# Estimate the probability distribution for the number of settlers originating at a site given a sample from a finite settler pool These functions calculate the probability mass function (`d.rel.conn.finite.settlement`), the cumulative distribution function (`p.rel.conn.finite.settlement`) and the quantile function (`q.rel.conn.finite.settlement`) for the true number of settlers at a site that originated in a particular site given a known fraction of marked eggs among the eggs originating at the source site, a sample of settlers at the destination site, a known fraction of which are marked, and a finite settler pool of known size. UTF-8 ```r d.rel.conn.finite.settlement( n.origin, p, k, n.obs, n.settlers, prior.n.origin = 1 ) p.rel.conn.finite.settlement( n.origin, p, k, n.obs, n.settlers, prior.n.origin = 1 ) q.rel.conn.finite.settlement(q, p, k, n.obs, n.settlers, prior.n.origin = 1) ``` ## Arguments - `n.origin`: Vector of integers of possible numbers of settlers in the cohort that originated at the site of marking. All values should be integers `<=n.settlers`. - `p`: Fraction of individuals (i.e., eggs) marked in the source population - `k`: Number of marked settlers in sample - `n.obs`: Total number of settlers collected - `n.settlers`: Total number of settlers at the destination site from which the `n.obs` (`<=n.settlers`) settlers are collected - `prior.n.origin`: A prior probability mass function for the number of settlers in the cohort originating at the site of marking. Must be a scalar or a vector of length `n.settlers+1`. Defaults to `1`. - `q`: Vector of quantiles ## Returns A vector of probabilities or quantiles. ## Details The relative connectivity between the source and destination sites is calculated as `n.origin/n.settlers`. ## Functions * `d.rel.conn.finite.settlement`: Returns the probability mass function for the numbers of settlers in the cohort that originated at the source site (i.e., site of marking). * `p.rel.conn.finite.settlement`: Returns the cumulative distribution function for the numbers of settlers in the cohort that originated at the source site (i.e., site of marking). * `q.rel.conn.finite.settlement`: Returns quantiles of the cumulative distribution function for the numbers of settlers in the cohort that originated at the source site (i.e., site of marking). ## Examples ```r 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") ``` ## 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. ## See Also Other connectivity estimation: `d.rel.conn.beta.prior()`, `d.rel.conn.dists.func()`, `d.rel.conn.multinomial.unnorm()`, `d.rel.conn.multiple()`, `d.rel.conn.unif.prior()`, `dual.mark.transmission()`, `optim.rel.conn.dists()`, `r.marked.egg.fraction()` ## Author(s) David M. Kaplan dmkaplan2000@gmail.com

d.rel.conn.finite.settlement function