We describe a statistical reconstruction methodology for the GLAST LAT. The
methodology incorporates in detail the statistics of the interactions of
photons and charged particles with the tungsten layers in the LAT, and uses the
scattering distributions to compute the full probability distribution over the
energy and direction of the incident photons. It uses model selection methods
to estimate the probabilities of the possible geometrical configurations of the
particles produced in the detector, and numerical marginalization over the
energy loss and scattering angles at each layer. Preliminary results show that
it can improve on the tracker-only energy estimates for muons and electrons
incident on the LAT.Comment: To appear in the proceedings of the First GLAST Symposium (held at
  Stanford University, 5-8 February 2007

Cohen-Tanugi, Johann

Morris, Robin D.

English

arXiv

Robin D. Morris

Johann Cohen-Tanugi

Crossref

Modern Statistical Methods for GLAST Event Analysis

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Modern Statistical Methods for GLAST Event Analysis
Robin D. Morris∗ and Johann Cohen-Tanugi†
∗USRA-RIACS, 444, Castro St, Suite 320, Mountain View, CA 94041
†SLAC/KIPAC, Stanford University, Stanford, CA
Abstract. We describe a statistical reconstruction methodology for the GLAST LAT. The methodology incorporates in detail
the statistics of the interactions of photons and charged particles with the tungsten layers in the LAT, and uses the scattering
distributions to compute the full probability distribution over the energy and direction of the incident photons. It uses model
selection methods to estimate the probabilities of the possible geometrical configurations of the particles produced in the
detector, and numerical marginalization over the energy loss and scattering angles at each layer. Preliminary resultsshow that
it can improve on the tracker-only energy estimates for muons and electrons incident on the LAT.
Keywords: Event Analysis; Bayes Theorem; Markov chain Monte Carlo
PACS: 95.55.Ka; 95.75.Pq;02.50.Tt
INTRODUCTION
The Large Area Telescope (LAT) [1] is the primary instrumento GLAST, and so it is of utmost importance to extract
as much information as possible from the response of the LAT to incident photons and particles. While the quantities
of primary interest for each event are few (namely the azimuth, elevation and energy of the incident photon/particle),
the rich physics of the interactions of the particles/photons with the LAT makes a principled reconstruction algorithm
complex. Whilst the objects of primary interest are photons, the interaction of a photon with the LAT is essentially
that of an electron-positron pair. Therefore we concentrate on the basic building blocks of event reconstruction, the
analysis of the interaction of charged particles with the LAT.
Figure 1 (left) shows the schematic of an interaction between a charged particle and the LAT. Visible in the
figure are 1) multiple Coulomb scattering in the tungsten foils; 2) the production of secondary photons; and 3) the
production of secondary charged particles. The GEANT4 toolkit [2] is designed to simulate these physics processes
in the forward direction. The task in event reconstruction is theinverse problem – estimating, from the data of the
microstrip responses, which physics processes actually occurred in a particular event. The result is an estimate not
only for the original particle and its properties, but also of all secondary particles and photons. To accurately estimate
the primary particle, it is necessary to estimate accurately ll secondaries.
Figure 1 (right) shows the tree of hypotheses for the physical processes at the first two layers of interaction of
a charged particle with the LAT. The final leaves describe thes ructure of the hypothesized event reconstructions.
Clearly as we descend the layers the number of branches in this tree explodes. But only a small number of their leaves
will be consistent with the instrument data - with the pattern of microstrips that fired - and, further, this consideration
can be used to exclude branches as the tree is descended, further limiting the number of leaves that must be considered.
A full event reconstruction consists of two stages:
1. The enumeration of the possible event structures consiste t with the microstrip data.
2. The computation of the parameters of each event structureand their relative probabilities.
Computation of the relative probabilities allows the final event reconstruction to be a weighted average of event
structures, weighted according to the probability that themicrostrip data actually came from that structure.
METHODOLOGY
We concentrate first on the simplest type of event. If the onlyph sical process that actually occurred was multiple
Coulomb scattering, then there is only a single (x-y) pair ofmicrostrips at each layer (excluding noisy strips). This
is the case for muons of moderate energy (up to a few hundred MeV), where the probability of producing secondary
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charged particle
mcs +
photon
mcs +
charged
mcs
mcs mcs +
photon
mcs +
charged
mcs +
photon
physics
mcs mcs +
photon
mcs +
charged
9 options − combinations of
mcs, mcs + phton,
mcs + charged
for each particle incidentLayer 2
Layer 1
etc.
FIGURE 1. LEFT: A schematic of a charged particle interacting with theLAT. The solid line is the incident charged particle.
Long-dashed lines indicate secondary charged particles. The short-dashed line is a secondary photon. The solid blobs indicate
microstrips that fired. RIGHT: The tree of possible event structures
electrons or photons is extremely small, and hence can be negl ct d. We parameterize the trajectory of the particle by
1) its origin (a point outside the LAT); 2) the position at whic it traverses each conversion layer; and 3) its endpoint
(also outside the LAT); and from these we derive the incidentdirections(θ ,φ) and the scattering angles,θi at each
layer. Finally, we add the incident energy,E, and the energy deposited in each conversion layerδEi. Denoting bysi
the microstrips at each layer, we can write
