[R] nls fit to exponential decay with unknown time origin

[Peter Dalgaard BSA]

"Robert D. Merithew" <merithew at ccmr.cornell.edu> writes:

> I'm trying to use nls() to fit an exponential decay with an unknown offset
> in the time (independent variable).  (Perhaps this is inherently very
> difficult?).
> 
> 
> > decay.pl <- nls (amp ~ expn(b0,b1,tau,t0,t), data = decay,
> +                  start = c(b0=1, b1=7.5, tau=3.5, t0=0.1), trace=T)
> Error in nlsModel(formula, mf, start) : singular gradient matrix at
> initial parameter estimates
> > 
> > expn(1, 7.5, 3.5, 0.1, decay$t)
> [1] 6.636080 5.259657 4.164661 2.797383 1.760584 1.250300
> attr(,"gradient")
>      b0         b1        tau         t0
> [1,]  1 0.75147729 -0.4600881 1.61030849
> [2,]  1 0.56795432 -0.6884997 1.21704497
> [3,]  1 0.42195477 -0.7801858 0.90418880
> [4,]  1 0.23965104 -0.7336256 0.51353794
> [5,]  1 0.10141123 -0.4973290 0.21730978
> [6,]  1 0.03337327 -0.2431481 0.07151415
> > 
> 
> 
> Is the gradient matrix really singular?  It works without the t0
> parameter.
> 

Yes. b1 and t0 are not simultaneously identifiable.  Shifting an
exponential curve in time is the same as multiplying by a constant.

Try calculating the ratio of the t0 and b1 columns above.
-- 
   O__  ---- Peter Dalgaard             Blegdamsvej 3  
  c/ /'_ --- Dept. of Biostatistics     2200 Cph. N   
 (*) \(*) -- University of Copenhagen   Denmark      Ph: (+45) 35327918
~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk)             FAX: (+45) 35327907
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