etienne: Etienne's sampling formula

Description

Function etienne() returns the probability of a given dataset given theta and m according to the Etienne's sampling formula. Function optimal.params() returns the maximum likelihood estimates for theta and m using numerical optimization

Usage

etienne(theta, m, D, log.kda = NULL, give.log = TRUE, give.like = TRUE)
optimal.params(D, start = NULL, give = FALSE, ...)

Arguments

theta

Fundamental biodiversity parameter

Immigration probability

Dataset; a count object

log.kda

The KDA as defined in equation A11 of the reference

give.log

Boolean, with default TRUE meaning to return the logarithm of the value

give.like

Boolean, with default TRUE meaning to return the likelihood and FALSE meaning to return the probability

start

In function optimal.params(), the start point for the optimization routine $(\theta,m)$.

give

In function optimal.params(), Boolean, with TRUE meaning to return all output of the optimization routine, and default FALSE meaning to return just the point estimate

...

In function optimal.params(), further arguments passed to optim()

Details

Function etienne() is just Etienne's formula 6: $$P[D|\theta,m,J]= \frac{J!}{\prod_{i=1}^Sn_i\prod_{j=1}^J{\Phi_j}!} \frac{\theta^S}{(\theta)_J}\times \sum_{A=S}^J\left(K(D,A) \frac{(\theta)_J}{(\theta)_A} \frac{I^A}{(I)_J} \right)$$

where $K(D,A)$ is given by function logkda (qv). It might be useful to know the (trivial) identity for the Pochhammer symbol documented in theta.prob.Rd. For convenience, Etienne's

Function optimal.params() uses optim() to return the maximum likelihood estimate for theta and m.

Compare function optimal.theta(), which is restricted to no dispersal limitation, ie $m=1$x

References

R. S. Etienne 2005. A new sampling formula for biodiversity. Ecology letters, 8:253-260

Examples

Run this code

data(butterflies)
optimal.params(butterflies)


#Now the one from Etienne 2005, supplementary online info:

zoo <- count(c(pigs=1, dogs=1, cats=2, frogs=3, bats=5, slugs=8))
optimal.params(zoo)   #compare his answer of 7.0479586 and 0.22635923.