d.rel.conn.multinomial.unnorm {ConnMatTools}

Calculates unnormalized probability density for relative connectivity values from multiple distinct sites

Description

This functions calculates the unnormalized probability density function for the relative (to all settlers at the destination site) connectivity value for larval transport between multiple source sites to a destination site. An arbitrary number of source sites can be evaluated.

Usage

d.rel.conn.multinomial.unnorm(
  phis,
  ps,
  ks,
  n.sample,
  log = FALSE,
  dirichlet.prior.alphas = 1/(length(phis) + 1)
)

Arguments

phis

Vector of fractions of individuals (i.e., eggs) from the source populations settling at the destination population

ps

Vector of fractions of individuals (i.e., eggs) marked in each of the source populations

ks

Vector of numbers of marked settlers from each source population found in the sample

n.sample

Vector of total numbers of settlers collected

log

Boolean indicating whether or not to return the log probability density. Defaults to FALSE.

dirichlet.prior.alphas

Parameter value for a Dirichlet prior distribution for the phis. Can be a single value for a Dirichlet prior with uniform parameters, or a vector of length = length(phis)+1. Defaults to 1/(length(phis)+1), the value for the "reference distance" non-informative prior of Berger et al. 2015.

Details

As this function returns the unnormalized probability density, it must be normalized somehow to be produce a true probability density. This can be acheived using a variety of approaches, including brute force integration of the unnormalized probability density and MCMC algorithms.

Value

The unnormalized probability density value. If log=TRUE, then the logarithm of the probability density value will be returned.

Author(s)

David M. Kaplan [e-mail]

References

Kaplan DM, Cuif M, Fauvelot C, Vigliola L, Nguyen-Huu T, Tiavouane J and Lett C (in press) Uncertainty in empirical estimates of marine larval connectivity. ICES Journal of Marine Science. doi:10.1093/icesjms/fsw182.

Berger JO, Bernardo JM, Sun D (2015) Overall Objective Priors. Bayesian Analysis 10:189-221. doi:10.1214/14-BA915

See Also

Other connectivity estimation: d.rel.conn.beta.prior(), d.rel.conn.dists.func(), d.rel.conn.finite.settlement(), d.rel.conn.multiple(), d.rel.conn.unif.prior(), dual.mark.transmission(), optim.rel.conn.dists(), r.marked.egg.fraction()

Examples

library(ConnMatTools)

ps <- c(0.7,0.5) # Fraction of eggs "marked" at each source site
ks <- c(4,5) # Number of marked settlers among sample from each source site
n.sample <- 20 # Total sample size.  Must be >= sum(ks)


phis0 = runif(3,min=0.05)
phis0 = phis0 / sum(phis0)
phis0 = phis0[1:2] # Don't include relative connectivity of unknown sites

nbatch=1e4

library(mcmc)
ans = metrop(d.rel.conn.multinomial.unnorm,
             initial=phis0,nbatch=nbatch,scale=0.1,
             log=TRUE,ps=ps,ks=ks,n.sample=n.sample)
# A more serious test would adjust blen and scale to improve results, and would repeat
# multiple times to get results from multiple MCMC chains.

# Plot marginal distribution of relative connectivity from first site
h=hist(ans$batch[,1],xlab="Rel. Conn., Site 1",
       main="Relative Connectivity for Source Site 1")

# For comparison, add on curve that would correspond to single site calculation
phi = seq(0,1,length.out=40)
d1 = d.rel.conn.beta.prior(phi,ps[1],ks[1],n.sample)

lines(phi,d1*nbatch*diff(h$breaks)[1],col="red",lwd=5)

# Image plot of bivariate probability density
t=table(cut(ans$batch[,1],phi),cut(ans$batch[,2],phi))
image(t,col=heat.colors(12)[12:1],xlab="Rel. Conn., Site 1",ylab="Rel. Conn., Site 2")

# Add line indicate region above which one can never find results as that would 
# lead to a total connectivity great than 1
abline(1,-1,col="black",lty="dashed",lwd=3)