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

Source:`R/WalkerBivarDirichlet.R`

`WalkerBivarDirichlet.Rd`

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 (the weights, 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 slice sampling
(Walker 2007). We recommend use of the alternative to this, implemented at PolyaUrnBivarDirichlet

**Reference:**

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.

Walker, SG. 2007. “Sampling the Dirichlet Mixture Model with Slices.” *Communications in
Statistics - Simulation and Computation* **36** (1):45-54. https://doi.org/10.1080/03610910601096262.

## Usage

```
WalkerBivarDirichlet(
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,
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,
n_clust = min(10, length(rc_determinations))
)
```

## 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`

.- 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`

.- 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.

## Value

A list with 11 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`

An \(n_{\textrm{out}}\) by \(n_{\textrm{obs}}\) integer matrix. Provides 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, precisions, and weights) 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 the means of the current calendar age clusters \(\phi_j\).

`tau`

A list of length \(n_{\textrm{out}}\) each entry giving a vector of the precisions of the current calendar age clusters \(\tau_j\).

`weight`

A list of length \(n_{\textrm{out}}\) each entry giving the mixing weights of each calendar age 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 "Walker".

`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`

, and`alpha_rate`

, and the slice parameters`slice_width`

and`slice_multiplier`

.

## See also

PolyaUrnBivarDirichlet for our preferred MCMC method to update the Bayesian DPMM
(otherwise an identical model); and PlotCalendarAgeDensityIndividualSample,
PlotPredictiveCalendarAgeDensity and PlotNumberOfClusters
to access the model output and estimate the calendar age information.

See also PPcalibrate for an an alternative (similarly rigorous) approach to
calibration and summarisation of related radiocarbon determinations using a variable-rate Poisson process

## 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).
walker_output <- WalkerBivarDirichlet(
two_normals$c14_age,
two_normals$c14_sig,
intcal20,
n_iter = 100,
show_progress = FALSE)
# The radiocarbon determinations can be given as F14C concentrations
walker_output <- WalkerBivarDirichlet(
two_normals$f14c,
two_normals$f14c_sig,
intcal20,
F14C_inputs = TRUE,
n_iter = 100,
show_progress = FALSE)
```