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

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.

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

`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 |

`dirichlet.prior.alphas` |
Parameter value for a Dirichlet prior
distribution for the |

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.

The unnormalized probability density value. If `log=TRUE`

, then
the logarithm of the probability density value will be returned.

David M. Kaplan dmkaplan2000@gmail.com

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()`

```
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)
```

[Package *ConnMatTools* version 0.3.5 Index]