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This commit speeds up the likelihood function by about ~20% by using the
precomputed track positions, directions, times, etc. instead of interpolating
them on the fly.
It also switches to computing the number of points to integrate along the track
by dividing the track length by a specified distance, currently set to 1 cm.
This should hopefully speed things up for lower energies and result in more
stable fits at high energies.
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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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Since we only have the range and dE/dx tables for light water for electrons and
protons it's not correct to use the heavy water density. Also, even though we
have both tables for muons, currently we only load the heavy water table, so we
hardcode the density to that of heavy water.
In the future, it would be nice to load both tables and use the correct one
depending on if we are fitting in the heavy or light water.
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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 updates the calculation of the muon kinetic energy as a function of
distance along the track. Previously I was using an approximation from the PDG,
but it doesn't seem to be very accurate and won't generalize to the case of
electrons. The kinetic energy is now calculated using the tabulated values of
dE/dx as a function of energy.
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This commit makes sure that when we conolve the single PE charge distribution
with a gaussian we integrate starting at zero since the PDF is zero for q < 0.
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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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spaced
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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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To fit the path of muons and electrons I use the Karhunen-Loeve expansion of a
random 2D walk in the polar angle in x and y. This allows you to decompose the
path into a sum over sine functions whose coefficients become random variables.
The nice thing about fitting the path in this way is that you can capture
*most* of the variation in the path using a small number of variables by only
summing over the first N terms in the expansion and it is easy to calculate the
probability of the coefficients since they are all uncorrelated.
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