eratosthenes: Archaeological Synchronism

The R package eratosthenes aims to provide a coherent foundation for archaeological chronology-building by incorporating, computationally, all relevant sources of information on uncertain archaeological or historical dates. Archaeological dates are often subject to relational conditions (via seriation or stratigraphic relationships) and absolute constraints (such as radiocarbon dates, datable artifacts, or other known historical events, as termini post or ante quem), which prompt the use of a joint conditional probability density to convey those relationships. The date of any one event can then be marginalized from that full, joint conditional distribution, which is achieved using a Gibbs sampler to draw estimates uniformly between potential earliest and latest bounds. Ancillary functions include checking for discrepancies in sequences of events and constraining optimal seriations to known sequences.

While software exists for calibrating and conditioning radiocarbon dates upon relative constraints, such as BCal (Buck, Christen, and James 1999) and OxCal (Bronk Ramsey 2009), the aim of eratosthenes is to extend the application of probability theory more generally to dating all archaeological phenomena, especially the production dates of artifact types. Rcpp is required for faster Gibbs sampling.

The package is named after Eratosthenes of Cyrene, author of the Chronographiai.

Installation

To obtain the current development version of eratosthenes from GitHub, install the package in the R command line with devtools:

library(devtools)
install_github("scollinselliott/eratosthenes", dependencies = TRUE, build_vignettes = TRUE) 

Usage

The following comments are intended as a general introduction. See vignettes for more information on the package functionality.

The basic objects of interest in eratosthenes are:

• sequences of relative events, typically stratigraphic deposits, but also isolated contexts such as may be part of a frequency or contextual seriation
• finds, elements which belong to those events, typically artifacts
• absolute constraints, as either termini post or ante quem, expressed as samples from a probability density

Some functions related to relative sequences:

• seq_check() sees whether partial sequences agree in their relative ordering of elements.
• seq_adj() provides the means to coerce an “input” sequence to a discrepant “target” sequence which contains fewer elements. E.g., if one has obtained an optimal seriation of contexts (of both single, unrelated deposits and stratigraphic deposits) as determined by the presence/absence of find-types, which conflicts with a sequence obtained from a stratigraphic sequence whose physical relationships are certain, this function will reorder the optimal seriation, fitting any single deposits missing from the stratigraphic sequence accordingly.

The package eratosthenes does not have functionality to produce serations, as packages seriation, vegan, and lakhesis can perform this task already.

At the core of eratosthenes is a Gibbs sampler, a common Markov Chain Monte Carlo (MCMC) techinque (Geman and Geman 1984; Buck, Cavanagh, and Litton 1996; Lunn et al. 2013). Estimating marginal densities is accomplished by the function gibbs_ad(), which will yield samples for dates of deposition, production, and any absolute constraints themselves (that is, the density of that extrinsic date as impacted by all other events in the joint distribution).

Input

The function gibbs_ad() takes as inputs the following objects:

• sequences: A list of relative sequences of contexts or events.
• finds: A list of any elements which belong to a context or event, which may be assigned a given type.
• tpq and taq: Separate lists that indicate any elements that provide extrinsic (i.e., absolute) chronological information, as termini post and ante quem.
• alpha and omega: lowest and highest bounds within which to sample.
• trim: whether to remove contexts from the output that are before or after user-provided t.p.q. and t.a.q. (i.e., those which depend on alpha and omega).
• rule: the rule for determining the earliest date of production of an artifact type. Initial threshold boundaries are first established between the earliest depositional context containing an aritfact of that type and the next earliest context which lacks it. Then, the following rules will sample a date accordingly:
  • naive: samples are drawn between the initial threshold sample and the depositional date of that artifact
  • earliest: samples are drawn within the initial threshold boundaries

Absolute dates can take any form:

• Single dates, e.g., 79 for 79 CE.
• Samples between two potential dates for a date range, e.g., runif(10^5, -91, -88) for 91-88 BCE.
• Samples from a bespoke density, e.g., from a calibrated radiocarbon date. eratosthenes does not provide functionality for calibrating dates, which can be accomplished using preexisting software or directly from a calibration curve. The R package Bchron (Haslett and Parnell 2008) provides functions for calibrating dates. As a brief example, given an uncalibrated date and its standard deviation, a crude sample of calibrated dates can be drawn from the IntCal20 curve data, available from IntCal here (Reimer et al. 2020), using the following script:

intcal20 <- read.csv("../path/to/intcal20.14c")

# 14c date mean and st.dev.
mu <- 2040  
sigma <- 30

# samples of 14c date
uncalib <- round(rnorm(10^5, mu, sigma))

calib <- c()

for (i in 1:length(uncalib)) {
  x <- intcal20$CAL.BP[ intcal20$X14C.age == uncalib[i] ] 
  #g <- intcal20$Sigma[ intcal20$X14C.age == uncalib[i] ]

  if (length(x) > 0) {
    for (j in 1:length(x)) {
      calib <- c(calib, x[j])  
    }
  }
}

# samples of cal BC date
calBC <- 1950 - calib
hist(calBC, breaks = 100)

Output

Results are given in a list object of class marginals containing the following objects:

• deposition: a list of the marginal densities of the date of the final deposition of contexts.
• externals: a list of the the marginal densities of date of any terminus post quem or terminus ante quem, as affected by depositional variates in the joint conditional distribution.
• production: a list of the marginal densities of the production date of a given type or class of artifact, given the rule stipulated in the input.

In order to estimate a date of use for any one artifact, one can use the gibbs_ad() function again, taking the production date of the artifact type from the marginals object as a t.p.q. and its depositional context as a t.a.q.

Bibliography

Bronk Ramsey, C. 2009. “Bayesian Analysis of Radiocarbon Dates.” Radiocarbon 51: 337–60.

Buck, C. E., W. G. Cavanagh, and C. D. Litton. 1996. Bayesian Approach to Interpreting Archaeological Data. Chichester: John Wiley & Sons.

Buck, C. E., J. A. Christen, and G. N. James. 1999. “BCal: An On-Line Bayesian Radiocarbon Calibration Tool.” Internet Archaeology 7. https://intarch.ac.uk/journal/issue7/buck/.

Geman, S., and D. Geman. 1984. “Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images.” IEEE Transactions on Pattern Analysis and Machine Intelligence 6: 721–41.

Haslett, J., and A. C. Parnell. 2008. “A Simple Monotone Process with Application to Radiocarbon-Dated Depth Chronologies.” Journal of the Royal Statistical Society: Series C (Applied Statistics) 57 (4): 399–418.

Lunn, D., C. Jackson, N. Best, A. Thomas, and D. Spiegelhalter. 2013. The BUGS Book: A Practical Introduction to Bayesian Analysis. Boca Raton, FL: CRC Press.

Reimer, P. J., W. E. N. Austin, E. Bard, A. Bayliss, P. G. Blackwell, C. Bronk Ramsey, M. Butzin, et al. 2020. “The IntCal20 Northern Hemisphere Radiocarbon Age Calibration Curve (0–55 Cal kBP).” Radiocarbon 62: 725–57.