Bayesian classification for dating archaeological sites via projectile points

SORT 45 (1) January-June 2021, 33-46 DOI: 10.2436/20.8080.02.108 Bayesian classification for dating archaeological sites via projectile points Carmen Armero1, Gonzalo Garc´ıa-Donato2, Joaqu´ın Jim´enez-Puerto3, Salvador Pardo-Gord´o3and Joan Bernabeu3 Abstract Dating is a key element for archaeologists. We propose a Bayesian approach to provide chronology to sites that have neither radiocarbon dating nor clear stratigraphy and whose only information comes from lithic arrowheads. This classifier is based on the Dirichlet-multinomial inferential process and posterior predictive distributions. The procedure is applied to predict the period of a set of undated sites located in the east of the Iberian Peninsula during the 4th and 3rd millennium cal BC. MSC: 62F15, 62H30, 01A10. Keywords: Bifacial flint arrowheads, chronological model, Dirichlet-multinomial process, posterior predictive distribution, radiocarbon dating. 1 Introduction Dating is a key element for archaeologists. A time scale to locate the information collected from excavations and field work is always necessary in order to build, albeit with uncertainty, our most remote past. Archaeological scientists generally use stratigraphic expert information and dating techniques for examining the age of the relevant artifacts. Bayesian inference is commonly used in archaeology as a tool to construct robust chronological models based on information from scientific data as well as expert knowledge (e.g. stratigraphy) (Buck, Cavanagh and Litton, 1996). Radiocarbon dating is one of the most popular techniques for obtaining data due to carbon’s presence in any being that has lived on Earth. However, it is not always possible in all studies to collect organic material and obtain that type of information or to have good stratigraphic references. In these cases, the challenge is to be able to assign non 1Departament d’Estad´ıstica i IO. Universitat de Val`encia, Carrer Doctor Moliner 50, 46100, Burjassot, Spain. 2Department of Economics and Finance. Universidad Castilla-La Mancha. Edificio Jur´ıdico-Empresarial “Melchor de Macanaz”, Plaza de la Universidad, 1, 02071 Albacete, Spain. 3Department of Prehistory, Archaeology and Ancient History, Universitat de Val`encia, Avda. Blasco Ib´a˜nez 28, 46010 Val`encia, Spain. Received: November 2020 Accepted: March 2021 34 Bayesian classification for dating archaeological sites via projectile points radiocarbon dated collections to specific chronological times. The relevant information is based on cultural material that includes elements with markers that point out the different cultural traits of the social groups involved as well as the social relationships between them. One of these useful items is the lithic productions, and more specifically the arrowheads. During the 4th and 3rd millennium cal BC bifacial flint arrowheads appear and spread in the east of the Iberian Peninsula. Archaeological research suggests that the shape of these arrowheads could be related with specific period and/or geographical social units spatially defined. In this context, we propose an automatic Bayesian procedure, very popular in text classification (Wang, Hodges and Tang, 2003), based on predictive probability distributions for classifying the period to which an undated site belongs according to the type and number of arrows found in it. This proposal takes into account the Dirichletmultinomial inferential process for learning about the proportion of different types of arrowheads in each chronological period, and the concept of posterior predictive distribution for a new undated site. This paper is organized in five sections. Following this introduction, Section 2 briefly introduces the archaeological framework and the lithic material that will be the basis for the classification process. Section 3 describes the two stages of the Bayesian statistical analysis. The first is of an inferential type and focuses on the study of the abundance of different types of arrows in the different periods considered. The second uses the information from the first stage to predict the period of an undated site from the number and type of arrowheads encountered. Section 4 applies the methodological procedure from the previous section to a set of sites in the east of the Iberian Peninsula during Late Neolithic and Chalcolithic (4th-3rd millennium BC). Finally, Section 5 concludes. 