p(θ ,φ ,E,θ1,δE1, . . .θn,δEn|s1 . . . sn) ∝ p(s1 . . . sn|θ ,φ ,E,θ1,δE1, . . .θn,δEn)p(θ ,φ ,E,θ1,δE1, . . .θn,δEn) (1)
The first term on the right hand side is the likelihood. It takes one of two values – one if the trajectory described by
θ ,φ ,θ1, . . .θn intersects all the microstrips that fired, and zero otherwise. It serves to limit the region of the state space
that is of interest. The second term contains all the physicsof the interactions of the particle with the LAT. We use
conditional independence and decompose it as follows.
p(θ ,φ ,E) Priors on azimuth, elevation and energy.
×p(θ1,δE1|E) Distribution of scattering angle and energy loss for a particle of energyE.
×p(θ2,δE2|E,δE1) Same for the particle at layer 2, which has energyE− δE1.
× . . .
×p(θn,δEn|E,δE1, . . . ,δEn) At layern the particle has energyE− δE1− . . .− δEn−1. (2)
These scattering distributions are the known distributions for particles of a specified energy incident on a LAT foil
[3]. The non-Gaussian tails of the scattering angle distribu ions were modeled by a second Gaussian component. For
muons, the energy loss was parameterized as a Landau distribution, with the distribution’s parameters being functions
of energy.
The parameters of primary interest, however, are the azimuth, elevation and energy, and so the distribution of primary
interest isp(θ ,φ ,E). This is obtained from (1) by marginalization. This is performed numerically using Markov chain
Monte Carlo (MCMC). Originally developed in physics [4], ithas been extensively developed in statistics in the past
20 years, and is now a standard tool for use in the analysis of complex, high-dimensional probability distributions [5].
It works by simulating a Markov chain whose equilibrium distribution is constructed to be the distribution of interest
(in this case, the distribution in (1)). Averages over the distribution can then be made by forming averages over the
states of the simulated Markov chain. For example, the mean energy is estimated by forming the mean of the energy
variables over a length of the simulated chain, while ignorig all the other variables. Collecting all the variables into
x, and initializingx← x0 the MCMC algorithm is iteration of
1. propose a change,x← x′ with some proposal distributionπ(x′;x)
2. accept the change withpa =
p(x′)π(x;x′)
p(x)π(x′;x) , and setx← x
′, else retainx
The proposal distribution at each stage may be chosen to onlychange some of the elements ofx. For this work, we use
a cycle of proposals that successively proposes changes toθ ,φ ,E,θ1,δE1, . . . ,θn,δEn.
0 100 200 300 400 500
0
100
200
300
400
Energy (MeV)
0 100 200 300 400 500
0
50
100
150
200
Energy (MeV)
0 100 200 300 400 500
0
50
100
150
Energy (MeV)
0 100 200 300 400 500
0
20
40
60
80
100
Energy (MeV)
FIGURE 2. Energy estimates for muons (2 left panes) and electrons (2 right panes)
For electrons incident on the LAT foils, as well as multiple Coulomb scattering, there is appreciable probability
of producing a secondary photon, and a small probability of pr ducing a secondary electron. (At 100MeV these
probabilities are≃ 0.25 and≃ 0.01 respectively.) We restrict the discussion here to eventswhich contain at most
secondary photons. Typically, for electrons of a few hundred M V the secondary photons are not detected. They carry
energy away from the electron which is “lost” to the tracker.
In the forward direction this is modeled by a mixture distribut on. With probabilitypns(E) no secondary is produced,
and the energy loss follows a Landau distribution. With probability ps(E), a photon is produced and the energy loss
has two components, a Landau distributed component from multiple Coulomb scattering, plus a component distributed
as 1/E representing the energy carried away by the photon.
The samples generated by the MCMC algorithm represent the distribution over the trajectories’ parameters. To
compute the probability for an event structure it is necessary to compute the normalizing factor that was omitted from
equation (1). This can be done by using the MCMC output to construct an importance sampling distribution, and using
samples from that distribution to compute the normalizing factor. This will be discussed elsewhere.
RESULTS
One thousand events were simulated [6] for each of four sources – muons and electrons of 100 and 200 MeV. The
incidence directions were chosen randomly within a 45 degree cone. Figure 2 shows the energy estimates of the
reconstruction. We do not show direction estimates for reasons of space and also because, for charged particles,
the accuracy of the direction estimate is determined almostentirely by multiple Coulomb scattering in the top foil.
For electron events, those events where two charged tracks were detected by the reconstruction algorithm were not
analyzed. In all four cases the estimates are unbiased; the histograms are centered accurately on either 100MeV or
200MeV. The histograms for electrons show more dispersion tha hose for muons, due to the effect of energy being
transferred into photons which are not detected. Note that these estimates were made using only the first 12 regular
GLAST layers of the tracker and did not use any information from the calorimeter. In an upcoming paper we will
present detailed results and comparisons with current methodology.
ACKNOWLEDGMENTS
Our analysis methodology is built upon the software provided by the LAT Science Analysis Software Team. RDM is
supported by a grant from the NASA AISR program.
REFERENCES
1. http://www-glast.slac.stanford.edu
2. S. Agostinelli et al., “GEANT4 - a simulation toolkit”, NIM-A506 3, pp 250-303 (2003)
3. W.-M Yao et al., Journal of Physics G,33, 1 (2006)
4. N. Metropolis et al. “Equation of state calculations by fast computing machines”. J. Chem. Phys.21, 1087-1092 (1953).
5. W.R. Gilks et al (eds), Markov Chain Monte Carlo in Practice, Chapman and Hall/CRC (1995)
6. W. Atwood et al. “The full simulation of the GLAST LAT high en rgy gamma ray telescope”. InPerugia 2004, Calorimetry
in particle physics, pp 329-336 (2004)


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