Adjust Bayesian priors - Two Source Trophic Position with \(\alpha_r\)

Create priors for trophic position using a two source model with \(\alpha_r\) derived from Post 2002 and Heuvel et al. 2024.

Usage

two_source_priors_params_ar(
  a = NULL,
  b = NULL,
  n1 = NULL,
  n1_sigma = NULL,
  n2 = NULL,
  n2_sigma = NULL,
  dn = NULL,
  dn_sigma = NULL,
  tp_lb = NULL,
  tp_ub = NULL,
  sigma_lb = NULL,
  sigma_ub = NULL,
  bp = FALSE
)

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.

dn

mean (\(\mu\)) prior value for \(\Delta\)N. Defaults to 3.4.

dn_sigma

variance (\(\sigma\)) for \(\delta^{15}\)N. Defaults to 0.5.

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)

Value

stanvars object to be used with brms() call.

Details

We will use the following equations derived from Post 2002 and Heuvel et al. 2024:

1. $$\alpha = (\delta^{13} C_c - \delta ^{13}C_2) / (\delta ^{13}C_1 - \delta ^{13}C_2)$$

2. $$\alpha = \alpha_r \times (\alpha_{max} - \alpha_{min}) + \alpha_{min}$$

3. $$\delta^{15}N = \Delta N \times (tp - \lambda_1) + n_1 \times \alpha_r + n_2 \times (1 - \alpha_r)$$

4. $$\delta^{15}N = \Delta N \times (tp - (\lambda_1 \times \alpha_r + \lambda_2 \times (1 - \alpha_r))) + n_1 \times \alpha_r + n_2 \times (1 - \alpha_r)$$

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. This equation is added to the data frame using add_alpha().

For equation 2)

\(\alpha\) is being corrected using equations in Heuvel et al. 2024 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) and 4)

\(\delta^{15}\)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.

See also

two_source_priors_ar(), two_source_model_ar(), and brms::brms()

Examples

two_source_priors_params_ar()
#> $a
#> $a$name
#> [1] "a"
#> 
#> $a$sdata
#> [1] 1
#> 
#> $a$scode
#> [1] "real a;"
#> 
#> $a$block
#> [1] "data"
#> 
#> $a$position
#> [1] "start"
#> 
#> $a$pll_args
#> [1] "data real a"
#> 
#> 
#> $b
#> $b$name
#> [1] "b"
#> 
#> $b$sdata
#> [1] 1
#> 
#> $b$scode
#> [1] "real b;"
#> 
#> $b$block
#> [1] "data"
#> 
#> $b$position
#> [1] "start"
#> 
#> $b$pll_args
#> [1] "data real b"
#> 
#> 
#> $dn
#> $dn$name
#> [1] "dn"
#> 
#> $dn$sdata
#> [1] 3.4
#> 
#> $dn$scode
#> [1] "real dn;"
#> 
#> $dn$block
#> [1] "data"
#> 
#> $dn$position
#> [1] "start"
#> 
#> $dn$pll_args
#> [1] "data real dn"
#> 
#> 
#> $dn_sigma
#> $dn_sigma$name
#> [1] "dn_sigma"
#> 
#> $dn_sigma$sdata
#> [1] 0.5
#> 
#> $dn_sigma$scode
#> [1] "real dn_sigma;"
#> 
#> $dn_sigma$block
#> [1] "data"
#> 
#> $dn_sigma$position
#> [1] "start"
#> 
#> $dn_sigma$pll_args
#> [1] "data real dn_sigma"
#> 
#> 
#> $tp_lb
#> $tp_lb$name
#> [1] "tp_lb"
#> 
#> $tp_lb$sdata
#> [1] 2
#> 
#> $tp_lb$scode
#> [1] "real tp_lb;"
#> 
#> $tp_lb$block
#> [1] "data"
#> 
#> $tp_lb$position
#> [1] "start"
#> 
#> $tp_lb$pll_args
#> [1] "data real tp_lb"
#> 
#> 
#> $tp_ub
#> $tp_ub$name
#> [1] "tp_ub"
#> 
#> $tp_ub$sdata
#> [1] 10
#> 
#> $tp_ub$scode
#> [1] "real tp_ub;"
#> 
#> $tp_ub$block
#> [1] "data"
#> 
#> $tp_ub$position
#> [1] "start"
#> 
#> $tp_ub$pll_args
#> [1] "data real tp_ub"
#> 
#> 
#> $sigma_lb
#> $sigma_lb$name
#> [1] "sigma_lb"
#> 
#> $sigma_lb$sdata
#> [1] 0
#> 
#> $sigma_lb$scode
#> [1] "real sigma_lb;"
#> 
#> $sigma_lb$block
#> [1] "data"
#> 
#> $sigma_lb$position
#> [1] "start"
#> 
#> $sigma_lb$pll_args
#> [1] "data real sigma_lb"
#> 
#> 
#> $sigma_ub
#> $sigma_ub$name
#> [1] "sigma_ub"
#> 
#> $sigma_ub$sdata
#> [1] 10
#> 
#> $sigma_ub$scode
#> [1] "real sigma_ub;"
#> 
#> $sigma_ub$block
#> [1] "data"
#> 
#> $sigma_ub$position
#> [1] "start"
#> 
#> $sigma_ub$pll_args
#> [1] "data real sigma_ub"
#> 
#> 
#> attr(,"class")
#> [1] "stanvars"