# Functions for estimating the probability distribution for relative connectivity given a pair of distributions for scores for marked and unmarked individuals These functions return functions that calculate the probability density function (`d.rel.conn.dists.func`), the probability distribution function (aka the cumulative distribution function; `p.rel.conn.dists.func`) and the quantile function (`q.rel.conn.dists.func`) for relative connectivity given a set of observed score values, distributions for unmarked and marked individuals, and an estimate of the fraction of all eggs marked at the source site, `p`. UTF-8 ```r d.rel.conn.dists.func( obs, d.unmarked, d.marked, p = 1, N = max(100, min(5000, 2 * length(obs))), prior.shape1 = 0.5, prior.shape2 = prior.shape1, prior.func = function(phi) dbeta(phi, prior.shape1, prior.shape2), ... ) p.rel.conn.dists.func( obs, d.unmarked, d.marked, p = 1, N = max(100, min(5000, 2 * length(obs))), prior.shape1 = 0.5, prior.shape2 = prior.shape1, prior.func = function(phi) dbeta(phi, prior.shape1, prior.shape2), ... ) q.rel.conn.dists.func( obs, d.unmarked, d.marked, p = 1, N = max(100, min(5000, 2 * length(obs))), prior.shape1 = 0.5, prior.shape2 = prior.shape1, prior.func = function(phi) dbeta(phi, prior.shape1, prior.shape2), ... ) ``` ## Arguments - `obs`: Vector of observed score values for potentially marked individuals - `d.unmarked`: A function representing the PDF of unmarked individuals. Must be normalized so that it integrates to 1 for the function to work properly. - `d.marked`: A function representing the PDF of marked individuals. Must be normalized so that it integrates to 1 for the function to work properly. - `p`: Fraction of individuals (i.e., eggs) marked in the source population. Defaults to 1. - `N`: number of steps between 0 and 1 at which to approximate likelihood function as input to `approxfun`. Defaults to `2*length(obs)` so long as that number is comprised between `100` and `5000`. - `prior.shape1`: First shape parameter for Beta distributed prior. Defaults to 0.5. - `prior.shape2`: Second shape parameter for Beta distributed prior. Defaults to being the same as `prior.shape1`. - `prior.func`: Function for prior distribution. Should take one parameter, `phi`, and return a probability. Defaults to `function(phi) dbeta(phi,prior.shape1,prior.shape2)`. If this is specified, then inputs `prior.shape1` and `prior.shape2` are ignored. - ``...``: Additional arguments for the `integrate` function. ## Returns A function that takes one argument (the relative connectivity for `d.rel.conn.dists.func` and `p.rel.conn.dists.func`; the quantile for `q.rel.conn.dists.func`) and returns the probability density, cumulative probability or score value, respectively. The returned function accepts both vector and scalar input values. ## Details The normalization of the probability distribution is carried out using a simple, fixed-step trapezoidal integration scheme. By default, the number of steps between relative connectivity values of 0 and 1 defaults to `2*length(obs)` so long as that number is comprised between `100` and `5000`. ## Functions * `d.rel.conn.dists.func`: Returns a function that is PDF for relative connectivity * `p.rel.conn.dists.func`: Returns a function that is cumulative probability distribution for relative connectivity * `q.rel.conn.dists.func`: Returns a function that is quantile function for relative connectivity ## Examples ```r library(ConnMatTools) data(damselfish.lods) # Histograms of simulated LODs l <- seq(-1,30,0.5) h.in <- hist(damselfish.lods$in.group,breaks=l) h.out <- hist(damselfish.lods$out.group,breaks=l) # PDFs for marked and unmarked individuals based on simulations d.marked <- stepfun.hist(h.in) d.unmarked <- stepfun.hist(h.out) # Fraction of adults genotyped at source site p.adults <- 0.25 # prior.shape1=1 # Uniform prior prior.shape1=0.5 # Jeffreys prior # Fraction of eggs from one or more genotyped parents p <- dual.mark.transmission(p.adults)$p # PDF for relative connectivity D <- d.rel.conn.dists.func(damselfish.lods$real.children, d.unmarked,d.marked,p, prior.shape1=prior.shape1) # Estimate most probable value for relative connectivity phi.mx <- optim.rel.conn.dists(damselfish.lods$real.children, d.unmarked,d.marked,p)$phi # Estimate 95% confidence interval for relative connectivity Q <- q.rel.conn.dists.func(damselfish.lods$real.children, d.unmarked,d.marked,p, prior.shape1=prior.shape1) # Plot it up phi <- seq(0,1,0.001) plot(phi,D(phi),type="l", xlim=c(0,0.1), main="PDF for relative connectivity", xlab=expression(phi), ylab="Probability density") abline(v=phi.mx,col="green",lty="dashed") abline(v=Q(c(0.025,0.975)),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.finite.settlement()`, `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.dists.func function