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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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This commit updates path_init() to check that beta > 0 before dividing by it to
compute the time. Previously when fitting electrons it would occasionally
divide by zero which would cause the inf to propagate all the way through the
likelihood function.
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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 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 updates path_eval() to calculate theta0 using the residual
scattering RMS for a truncated KL expansion. Since there isn't a nice closed
form solution for this, we instead compute a rough approximation by evaluating
the residual scattering RMS at the center of the track.
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This commit updates the initial guess for the energy using a simple heuristic
of ~6 hits/MeV. I also updated the initial phase where we do a bunch of "quick"
minimizations to loop over a series of starting positions and automatically
calculate the approximate direction and t0 for the event.
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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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