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| 1 | +--- |
| 2 | +title: 'eratosthenes: Synchronizing archaeological chronologies with special focus on artifact typologies' |
| 3 | +tags: |
| 4 | + - R |
| 5 | + - archaeology |
| 6 | + - chronology |
| 7 | +authors: |
| 8 | + - name: Stephen A. Collins-Elliott |
| 9 | + orcid: 0000-0002-5642-6903 |
| 10 | + affiliation: 1 |
| 11 | + corresponding: true |
| 12 | +affiliations: |
| 13 | + - name: Department of Classics, University of Tennessee, Knoxville, Tennessee, United States |
| 14 | + index: 1 |
| 15 | + ror: 020f3ap87 |
| 16 | +date: "30 June 2025" |
| 17 | +bibliography: ../inst/references.bib |
| 18 | +--- |
| 19 | + |
| 20 | +# Summary |
| 21 | + |
| 22 | +Archaeological chronology-building centers on two types of dates, absolute and relative. Absolute dates are associated with a calendrical date (even approximate), and relative refer to events only known via relationships to others (before/after). Relative events are then associated to absolute via contextual or logical associations. E.g, a _terminus post quem_ (_t.p.q._) of a datable coin found in a soil deposit ensures the deposit came after the production of the coin. Applying probabilistic methods to chronology has traditionally centered on radiocarbon dating, but the broader goal of extending formal methods to all aspects of chronology necessitate a way to address artifact typologies. Artifacts largely persist in being dated by qualitative judgment (e.g, "around the start of the 4th century BCE"). Typologies yet constitute a major chronological component, and their production and use transcend assumptions about depositional sequences (e.g., finds can be made elsewhere and well before a site is occupied). Moreover, typologies are in a constant state of revision and adjustment, such that the dates of types can differ according to authority. The reasons why a type is dated the way it is (i.e., which sites, deposits, and their comparisons inform its date) can also become opaque, such that investigators inherently resist chronological revisions in order to assess the bibliography on any one type (on this conservatism, see @rotroff_four_2005, 20). This situation further presents a challenge for any researcher applying statistical methods to archaeological chronologies with artifacts, as one must make _ad hoc_ and often awkward decisions about dates of types, resulting in what @lavan_checklist_2021[15] called "abonimable" chronologies. |
| 23 | + |
| 24 | +The `R` package `eratosthenes` (named after Eratosthenes of Cyrene, author of the _Chronographiai_) provides functions for chronology-building, above all to bring artifact dating within the scope of formal mathematical estimation. Hence, an investigator can obtain separate probability density functions (p.d.f.) for the production, use, and deposition of an artifact type, in addition to marginal p.d.f.s for all relative sequential events (howsover determined) and absolute constraints (howsoever defined). It uses Gibbs sampling, by now a conventional Markov Chain Monte Carlo method in archaeological chronology [@geman_stochastic_1984; @buck_bayesian_1996]. In particular, a two-step routine of Gibbs sampling is performed, the first a preliminiary sampler to select a starting date, and a second main sampler that uses consistent batch means (CBM) as a stopping rule, given that convergence in distribution is assured [@jones_fixed-width_2006; @flegal_markov_2008]. Full reporting on the Monte Carlo standard errors (MCSE) is provided, giving a concrete error in +/- years for each marginal density. Finally, `eratosthenes` provides functions for assessing the level of dependence of events upon each other within the structure of joint conditional density. Changes in artifact dates brought about by any change in the structure of a chronology can therefore be readily evaluted. |
| 25 | + |
| 26 | +# Statement of Need |
| 27 | + |
| 28 | +Constructing chronologies via formal means is a major disciplinary goal, with a broad array of software developed to those ends. The [CRAN Task View: Archaeological Science](https://github.com/benmarwick/ctv-archaeology) maintained by Ben Marwick provides a section on chronological dating software in `R` [@r_core_team_r:_2024]. The calibration of $^{14}$C dates and the estimation of their posterior probability densities given constraints [@buck_bayesian_2025] are well served by `OxCal` [@bronk_ramsey_bayesian_2009] and `BCal` [@buck_bcal_1999], as well as `R` packages `oxcAAR` [@hinz_oxcaar_2021], `Bchron` [@haslett_simple_2008], and `rcarbon` [@crema_spatio-temporal_2017]. General chronological modeling is served by `Chronomodel` [@lanos_hierarchical_2017], which also centers on radiocarbon dating, and `ChronoLog` [@levy_chronological_2021], which focuses on discretized time intervals. The R package `ArchaeoPhases` [@philippe_analysis_2020] handles post-processing of samples form `OxCal`, `BCal`, and `ChronoModel`. Other chronological software has been focused on modeling count data over time, such as `kairos` [@frerebeau_kairos_2025] and `baorista` [@crema_bayesian_2025], which also focuses on counts of durative/interval events. |
