Bayesian classification for dating archaeological sites via projectile points

SORT 45 (1) Janua y-June 2021, 33-46 DOI: 10.2436/20.8080.02.108 Bayesian classi ica ion o da ing a chaeological si es ia p ojec ile poin s Ca men A me o1, Gonzalo Ga c´ıa-Dona o2, Joaqu´ın Jim´enez-Pue o3, Sal ado Pa do-Go d´o3and Joan Be nabeu3 Abs ac Da ing is a key elemen o a chaeologis s. We p opose a Bayesian app oach o p o ide ch onol- ogy o si es ha ha e nei he adioca bon da ing no clea s a ig aphy and whose only in o ma ion comes om li hic a owheads. This classi ie is based on he Di ichle -mul inomial in e en ial p o- cess and pos e io p edic i e dis ibu ions. The p ocedu e is applied o p edic he pe iod o a se o unda ed si es loca ed in he eas o he Ibe ian Peninsula du ing he 4 h and 3 d millennium cal BC. MSC: 62F15, 62H30, 01A10. Keywo ds: Bi acial lin a owheads, ch onological model, Di ichle -mul inomial p ocess, pos e io p edic i e dis ibu ion, adioca bon da ing. 1 In oduc ion Da ing is a key elemen o a chaeologis s. A ime scale o loca e he in o ma ion col- lec ed om exca a ions and ield wo k is always necessa y in o de o build, albei wi h unce ain y, ou mos emo e pas . A chaeological scien is s gene ally use s a i- g aphic expe in o ma ion and da ing echniques o examining he age o he ele an a i ac s. Bayesian in e ence is commonly used in a chaeology as a ool o cons uc obus ch onological models based on in o ma ion om scien i ic da a as well as expe knowledge (e.g. s a ig aphy) (Buck, Ca anagh and Li on, 1996). Radioca bon da ing is one o he mos popula echniques o ob aining da a due o ca bon’s p esence in any being ha has li ed on Ea h. Howe e , i is no always possible in all s udies o collec o ganic ma e ial and ob ain ha ype o in o ma ion o o ha e good s a ig aphic e e ences. In hese cases, he challenge is o be able o assign non 1Depa amen d’Es ad´ıs ica i IO. Uni e si a de Val`encia, Ca e Doc o Moline 50, 46100, Bu jasso , Spain. 2Depa men o Economics and Finance. Uni e sidad Cas illa-La Mancha. Edi icio Ju ´ıdico-Emp esa ial “Mel- cho de Macanaz”, Plaza de la Uni e sidad, 1, 02071 Albace e, Spain. 3Depa men o P ehis o y, A chaeology and Ancien His o y, Uni e si a de Val`encia, A da. Blasco Ib´a˜nez 28, 46010 Val`encia, Spain. Recei ed: No embe 2020 Accep ed: Ma ch 2021 34 Bayesian classi ica ion o da ing a chaeological si es ia p ojec ile poin s adioca bon da ed collec ions o speci ic ch onological imes. The ele an in o ma ion is based on cul u al ma e ial ha includes elemen s wi h ma ke s ha poin ou he di - e en cul u al ai s o he social g oups in ol ed as well as he social ela ionships be ween hem. One o hese use ul i ems is he li hic p oduc ions, and mo e speci ically he a owheads. Du ing he 4 h and 3 d millennium cal BC bi acial lin a owheads appea and sp ead in he eas o he Ibe ian Peninsula. A chaeological esea ch sugges s ha he shape o hese a owheads could be ela ed wi h speci ic pe iod and/o geog aphical social uni s spa ially de ined. In his con ex , we p opose an au oma ic Bayesian p ocedu e, e y popula in ex classi ica ion (Wang, Hodges and Tang, 2003), based