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Re: [Phys-L] re Analysis of Half-Life measurement using PyStan

Thanks for catching the missing data, Brian. That happened because of an
earlier attempt to improve the fit by removing a couple of outliers. I
neglected to put them back. It wasn't malicious. In fact, it didn't
improve the earlier models.

Curiously, putting those two data points back does bring the model into
closer agreement with the accepted value.

Cu64 half-life now works out as:
4.987011 +/- 0.402007 minutes
(The accepted value is 5.12 minutes.)
The error reported here is what the Stan library calls sd, or standard
deviation.
It also reports a se_mean of 0.012346. Using this, the Z-value is 10. But
using the quoted error above the Z-value is just 0.33.
All this is probably too much fuss over this little data set.

I've updated github with the missing data.

Paul


On Tue, Oct 12, 2021 at 3:45 PM Brian Whatcott <betwys1@sbcglobal.net>
wrote:


-
- Paul NordI was hoping for some feedback on this analysis.
Specifically, what did you think of my conclusion: "All of these models
generate curves which are very close to the data. While the errors seem
very large, they are actually a better representation of the true
uncertainty in applying this model to this data. Many least-squares fitting
functions will give uncertainties which give too much confidence in the
model predictions." Paul On Fri, Oct 8, 2021 at 3:07 PMTue, Oct 12 at 1:35
PM
-

- Paul Nord <Paul.Nord@valpo.edu
- > wrote:
- > I did a thing:
- > <https://sites.google.com/valpo.edu/double-exponential-decay/ >
- > Surprisingly, the resulting uncertainties are bad while the fit
looks
- > quite good. I've read that this sort of analysis gives a better
estimation
- > of the true uncertainty than one often gets with least squares
fitting.
- > Plotting a selection of 1000 models generated from this analysis
shows that
- > they all lie very close to the mean. The assumptions of this model
do not
- > constrain the model parameters given the particular data collected.
- > > Said again: any of the terms in this model can be adjusted about
10%. You
- > can still get a similar and reasonable fit. You just need the right
- > tweaking of the other parameters.
- > > Bad model assumptions here almost certainly include:
- > A) The background rate is constant
- > B) The stability of the detector > > Paul
-

- > __________ Forum for Physics Educators Phys-l@mail.phys-l.org
https://www.phys-l.org/mailman/listinfo/phys-l
-

-

- John Denker via Phys-l <phys-l@mail.phys-l.org
UnsubscribeTo:phys-l@phys-l.orgCc:John DenkerTue, Oct 12 at 2:41 PMOn
10/12/21 11:34 AM, Paul Nord wrote:

I was hoping for some feedback on this analysis. Specifically, what
did
you think of my conclusion:
"All of these models generate curves which are very close to the data.
While the errors seem very large, they are actually a better
representation
of the true uncertainty in applying this model to this data. Many
least-squares fitting functions will give uncertainties which give too
much
confidence in the model predictions."

I have been following this with interest.

Here's why this is important: AFAICT there are very few examples of
assignments where students are expected to measure the uncertainty
of the actual data set

In contrast, there are eleventy squillion assigments where they
are required to calculate a predicted uncertainty, but then don't
check it against experiment, which is IMHO insane.

So my zeroth-order feedback is: You're on the right track.
AFAICT you are making solid progress in an important direction.
I'll help if I can.

=============

At the next level of detail, I don't know enough to say anything
definite about the conclusion quoted above. However I will say:

-- As a rule of thumb, it's true that:
a) most least-squares routines are trash, even when applied to
Gaussians.
b) applying least squares to Poisson data is begging for trouble.
c) when there are 5 fitting parameters, it's likely that there are
correlations, whereupon the whole notion of "error bars" becomes
problematic (to put it politely).

-- If you post the raw data somewhere (google docs or whatever) I might
find time to take a whack at it with my tools. No promises.

I assume each row in the data set is of the form
bin start time, bin end time, counts in the bin

or something like that. Time-stamped events would be better, but I
can cope with binned data if necessary.

********************************************************************************************************
I expect John was looking for the data set which you provided at
https://github.com/paulnord/bayesian_analysis/blob/main/my_data.csv

and which to the jaundiced eye seems to be gardened at two time stamps:
2.5 and 3 minutes.


| t | count |
| | 0.5 | 87 |
| | 1 | 164 |
| | 1.5 | 259 |
| | 2 | 331 |
| | 3.5 | 494 |
| | 4 | 551 |
| | 4.5 | 604 |
| | 5 | 659 |
| | 5.5 | 714 |
| | 6 | 766 |
| | 6.5 | 825 |
| | 7 | 875 |
| | 7.5 | 924 |
| | 8 | 971 |
| | 8.5 | 1007 |
| | 9 | 1054 |
| | 9.5 | 1100 |
| | 10 | 1143 |
| | 10.5 | 1179 |
| | 11 | 1221 |
| | 11.5 | 1260 |
| | 12 | 1305 |
| | 12.5 | 1346 |
| | 13 | 1388 |
| | 13.5 | 1418 |
| | 14 | 1456 |
| | 14.5 | 1490 |
| | 192.3833333 | 11184 |
| | 196.8333333 | 11407 |
| | 307.15 | 16679 |
| | 312.1833333 | 16919 |
| | 916.6666667 | 40583 |
| | 918.6666667 | 40660 |
| | 1086.633333 | 46063 |
| | 1089.583333 | 46138 |
| | 1271.666667 | 51484 |
| | 1273.666667 | 51538 |
| | 1446.666667 | 56114 |
| | 1448.666667 | 56166 |
| | 1629.1 | 60640 |
| | 1631.85 | 60697 |
| | 1747.716667 | 63394 |
| | 1748.716667 | 63414 |


I want to begin by congratulating you on finding a tool which offers the
desired Bayesian probabilities for the parameters in question.
I spent some time in modeling your data with count = D0 (1 - exp (- t/tau)
) + BG*t and then with count = D1(1 - exp( -t/tau1) + D2(1 - exp(-t/tau2)
+ BG*tusing a non linear regression app (NLREG) with lightly constrained
bounds for tau1 and tau2. ( then converting to 1/2 life = ln(2)*tau as
a convenience)
I found judicious bounds could elliminate failures to converge on an
explanatory parameter set due to correlation of terms, and it was not
difficult to arrive atparameters which could explain 99.99% of data
variance but which were far from your values and far from the published
values.This is of course the usual fate of curve fitting to exponentials.
It was amusing to see that if I set the time for a zero count at -11 min
for zero counts.my estimation of the longer mean life time looked more
respectable This is just gardening the data, of course!
I expect to expend more effort in processing your dataset. It was not
clear if this set is synthetic, or in fact, the data from experimental
observations.
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