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This commit adds the function ln() to compute log(n) for integer n. It uses a
lookup table for n < 100 to speed things up.
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Previously to avoid computing P(q,t|n)*P(n|mu) for large n when they were very
unlikely I was using a precomputed maximum n value based only on the expected
number of PE. However, this didn't take into account P(q|n).
This commit updates the likelihood function to dynamically decide when to quit
computing these probabilities when the probability for a given n divided by the
most likely probability is less than some threshold.
This threshold is currently set to 10**(-10) which means we quit calculating
these probabilities when the probability is 10 million times less likely than
the most probable value.
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This commit adds a fast function to calculate the expected number of PE at a
PMT without numerically integrating over the track. This calculation is *much*
faster than integrating over the track (~30 ms compared to several seconds) and
so we use it during the "quick" minimization phase of the fit to quickly find
the best position.
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For some reason the fit seems to have trouble with the kinetic energy.
Basically, it seems to "converge" even though when you run the minimization
again it finds a better minimum with a lower energy. I think this is likely due
to the fact that for muons the kinetic energy only really affects the range of
the muon and this is subject to error in the numerical integration.
I also thought that maybe it could be due to roundoff error in the likelihood
calculation, so I implemented the Kahan summation to try and reduce that. No
idea if it's actually improving things, but I should benchmark it later to see.
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I found when simulating high energy muons that the expected charge for some
PMTs which should be getting hit was zero. The reason for this is that the
integrand was very sharply peaked at the Cerenkov angle which makes it
difficult to integrate for numerical integration routines like cquad. To solve
this I split up the integral at the point when the track was at the Cerenkov
angle from the PMT to make sure that cquad didn't miss the peak. However,
calling cquad twice takes a lot of time so it's not necessarily good to do this
for all fits. Also, it's not obvious if it is necessary any more now that the
angular distribution calculation was fixed.
I think the real reason that cquad was missing those integrals was that for a
high energy muon the range is going to be very large (approximately 40 meters
for a 10 GeV muon). In this case, I should really only integrate up to the edge
of the cavity or PSUP and hopefully cquad picks enough points in there to get a
non zero value.
I also added a check to only compute tmean when at least one PMT has a valid
time. This prevents a divide by zero which causes the likelihood function to
return nan.
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This commit updates the likelihood fit to use the KL path expansion. Currently,
I'm just using one coefficient for the path in both x and y.
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refraction
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The RMS scattering angle calculation comes from Equation 33.15 in the PDG
article on the passage of particles through matter. It's not entirely obvious
if this equation is correct for a long track. It seems like it should be
integrated along the track to add up the contributions at different energies,
but it's not obvious how to do that with the log term.
In any case, the way I was previously calculating it (by using the momentum and
velocity at each point along the track) was definitely wrong.
I will try this out and perhaps try to integrate it later.
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