In joechip90/PaGAn: Paleo-Geography Analysis Package

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The PaGAn package supports the implementation of a variation of the plateau relationships outlined in Brewer et al. (2016) through its inclusion as a type of hierarchical effect. The univariate version of the plateau relationship can be described through the use of the following parameters:

$\beta_1$ (where $\beta_1 > 0$) - Describing the magnitude of the ascending part of the plateau relationship
$\beta_2$ (where $\beta_2 > 0$) - Describing the magnitude of the descending part of the plateau relationship
$\hat{x}$ - Describing the location of the hypothetical apex of the plateau relationship on the x-axis
$\delta$ (where $\delta > 0$) - Describing the offset (on the $y$-axis of the plateau peak from the hypothetical apex

The marginal response, $y$, then takes the following functional form:

$$ y \left( x \right) = \textrm{min} \left{\left( x - \hat{x} \right) \beta_1, \left( \hat{x} - x \right) \beta_2, -\delta \right} $$

Plateau relationships can be ill-behaved in situations where there is little or no data to inform the posterior distribution of the ascending or descending components of the relationship. The locations on the $x$-axis where the ascending or descending components drive the relationship are controlled by the parameterisation of the relationship and so this can result in poor mixing of MCMC samplers under certain conditions. To avoid this we enforce some constraints on the plateau parameters such that, in situations where the ascending or descending components fall outside the range of the data, we enforce a conservative bias towards symmetric parameterisations. This is done by defining an intermediate stochastic node, $\mathbf{z}$, that has the following distribution:

$$ \mathbf{z} \sim \mathcal{N}_2 \left( \mathbf{0}, \frac{1}{\tau} \left[\begin{matrix} 1 & \rho \ \rho & 1 \end{matrix}\right] \right) $$ and define

$$ \begin{align} \beta_1 &= \left| z_1 \right| \ \beta_2 &= \left| z_2 \right| \end{align} $$ where $\tau$ ($\tau > 0$) and $\rho$ ($0 \leq \rho < 1$) are hyperparameters that define the scale of the ascending and descending coefficients and the correlation between them respectively. The Cholesky decomposition of the variance-covariance matrix, $\mathbf{L}$ is therefore

$$ \mathbf{L} = \frac{1}{\sqrt{\tau}} \left[ \begin{matrix} 1 & 0 \ \rho & \sqrt{1 - \rho^2} \end{matrix} \right] $$