on p edic i e p obabili y dis i- bu ions o classi ying he pe iod o which an unda ed si e belongs acco ding o he ype and numbe o a ows ound in i . This p oposal akes in o accoun he Di ichle - mul inomial in e en ial p ocess o lea ning abou he p opo ion o di e en ypes o a owheads in each ch onological pe iod, and he concep o pos e io p edic i e dis i- bu ion o a new unda ed si e. This pape is o ganized in i e sec ions. Following his in oduc ion, Sec ion 2 b ie ly in oduces he a chaeological amewo k and he li hic ma e ial ha will be he basis o he classi ica ion p ocess. Sec ion 3 desc ibes he wo s ages o he Bayesian s a is ical analysis. The i s is o an in e en ial ype and ocuses on he s udy o he abundance o di e en ypes o a ows in he di e en pe iods conside ed. The second uses he in o ma ion om he i s s age o p edic he pe iod o an unda ed si e om he numbe and ype o a owheads encoun e ed. Sec ion 4 applies he me hodological p ocedu e om he p e ious sec ion o a se o si es in he eas o he Ibe ian Peninsula du ing La e Neoli hic and Chalcoli hic (4 h-3 d millennium BC). Finally, Sec ion 5 concludes. 2 Ch onological pe iods and li hic in o ma ion One o he main goals in a chaeological esea ch is ocused on he way he membe s o he p ehis o ic cul u es in e ac wi h he landscape and he objec s. F om an e olu i e pe spec i e, he way human cul u es change h ough space- ime is de e mined by inhe - i ance pa e ns, adap a ion and in e ac ion (Shennan, C ema, and Ke ig, 2015). The e- o e, he analysis o i ems om he a chaeological eco ds, able o cap u e he cul u al e olu ion o he human g oups, would be a main goal o he esea che . The concep o “cul u e” co e s many ac o s. Hence, we will use he ma e ial cul u e as an a chaeologic p oxy in o de o analyse he e olu ion and dispe sion o he cul u al ai s in he s udy a ea. No all he i ems included in ma e ial cul u e a e use ul o ha . Those which show a wide geog aphic and cul u al dispe sion o whose a iabili y is low a e no con enien o de ec changes. This is no he case wi h li hic p oduc ions, and mo e speci ically a owheads, which p o ide in o ma ion no only o unde s anding Ca men A me o, Gonzalo Ga c´ıa-Dona o, Joaqu´ın Jim´ enez-Pue o... 35 he socio-economic and cul u al s uc u es o human g oups, bu hey can be used as a aluable ool o ch onological da ing. The a i al o he neoli hic economy, based on domes ic esou ces, in he Ibe ian Peninsula is da ed on he i s hal o he 6 h millennium cal BC. We will ha e o wai un il he 4 h-3 d millennium o be able o wi ness clea winds o change. This is he momen o he appea ance o a highe le el o hie a chy in some socie ies. The La e Ne- oli hic (4 h-3 d millennium cal BC) in he o ien al Ibe ian ac¸ade is he ime o he ansi o a highe complexi y in social and economic e ms. This p ocess will las long and i will c ys allize by he end o he 3 d millenium cal BC (Be nabeu and O ozco, 2014). The e alua ion o his p ocess in such a huge ame aces some p oblems which need o be add essed. One o hese di icul ies is closely associa ed wi h he ch onological a ibu ion o a big pa o he pe iod’s a chaeological eco d due o