[R] Problem with Integrate

[Anon.]

The background: I'm trying to fit a Poisson-lognormal distrbutuion to 
some data.  This is a way of modelling species abundances:
N ~ Pois(lam)
log(lam) ~ N(mu, sigma2)
The number of individuals are Poisson distributed with an abundance 
drawn from a log-normal distrbution.

To fit this to data, I need to integrate out lam.  In principle, I can 
do it this way:

PLN1 <- function(lam, Count, mu, sigma2) {
   dpois(Count, exp(lam), log=F)*dnorm(LL, mu, sqrt(sigma2))
}

and integrate between -Inf and Inf.  For example, with mu=2, and 
sigma2=2.8 (which are roughly right for the data), and Count=73, I get this:

 > integrate(PLN1, -10, 10, Count=73, mu=2, sigma2=2.8)
0.001289726 with absolute error < 2.5e-11
 > integrate(PLN1, -20, 20, Count=73, mu=2, sigma2=2.8)
0.001289726 with absolute error < 2.5e-11
 > integrate(PLN1, -100, 100, Count=73, mu=2, sigma2=2.8)
2.724483e-10 with absolute error < 5.3e-10
 > integrate(PLN1, -500, 500, Count=73, mu=2, sigma2=2.8)
1.831093e-73 with absolute error < 3.6e-73
 > integrate(PLN1, -1000, 1000, Count=73, mu=2, sigma2=2.8)
Error in integrate(PLN1, -1000, 1000, Count = 73, mu = 2, sigma2 = 2.8):
         non-finite function value
In addition: Warning message:
NaNs produced in: dpois(x, lambda, log)

So, the integral gets smaller, and then gives an error.

I then tried entering the formula directly:
PLN2 <- function(LL, Count, mu, sigma2) {
exp(-LL-(log(LL)-mu)^2/(2*sigma2))*LL^(Count-1)/(gamma(Count+1)*sqrt(2*pi*sigma2))
}

 > integrate(PLN2, 0, 100, Count=73, mu=2, sigma2=2.8)
0.001287821 with absolute error < 2.6e-10
 > integrate(PLN2, 0, 1000, Count=73, mu=2, sigma2=2.8)
0.001289726 with absolute error < 2.9e-08
 > integrate(PLN2, 0, 10000, Count=73, mu=2, sigma2=2.8)
0.001289726 with absolute error < 9.7e-06
 > integrate(PLN2, 0, 19100, Count=73, mu=2, sigma2=2.8)
1.160307e-08 with absolute error < 2.3e-08
 > integrate(PLN2, 0, 19200, Count=73, mu=2, sigma2=2.8)
Error in integrate(PLN2, 0, 19200, Count = 73, mu = 2, sigma2 = 2.8) :
         non-finite function value

And the same thing happens.

I assume that this is because for much of the range, the integral is 
basically zero.

Can anyone suggest a fix?  Preferably one that will work with Count=320 
and Count=0 (both of which I have in the data).

Bob

-- 
Bob O'Hara

Dept. of Mathematics and Statistics
P.O. Box 4 (Yliopistonkatu 5)
FIN-00014 University of Helsinki
Finland
Telephone: +358-9-191 23743
Mobile: +358 50 599 0540
Fax:  +358-9-191 22 779
WWW:  http://www.RNI.Helsinki.FI/~boh/
Journal of Negative Results - EEB: http://www.jnr-eeb.org