Etienne's sampling formula

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

References

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

Author

Robin K. S. Hankin

Examples

data(butterflies)
if (FALSE) optimal.params(butterflies) # \dontrun{}  #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.
#>     theta         m 
#> 7.0577736 0.2261709