2 Chronological periods and lithic information One of the main goals in archaeological research is focused on the way the members of the prehistoric cultures interact with the landscape and the objects. From an evolutive perspective, the way human cultures change through space-time is determined by inheritance patterns, adaptation and interaction (Shennan, Crema, and Kerig, 2015). Therefore, the analysis of items from the archaeological records, able to capture the cultural evolution of the human groups, would be a main goal for the researcher. The concept of “culture” covers many factors. Hence, we will use the material culture as an archaeologic proxy in order to analyse the evolution and dispersion of the cultural traits in the study area. Not all the items included in material culture are useful for that. Those which show a wide geographic and cultural dispersion or whose variability is low are not convenient to detect changes. This is not the case with lithic productions, and more specifically arrowheads, which provide information not only for understanding Carmen Armero, Gonzalo Garc´ıa-Donato, Joaqu´ın Jim´ enez-Puerto... 35 the socio-economic and cultural structures of human groups, but they can be used as a valuable tool for chronological dating. The arrival of the neolithic economy, based on domestic resources, in the Iberian Peninsula is dated on the first half of the 6th millennium cal BC. We will have to wait until the 4th-3rd millennium to be able to witness clear winds of change. This is the moment of the appearance of a higher level of hierarchy in some societies. The Late Neolithic (4th-3rd millennium cal BC) in the oriental Iberian fac¸ade is the time of the transit to a higher complexity in social and economic terms. This process will last long and it will crystallize by the end of the 3rd millenium cal BC (Bernabeu and Orozco, 2014). The evaluation of this process in such a huge frame faces some problems which need to be addressed. One of these difficulties is closely associated with the chronological attribution of a big part of the period’s archaeological record due to scarce radiocarbon data. Type 1 with rhomboid or rhombus-eye shape Type 2 with side appendages or cruciform Type 3 leaf-like Type 4 with peduncle but without flints Type 5 with a concave base Type 6 asymmetric Type 7 with peduncle and flints Figure 1:Arrowhead types used for the study. The classification of the arrowheads in this period is based on the previous works performed around the typological formalization for the study area. They are mainly inspired 36 Bayesian classification for dating archaeological sites via projectile points by morpho-descriptive typologies. Therefore, the classification contains a functional and morphological meaning. Arrowheads constitute a very representative tool group of the Late Neolithic and Chalcolithic. Their function is quite proved thanks to the studies in traceology, experimental archaeology and etnoarchaeology. Some well known examples are the spectacular findings of arrowheads still nailed into the victim bones, present in many burials from the 4th and 3rd millennium BC (i.e. San Juan ante Portam Latinam: Vegas 2007). We cannot forget the awesome finding of a full equipment ¨ Otzi, the “Iceman”, discovered in the Alps (Cave-Browne, 2016), and exceptionally conserved. Moreover, the existence of excavated sites (Ereta del Pedregal) in which the whole arrowhead operative chain process can be observed, has provided additional information (Juan-Cabanilles, 1994). The arrowhead types present in the archaeological records have been classified in seven types following a morphological criterion, based on previous typologies for the study area (Juan-Cabanilles, 2008) (See Figure 1). 3 Bayesian classification process Bayesian classification within the framework of archaelogical datation with lithic information will provide a probability distribution for the period to which an undated site belongs in which a given set of different types of arrowheads has been found. This probability distribution depends on the knowledge of the abundance of each type of arrowheads in each period, expressed via the posterior distribution for the probability associated with each type of arrowhead, and the posterior predictive distribution for the period of that particular updated site. 3.1 Dirichlet-multinomial inferential process Let Yi j be the random variable that describes the number of type j,j=1,...,Jarrowheads, of the total nicollected in the sites belonging to period i,i=1,...,I. We define the randomvectorYi= (Yi1,Yi2,...,Yi,J−1)Tand the probability vector θ θ θi= (θi1,θi2,...,θi,J−1)T, where θi j is the probability that an arrowhead of period iis of type j. A probabilistic model for Yi|θ θ θiis the multinomial distribution, Mn(θ θ θi,ni), with probability distribution f(yi|θ θ θi) = ni! ∏J−1 j=1yi j!yiJ!J−1 ∏ j=1 θyi j i j θyi j iJ ,(1) where yiis an observation of Yi,yiJ =ni−∑J−1 j=1yi j is the total number of arrowheads of type Jin the sites of period i, and θiJ =1−∑J−1 j=1θi j is the probability that an arrowhead of period iis of type J. Carmen Armero, Gonzalo Garc´ıa-Donato, Joaqu´ın Jim´ enez-Puerto... 