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

## 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 dmkaplan2000@gmail.com

### 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

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,ks,n.sample)

lines(phi,d1*nbatch*diff(h\$breaks),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)
```

[Package ConnMatTools version 0.3.5 Index]