sca ce adioca bon da a. Type 1 wi h homboid o hombus-eye shape Type 2 wi h side appendages o c uci o m Type 3 lea -like Type 4 wi h peduncle bu wi hou lin s Type 5 wi h a conca e base Type 6 asymme ic Type 7 wi h peduncle and lin s Figu e 1:A owhead ypes used o he s udy. The classi ica ion o he a owheads in his pe iod is based on he p e ious wo ks pe - o med a ound he ypological o maliza ion o he s udy a ea. They a e mainly inspi ed 36 Bayesian classi ica ion o da ing a chaeological si es ia p ojec ile poin s by mo pho-desc ip i e ypologies. The e o e, he classi ica ion con ains a unc ional and mo phological meaning. A owheads cons i u e a e y ep esen a i e ool g oup o he La e Neoli hic and Chalcoli hic. Thei unc ion is qui e p o ed hanks o he s udies in aceology, expe imen al a chaeology and e noa chaeology. Some well known exam- ples a e he spec acula indings o a owheads s ill nailed in o he ic im bones, p esen in many bu ials om he 4 h and 3 d millennium BC (i.e. San Juan an e Po am La i- nam: Vegas 2007). We canno o ge he awesome inding o a ull equipmen ¨ O zi, he “Iceman”, disco e ed in he Alps (Ca e-B owne, 2016), and excep ionally conse ed. Mo eo e , he exis ence o exca a ed si es (E e a del Ped egal) in which he whole a - owhead ope a i e chain p ocess can be obse ed, has p o ided addi ional in o ma ion (Juan-Cabanilles, 1994). The a owhead ypes p esen in he a chaeological eco ds ha e been classi ied in se en ypes ollowing a mo phological c i e ion, based on p e ious ypologies o he s udy a ea (Juan-Cabanilles, 2008) (See Figu e 1). 3 Bayesian classi ica ion p ocess Bayesian classi ica ion wi hin he amewo k o a chaelogical da a ion wi h li hic in o - ma ion will p o ide a p obabili y dis ibu ion o he pe iod o which an unda ed si e belongs in which a gi en se o di e en ypes o a owheads has been ound. This p obabili y dis ibu ion depends on he knowledge o he abundance o each ype o a owheads in each pe iod, exp essed ia he pos e io dis ibu ion o he p obabili y associa ed wi h each ype o a owhead, and he pos e io p edic i e dis ibu ion o he pe iod o ha pa icula upda ed si e. 3.1 Di ichle -mul inomial in e en ial p ocess Le Yi j be he andom a iable ha desc ibes he numbe o ype j,j=1,...,Ja ow- heads, o he o al nicollec ed in he si es belonging o pe iod i,i=1,...,I. We de ine he andom ec o Yi= (Yi1,Yi2,...,Yi,J−1)Tand he p obabili y ec o θ θ θi= (θi1,θi2,...,θi,J−1)T, whe e θi j is he p obabili y ha an a owhead o pe iod iis o ype j. A p obabilis ic model o Yi|θ θ θiis he mul inomial dis ibu ion, Mn(θ θ θi,ni), wi h p obabili y dis ibu ion (yi|θ θ θi) = ni! ∏J−1 j=1yi j!yiJ!J−1 ∏ j=1 θyi j i j θyi j iJ ,(1) whe e yiis an obse a ion o Yi,yiJ =ni−∑J−1 j=1yi j is he o al numbe o a owheads o ype Jin he si es o pe iod i, and θiJ =1−∑J−1 j=1θi j is he p obabili y ha an a owhead o pe iod iis o ype J. Ca men A me o, Gonzalo Ga c´ıa-Dona o, Joaqu´ın Jim´ enez-Pue o... 37 The combina ion o a mul inomial sampling model wi h a conjuga e Di ichle p io dis ibu ion was p oposed by Lindley (1964) and Good (1967) as he gene alisa ion o he be a-binomial model. The Di ichle dis ibu