Functions for estimating the probability distribution of relative connectivity values as a weighted sum over possible input parameters
Description
These functions calculate the probability density function
(d.rel.conn.multiple
), the probability distribution function (aka the
cumulative distribution function; p.rel.conn.multiple
) and the
quantile function (q.rel.conn.multiple
) for the relative (to all
settlers at the destination site) connectivity value for larval transport
between a source and destination site. This version allows one to input
multiple possible fractions of individuals (i.e., eggs) marked at the source
site, multiple possible numbers of settlers collected and multiple possible
marked individuals observed in the sample. This gives one the possibility to
produce ensemble averages over different input parameter values with
different probabilities of being correct.
Usage
1 2 3 4 5 6 7 8 9 10 11  d.rel.conn.multiple(phi, ps, ks, ns, weights = 1,
d.rel.conn = d.rel.conn.beta.prior, ...)
p.rel.conn.multiple(phi, ps, ks, ns, weights = 1,
p.rel.conn = p.rel.conn.beta.prior, ...)
q.rel.conn.multiple.func(ps, ks, ns, weights = 1,
p.rel.conn = p.rel.conn.beta.prior, N = 1000, ...)
q.rel.conn.multiple(q, ps, ks, ns, weights = 1,
p.rel.conn = p.rel.conn.beta.prior, N = 1000, ...)

Arguments
phi 
Vector of fractions of individuals (i.e., eggs) from the source population settling at the destination population 
ps 
Vector of fractions of individuals (i.e., eggs) marked in the source population 
ks 
Vector of numbers of marked settlers found in sample 
ns 
Vector of total numbers of settlers collected 
weights 
Vector of weights for each set of p, k and n values 
d.rel.conn 
Function to use to calculate probability density for
individual combinations of 
p.rel.conn 
Function to use to calculate cumulative probability
distribution for individual combinations of 
N 
Number of points at which to estimate cumulative probability
function for reverse approximation of quantile distribution. Defaults to

q 
Vector of quantiles 
... 
Additional arguments for the function 
Details
If ps
, ks
, ns
and weights
can be scalars or
vectors of the same length (or lengths divisible into that of the largest
input parameter). weights
are normalized to sum to 1 before being
used to sum probabilities from each individual set of input parameters.
Value
Vector of probabilities or quantiles, or a function in the case of
q.rel.conn.multiple.func
Functions

d.rel.conn.multiple
: Estimates quantiles for the probability distribution function for relative connectivity between a pair of sites for multiple possiblep
,k
andn
values. 
p.rel.conn.multiple
: Estimates the cumulative probability distribution for relative connectivity between a paire of sites for multiple possiblep
,k
andn
values. 
q.rel.conn.multiple.func
: Returns a function to estimate quantiles for the probability distribution function for relative connectivity between a pair of sites for multiple possiblep
,k
andn
values. 
q.rel.conn.multiple
: Estimates quantiles for the probability distribution function for relative connectivity between a pair of sites for multiple possiblep
,k
andn
values.
Author(s)
David M. Kaplan dmkaplan2000@gmail.com
References
Kaplan DM, Cuif M, Fauvelot C, Vigliola L, NguyenHuu 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.finite.settlement
,
d.rel.conn.multinomial.unnorm
,
d.rel.conn.unif.prior
,
dual.mark.transmission
,
optim.rel.conn.dists
,
r.marked.egg.fraction
Examples
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)
# p values have uniform probability between 0.1 and 0.4
p < seq(0.1,0.8,length.out=100)
# Weights the same for all except first and last, which are halved
w < rep(1,length(p))
w[1]<0.5
w[length(w)]<0.5
n < 20 # Sample size
k < 2 # Marked individuals in sample
# phi values to use for plotting distribution
phi < seq(0,1,0.01)
prior.shape1 = 1 # Uniform prior
# prior.shape1 = 0.5 # Jeffreys prior
# Plot distribution
plot(phi,d.rel.conn.multiple(phi,p,k,n,w,prior.shape1=prior.shape1),
main="Probability density for relative connectivity",
xlab=expression(phi),
ylab="Probability density",
type="l")
# Add standard distributions for max and min p values
lines(phi,d.rel.conn.beta.prior(phi,min(p),k,n,prior.shape1=prior.shape1),
col="red",lty="dashed")
lines(phi,d.rel.conn.beta.prior(phi,max(p),k,n,prior.shape1=prior.shape1),
col="red",lty="dashed")
# Add some quantiles
q = q.rel.conn.multiple(c(0.025,0.25,0.5,0.75,0.975),
p,k,n,w,prior.shape1=prior.shape1)
abline(v=q,col="green")
