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Clarks 2Dt function in R

2 messages · Ronald Bialozyt, Ben Bolker

Ronald Bialozyt
# 
Dear Ben,

you answerd to Nancy Shackelford about Clarks 2Dt function.
Since the thread ended just after your reply,
I would like to ask, if you have an idea how to use this function in R

I defined it the following way:

function(x , p, u) {
  (p/(pi*u))*(1+(x^2/u))^(p+1)
}

and would like to fit this one to my obeservational data (count)

       [,1] [,2]
 [1,]   15   12
 [2,]   45   13
 [3,]   75   10
 [4,]  105    8
 [5,]  135   16
 [6,]  165    5
 [7,]  195   15
 [8,]  225    8
 [9,]  255    9
[10,]  285   12
[11,]  315    5
[12,]  345    4
[13,]  375    1
[14,]  405    1
[15,]  435    1
[16,]  465    0
[17,]  495    1
[18,]  525    2
[19,]  555    0
[20,]  585    0
[21,]  615    0
[22,]  645    0
[23,]  675    0

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

Thank you for reading 

Ciao
Ronald Bialozyt
Ben Bolker
# 
<bialozyt <at> biologie.uni-marburg.de> writes:
Dear Ronald,

  I got started on your problem, but I didn't finish it.
I got a plausible answer to start with, but when checking
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.
You were multiplying instead of dividing by the second
term (I changed it by raising the term to a negative
power instead.

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)
curve(clark2Dt(x,p=4),col=2,add=TRUE)
curve(clark2Dt(x,p=2),col=4,add=TRUE)
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)),
           method="Nelder-Mead")

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)