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Re: [Phys-L] Half-Life measurement : uncertainties, correlations, SVD

John,

One more thing:
I'm bothered by something else in my fitting function.
Each data point contributes to the log likelihood of having been produced
by the function of the parameters. We calculate the probability of that
function having produced that data point as a Poisson distribution around
the purely theoretical value given by the function.

My problem with that is that the data are correlated to each other. Say,
for example, that we have data points A, and B with some number of counts
at time t1 and t2. I plug t1 into my function and get a set of
probabilities which (hopefully) are near A. But point B is strictly larger
than A. The Poisson distribution around B gives some probability for
values that are less than A. There's correlation between points that we've
failed to consider. If we get a few extra counts early on, those stay with
us in succeeding measurements. We should get fewer counts in at least some
of the later steps to bring us back in line with the model.

Would it be better to consider only the differential counts between time t1
and t2?
Would we gain more information from considering the differential
counts between each pair of measurements?
Or perhaps just using a Poisson probability distribution at each measured
time step takes care of all of that. Afterall, the sum of Poisson
distributions is still a Poisson distribution.

Paul


On Tue, Oct 19, 2021 at 12:21 PM Paul Nord <Paul.Nord@valpo.edu> wrote:

John,

I noticed the higher activity in the second data set. There are a number
of other possible causes.
- distance to the detector
- geometric efficiency of the detector
- absolute efficiency of the detector
- placement of the sample in the radiation bath (we know that there is a
gradient in the neutron flux)

Students used two different setups. I don't know if the efficiency of
each detector was similar. We give them a little freedom to screw this
up. So they may not have been careful about getting the sample close to
the detector.

I suspect that the instructor removed the samples from the neutron bath at
the same time. The lower activity was observed in the lab group that was
in the room next door (a little farther from the starting point). It
shouldn't make much difference, but it's not difficult to imagine two or
three minutes of lost time that could have been avoided.

I'll see if I can reproduce this measurement with better data collection.
I'm wondering if the parameter fit will still have the same covariance
problems. I suspect that we will find that it's easy to find reasonable
fits over a range of parameters that make remarkably similar graphs.

Paul

On Tue, Oct 19, 2021 at 11:55 AM John Denker via Phys-l <
phys-l@mail.phys-l.org> wrote:

On 10/18/21 5:33 PM, I discussed the uncertainties in the
half-life data. I should have mentioned that was for the
my_data.csv observations.

Just now I did the same analysis for the sean_data.csv observations.

sean_data
cost² errorbar fast.amp fast.dk slow.amp
slow.dk bl turn
55387.0 0.425% -0.007962 0.071437 0.786899
-0.60846 0.073478 15.0°
4287.48 1.527% 0.011017 -0.090177 -0.603719
-0.768252 0.192509 22.7°
659.969 3.893% -0.790928 0.583555 -0.085985
-0.05101 -0.154603 5.0°
87.595 10.685% 0.472814 0.75671 -0.051547
0.069502 0.443113 13.2°
33.393 17.305% -0.388195 -0.271329 0.079134
0.179258 0.858656 23.6°

To a first approximation this is similar to the other data set
(see below).
++ The big uncertainty is still more than 40× larger than the small
uncertainty. The log-improbability landscape is taco-shaped.

Even so, there are some differences we can notice.
−− The eigenvector with the most uncertainty has rotated about 24°
relative to the other data set, becoming more nearly aligned with
the "baseline" parameter.
−− The eigenvectors with the least and next-to-least uncertainty
have both rotated to become much less sensitive to the "baseline"
parameter. They are now mostly the gerade and ungerade combination
of the slow amplitude and slow decay constant.

my_data
cost² errorbar fast.amp fast.dk slow.amp
slow.dk bl
42177.2 0.487% -0.00775 0.065394 0.837471
-0.46193 0.284476
3165.41 1.777% 0.010599 -0.078896 -0.532074
-0.633576 0.556005
375.998 5.157% -0.772627 0.60252 -0.082759
-0.130203 -0.12734
40.978 15.622% 0.459398 0.734074 -0.028012
0.293975 0.403588
23.718 20.533% -0.437983 -0.29598 0.088915
0.530875 0.656379


Looking farther upstream, the main difference between the two data
sets is that sean started out with a markedly larger amount of the
fast component.

Action item: This suggests that technique is important. If I were
doing it, I would /practice/ with a blank sample, carrying it across
the room, emplacing it, slamming the shields closed, and starting
the clock and counter. Minimize the time it takes for all that.

This is what we call a /pre-thought/ process. I never advocate
doing anything thoughtlessly ... but often it helps to think
things through in advance, and to practice, so that you don't
need to spend much time on /additional/ thinking when dealing
with the live sample.

Also I renew the suggestion to split the experiment: Do it once
optimized for the fast component, and again optimized for the slow
component.
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