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Thanks, Dave; as you obviously know, this is a common issue, even
with simple models. The VBGF include an unobserved (asymptotic)
length and an unobserved age (t0). Production models, before smart
people started using Fmsy and MSY as leading parameters, were
parameterized with the unit growth rate at B=0 and the asymptotic
population size, neither directly observable. I think it comes down
to this: it makes sense when thinking about models to put them in
terms of asymptotic quantities, but that doesn't make sense when
fitting them!<br>
<br>
<br>
dave fournier wrote on 8/19/2013 4:48 PM:<br>
<blockquote cite="mid:5212AEDF.7090304@otter-rsch.com" type="cite">I
came across the following simple 3 parameter logistics problem
describing
<br>
how difficult it was to fit a small "weeds" data set.
<br>
<br>
<a class="moz-txt-link-freetext" href="https://groups.nceas.ucsb.edu/non-linear-modeling/projects/weeds/WRITEUP/weeds.pdf">https://groups.nceas.ucsb.edu/non-linear-modeling/projects/weeds/WRITEUP/weeds.pdf</a>
<br>
<br>
there are 12 data points
<br>
<br>
t weeds
<br>
1 5.308
<br>
2 7.24
<br>
3 9.638
<br>
4 12.866
<br>
5 17.069
<br>
6 23.192
<br>
7 31.443
<br>
8 38.558
<br>
9 50.156
<br>
10 62.948
<br>
11 75.995
<br>
12 91.972
<br>
<br>
The model to be fit is
<br>
<br>
y = b_1/(1+b_2*exp(-b_3*t))
<br>
<br>
We are told several times how difficult this data set is to fit.
<br>
<br>
In fact is is a trivial data set to fit as I shall show. The
<br>
problem is that the parametrization used is a very poor one.
<br>
<br>
Consider the parameter b_1. This is the number of weeds at
t=infinity.
<br>
<br>
It appears to be somewhat greater than 91, but how much? It is
not really obvious.
<br>
the parameter b_2 is even less intuitive.
<br>
<br>
we shall replace b_1 and b_2 with two parameters whose value is
almost obvious,
<br>
namely the number of weeds at time t_1=1, and t_n=12. Call thes
parameter y_1 and y_n.
<br>
<br>
clearly y_1 is close to 5.308 and y_n is close to 91.9
<br>
<br>
Given values for y_1, y_2, and b_3 it is a simple matter to solve
for b_2 and b_1.
<br>
<br>
The code in the tpl file is
<br>
<br>
b3=exp(logb3);
<br>
dvariable gamma1=exp(-b3*t(1));
<br>
dvariable gamman=exp(-b3*t(noObs));
<br>
b2=(yn-y1)/(y1*gamma1-yn*gamman); // solve for b2
<br>
b1=y1*(1.0+b2*gamma1); // solve for b1
<br>
<br>
For both models the sum of squares is rss 2.58728
<br>
<br>
Now lets make the data a bit more challenging. we change the
12'th data point to
<br>
110.
<br>
<br>
For Ander's verion the sum of squares is rss 60.6799
<br>
<br>
while for my version it is rss 55.8948
<br>
<br>
Now change the 12'th data point to 125.
<br>
<br>
For Ander's version the sum of squares is rss 237.054
<br>
and for mine rss 133.582
<br>
<br>
So maybe we should think more about how to parameterize these
models.
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<br>
<div class="moz-signature">-- <br>
<b>Michael Prager, Ph.D.</b><br>
d/b/a Prager Consulting<br>
Portland, Oregon, USA<br>
<a class="moz-txt-link-freetext" href="http://www.mhprager.com/">http://www.mhprager.com/</a><br>
mobile (252) 269-7005<br>
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