37 The combination of a multinomial sampling model with a conjugate Dirichlet prior distribution was proposed by Lindley (1964) and Good (1967) as the generalisation of the beta-binomial model. The Dirichlet distribution for θ θ θiwith parameters α α αi= (αi1,...,αiJ)T,αi j >0,j=1,...,J, Dir(α α αi), is a multivariate continuous distribution with joint density function π(θ θ θi) = Γ(αi+) ∏J j=1Γ(αim)J−1 ∏ j=1 θαi j−1 i j θαiJ−1 iJ ,(2) where Γ(·)represents the gamma function and αi+=∑J j=1αi j. We assume an inferential process for each θ θ θi,i=1,...,Iin the framework of the Dirichlet-multinomial process with a non-informative prior distribution for θ θ θithat gives all the protagonism of the process to the data. There are many proposals for elicit the parameters α α αiin a non-informative way: Haldane’s prior, Perks’ prior or reference distance prior, hierarchical approach prior and Jeffreys’ prior or common reference prior, and Bayes-Laplace prior. All them have good theoretical properties but they also have some small shortcomings. We choose the Perks’ prior as a result of Alvares, Armero and Forte (2018). This prior was firstly proposed by Perks (1947), but recently it has been also obtained as the reference distance prior by Berger et al. (2015). This is a Dirichlet distribution with all parameters equal to 1/J, where Jis the number of arrow types. Figure 2 shows the density and other characteristics of a Perk’s distribution with three categories. Figure 2:Perks’ distribution when the number of types of arrowheads is J =3(a), its projection onto the simplex triangle (b), and the marginal prior distribution for each individual component, a beta distribution with parameters 1/3and 2/3, Be(1/3,2/3), which maintains high density values close to 0 and 1(c). The posterior distribution for θ θ θiwhen data yiare observed is also a Dirichlet distribution (Lindley, 1964; Good, 1967), π(θ θ θi|yi) = Dir(αi1=yi1+ (1/J),...,αiJ =yiJ +(1/J)).(3) 38 Bayesian classification for dating archaeological sites via projectile points This posterior distribution has an important and positive feature: never assigns absolute probabilities 1 or 0 to the presence of any type of headarrows. This fact avoids working with absolute values of the probabilities, 0 and 1, which would prevent future updates of their values generated by new data. The marginal posterior distribution for each probability θi j is the beta distribution (Gelman et al. , 2014) π(θi j |yi) = Be(αi j,αi+−αi j),(4) with posterior mean and variance αi j/αi+and αi j(αi+−αi j)/(α2 i+(αi++1)), respectively. 3.2 Predictive process After learning about the distribution of the proportion of arrowheads types in each site, we have to assign a probability distribution to the random variable that describes the period m∗to which a new undated site s∗belongs given that a total of n∗arrowheads y∗= (y∗ 1,...,y∗ J)Thave been observed in it. Following Bayes’ theorem: P(m∗=mi|y∗,y)∝P(Y∗=y∗|m∗=mi,y)P(m∗=mi|y),i=1,...,I,(5) where y= (y1,...,yI)Tare the observed data in the previous estimation process and Y∗= (Y∗ 1,...,Y∗ J)Tis the random vector that describes the number of arrowheads of the different types that will be recorded in that new site. It is important to note that Yand Y∗ in capital letters refer to the random vector that generate or will generate the data yand y∗, respectively, which we always represent by lower case letters. The asterisk is used to represent the subsequent random variables and observations of the prediction process. The posterior predictive distribution in (5) is proportional to the product of two terms. The first 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 first probability in the integrand, P(Y∗=y∗|θ θ θi,m∗=mi,y), is associated with new experimental results in the presence of θ θ θiand the data yfrom the estimation process Carmen Armero, Gonzalo Garc´ıa-Donato, Joaqu´ın Jim´ enez-Puerto... 39 which are irrelevant due to the presence of θ θ θi. It is a multinomial probability computed from (1). The second term, π(θ θ θi|m∗=mi,y), is the Dirichlet posterior distribution for θ θ θigiven in (3). The second element in the product in (5), P(m∗=mi|y), can be estimated as the proportion of sites in the sample for each of the periods under consideration (Barber, 2012). 4 East of the Iberian Peninsula sites during the 4th and 3rd millennium cal BC We apply the classification procedure above to a set of undated sites in the East of the Iberian Peninsula during the 4th and 3rd millennium cal BC. Data for the inferential process of the study come from 31 archaeological sites radiocarbon dated with arrowheads, clear contexts and stratigraphy. 