These functions calculate the probability density function
(d.rel.conn.beta.prior
), the probability distribution function (aka
the cumulative distribution function; p.rel.conn.beta.prior
) and the
quantile function (q.rel.conn.beta.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 nonuniform prior is
used for the relative connectivity value.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  d.rel.conn.beta.prior(phi, p, k, n, prior.shape1 = 0.5,
prior.shape2 = prior.shape1, prior.func = function(phi) dbeta(phi,
prior.shape1, prior.shape2), ...)
p.rel.conn.beta.prior(phi, p, k, n, prior.shape1 = 0.5,
prior.shape2 = prior.shape1, prior.func = function(phi) dbeta(phi,
prior.shape1, prior.shape2), ...)
q.rel.conn.beta.prior.func(p, k, n, prior.shape1 = 0.5,
prior.shape2 = prior.shape1, prior.func = function(phi) dbeta(phi,
prior.shape1, prior.shape2), N = 1000, ...)
q.rel.conn.beta.prior(q, p, k, n, prior.shape1 = 0.5,
prior.shape2 = prior.shape1, prior.func = function(phi) dbeta(phi,
prior.shape1, prior.shape2), N = 1000, ...)

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 
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.func 
Function for prior distribution. Should take one
parameter, 
N 
Number of points at which to estimate cumulative probability
function for reverse approximation of quantile distribution. Defaults to

q 
Vector of quantiles 
... 
Extra arguments for the 
The prior distribution for relative connectivity phi
defaults to a
Beta distribution with both shape parameters equal to 0.5. This is the
Reference or Jeffreys prior for a binomial distribution parameter. Both
shape parameters equal to 1 corresponds to a uniform prior.
Estimations of the probability distribution are based on numerical
integration using the integrate
function, and therefore are
accurate to the level of that function. Some modification of the default
arguments to that function may be necessary to acheive good results for
certain parameter values.
Vector of probabilities or quantiles, or a function in the case of
q.rel.conn.beta.prior.func
.
d.rel.conn.beta.prior
: Returns the probability density for
relative connectivity between a pair of sites
p.rel.conn.beta.prior
: Returns the cumulative probability
distribution for relative connectivity between a paire of sites
q.rel.conn.beta.prior.func
: Returns a function to estimate quantiles for
the probability distribution function for relative connectivity between a
pair of sites.
q.rel.conn.beta.prior
: Estimates quantiles for the probability
distribution function for relative connectivity between a pair of sites
David M. Kaplan dmkaplan2000@gmail.com
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.
Other connectivity.estimation: d.rel.conn.dists.func
,
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
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  library(ConnMatTools)
k < 10 # Number of marked settlers among sample
n.obs < 87 # Number of settlers in sample
p < 0.4 # Fraction of eggs that was marked
phi < seq(0.001,10.001,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)
qrc < q.rel.conn.unif.prior(c(0.025,0.975),p,k,n.obs) # 95% confidence interval
# Probability distribution assuming infinite settler pool and using reference/Jeffreys prior
drp < d.rel.conn.beta.prior(phi,p,k,n.obs)
prp < p.rel.conn.beta.prior(phi,p,k,n.obs)
qrp < q.rel.conn.beta.prior(c(0.025,0.975),p,k,n.obs) # 95% confidence interval
# Make a plot of different distributions
# black = Jeffreys prior; red = uniform prior
# Jeffreys prior draws distribution slightly towards zero
plot(phi,drp,type="l",main="Probability of relative connectivity values",
xlab=expression(phi),ylab="Probability density")
lines(phi,drc,col="red")
abline(v=qrp,col="black",lty="dashed")
abline(v=qrc,col="red",lty="dashed")

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