d.rel.conn.dists.func {ConnMatTools} R Documentation

## Functions for estimating the probability distribution for relative connectivity given a pair of distributions for scores for marked and unmarked individuals

### Description

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`.

### Usage

```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.

### 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`.

### Value

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.

### 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

### 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.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()`

### Examples

```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

# prior.shape1=1 # Uniform prior
prior.shape1=0.5 # Jeffreys prior

# Fraction of eggs from one or more genotyped parents

# 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")
```

[Package ConnMatTools version 0.3.5 Index]