ion o θ θ θiwi h pa ame e s α α αi= (αi1,...,αiJ)T,αi j >0,j=1,...,J, Di (α α αi), is a mul i a ia e con inuous dis ibu ion wi h join densi y unc ion π(θ θ θi) = Γ(αi+) ∏J j=1Γ(αim)J−1 ∏ j=1 θαi j−1 i j θαiJ−1 iJ ,(2) whe e Γ(·) ep esen s he gamma unc ion and αi+=∑J j=1αi j. We assume an in e en ial p ocess o each θ θ θi,i=1,...,Iin he amewo k o he Di ichle -mul inomial p ocess wi h a non-in o ma i e p io dis ibu ion o θ θ θi ha gi es all he p o agonism o he p ocess o he da a. The e a e many p oposals o elici he pa ame e s α α αiin a non-in o ma i e way: Haldane’s p io , Pe ks’ p io o e e ence dis- ance p io , hie a chical app oach p io and Je eys’ p io o common e e ence p io , and Bayes-Laplace p io . All hem ha e good heo e ical p ope ies bu hey also ha e some small sho comings. We choose he Pe ks’ p io as a esul o Al a es, A me o and Fo e (2018). This p io was i s ly p oposed by Pe ks (1947), bu ecen ly i has been also ob ained as he e e ence dis ance p io by Be ge e al. (2015). This is a Di ich- le dis ibu ion wi h all pa ame e s equal o 1/J, whe e Jis he numbe o a ow ypes. Figu e 2 shows he densi y and o he cha ac e is ics o a Pe k’s dis ibu ion wi h h ee ca ego ies. Figu e 2:Pe ks’ dis ibu ion when he numbe o ypes o a owheads is J =3(a), i s p ojec ion on o he simplex iangle (b), and he ma ginal p io dis ibu ion o each indi idual componen , a be a dis ibu ion wi h pa ame e s 1/3and 2/3, Be(1/3,2/3), which main ains high densi y alues close o 0 and 1(c). The pos e io dis ibu ion o θ θ θiwhen da a yia e obse ed is also a Di ichle dis i- bu ion (Lindley, 1964; Good, 1967), π(θ θ θi|yi) = Di (αi1=yi1+ (1/J),...,αiJ =yiJ +(1/J)).(3) 38 Bayesian classi ica ion o da ing a chaeological si es ia p ojec ile poin s This pos e io dis ibu ion has an impo an and posi i e ea u e: ne e assigns absolu e p obabili ies 1 o 0 o he p esence o any ype o heada ows. This ac a oids wo king wi h absolu e alues o he p obabili ies, 0 and 1, which would p e en u u e upda es o hei alues gene a ed by new da a. The ma ginal pos e io dis ibu ion o each p obabili y θi j is he be a dis ibu ion (Gelman e al. , 2014) π(θi j |yi) = Be(αi j,αi+−αi j),(4) wi h pos e io mean and a iance αi j/αi+and αi j(αi+−αi j)/(α2 i+(αi++1)), espec- i ely. 3.2 P edic i e p ocess A e lea ning abou he dis ibu ion o he p opo ion o a owheads ypes in each si e, we ha e o assign a p obabili y dis ibu ion o he andom a iable ha desc ibes he pe iod m∗ o which a new unda ed si e s∗belongs gi en ha a o al o n∗a owheads y∗= (y∗ 1,...,y∗ J)Tha e been obse ed in i . Following Bayes’ heo em: P(m∗=mi|y∗,y)∝P(Y∗=y∗|m∗=mi,y)P(m∗=mi|y),i=1,...,I,(5) whe e y= (y1,...,yI)Ta e he obse ed da a in he p e ious es ima ion p ocess and Y∗= (Y∗ 1,...,Y∗ J)Tis he andom ec o ha desc ibes he numbe o a owheads o he di e en ypes ha will be eco ded in ha new si e. I is impo an o no e ha Yand Y∗ in capi al le e s e e o he andom ec o ha gene a e o will gene a e he da a yand y∗, espec i ely, which we always ep esen by lowe case le e s. The as e isk is used o ep esen he subsequen andom a iables and obse a ions o he p edic ion p ocess. The pos e io p edic i e dis ibu ion in (5) is p opo ional o he p oduc o wo e ms. The i s one is: P(Y∗=y∗|m∗=mi,y) = ZP(Y∗=y∗|θ θ θi,m∗=mi,y)π(θ θ θi|m∗=mi,y)dθ θ θi =Zn∗! y∗ 1!y∗ 2!