Bronk Ramsey 2009 - Deposition Models and Sequences

Summary

When samples come from a depositional record, depth carries information. The paper describes: simple Sequence order only; defined deposition rate (exact gaps $t_{i+1}=t_i+g_i$, one free variable, posterior computed without MCMC); constant rate (U_Sequence); piecewise-constant rates with change points; and random-variability models (P_Sequence). Bchron-style monotone-process models (used in example E39 for the Sluggan Bog core) are in the same family but are not described in this paper.

Overview

The generic group priors have no notion of depth. For tree rings or sediment cores, the spacing between samples is known or constrained, which sharpens the posterior.

Main Content

Defined deposition rate (Eq. 32)

For dated events $t_i$ with known gaps $g_i$ (e.g. counted tree rings): $t_{i+1}=t_i+g_i$ for $a\le i<b$. There is one independent variable (typically the felling date $t_b$), so “this is one of the few Bayesian models where the posterior probability densities can be directly calculated without having to use Monte Carlo methods.” Implemented as D_Sequence; used to fit floating tree-ring chronologies to the calibration data (Christen and Litton 1995; Bronk Ramsey et al. 2001).

Other deposition models (listed, not derived)

• Sequence: order only, “surprisingly effective” but ignores depth.
• Constant rate (U_Sequence): Christen et al. 1995; Kilian et al. 1995.
• Piecewise-constant rate with periodic changes: Aguilar et al. 2002; Blaauw et al. 2003; Blaauw and Christen 2005.
• Non-linear rates: Walker et al. 2003; Telford et al. 2004; Heegaard et al. 2005.
• Randomly variable rate (P_Sequence, Poisson-type): Bronk Ramsey 2008. The priors “are mathematically more complicated” than group priors; ideally they encode knowledge about the deposition process.

Relation to Bchron and E39 (outside the paper). Bchron (Haslett and Parnell 2008; Parnell et al. 2008) models cumulative age as a monotone stochastic process in depth with gamma-distributed increments, with each date calibrated by an IntCal-type curve. E39 fits 31 dates from the Sluggan Bog peat core this way, treating depth-ordered ages as a monotone process and estimating age at unsampled depths.

Examples

Why the defined-rate model needs no MCMC

Setup: 30 tree rings, so $g_i=1$ year. Only the final-ring date $t_b$ is unknown; each earlier ring is $t_b-(b-i)$. The posterior of $t_b$ is $\propto {\prod }_{i}p(y_i\mid t_b-(b-i))$ evaluated on a grid of $t_b$ and multiplied by the prior. The wiggle-match is “sharp” because the calibration curve’s wiggles must fit all rings simultaneously.

import numpy as np
def wiggle_match(r_obs, s_obs, gaps, cal, r_curve, s_curve):
    """posterior over final-ring date on grid `cal` (untested toy)"""
    offs = np.r_[np.cumsum(gaps[::-1])[::-1], 0]    # years before final ring
    logp = np.zeros_like(cal, dtype=float)
    for k, tb in enumerate(cal):
        idx = np.searchsorted(cal, tb - offs)       # clip to grid in real code
        sd2 = s_obs**2 + s_curve[idx.clip(0, len(cal)-1)]**2
        logp[k] = (-0.5*(r_obs - r_curve[idx.clip(0, len(cal)-1)])**2/sd2 - 0.5*np.log(sd2)).sum()
    p = np.exp(logp - logp.max()); return p/p.sum()

Connections

Bronk Ramsey 2009 - Other Group Priors Multiphase Models and Cross-Referencing, Bronk Ramsey 2009 - Calibration Likelihood and the Calibration Curve, Hilbert Space Gaussian Processes (alternative smooth-in-depth model).

See Also

Bronk Ramsey 2009 - Bayesian Analysis of Radiocarbon Dates - Overview