# [R] Clarks 2Dt function in R

Ben Bolker bbolker at gmail.com
Mon Apr 4 16:42:25 CEST 2011

```<bialozyt <at> biologie.uni-marburg.de> writes:

>
> Dear Ben,
>
> I would like to ask, if you have an idea how to use this function in R
>

Dear Ronald,

I got started on your problem, but I didn't finish it.
the answer I ran into some trouble.  Unfortunately, fitting
these functions is a bit harder than one might expect ...
it takes quite a bit of fussing to get a good, reliable answer.

My partly-worked solution is below.

> I defined it the following way:

You were multiplying instead of dividing by the second
term (I changed it by raising the term to a negative

The lesson here: *always* do some sanity checks (graphical or otherwise)
of your functions.  I actually did the whole fit before I tried to plot
the curves and found that they were increasing rather than decreasing ...

## fixed
clark2Dt <- function(x , p, u=1) {
(p/(pi*u))/(1+(x^2/u))^(p+1)
}

It might be preferable to define this in terms of s=sqrt(u)
instead (then s would be a scale parameter with the same units
as x, more easily interpretable ...

Sanity checks:

par(las=1,bty="l") ## personal preferences
curve(clark2Dt(x,p=6),from=0,to=5)
legend("topright",paste("p",c(6,4,2),sep="="),col=c(1,2,4),lty=1)

Grab data (in the future, if possible, please use dput(), which puts
your data in the most convenient form, or write out a statement like
this to define your data ...)

X <- as.data.frame(matrix(
c(15,12,
45,13,
75,10,
105,8,
135,16,
165,5,
195,15,
225,8,
255,9,
285,12,
315,5,
345,4,
375,1,
405,1,
435,1,
465,0,
495,1,
525,2,
555,0,
585,0,
615,0,
645,0,
675,0),
ncol=2,byrow=TRUE,
dimnames=list(NULL,c("dist","count"))))

## assume these are traps/samples with unit size
## (if not, it will get absorbed into the "fecundity" constant

library(bbmle)
m1 <- mle2(count~dnbinom(mu=f*clark2Dt(dist,p,u),size=k),
data=X,start=list(f=20,u=10,p=5,k=2),
lower=rep(0.002,4),method="L-BFGS-B")

## we get a plausible-looking fit ...
with(X,plot(count~dist,pch=16,las=1,bty="l"))
newdat <- data.frame(dist=1:700)  ## overkill but harmless
lines(newdat\$dist,predict(m1,newdata=newdat))

## but the coefficients look funny, especially f

coef(m1)

## tried resetting parscale but it's bogus (gets stuck at a worse likelihood)
m2 <- mle2(count~dnbinom(mu=f*clark2Dt(dist,p,u),size=k),
data=X,start=list(f=20,u=10,p=5,k=2),
control=list(parscale=abs(coef(m1))),
lower=rep(0.002,4),method="L-BFGS-B")

m3 <- mle2(count~dnbinom(mu=exp(logf)*clark2Dt(dist,exp(logp),exp(logu)),
size=exp(logk)),
data=X,start=list(logf=log(20),logu=log(10),logp=log(5),
logk=log(2)),

exp(coef(m3))
coef(m1)
summary(m1)

## hmm.  Redefine in terms of s instead of u and (more importantly)
## with f = seed density at r=0 rather the

cov2cor(vcov(m1)) ## shows that f and u are horribly correlated

newclark2Dt <- function(x , p, s=1, eps=1e-70) {
d <- (1+(x/s)^2)
r <- 1/d^(p+1)
if (any(!is.finite(r))) browser()
r
}

dnbinom_pen <- function(x,mu,size,pen=1000,log=TRUE) {
mu <- rep(mu,length.out=length(x))
logval <- ifelse(mu==0 && x==0,pen*x^2,dnbinom(x,mu=mu,size=size,log=TRUE))
if (log) logval else exp(logval)
}

## needed for predict()
snbinom_pen <- snbinom

m4 <- mle2(count~dnbinom(mu=f*newclark2Dt(dist,p,s),size=k),
data=X,start=list(f=20,s=10,p=5,k=2),
lower=rep(0.002,4),method="L-BFGS-B")

m5 <- mle2(count~dnbinom_pen(mu=f*newclark2Dt(dist,1/(pinv),s),size=exp(logk)),
data=X,start=list(f=15,s=10,pinv=100,logk=1),trace=TRUE,
##           control=list(parscale=c(200,0.002,1.66,3600)),
lower=rep(0.002,4),method="L-BFGS-B")

with(X,plot(count~dist,pch=16,las=1,bty="l"))
newdat <- data.frame(dist=1:700)  ## overkill but harmless
lines(newdat\$dist,predict(m1,newdata=newdat))
lines(newdat\$dist,predict(m5,newdata=newdat),col=2)

> but I am not able to fit anything.
> Do you have an idea?
> I guess there is something wrong in my formula for Clarks 2Dt
>