···y∗ J!θy∗ 1 i1θy∗ 2 i2···θyJ∗ iJ Γ(αi+) ∏J j=1Γ(αi j)θαi1−1 i1θαi2−1 i2···θαiJ −1 iJ dθ θ θi =n∗! y∗ 1!y∗ 2!···y∗ J! Γ(αi+) ∏J j=1Γ(αi j)Zθαi1+y∗ 1−1 i1θαi2+y∗ 2−1 i2···θαiJ+yJ∗−1 iJ dθ θ θi =n∗! y∗ 1!y∗ 2!···yJ∗! Γ(αi+) Γ(αi++n∗) J ∏ j=1 Γ(αi j +y∗ j) Γ(αi j). The i s p obabili y in he in eg and, P(Y∗=y∗|θ θ θi,m∗=mi,y), is associa ed wi h new expe imen al esul s in he p esence o θ θ θiand he da a y om he es ima ion p ocess Ca men A me o, Gonzalo Ga c´ıa-Dona o, Joaqu´ın Jim´ enez-Pue o... 39 which a e i ele an due o he p esence o θ θ θi. I is a mul inomial p obabili y compu ed om (1). The second e m, π(θ θ θi|m∗=mi,y), is he Di ichle pos e io dis ibu ion o θ θ θigi en in (3). The second elemen in he p oduc in (5), P(m∗=mi|y), can be es ima ed as he p opo ion o si es in he sample o each o he pe iods unde conside a ion (Ba be , 2012). 4 Eas o he Ibe ian Peninsula si es du ing he 4 h and 3 d millennium cal BC We apply he classi ica ion p ocedu e abo e o a se o unda ed si es in he Eas o he Ibe ian Peninsula du ing he 4 h and 3 d millennium cal BC. Da a o he in e en ial p o- cess o he s udy come om 31 a chaeological si es adioca bon da ed wi h a owheads, clea con ex s and s a ig aphy. 4.1 In e en ial p ocess All 14C da ed si es ha e been il e ed using only hose whose adioca bon da es come om sho -li ed singula samples. The inal le els used o he pe iodiza ion a e: A e- nal de la Cos a (Be nabeu, 1993), Ba anc del Migdia (Sole D´ıaz e al., 2016), Beni- eixi (Pascual Beney o, 2010), Cam´ı de Missena (Pascual Beney o, Ba be `a and Ribe a, 2005), Cola a (G´omez Puche e al., 2004), Co a del Rande o (Sole D´ıaz e al., 2016), Co a dels Diable s (Aguilella, Ola ia Puyoles and Gusi Jene , 1999), Jo ades (Be nabeu, 1993), La Vi al (P´e ez-Jo d`a e al., 2011), Niue (Be nabeu, Pascual Beni o, O ozco K¨ohle , Badal Ga c´ıa, Fumanal Ga c´ıa and Ga c´ıa Puchol, 1994), and Quin a e (Ga c´ıa Puchol e al., 2014). These si es a e loca ed in he eas e n Medi e anean a ea. Figu e 3 Figu e 3:Si ua ion map o he si es wi h a owheads p esen in he s udy a ea. 40 Bayesian classi ica ion o da ing a chaeological si es ia p ojec ile poin s shows a map wi h he da ed si ed as well as he si es wi hou 14C da a ion whose ch ono- logical classi ica ion is he inal objec o his s udy. Based on he ch ono-s a ig aphic and a ailable expe in o ma ion, we ha e p o- posed i e in e als o ch onological pe iods o ganiza ion comp ised be ween ca. 4600- 3200 cal BC. Table 1 includes he pe iod o each o he pe iods conside ed as well as he si es included in each o hem. Each si e usually con ains many di e en a chaeological le els a ached o di e - en momen s o occupa ion. In his speci ic case, a chaeological con ex s con aining a owheads ha e been da ed h ough adioca bon de e mina ions. Some o hese si es con ain di e en da ed le els in which a owheads we e p esen . Hence we ha e de- sc ibed hem wi h he name o he si e and a numbe o di e en ia e hem. Based