[R] Modified Bessel Function - 2nd kind

[Dr Andrew Wilson]

Many thanks for this pointer.

Using the formula from the page you referenced, I now have the formula
with the modified Bessel function of the second kind:

> x <- c(1,2,3,4,5,6,7,8)
> y <- c(1,4,5,7,5,4,1,1)
> library(nls)
> library(gregmisc)
> y2 <- nls(y ~
sqrt((2*a)/pi)*exp(a*sqrt(1-q))*((((a*q)/2)^x)/factorial(x))* ((pi/2) *
(besselI(a,-(x-0.5)) - besselI(a,(x-0.5)))/sin((x-0.5) * pi)),
start=list(a=0.1,q=0.1),trace=TRUE)

However, for some reason, I get the following error message:

133.9999 :  0.100 0.001 
Error in numericDeriv(form[[3]], names(ind), env) : 
        Missing value or an Infinity produced when evaluating the model
In addition: Warning messages: 
1: NaNs produced in: sqrt((2 * a)/pi) 
2: NaNs produced in: sqrt(1 - q) 
3: NaNs produced in: besselI(x, nu, 1 + as.logical(expon.scaled)) 
4: NaNs produced in: besselI(x, nu, 1 + as.logical(expon.scaled)) 

Could anyone tell me what I'm doing wrong (and how to fix it)?

The following constraints should apply to the parameters, but I'm not
aware of a "constrain" option in nls that allows me to set these minima
and maxima:

0 < q < 1
a > 0

Many thanks,
Andrew Wilson