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delta rays
This commit introduces a new method for integrating over the particle track to
calculate the number of shower and delta ray photons expected at each PMT. The
reason for introducing a new method was that the previous method of just using
the trapezoidal rule was both inaccurate and not stable. By inaccurate I mean
that the trapezoidal rule was not producing a very good estimate of the true
integral and by not stable I mean that small changes in the fit parameters
(like theta and phi) could produce wildly different results. This meant that
the likelihood function was very noisy and was causing the minimizers to not be
able to find the global minimum.
The new integration method works *much* better than the trapezoidal rule for
the specific functions we are dealing with. The problem is essentially to
integrate the product of two functions over some interval, one of which is very
"peaky", i.e. we want to find:
\int f(x) g(x) dx
where f(x) is peaked around some region and g(x) is relatively smooth. For our
case, f(x) represents the angular distribution of the Cerenkov light and g(x)
represents the factors like solid angle, absorption, etc.
The technique I discovered was that you can approximate this integral via a
discrete sum:
constant \sum_i g(x_i)
where the x_i are chosen to have equal spacing along the range of the integral
of f(x), i.e.
x_i = F^(-1)(i*constant)
This new method produces likelihood functions which are *much* more smooth and
accurate than previously.
In addition, there are a few other fixes in this commit:
- switch from specifying a step size for the shower integration to a number of
points, i.e. dx_shower -> number of shower points
- only integrate to the PSUP
I realized that previously we were integrating to the end of the track even
if the particle left the PSUP, and that there was no code to deal with the
fact that light emitted beyond the PSUP can't make it back to the PMTs.
- only integrate to the Cerenkov threshold
When integrating over the particle track to calculate the expected number
of direct Cerenkov photons, we now only integrate the track up to the point
where the particle's velocity is 1/index. This should hopefully make the
likelihood smoother because previously the estimate would depend on exactly
whether the points we sampled the track were above or below this point.
- add a minimum theta0 value based on the angular width of the PMT
When calculating the expected number of Cerenkov photons we assumed that
the angular distribution was constant over the whole PMT. This is a bad
assumption when the particle is very close to the PMT. Really we should
average the function over all the angles of the PMT, but that would be too
computationally expensive so instead we just calculate a minimum theta0
value which depends on the distance and angle to the PMT. This seems to
make the likelihood much smoother for particles near the PSUP.
- add a factor of sin(theta) when checking if we can skip calculating the
charge in get_expected_charge()
- fix a nan in beta_root() when the momentum is negative
- update PSUP_RADIUS from 800 cm -> 840 cm
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This commit updates the test-find-peaks script to plot Cerenkov rings for each
of the peaks. It also updates the script to use quad to find the position
instead of using the MC information. Finally, I added a -n argument to the
script to specify how many peaks to draw.
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Previously, the algorithm used to find peaks was to search for all peaks in the
Hough transform above some constant fraction of the highest peak. This
algorithm could have issues finding smaller peaks away from the highest peak.
The new algorithm instead finds the highest peak in the Hough transform and
then recomputes the Hough transform ignoring all PMT hits within the Cerenkov
cone of the first peak. The next peak is found from this transform and the
process is iteratively repeated until a certain number of peaks are found.
One disadvantage of this new system is that it will *always* find the same
number of peaks and this will usually be greater than the actual number of
rings in the event. This is not a problem though since when fitting the event
we loop over all possible peaks and do a quick fit to determine the starting
point and so false positives are OK because the real peaks will fit better
during this quick fit.
Another potential issue with this new method is that by rejecting all PMT hits
within the Cerenkov cone of the first peak we could miss a second peak very
close to the first peak. This is partially mitigated by the fact that when we
loop over all possible combinations of the particle ids and directions we allow
each peak to be used more than once. For example, when fitting for the
hypothesis that an event is caused by two electrons and one muon and given two
possible directions 1 and 2, we will fit for the following possible direction
combinations:
1 1 1
1 1 2
1 2 1
1 2 2
2 2 1
2 2 2
Therefore if there is a second ring close to the first it is possible to fit it
correctly since we will seed the quick fit with two particles pointing in the
same direction.
This commit also adds a few tests for new functions and changes the energy step
size during the quick fit to 10% of the starting energy value.
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To characterize the angular distribution of photons from an electromagnetic
shower I came up with the following functional form:
f(cos_theta) ~ exp(-abs(cos_theta-mu)^alpha/beta)
and fit this to data simulated using RAT-PAC at several different energies. I
then fit the alpha and beta coefficients as a function of energy to the
functional form:
alpha = c0 + c1/log(c2*T0 + c3)
beta = c0 + c1/log(c2*T0 + c3).
where T0 is the initial energy of the electron in MeV and c0, c1, c2, and c3
are parameters which I fit.
The longitudinal distribution of the photons generated from an electromagnetic
shower is described by a gamma distribution:
f(x) = x**(a-1)*exp(-x/b)/(Gamma(a)*b**a).
This parameterization comes from the PDG "Passage of particles through matter"
section 32.5. I also fit the data from my RAT-PAC simulation, but currently I
am not using it, and instead using a simpler form to calculate the coefficients
from the PDG (although I estimated the b parameter from the RAT-PAC data).
I also sped up the calculation of the solid angle by making a lookup table
since it was taking a significant fraction of the time to compute the
likelihood function.
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I noticed when fitting electrons that the cquad integration routine was not
very stable, i.e. it would return different results for *very* small changes in
the fit parameters which would cause the fit to stall.
Since it's very important for the minimizer that the likelihood function not
jump around, I am switching to integrating over the path by just using a fixed
number of points and using the trapezoidal rule. This seems to be a lot more
stable, and as a bonus I was able to combine the three integrals (direct
charge, indirect charge, and time) so that we only have to do a single loop.
This should hopefully make the speed comparable since the cquad routine was
fairly effective at only using as many function evaluations as needed.
Another benefit to this approach is that if needed, it will be easier to port
to a GPU.
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path coefficients
Previously I was adding the log likelihood of the path coefficients instead of
the *negative* log likelihood! When fitting electrons this would sometimes
cause the fit to become unstable and continue increasing the path coefficients
without bound since the gain in the likelihood caused by increasing the
coefficients was more than the loss caused by a worse fit to the PMT data.
Doh!
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This commit fixes a bug in the calculation of the average rms width of the
angular distribution for a path with a KL expansion. I also made a lot of
updates to the test-path program:
- plot the distribution of the KL expansion coefficients
- plot the standard deviation of the angular distribution as a function of
distance along with the prediction
- plot the simulated and reconstructed path in 3D
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This commit adds a function called get_path_length() which computes the path
length inside and outside a sphere for a line segment between two points. This
will be useful for calculating the photon absorption for paths which cross the
AV and for computing the time of flight of photons from a track to a PMT.
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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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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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