on he ch ono-s a ig aphic and a ailable expe in o ma ion, we ha e p oposed i e successi e in e als o ch onological pe iods comp ised be ween ca. 4600-2150 cal BC. These pe- iods ha e esul ed om he applica ion o Bayesian adioca bon modeling me hods o he a chaeologic in o ma ion a ailable o each pe iod. Table 1:Pe iods and si es ex ac ed om clea a chaeological con ex s wi h adioca bon de e mina ions. Si es 14C da ed Pe iod Jo ades 1, Jo ades 2, and Niue 1 1 Cola a 1, Cola a 2, Jo ades 3, Jo ades 4, Niue 2, 2 and Quin a e Beni eixi , Diable s 1, Diable s 2, Diable s 3, 3 Jo ades 5, La Vi al 1, La Vi al 2, Migdia 1, Missena 1, Niue 3, Niue 4, Rande o 1, and Rande o 2 La Vi al 3, Migdia 2, Missena 2, and Missena 3 4 A enal Cos a, La Vi al 3, Missena 4, Missena 5, 5 and Missena 6 Table 2:Pos e io Di ichle dis ibu ion o he p opo ion o a owheads om ype 1 o ype 7 in each o he pe iods conside ed. Pe iod Pos e io dis ibu ion 1 Di (15/7, 22/7, 8/7, 1/7, 1/7, 1/7, 1/7) 2 Di (29/7, 36/7, 15/7, 8/7, 1/7, 1/7, 1/7) 3 Di (43/7, 1/7, 43/7, 64/7, 29/7, 1/7, 71/7) 4 Di (15/7, 1/7, 15/7, 8/7, 15/7, 1/7, 43/7) 5 Di (1/7, 1/7, 1/7, 15/7, 1/7, 8/7, 36/7) Table 2 includes he pos e io dis ibu ion o he di e en ypes o a owheads in each o he pe iods conside ed. In all o hem he selec ed p io dis ibu ion is he Pe k Ca men A me o, Gonzalo Ga c´ıa-Dona o, Joaqu´ın Jim´ enez-Pue o... 41 pe iod 1 pe iod 2 pe iod 3 pe iod 4 pe iod 5 ype.1 ype.2 ype.3 ype.4 ype.5 ype.6 ype.7 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.0 2.5 5.0 7.5 10.0 0.0 2.5 5.0 7.5 10.0 0.0 2.5 5.0 7.5 10.0 0.0 2.5 5.0 7.5 10.0 0.0 2.5 5.0 7.5 10.0 0.0 2.5 5.0 7.5 10.0 0.0 2.5 5.0 7.5 10.0 P obabili y Figu e 4:Pos e io ma ginal dis ibu ion o he p obabili y associa ed wi h each ype o a owhead in each o he pe iods in he s udy. dis ibu ion Di (1/7, 1/7, 1/7, 1/7, 1/7, 1/7, 1/7). The e o e, hose pa ame e s o he co - esponding pos e io dis ibu ion ha con inue o be wo h 1/7 co espond o hose ypes o a ows ha ha e no been obse ed in he sample. Table 3 shows he pos e io mean o he p obabili y associa ed wi h each ype o a owhead in each o he pe iods in he s udy. Figu e 4 shows he pos e io ma ginal dis ibu ion o he p obabili y o he di e en ypes o a owheads in each o he i e ch onological pe iods conside ed. Resul s in Table 3 and Figu e 4 indica e ha he dis- ibu ion o he di e en ypes o a owheads is e y simila in Pe iods 1 and 2: Type 1 and 2 a owheads a e he mos abundan and abou he 75% and 70% o he o al o a owheads in bo h pe iods a e ype 1 o 2. Type 3 a owheads ha e poo ele ance in bo h Pe iods and ypes 4, 5, 6, and 7 a e i ually nonexis en . In Pe iod 3, we ind p ac ically no ype 2 and 6 a owheads. The emaining a owheads in his pe iod ha e a p esence qui e simila bu ype 4 and 7 ha e a sligh ly highe p esence. Pe iod 4 shows a la ge p esence o ype 7 a ows and, o a lesse ex en , o ype 1, 3 and 5 a ows (p o- babili ies o abou 0.15). A owheads o ype 2 and 6 ha e no ele ance. App oxima ely 57% and 24% o he a ows o Pe iod 5 a e o ype 7 and 4, espec i ely. The emaining a owhead ypes, excep possibly hose o ype 6, a e essen ially i ele an .