etienne {untb}

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, log.kda = NULL, start = NULL, give = FALSE, ...) 

Arguments

theta

Fundamental biodiversity parameter

m

Immigration probability

D

Dataset; a count object

log.kda

The KDA as defined in equation A11 of Etienne 2005. See details section

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 \log K(D,A) is given by function logkda() (qv). It might be useful to know the (trivial) identity for the Pochhammer symbol [written (z)_n] 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.

Argument log.kda is optional: this is the K(D,A) as defined in equation A11 of Etienne 2005; it is computationally expensive to calculate. If it is supplied, the functions documented here will not have to calculate it from scratch: this can save a considerable amount of time

Author(s)

Robin K. S. Hankin

References

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

See Also

logkda,optimal.theta

Examples

data(butterflies)
## Not run: optimal.params(butterflies)  #takes too long without PARI/GP


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

zoo <- count(c(pigs=1, dogs=1, cats=2, frogs=3, bats=5, slugs=8))
l <- logkda.R(zoo, use.brob=TRUE)  # Use logkda() if pari/gp is available
optimal.params(zoo, log.kda=l)  #compare his answer of 7.047958 and 0.22635923.