[R] nls() fails on a simple exponential fit, when lm() gets it right?

[Toby Marthews]

Dear R-help,

Here's a simple example of nonlinear curve fitting where nls seems to get
the answer wrong on a very simple exponential fit (my R version 2.7.2).

Look at this code below for a very basic curve fit using nls to fit to (a)
a logarithmic and (b) an exponential curve. I did the fits using
self-start functions and I compared the results with a more simple fit
using a straight lm() command.

The logarithmic fit works 100% correctly below. The problem is with the
exponential fit: the nls command gives the wrong values and I have
completely failed to see why this should happen. I can't see any misake in
my self-start function, so why the mismatch?? Even worse, if I give nls a
trivial fit by removing the "#&#&" on line 6, it fails to converge at all
when the 'simpler' method with lm() easily gives the right answer (r=0.03
and A=5).

I did the same exp and ln fits using MS Excel and the numbers returned are
exactly as for the 'simpler ways', so nls is definitely in the wrong, but
the self-start is almost identical to the one for the logarithmic fit,
which works perfectly ....

I can't see my mistake at all. Can anyone help??

Thanks,
Toby Marthews


traceson=FALSE;maxint=10000;minstepfactor=2^(-30);tolerance=1e-7
xtxt="DBH in mm";ytxt="Height in m"
dbh=c(0.9,1.0,1.1,1.2,4.8,4.9,5.0,5.1,8.9,9.0,9.1,9.2,11.8,11.9,12.0,12.1,14.9,15.1,15.2,15.5)
height=c(5.770089,5.154794,4.888847,6.356770,14.849109,14.973146,15.423816,14.865038,21.335985,20.477624,20.915823,21.886004,23.714724,24.182210,23.954929,23.784659,25.334360,25.411320,26.218614,25.612478)
#&#&#height=5*exp(0.03*dbh)

logarithmicInit=function(mCall,LHS,data) {
 xy=sortedXyData(mCall[["x"]],LHS,data)
 aaa=0.01			#Just guess aaa=0.01 to start the fit
 bbb=min(xy$y)		#Use the minimum y value as an initial guess for bbb
 value=c(aaa,bbb)
 names(value)=mCall[c("aaa","bbb")]
 return(value)
}
fmla=as.formula("~(aaa*log(x)+bbb)")
SSlogarithmic=selfStart(fmla,initial=logarithmicInit,parameters=c("aaa","bbb"))

exponenInit=function(mCall,LHS,data) {
 xy=sortedXyData(mCall[["x"]],LHS,data)
 r=0.01				#Just guess r=0.01 to start the fit
 A=min(xy$y)		#Use the minimum y value as an initial guess for A
 value=c(r,A)
 names(value)=mCall[c("r","A")]
 return(value)
}
fmla=as.formula("~(A*exp(r*x))")
SSexponen=selfStart(fmla,initial=exponenInit,parameters=c("r","A"))

cat("Logarithmic fit (here, the self-start and the 'simpler' way match):\n")
tmp=getInitial(height~SSlogarithmic(dbh,aaa,bbb),data=data.frame(dbh,height))
cat("(Starting values for the logarithmic fit were: aaa=",tmp[1],"and
bbb=",tmp[2],")\n")
modelfit=nls(height~SSlogarithmic(x=dbh,aaa,bbb),trace=traceson,control=list(maxiter=maxint,tol=tolerance,minFactor=minstepfactor))
bfaaa=summary(modelfit)$coefficients[1];bfbbb=summary(modelfit)$coefficients[2]
cat("Best logarithmic fit is (",ytxt,")=(aaa*log(",xtxt,")+bbb) with
parameters aaa=",bfaaa,"and bbb=",bfbbb,"\n")
#Produces: Best logarithmic fit is ( Height in m )=(aaa*log( DBH in mm
)+bbb) with parameters aaa= 7.551666 and bbb= 4.594367

lnfit=lm(height~log(dbh))	#This is doing the ln fit without a self-start
function
cat("Done another, simpler, way, the best logarithmic fit has parameters:
aaa=",lnfit$coefficients[2],"and bbb=",lnfit$coefficients[1],"\n")
#Produces: Done another, simpler, way, the best logarithmic fit has
parameters: aaa= 7.551666 and bbb= 4.594367

cat("Exponential fit (here, the self-start and the 'simpler' way DON'T
match):\n")
tmp=getInitial(height~SSexponen(dbh,r,A),data=data.frame(dbh,height))
cat("(Starting values for the exponential fit: r=",tmp[1],"and
A=",tmp[2],")\n")
modelfit=nls(height~SSexponen(x=dbh,r,A),trace=traceson,control=list(maxiter=maxint,tol=tolerance,minFactor=minstepfactor))
bfr=summary(modelfit)$coefficients[1];bfA=summary(modelfit)$coefficients[2]
cat("Best exponential fit is (",ytxt,")=(A*exp(r*(",xtxt,"))) with
parameters r=",bfr,"and A=",bfA,"\n")
#Produces: Best exponential fit is ( Height in m )=(A*exp(r*( DBH in mm
))) with parameters r= 0.07134055 and A= 9.47907

expfit=lm(log(height)~dbh)	#This is doing the exp fit without a self-start
function
cat("Done another, simpler, way, the best exponential fit has parameters:
r=",expfit$coefficients[2],"and A=",exp(expfit$coefficients[1]),"\n")
#Produces: Done another, simpler, way, the best exponential fit has
parameters: r= 0.1028805 and A= 6.75045