| 29 | + |
| 30 | +Given that there exist substantial software for calibrating radiocarbon dates, `eratosthenes` does not aim to perform this task. Likewise, the formal process of establishing relative sequences of contexts and finds via comparison, called seriation (or ordination), is a computationally difficult problem well served by many `R` packages, such as `seriation` [@hahsler_getting_2008], `vegan` [@oksanen_vegan_2024], `boral` [@hui_boral_2016], `ecoCopula` [@popovic_fast_2022], `VGAM` [@yee_new_2004], and `lakhesis` [@collins-elliott_lakhesis_underreview]. |
| 31 | + |
| 32 | +Two of the needs which `eratosthenes` aims to satisfy are (1) estimation of artifact dates (production, use, deposition) via formal means and (2) tools for evaluating the dependence of events on each other discussed in detail below. The package is further motivated by a philosophy of generalism and minimalism, eschewing the following: |
| 33 | + |
| 34 | +* intervals or durative events. If so desired, such instances can be asserted as two separate point events in sequences, e.g., `"X - Start"` and `"X - End"`. |
| 35 | +* periods and phases. Periods and phases are not actual material or behavioral events, but ideal (and often contested) constructs used to make sense of the past. If so desired, an investigator can always enter period-related events, e.g., `"Archaic Period - Start"`, into their list of sequences. |
| 36 | +* discretization of time into intervals. Samples are drawn along the continuum. |
| 37 | +* overly cumbersome chronological relationships. As `eratosthenes` samples points along the continuum, there is only before and after. If desired, overlaping events can be expressed in sequence construction: e.g., for sequences $[A, B, C]$ and $[A, D, E, C]$, $B$ and $D,E$ will overlap with each other. |
| 38 | + |
| 39 | +The focus of the package is on the structure of the joint conditional, rather than specific probability models. Hence, `eratosthenes` relies on the continuous uniform for estimating relative events. Any model can however be used for absolute constraints, from single points to customized densities. |
| 40 | + |
| 41 | +# Estimating Dates of Artifact Production, Use, and Deposition |
| 42 | + |
| 43 | +Dating artifact types is handled by working backwards from depositional contexts, which is their primary point of observation. If a _t.p.q._ (an asbolute constraint after which an event occurs) is given as an artifact type, it is taken to refer to that artifact's production date and is treated as such, but in general, such information is unavailable for most types. The date of artifact production must be estimated according to some stated rule, with two options given in `eratosthenes`. To start, initial threshold boundaries are established between the earliest depositional context containing that artifact-type and context immediately prior which lacks it. The `"earliest"` rule draws samples from within those initial thresholds. The default `"naive"` rule draws samples betwen a lower bound formed by a sample drawn from within the initial thresholds and an upper bound formed by depositional date of that artifact. After estimating production dates, a use date is sampled between its dates of production and deposition. Figure \autoref{fig1} illustrates these rules. |
| 44 | + |
| 45 | + |
| 46 | + |
| 47 | +Recognizing that artifact typologies are themselves flexible categories, "types" can be defined either by specifying a set of artifacts in terms of individual id numbers, or by specifying one or more (sub)types or (sub)classes. Reporting on the Monte Carlo mean and standard error is included for all densitites. |
| 48 | + |
| 49 | +# Evaluating Displacement of Events |
| 50 | + |
