Like Francois, I played with a MATLAB code for exponential decay. Galloping off rapidly in all directions, I generated two exponential decay series with noise, to address the issue of extracting data for two decay species, like this.....[spoiler: extracting multiple decay parameters from observations is an on going topic in the literature]>> >> x = (0:0.2:5)';y = 2*exp(-0.2*x) + 0.01*randn(size(x));y2 = 3*exp(-0.3*x) + 0.01*randn(size(x));
f = fit(x,y,'exp1')plot(f,x,y)
f =
General model Exp1: f(x) = a*exp(b*x) Coefficients (with 95% confidence bounds): a = 2.06 (1.941, 2.179) b = -0.1894 (-0.2159, -0.1629)
f2 = fit(x,y2,'exp2')plot(f2,x,y2)
f2 =
General model Exp2: f2(x) = a*exp(b*x) + c*exp(d*x) Coefficients (with 95% confidence bounds): a = -0.197 (-7.766e+04, 7.766e+04) b = -0.2926 (-1567, 1566) c = 3.203 (-7.766e+04, 7.767e+04) d = -0.3006 (-98.32, 97.72)>>
[Conclusion: stop wasting time!]
Next I plotted Francois' 'nice-shaped posterior' (forgive the mental image) whichI show here:https://imgur.com/gallery/TpP4ZdY
...to confirm his code for a noiseless error-free data set.
On Tuesday, September 21, 2021, 04:24:44 PM CDT, John Denker via Phys-l <phys-l@mail.phys-l.org> wrote:
On 9/21/21 10:28 AM, Paul Nord wrote:
That title doesn't google well.
Agreed.
1) I reckon it's likely that you are generally on the right
track. This is something a lot of textbooks get wrong.
2) It would help to spell out in plain English the background
of what you are doing, and the objective.
3a) If I had to guess, I'd say there's an experiment to measure
the half life by counting decays, and it's tricky because:
— The number of decays in any reasonable interval is small.
— The observed numbers are subject large-percentage fluctuations
due to Poisson statistics, even though the underlying physics
is not changing.
— Ordinary "textbook style" least-squares curve fitting to
extract the rate fails miserably. That's because "least
squares" usually means maximum likelihood, which is jargon
for maximum a priori, but any sane person would want maximum
a posterori.
3b) But I'd rather not guess. Please clarify.
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