Calibrate and Summarise Multiple Radiocarbon Samples via a Bayesian Non-Parametric DPMM (with Polya Urn Updating)

This function calibrates sets of multiple radiocarbon (${}^{14}$C) determinations, and simultaneously summarises the resultant calendar age information. This is achieved using Bayesian non-parametric density estimation, providing a statistically-rigorous alternative to summed probability distributions (SPDs).

It takes as an input a set of ${}^{14}$C determinations and associated $1\sigma$ uncertainties, as well as the radiocarbon age calibration curve to be used. The samples are assumed to arise from an (unknown) shared calendar age distribution $f(\theta)$ that we would like to estimate, alongside performing calibration of each sample.

The function models the underlying distribution $f(\theta)$ as a Dirichlet process mixture model (DPMM), whereby the samples are considered to arise from an unknown number of distinct clusters. Fitting is achieved via MCMC.

It returns estimates for the calendar age of each individual radiocarbon sample; and broader output (including the means and variances of the underpinning calendar age clusters) that can be used by other library functions to provide a predictive estimate of the shared calendar age density $f(\theta)$.

For more information read the vignette:
vignette("Non-parametric-summed-density", package = "carbondate")

Note: The library provides two slightly-different update schemes for the MCMC. In this particular function, updating of the DPMM is achieved by a Polya Urn approach (Neal 2000) This is our recommended updating approach based on testing. The alternative, slice-sampled, approach can be found at WalkerBivarDirichlet

References:
Heaton, TJ. 2022. “Non-parametric Calibration of Multiple Related Radiocarbon Determinations and their Calendar Age Summarisation.” Journal of the Royal Statistical Society Series C: Applied Statistics 71 (5):1918-56. https://doi.org/10.1111/rssc.12599.
Neal, RM. 2000. “Markov Chain Sampling Methods for Dirichlet Process Mixture Models.” Journal of Computational and Graphical Statistics 9 (2):249 https://doi.org/10.2307/1390653.

Usage

PolyaUrnBivarDirichlet(
  rc_determinations,
  rc_sigmas,
  calibration_curve,
  F14C_inputs = FALSE,
  n_iter = 1e+05,
  n_thin = 10,
  use_F14C_space = TRUE,
  slice_width = NA,
  slice_multiplier = 10,
  n_clust = min(10, length(rc_determinations)),
  show_progress = TRUE,
  sensible_initialisation = TRUE,
  lambda = NA,
  nu1 = NA,
  nu2 = NA,
  A = NA,
  B = NA,
  alpha_shape = NA,
  alpha_rate = NA,
  mu_phi = NA,
  calendar_ages = NA
)

Arguments

rc_determinations: A vector of observed radiocarbon determinations. Can be provided either as ${}^{14}$C ages (in ${}^{14}$C yr BP) or as F${}^{14}$C concentrations.
rc_sigmas: A vector of the (1-sigma) measurement uncertainties for the radiocarbon determinations. Must be the same length as rc_determinations and given in the same units.
calibration_curve: A dataframe which must contain one column calendar_age_BP, and also columns c14_age and c14_sig or f14c and f14c_sig (or both sets). This format matches the curves supplied with this package, e.g., intcal20, intcal13, which contain all 5 columns.
F14C_inputs: TRUE if the provided rc_determinations are F${}^{14}$C concentrations and FALSE if they are radiocarbon ages. Defaults to FALSE.
n_iter: The number of MCMC iterations (optional). Default is 100,000.
n_thin: How much to thin the MCMC output (optional). Will store every n_thin${}^\textrm{th}$ iteration. 1 is no thinning, while a larger number will result in more thinning. Default is 10. Must choose an integer greater than 1. Overall number of MCMC realisations stored will be $n_{\textrm{out}} = \textrm{floor}( n_{\textrm{iter}}/n_{\textrm{thin}}) + 1$ so do not choose n_thin too large to ensure there are enough samples from the posterior to use for later inference.
use_F14C_space: If TRUE (default) the calculations within the function are carried out in F${}^{14}$C space. If FALSE they are carried out in ${}^{14}$C age space. We recommend selecting TRUE as, for very old samples, calibrating in F${}^{14}$C space removes the potential affect of asymmetry in the radiocarbon age uncertainty. Note: This flag can be set independently of the format/scale on which rc_determinations were originally provided.
slice_width: Parameter for slice sampling (optional). If not given a value is chosen intelligently based on the spread of the initial calendar ages. Must be given if sensible_initialisation is FALSE.
slice_multiplier: Integer parameter for slice sampling (optional). Default is 10. Limits the slice size to slice_multiplier * slice_width.
n_clust: The number of clusters with which to initialise the sampler (optional). Must be less than the length of rc_determinations. Default is 10 or the length of rc_determinations if that is less than 10.
show_progress: Whether to show a progress bar in the console during execution. Default is TRUE.
sensible_initialisation: Whether to use sensible values to initialise the sampler and an automated (adaptive) prior on $\mu_{\phi}$ and (A, B) that is informed by the observed rc_determinations. If this is TRUE (the recommended default), then all the remaining arguments below are ignored.
lambda, nu1, nu2: Hyperparameters for the prior on the mean $\phi_j$ and precision $\tau_j$ of each individual calendar age cluster $j$: $$(\phi_j, \tau_j)|\mu_{\phi} \sim \textrm{NormalGamma}(\mu_{\phi}, \lambda, \nu_1, \nu_2)$$ where $\mu_{\phi}$ is the overall cluster centering. Required if sensible_initialisation is FALSE.
A, B: Prior on $\mu_{\phi}$ giving the mean and precision of the overall centering $\mu_{\phi} \sim N(A, B^{-1})$. Required if sensible_initialisation is FALSE.
alpha_shape, alpha_rate: Shape and rate hyperparameters that specify the prior for the Dirichlet Process (DP) concentration, $\alpha$. This concentration $\alpha$ determines the number of clusters we expect to observe among our $n$ sampled objects. The model places a prior on $\alpha \sim \Gamma(\eta_1, \eta_2)$, where $\eta_1, \eta_2$ are the alpha_shape and alpha_rate. A small $\alpha$ means the DPMM is more concentrated (i.e. we expect fewer calendar age clusters) while a large alpha means it is less less concentrated (i.e. many clusters). Required if sensible_initialisation is FALSE.
mu_phi: Initial value of the overall cluster centering $\mu_{\phi}$. Required if sensible_initialisation is FALSE.
calendar_ages: The initial estimate for the underlying calendar ages (optional). If supplied, it must be a vector with the same length as rc_determinations. Required if sensible_initialisation is FALSE.

