# d.rel.conn.unif.prior: Estimate the probability distribution of relative... In ConnMatTools: Tools for Working with Connectivity Data

## Description

These functions calculate the probability density function (`d.rel.conn.unif.prior`), the probability distribution function (aka the cumulative distribution function; `p.rel.conn.unif.prior`) and the quantile function (`q.rel.conn.unif.prior`) for the relative (to all settlers at the destination site) connectivity value for larval transport between a source and destination site given a known fraction of marked individuals (i.e., eggs) in the source population. A uniform prior is used for the relative connectivity value.

## Usage

 ```1 2 3 4 5``` ```d.rel.conn.unif.prior(phi, p, k, n, log = FALSE, ...) p.rel.conn.unif.prior(phi, p, k, n, log = FALSE, ...) q.rel.conn.unif.prior(q, p, k, n, log = FALSE, ...) ```

## Arguments

 `phi` Vector of fractions of individuals (i.e., eggs) from the source population settling at the destination population `p` Fraction of individuals (i.e., eggs) marked in the source population `k` Number of marked settlers found in sample `n` Total number of settlers collected `log` If `TRUE`, returns natural logarithm of probabilities, except for `q.rel.conn.unif.prior`, which expects log of probabilities as inputs `...` Extra arguments to Beta distribution functions. See `dbeta` for details. For expert use only. `q` Vector of quantiles

## Details

Estimations of the probability distribution are derived from the Beta distribution (see `dbeta`) and should be exact to great precision.

## Value

Vector of probabilities or quantiles.

## Functions

• `d.rel.conn.unif.prior`: Returns the probability density for relative connectivity between a pair of sites

• `p.rel.conn.unif.prior`: Returns the cumulative probability distribution for relative connectivity between a paire of sites

• `q.rel.conn.unif.prior`: Estimates quantiles for the probability distribution function for relative connectivity between a pair of sites

## 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.

Other connectivity estimation: `d.rel.conn.beta.prior()`, `d.rel.conn.dists.func()`, `d.rel.conn.finite.settlement()`, `d.rel.conn.multinomial.unnorm()`, `d.rel.conn.multiple()`, `dual.mark.transmission()`, `optim.rel.conn.dists()`, `r.marked.egg.fraction()`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36``` ```library(ConnMatTools) k <- 10 # Number of marked settlers among sample n.obs <- 87 # Number of settlers in sample n.settlers <- 100 # Total size of settler pool p <- 0.4 # Fraction of eggs that was marked phi <- seq(0,1,length.out=101) # Values for relative connectivity # Probability distribution assuming infinite settler pool and uniform prior drc <- d.rel.conn.unif.prior(phi,p,k,n.obs) prc <- p.rel.conn.unif.prior(phi,p,k,n.obs) qrc <- q.rel.conn.unif.prior(c(0.025,0.975),p,k,n.obs) # 95% confidence interval # Test with finite settlement function and large (approx. infinite) settler pool # Can be a bit slow for large settler pools dis <- d.rel.conn.finite.settlement(0:(7*n.obs),p,k,n.obs,7*n.obs) # Quantiles qis <- q.rel.conn.finite.settlement(c(0.025,0.975),p,k,n.obs,7*n.obs) # Finite settler pool dfs <- d.rel.conn.finite.settlement(0:n.settlers,p,k,n.obs,n.settlers) # Quantiles for the finite settler pool qfs <- q.rel.conn.finite.settlement(c(0.025,0.975),p,k,n.obs,n.settlers) # Make a plot of different distributions plot(phi,drc,type="l",main="Probability of relative connectivity values", xlab=expression(phi),ylab="Probability density") lines(phi,prc,col="blue") lines((0:(7*n.obs))/(7*n.obs),dis*(7*n.obs),col="black",lty="dashed") lines((0:n.settlers)/n.settlers,dfs*n.settlers,col="red",lty="dashed") abline(v=qrc,col="black") abline(v=qis/(7*n.obs),col="black",lty="dashed") abline(v=qfs/n.settlers,col="red",lty="dashed") ```