| 51 | +The impact or influence of events on one another within the conditional structure is assessed via the definition of "displacement," which is the shift in the date of a target event $i$ should another event $j$ (either a depositional event or an absolute constraint) be omitted from the joint conditional density. The squared displacement $\delta^2(i,j)$ of a target event $i$ caused by the omission of $j$ is computed as follows. Let $x_i$ be the estimated marginalized Monte Carlo mean date using all events within the full joint conditional, and then let $\tilde{x}_i^{(-j)}$ be its estimated date when event $j$ has been omitted. Squared displacement of $j$ upon $i$ is then defined as |
| 52 | + |
| 53 | +$$ |
| 54 | +\delta^2(i,j) = (\tilde{x}_i^{(-j)} - x_i)^2. |
| 55 | +$$ |
| 56 | + |
| 57 | +The omission of $j$ will then have has changed the date of $i$ by $\sqrt{\delta^2}$ amount of time. If squared displacement is low, then the omission of $j$ has not altered the date of $i$ much, but if it is high, it has had a greater impact. Squared displacement is measured in continuous time, whichever scale the investigator is using (typically years). |
| 58 | + |
| 59 | +Conversely, one can estimate the effective influence of an event $j$ upon all others by taking the mean squared displacement (MSD). This involves taking the mean of the squared displacements of all other events as each $j$ is omitted in a "jackknife" or "leave one out"-style of routine. Where $\Theta$ represents the set of all relative and absolute events, the MSD is defined as |
| 60 | + |
| 61 | +$$ |
| 62 | +\text{MSD}(j) = \frac{1}{n-1} \sum_{i \in \Theta, i \neq j} \delta^2 (i,j). |
| 63 | +$$ |
| 64 | + |
| 65 | +# Application |
| 66 | + |
| 67 | +For the purposes of illustration, an initial application was undertaken for a small sample of events in the Mediterranean from the last four centuries BCE (data available [here](https://github.com/scollinselliott/eratosthenes-data/tree/main/data/20250628)). Results are shown in Figure \autoref{fig2} for a selection of two depositional contexts and five shipwrecks. One can note, for example, the effect of using the sack of Carthage in 146 BCE as a _t.a.q_ for the depositonal date of context Byrsa II B 19.2 at that site [@lancel_byrsa_1982, 194]. |
| 68 | + |
| 69 | +. The wreck Grand Congloué A is earlier than the traditional date: future datasets will work to revise sequencing and constraints.\label{fig2}](fig2.png) |
| 70 | + |
| 71 | +The sack of Carthage has also been a key point for dating a particular type of ceramic transport container, the [Dressel 1 amphora type](https://archaeologydataservice.ac.uk/archives/view/amphora_ahrb_2005/details.cfm?id=324). Since it has not be found in pre-destruction layers of that site, the start of its production has been dated to the later part of the 2nd century BCE [@tchernia_vin_1986,42]. Rather than just relying on one site, however, the entire set of conditional events in `eda20250628` yield a density for its production, use, and deposition (Fig. \autoref{fig3}). One can however change the inputs to assess if and how its dates shfit around. The shipwreck at Isla Pedrosa, for example, evidenced Dressel 1B containers, which [@parker_ancient_1992,217-218] suggested were spurious. If one asserts a finds relationship between Dressel 1B the Isla Pedrosa wreck, the dates of its production, use, and deposition will be drawn heavily toward the mid-2nd c. BCE. For assessing which events are the most determinative in dating the Dressel 1B type, squared displacement is computed, which, for this dataset, show that the Madrague de Giens shipwreck has the greatest impact on shifting the type's date ($\delta^2 = 5.16\text{E}06; \alpha = -5000, \omega = 1950$; the full table is given [here](https://github.com/scollinselliott/eratosthenes-data/tree/main/data/20250628)). |
| 72 | + |
| 73 | +.\label{fig3}](fig3.png) |
| 74 | + |
| 75 | +# Future Work |
| 76 | + |
| 77 | +By providing a flexible, reliable framework for marginalizing the dates of archaeological events, as well as for evaluating changes in conditional structure of relative and absolute dates, `eratosthenes` allows for expedient revision of chronologies, transparency in the definition of the full joint conditional, and statistics on the certainty of estimates via Monte Carlo standard error. Current work in progress by the author which relies upon `eratosthenes` involves the synchronism of ceramics, coinage, radiocarbon dates, depositional/seriated sequences, and historical events for the central Mediterranean in the last four centuries BCE, with datasets uploaded to the GitHub repository [eratosthenes-data](https://github.com/scollinselliott/eratosthenes-data). |
| 78 | + |
| 79 | +# References |
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