 3: Mc=−0.0955Δ −0.4022Δb +0.0213Mp+3.2495 (9)
LM 4: Mc=−0.3696Δ −0.4206Δb +0.0096Mp+3.2060 (10)
The analysis o his model e eals ha he b– alue is he mos significa i e
a ibu e, as i appea s in he wo fi s le els o he ee. Also, he coefficien s
co esponding o b– alue ha e he g ea es weigh s in he linea models.
LM 1
LM 2 LM 3
LM 4
> -0.004<= -0.004
> 0.011<= 0.011
> 0.024<= 0.024
b
b
Fig. 1. T ee buil wi h M5P algo i hm
Thefi s cu offisse o Δb =−0.004. Thus, he fi s linea model, LM 1,
is ound when Δb ≤−0.004. This model has he bigges absolu e alue o he
Compu a ional In elligence Techniques o P edic ing Ea hquakes
Δb coefficien (a alue o -11.2781). Mo eo e , as his coefficien is nega i e, i
can be s a ed ha he smalle is he alue o Δb, he bigge is he ea hquake
magni ude. On he o he hand, he coefficien o Mpis posi i e (a alue o
0.3237), which leads o conclude ha Mcis di ec ly ela ed o Mp.Ino he
wo ds, he magni ude o he cu en ea hquake has a di ec ela ion wi h he
magni ude o he p e ious one.
The ea hquakes occu ed wi h Δb > −0.004 a e modeled by h ee linea
models(LM2,LM3andLM4).Allo hemp esen simila Δb coefficien s,
which in ol es in e se ela ion wi h he magni ude o he cu en ea hquake,
ha is, he bigge is Δb, he smalle is Mc. Ne e heless, i s influence is mo e
mode a e han ha o LM 1.
The second cu off is se o Δb =0.011.Thus,whenΔb > 0.011 he LM 4
model is p o ided (see equa ion (10). In his model, he mos significa i e coe -
ficien is ha co esponding o Δ wi h a weigh o -0.3696, e ealing ha he
longe is he ime elapsed, he smalle is he magni ude o he cu en ea h-
quake. I is also no able ha he magni ude o he p e ious ea hquake does no
influence much in his model as i is weigh ed by 0.0096.
When he b– alue a ies be ween -0.004 and 0.011, he model p oposes wo
diffe en linea models (LM 2 and LM 3), depending on he ime elapsed be ween
he p e ious and cu en ea hquake. Al hough bo h linea models a e qui e
simila , when he ime elapsed is less o equal han 0.024 (LM 2 model), he
magni ude o he p e ious ea hquake influences much mo e han when i is
g ea e han 0.024 (LM 3 model) as he coefficien o Mpis 0.1889 in LM 2
e sus 0.0213 in LM 3.
Finally, a measu e o he quali y o esul s is now discussed. The ee p esen s
a co ela ion coefficien o 0.67. The mean absolu e e o is 0.26 and he oo
mean squa ed e o is 0.35. These e o s a e conside ed sa is ac o y gi en he
s ochas ic na u e o he p oblem s udied.
4 Conclusions
Ea hquake da a om wo pa icula a eas o he Ibe ian Peninsula ha e been
success ully mined by means o wo diffe en echniques: QAR and he M5P
algo i hm. In pa icula , QAR wi h a confidence o 83.0% and a li o 5.6 on
a e age ha e been disco e ed and a eg ession- ee wi h an e o o 0.35 has
been buil . Bo h echniques ha e disco e ed he g ea influence ha he b– alue
has in ea hquakes occu ences as i s a ia ion along wi h he ime elapsed ha e
shown o be use ul o model diffe en ea hquakes. Thus, he pa e ns disco e ed
be o e an ea hquake akes place may be use ul in subsequen p edic ions.
Acknowledgmen s
The financial suppo om he Spanish Minis y o Science and Technology,
p ojec TIN2007-68084-C-02, and om he Jun a de Andalucía, p ojec P07-
TIC-02611, is acknowledged.
Ma ínez-Ál a ez e al.
F.
