carbondate (version 1.0.1)

two_normals: Example artificial data - Mixture of Normal Phases

Description

50 simulated radiocarbon determinations for which the underlying calendar ages are drawn from a mixture of two normals: $$f(\theta) = 0.45 N(3500, 200^2) + 0.55 N(5000, 100^2) $$ i.e., a mixture of a normal centred around 3500 cal yr BP; and another (slightly more concentrated/narrower) normal centred around 5000 cal yr BP.

The corresponding 50 radiocarbon ages were then simulated using the IntCal20 calibration curve incorporating both the uncertainty in the calibration curve and a hypothetical measurement uncertainty: $$X_i | \theta_i \sim N(m(\theta_i), \rho(\theta_i)^2 + \sigma_{i,\textrm{lab}}^2),$$ where \(m(\theta_i)\) and \(\rho(\theta_i)\) are the IntCal20 pointwise means and uncertainties; and \(\sigma_{i,\textrm{lab}}\), the simulated laboratory measurement uncertainty, was fixed at a common value of 25 \({}^{14}\)C yrs.

This dataset is included simply to give some quick-to-run examples.

Usage

two_normals

Format

two_normals

A data frame with 50 rows and 4 columns:

c14_age

The simulated \({}^{14}\)C age (in \({}^{14}\)C yr BP)

c14_sig

The (fixed) \({}^{14}\)C age measurement uncertainty used in the simulation (set at 25 \({}^{14}\)C yrs)

f14c

The corresponding simulated values of F\({}^{14}\)C concentration

f14c_sig

The (fixed) corresponding F\({}^{14}\)C measurement uncertainty used in the simulation

Examples

# Plotting calendar age density underlying two_normals
# Useful for comparisons against estimation techniques
weights_true <- c(0.45, 0.55)
cluster_means_true_calBP <- c(3500, 5000)
cluster_precisions_true <- 1 / c(200, 100)^2

# Create mixture density
truedens <- function(t, w, truemean, trueprec) {
  dens <- 0
  for(i in 1:length(w)) {
    dens <- dens + w[i] * dnorm(t, mean = truemean[i], sd = 1/sqrt(trueprec[i]))
  }
  dens
}

# Visualise mixture
curve(truedens(
  x,
  w = weights_true,
  truemean = cluster_means_true_calBP,
  trueprec = cluster_precisions_true),
  from = 2500, to = 7000, n = 401,
  xlim = c(7000, 2500),
  xlab = "Calendar Age (cal yr BP)",
  ylab = "Density",
  col = "red"
)