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
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