two_source_priors_params_arc: Adjust Bayesian priors - Two Source Trophic Position with $\alpha_r$ and carbon mixing model

Description

Adjust priors for trophic position using a two source model with $\alpha_r$ derived from Post 2002 and Heuvel et al. (2024) tools:::Rd_expr_doi("doi:10.1139/cjfas-2024-0028")

Usage

two_source_priors_params_arc(
  a = NULL,
  b = NULL,
  n1 = NULL,
  n1_sigma = NULL,
  n2 = NULL,
  n2_sigma = NULL,
  c1 = NULL,
  c1_sigma = NULL,
  c2 = NULL,
  c2_sigma = NULL,
  dn = NULL,
  dn_sigma = NULL,
  tp_lb = NULL,
  tp_ub = NULL,
  sigma_lb = NULL,
  sigma_ub = NULL,
  bp = FALSE
)

Value

stanvars object to be used with brms() call.

Arguments

a: ($\alpha$) exponent of the random variable for beta distribution. Defaults to 1. See beta distribution for more information.
b: ($\beta$) shape parameter for beta distribution. Defaults to 1. See beta distribution for more information.
n1: mean ($\mu$) prior for first $\delta^{15}$N baseline. Defaults to 8.0.
n1_sigma: variance ($\sigma$)for first $\delta^{15}$N baseline. Defaults to 1.
n2: mean ($\mu$) prior for second $\delta^{15}$N baseline. Defaults to 9.5.
n2_sigma: variance ($\sigma$) for second $\delta^{15}$N baseline. Defaults to 1.
c1: mean ($\mu$) prior for first $\delta^{13}$C baseline. Defaults to -21.
c1_sigma: variance ($\sigma$)for first $\delta^{13}$C baseline. Defaults to 1.
c2: mean ($\mu$) prior for second $\delta^{13}$C baseline. Defaults to -26.
c2_sigma: variance ($\sigma$) for second $\delta^{13}$C baseline. Defaults to 1.
dn: mean ($\mu$) prior value for $\Delta$N. Defaults to 3.4.
dn_sigma: variance ($\sigma$) for $\delta^{15}$N. Defaults to 0.25.
tp_lb: lower bound for priors for trophic position. Defaults to 2.
tp_ub: upper bound for priors for trophic position. Defaults to 10.
sigma_lb: lower bound for priors for $\sigma$. Defaults to 0.
sigma_ub: upper bound for priors for $\sigma$. Defaults to 10.
bp: logical value that controls whether informed baseline priors are supplied to the model for $\delta^{15}$N baselines. Default is FALSE meaning the model will use uninformed priors, however, the supplied data.frame needs values for both $\delta^{15}$N baseline (n1 and n2)

Details

We will use the following equations derived from Post 2002 and Heuvel et al. (2024) tools:::Rd_expr_doi("doi:10.1139/cjfas-2024-0028"):

$$\alpha = (\delta^{13} C_c - \delta ^{13}C_2) / (\delta ^{13}C_1 - \delta ^{13}C_2)$$
$$\alpha = \alpha_r \times (\alpha_{max} - \alpha_{min}) + \alpha_{min}$$
$$\delta^{13}C = c_1 \times \alpha_r + c_2 \times (1 - \alpha_r)$$
$$\delta^{15}N = \Delta N \times (tp - \lambda_1) + n_1 \times \alpha_r + n_2 \times (1 - \alpha_r)$$
$$\delta^{15}N = \Delta N \times (tp - (\lambda_1 \times \alpha_r + \lambda_2 \times (1 - \alpha_r))) + n_1 \times \alpha_r + n_2 \times (1 - \alpha_r)$$

For equation 1)

This equation is a carbon source mixing model with $\delta^{13}C_c$ is the isotopic value for consumer, $\delta^{13}C_1$ is the mean isotopic value for baseline 1 and $\delta^{13}C_2$ is the mean isotopic value for baseline 2.

For equation 2)

$\alpha$ is being corrected using equations in Heuvel et al. (2024) tools:::Rd_expr_doi("doi:10.1139/cjfas-2024-0028"). with $\alpha_r$ being the corrected value (bound by 0 and 1), $\alpha_{min}$ is the minimum $\alpha$ value calculated using add_alpha() and $\alpha_{max}$ being the maximum $\alpha$ value calculated using add_alpha().

For equation 3)

This equation is a carbon source mixing model with $\delta^{13}$C being estimated using c_1, c_2 and $\alpha_r$ calculated in equation 1.

For equation 4) and 5)

$\delta^{15}$N are values from the consumer, $n_1$ is $\delta^{15}$N values of baseline 1, $n_2$ is $\delta^{15}$N values of baseline 2, $\Delta$N is the trophic discrimination factor for N (i.e., mean of 3.4), tp is trophic position, and $\lambda_1$ and/or $\lambda_2$ are the trophic levels of baselines which are often a primary consumer (e.g., 2 or 2.5).

This function allows the user to adjust the priors for the following variables in the equation above:

The random exponent ($\alpha$; a) and shape parameters ($\beta$; b) for $\alpha_r$. This prior assumes a beta distribution.
The mean (n2;$\mu$) and variance (n2_sigma; $\sigma$) of the second $\delta^{15}$N for a given baseline. This prior assumes a normal distributions.
The mean (c1;$\mu$) and variance (c1_sigma; $\sigma$) of the second $\delta^{13}$C for a given baseline. This prior assumes a normal distributions.
The mean (c2;$\mu$) and variance (c2_sigma; $\sigma$) of the second $\delta^{13}$C for a given baseline. This prior assumes a normal distributions.
The mean (dn; $\mu$) and variance (dn_sigma; $\sigma$) of $\Delta$N (i.e, trophic enrichment factor). This prior assumes a normal distributions.
The lower (tp_lb) and upper (tp_ub) bounds for priors for trophic position. This prior assumes a uniform distributions.
The lower (sigma_lb) and upper (sigma_ub) bounds for variance ($\sigma$). This prior assumes a uniform distributions.

Examples

Run this code

two_source_priors_params_ar()

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