Re e ences
1. Ala as, B., Akin, E.: Rough pa icle swa m op imiza ion and i s applica ions in
da a mining. So Compu ing 12(12), 1205–1218 (2008)
2. Ala as, B., Akin, E., Ka ci, A.: MODENAR: Mul i-objec i e diffe en ial e olu ion
algo i hm o mining nume ic associa ion ules. Applied So Compu ing 8(1), 646–
656 (2008)
3. Albe g, D., Las , M., Ben-Yai , A.: Induc ion o mean ou pu p edic ion ees
om con inuous empo al me eo ological da a. Jou nal o Applied Quan i a i e
me hods 4(4), 485–494 (2009)
4. B eiman, L., F iedman, J.H., Olshen, R.A., S one, C.J.: Da a Mining. In: P ac i-
cal Machine Lea ning Tools and Techniques wi h Ja a Implemen a ions. Mo gan
Kau mann, San F ancisco (1999)
5. Fu, T.-C.: A e iew on ime se ies da a mining. Enginee ing Applica ions o A i-
ficial In elligence 24(1), 164–181 (2010)
6. Geng, L., Hamil on, H.J.: In e es ingness measu es o da a mining: A su ey. ACM
Compu ing Su eys 38(3), A icle 9 (2006)
7. Ha zikos, E.V., Tsoumakas, G., Tzanis, G., Bassiliades, N., Vlaha as, I.I.: An em-
pi ical s udy on sea wa e quali y p edic ion. Knowledge-Based Sys ems 21, 471–
478 (2008)
8. Kulhanek, O.: Semina on b– alue. Technical epo , Depa men o Geophysics,
Cha les Uni e si y, P ague (2005)
9. K. Lee and W. S. Yang. His o ical seismici y o Ko ea. Bulle in o he Seismological
Socie y o Ame ica, 71(3):846–855, 2006.
10. Ma ínez-Balles e os, M., T oncoso, A., Ma ínez-Ál a ez, F., Riquelme, J.C.: Min-
ing quan i a i e associa ion ules based on e olu iona y compu a ion and i s ap-
plica ion o a mosphe ic pollu ion. In eg a ed Compu e -Aided Enginee ing 17(3),
227–242 (2010)
11. Ma ínez-Ál a ez, F., Mo ales-Es eban, A., T oncoso, A., de Jus o, J.L., Rubio-
Escude o, C.: Pa e n ecogni ion o o ecas seismic ime se ies. Expe Sys ems
wi h Applica ions 37(12), 8333–8342 (2010)
12. Oh, M.S., Tan, A., T an, V., Yang, B.S.: Machine condi ion p ognosis based on
eg ession ees and one-s ep-ahead p edic ion. Mechanical Sys ems and Signal
P ocessing 22, 1179–1193 (2008)
13. O iols-Puig, A., Casillas, J., Be nadó-Mansilla, E.: Fi s app oach owa d online
e olu ion o associa ion ules wi h lea ning classifie sys ems. In: P oceedings o he
Compu a ion Con e ence Gene ic and E olu iona y GECCO 2008, pp. 2031–2038
(2008)
14. Quinlan, J.R.: Lea ning wi h con inous classes. In: P oceedings o he 5 h Aus-
alian Join Con e ence on A ificial In elligence, pp. 343–348. Wo ld Scien ific,
Singapo e (1992)
15. Wang, Y., Wi en, I.H.: Induc ion o model ees o p edic ing con inous classes.
In: P oocedings o he pos e pape s o he Eu opean Con e ence on Machine Lea n-
ing, pp. 128–137 (1997)
16. Yan, X., Zhang, C., Zhang, S.: Gene ic algo i hm-based s a egy o iden i ying
associa ion ules wi hou speci ying ac ual minimum suppo . Expe Sys ems wi h
Applica ions 36(2), 3066–3076 (2009)
17. Yin, Y., Zhong, Z., Wang, Y.: Mining quan i a i e associa ion ules by in e al
clus e ing. Jou nal o Compu a ional In o ma ion Sys ems 4(2), 609–616 (2008)