R help - Dec 2001 - nls fit to exponential decay with unknown time origin

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.


Code follows (after MASS 2nd ed. p. 271):

----------

require(nls)

t <- c(1.1,2.08,3.12,5.1,8.11,12)
amp <- c(7.4, 6.1, 4.95, 3.5, 2.3, 1.5)
decay <- data.frame (t = t, amp = amp)

expn <- function (b0, b1, tau, t0, t) {
  temp <- exp((t0-t)/tau)
  model.func <- b0 + b1 * temp
  Z <- cbind(1, temp, (b1 * (t0-t) * temp) / tau^2, b1 * temp / tau)
  dimnames(Z) <- list(NULL,
c("b0","b1","tau","t0"))
  attr(model.func, "gradient") <- Z
  model.func
}

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)

----------

thanks,
--
Robert Merithew
Laboratory of Atomic and Solid State Physics, Clark Hall
Cornell University, Ithaca, NY

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"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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