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This commit makes a few small changes to try and reduce the number of divisions
and multiplications done in get_expected_charge() to speed up the likelihood
function.
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Since we already calculate sin(theta) in get_expected_charge() there's no
reason to calculate it again in get_probability(). This *may* already be
optimized out by the compiler.
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gsl error
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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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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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sqrt(1-pow(cos_theta,2))
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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 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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Previously I had been assuming that a particle undergoing many small angle
Coulomb scatters had a track direction whose polar angle was a Gaussian.
However, this was just due to a misunderstanding of the PDG section "Multiple
scattering through small angles" in the "Passage of particles through matter"
article. In fact, what is described by a Gaussian is the polar angle projected
onto a plane. Therefore the distribution of the polar angle is actually:
(1/(sqrt(2*pi)*theta0**2))*theta*exp(-theta**2/(2*theta0))
This commit updates the code in scattering.c to correctly calculate the
probability that a photon is emitted at a particular angle. I also updated
test-likelihood.c to simulate a track correctly.
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photons
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