Force a starting point on exponential graph

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Hi,
I'm trying to fit an exponential curve to a process describing a radioactive decay. This is my data:
counts=[2423970,2171372,2065862,1830553,1100899,1037972,914015,752138,684123,606126];
normalized_counts=counts./counts(1);
time=[24.4,40.1,49.2,69.9,137.1,144.4,160.6,185.4,192.7,209.7];
At first I tried using the loveable cftool, which easily managed to fit a "classic" exponential function: y=a*exp(b*x). However, I'm interested in fitting a "pure" exponential [normalized_counts=exp(-a*time)]. When I tried to insert this custom function, I got this error:
Inf computed by model function, fitting cannot continue.
Try using or tightening upper and lower bounds on coefficients.
Then I tried to force an initial condition, so that the exponent will have to pass through the point (24.4, 1) using the code below, but to no avail as well:
x0=24.4;
y0=1;
g = @(p,time)y0*exp(-p*(time-x0));
f = fit(time,normalized_counts,g)
plot(f,time,normalized_counts)
[I got some matrix size errors, tried to add ' (to transpose) to the arrays but that didn't help as well].
Would apprecaite your help!

Answers (2)

Davide Masiello
Davide Masiello on 24 Sep 2022
Edited: Davide Masiello on 25 Sep 2022
See if this can help
counts = [2423970,2171372,2065862,1830553,1100899,1037972,914015,752138,684123,606126];
normalized_counts = counts./counts(1);
time = [24.4,40.1,49.2,69.9,137.1,144.4,160.6,185.4,192.7,209.7];
p0 = 1e-3;
g = @(p,x) exp(-p*(x-time(1)));
f = fit(time',normalized_counts',g,'StartPoint',p0)
f =
General model: f(x) = exp(-p*(x-time(1))) Coefficients (with 95% confidence bounds): p = 0.007151 (0.006894, 0.007408)
plot(f,time,normalized_counts)
  7 Comments
Davide Masiello
Davide Masiello on 27 Sep 2022
My pleasure, if that's the answer you were looking for don't forget to accept it!

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Torsten
Torsten on 25 Sep 2022
Edited: Torsten on 25 Sep 2022
You must normalize to time = 0, not time = 24.4.
The initial mass is the key, not the mass when already 24.4 (whatever) have passed.
I think what you should try to do is to introduce another unknown "counts_initial" and fit the curve as
counts_normalized = exp(-p(1)*time)
with
time=[24.4,40.1,49.2,69.9,137.1,144.4,160.6,185.4,192.7,209.7];
counts_normalized = counts/counts_initial
This means that the curve you have to fit is
counts/counts_initial = exp(-p(1)*time)
thus again the general exponential
counts = counts_initial*exp(-p(1)*time)
counts = [2423970,2171372,2065862,1830553,1100899,1037972,914015,752138,684123,606126];
time = [24.4,40.1,49.2,69.9,137.1,144.4,160.6,185.4,192.7,209.7];
p0 = [3e6,1e-3];
g = @(p,x) p(1)*exp(-p(2)*x);
g1 = @(p,x) g(p,x)-counts;
p = lsqnonlin(@(p)g1(p,time),p0);
Local minimum possible. lsqnonlin stopped because the size of the current step is less than the value of the step size tolerance.
hold on
plot(time,counts,'o')
plot([0,time],g(p,[0,time]))
hold off
grid

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