Value

A list with 10 items. The first 8 items contain output of the model, each of which has one dimension of size $n_{\textrm{out}} = \textrm{floor}( n_{\textrm{iter}}/n_{\textrm{thin}}) + 1$. The rows in these items store the state of the MCMC from every $n_{\textrm{thin}}$${}^\textrm{th}$ iteration:

cluster_identifiers: A list of length $n_{\textrm{out}}$ each entry gives the cluster allocation (an integer between 1 and n_clust) for each observation on the relevant MCMC iteration. Information on the state of these calendar age clusters (means and precisions) can be found in the other output items.
alpha: A double vector of length $n_{\textrm{out}}$ giving the Dirichlet Process concentration parameter $\alpha$.
n_clust: An integer vector of length $n_{\textrm{out}}$ giving the current number of clusters in the model.
phi: A list of length $n_{\textrm{out}}$ each entry giving a vector of length n_clust of the means of the current calendar age clusters $\phi_j$.
tau: A list of length $n_{\textrm{out}}$ each entry giving a vector of length n_clust of the precisions of the current calenadar age cluster $\tau_j$.
observations_per_cluster: A list of length $n_{\textrm{out}}$ each entry giving a vector of length n_clust of the number of observations for that cluster.
calendar_ages: An $n_{\textrm{out}}$ by $n_{\textrm{obs}}$ integer matrix. Gives the current estimate for the calendar age of each individual observation.
mu_phi: A vector of length $n_{\textrm{out}}$ giving the overall centering $\mu_{\phi}$ of the calendar age clusters.

where $n_{\textrm{obs}}$ is the number of radiocarbon observations i.e. the length of rc_determinations.

The remaining items give information about the input data, input parameters (or those calculated using sensible_initialisation) and the update_type

update_type: A string that always has the value "Polya Urn".
input_data: A list containing the ${}^{14}$C data used, and the name of the calibration curve used.
input_parameters: A list containing the values of the fixed hyperparameters lambda, nu1, nu2, A, B, alpha_shape, alpha_rate and mu_phi, and the slice parameters slice_width and slice_multiplier.

Examples

# Note these examples are shown with a small n_iter to speed up execution.
# When you run ensure n_iter gives convergence (try function default).

# Basic usage making use of sensible initialisation to set most values and
# using a saved example data set and the IntCal20 curve.
polya_urn_output <- PolyaUrnBivarDirichlet(
    two_normals$c14_age,
    two_normals$c14_sig,
    intcal20,
    n_iter = 100,
    show_progress = FALSE)

# The radiocarbon determinations can be given as F14C concentrations
polya_urn_output <- PolyaUrnBivarDirichlet(
    two_normals$f14c,
    two_normals$f14c_sig,
    intcal20,
    F14C_inputs = TRUE,
    n_iter = 100,
    show_progress = FALSE)

Calibrate and Summarise Multiple Radiocarbon Samples via a Bayesian Non-Parametric DPMM (with Polya Urn Updating)

Usage

Arguments

Value

See also

Examples