January 14th 2021

Reconstructing Relative Sea Level

Case Study: Newfoundland

A Bayesian Transfer Function (BTF)

\(\underbrace{P(parameters|Data)}_\text{Posterior}\propto \underbrace{P(Data|parameters)}_\text{Likelihood}\underbrace{P(parameters)}_\text{Prior}\)

The Process Model

  • For modelling puposes we consider a latent species response variable

  • Each latent species variable has a functional relationship with elevation:

\[{\bf s}_j = f_{\theta_j}(\text{elevation}) + {\bf\epsilon}_j\]

  • f is a penalised-spline function and \({\bf \theta}_j\) is a vector of parameters controlling the shape of the spline for species \(j\).

  • \(\epsilon_{ij} \sim N(0, \sigma_j^2)\) and is added to account for overdispersion (i.e, the data here are likely to exhibit more variation than may be capturped with the data model).

The Data Model

  • The observed species abundances are multinomial

    \[(y_{i1},....,y_{iM}) \sim Multi(p_{i1},....,p_{iM}, N_i)\]

  • The probabilites of the multinomial distribution are estimated as a function of the latent species variable via a softmax transformation where

\[{p}_{ij} = \frac{e^{s_{ij}}}{\sum_{j=1}^M e^{s_{ij}}}\]

Species Response Curves

A Bayesian Transfer Function (BTF)

\(\underbrace{P(parameters|Data)}_\text{Posterior}\propto \underbrace{P(Data|parameters)}_\text{Likelihood}\underbrace{P(parameters)}_\text{Prior}\)