4.1 Inferential process All 14C dated sites have been filtered using only those whose radiocarbon dates come from short-lived singular samples. The final levels used for the periodization are: Arenal de la Costa (Bernabeu, 1993), Barranc del Migdia (Soler D´ıaz et al., 2016), Beniteixir (Pascual Beneyto, 2010), Cam´ı de Missena (Pascual Beneyto, Barber`a and Ribera, 2005), Colata (G´omez Puche et al., 2004), Cova del Randero (Soler D´ıaz et al., 2016), Cova dels Diablets (Aguilella, Olaria Puyoles and Gusi Jener, 1999), Jovades (Bernabeu, 1993), La Vital (P´erez-Jord`a et al., 2011), Niuet (Bernabeu, Pascual Benito, Orozco K¨ohler, Badal Garc´ıa, Fumanal Garc´ıa and Garc´ıa Puchol, 1994), and Quintaret (Garc´ıa Puchol et al., 2014). These sites are located in the eastern Mediterranean area. Figure 3 Figure 3:Situation map of the sites with arrowheads present in the study area. 40 Bayesian classification for dating archaeological sites via projectile points shows a map with the dated sited as well as the sites without 14C datation whose chronological classification is the final object of this study. Based on the chrono-stratigraphic and available expert information, we have proposed five intervals or chronological periods organization comprised between ca. 46003200 cal BC. Table 1 includes the period of each of the periods considered as well as the sites included in each of them. Each site usually contains many different archaeological levels attached to different moments of occupation. In this specific case, archaeological contexts containing arrowheads have been dated through radiocarbon determinations. Some of these sites contain different dated levels in which arrowheads were present. Hence we have described them with the name of the site and a number to differentiate them. Based on the chrono-stratigraphic and available expert information, we have proposed five successive intervals or chronological periods comprised between ca. 4600-2150 cal BC. These periods have resulted from the application of Bayesian radiocarbon modeling methods to the archaeologic information available for each period. Table 1:Periods and sites extracted from clear archaeological contexts with radiocarbon determinations. Sites 14C dated Period Jovades 1, Jovades 2, and Niuet 1 1 Colata 1, Colata 2, Jovades 3, Jovades 4, Niuet 2, 2 and Quintaret Beniteixir, Diablets 1, Diablets 2, Diablets 3, 3 Jovades 5, La Vital 1, La Vital 2, Migdia 1, Missena 1, Niuet 3, Niuet 4, Randero 1, and Randero 2 La Vital 3, Migdia 2, Missena 2, and Missena 3 4 Arenal Costa, La Vital 3, Missena 4, Missena 5, 5 and Missena 6 Table 2:Posterior Dirichlet distribution for the proportion of arrowheads from type 1 to type 7 in each of the periods considered. Period Posterior distribution 1 Dir(15/7, 22/7, 8/7, 1/7, 1/7, 1/7, 1/7) 2 Dir(29/7, 36/7, 15/7, 8/7, 1/7, 1/7, 1/7) 3 Dir(43/7, 1/7, 43/7, 64/7, 29/7, 1/7, 71/7) 4 Dir(15/7, 1/7, 15/7, 8/7, 15/7, 1/7, 43/7) 5 Dir(1/7, 1/7, 1/7, 15/7, 1/7, 8/7, 36/7) Table 2 includes the posterior distribution of the different types of arrowheads in each of the periods considered. In all of them the selected prior distribution is the Perk Carmen Armero, Gonzalo Garc´ıa-Donato, Joaqu´ın Jim´ enez-Puerto... 41 period 1 period 2 period 3 period 4 period 5 type.1 type.2 type.3 type.4 type.5 type.6 type.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 Probability Figure 4:Posterior marginal distribution for the probability associated with each type of arrowhead in each of the periods in the study. distribution Dir(1/7, 1/7, 1/7, 1/7, 1/7, 1/7, 1/7). Therefore, those parameters of the corresponding posterior distribution that continue to be worth 1/7 correspond to those types of arrows that have not been observed in the sample. Table 3 shows the posterior mean for the probability associated with each type of arrowhead in each of the periods in the study. Figure 4 shows the posterior marginal distribution of the probability of the different types of arrowheads in each of the five chronological periods considered. Results in Table 3 and Figure 4 indicate that the distribution of the different types of arrowheads is very similar in Periods 1 and 2: Type 1 and 2 arrowheads are the most abundant and about the 75% and 70% of the total of arrowheads in both periods are type 1 or 2. Type 3 arrowheads have poor relevance in both Periods and types 4, 5, 6, and 7 are virtually nonexistent. In Period 3, we find practically no type 2 and 6 arrowheads. The remaining arrowheads in this period have a presence quite similar but type 4 and 7 have a slightly higher presence. Period 4 shows a large presence of type 7 arrows and, to a lesser extent, of type 1, 3 and 5 arrows (probabilities of about 0.15). Arrowheads of type 2 and 6 have no relevance. Approximately 57% and 24% of the arrows of Period 5 are of type 7 and 4, respectively. The remaining arrowhead types, except possibly those of type 6, are essentially irrelevant.