Reconstructing Muon Neutrino Induced Cascades

1
Reconstructing Muon Neutrino Induced Cascades
Introduction to a New Algorithm for
Reconstructing Composite Events
Rodín A. Porrata
2
Overview of Standard Analysis

• Muon Neutrino Channel
• Smooth upgoing events
• Good Directional Reconstruction
• Poor Neutrino Energy Reconstruction

• Electron Neutrino Cascade Channel
• Bright, not smooth events
• Good Cascade Energy Reconstruction
• Not highly directional

3
Reconstruction Channels
Complex
Composite
Bare
4
A Couple of Interesting Channels for Physics and
Astronomy

• Super-symmetric Tau pairs
• Disappear from analysis because they give bad
  linefit results.
• Upgoing Stau pairs are a clear signature of
  super-symmetry

• Muon Neutrino Induced Cascades
• and
• Disappear from analysis because they are too
  clumpy or are down going
• But good directional reconstruction
• Only neutrino interaction channel for which
  energy can be accurately and unambiguously
  calculated.
• See the galactic center at a few TeV

5
Overview of Standard Analysis
First Guess and Maximum Likelihood Methods

• Maximum Likelihood
• Utilize maximum information available i.e., ice
  properties, details of event phenomenology and
  photon propagation, -gt photon arrival time
  distribution.
• Requires independent method to search parameter
  space.
• Powells Method
• Simplex
• Simulated Annealing
• Very time consuming.
• Produces best possible parameter set given an
  archetypical event.

• First Guess
• Utilize partial information, i.e., extract main
  features of event topology
• Analytic or semi-analytic solutions
• Involves single sweep through data.
• Fast
• Used to select High Quality events.
• Two archetypes Muons and cascades, standard
  analysis stops here.

6
Reconstruction Methods for Muons
First Guess and Maximum Likelihood Methods

• Maximum Likelihood Method
• Read Muon photon tables directly
• Bulk tables
• Layered
• N.N. fit of tables
• Muon parameterized Pandel isotropic point source
  solution with extra parameters to fit a moving
  point source.
• New description of light from a muon
• Moving isotropic point source.
• Uses complete ice properties
• Gives muon energy and direction.
• Presently used only in searches for
• Monopoles (H. Wissing).
• Quark Nuggets (D. Hartke).
• Not utilized in standard analysis.

• First Guess Reconstruction Methods
• Direct Walk
• Jams
• Linefit
• Event Quality
• Smoothness.

7
Reconstruction Methods for Cascades
First Guess and Maximum Likelihood Methods

• Maximum Likelihood Method
• Read Cascade photon tables directly
• Bulk tables
• Layered
• N.N. fit of tables
• Pandel isotropic point source solution. This is
  what it was designed for!
• Likelihood derived in diffusive approximation.
• Energy Reconstruction
• Requires results of position reconstruction.
• New
• Direct photon distribution
• Uses complete ice properties

• First Guess Reconstruction Methods
• COG -gt position
• Event Quality
• Smoothness (not smooth).
• New First Guess
• Planewave fit -gt Direction.

8
Expectation - Maximization
A method to decompose complex events into
characteristic components

• Master function to be minimized
• The ath basis function is a F.G. method for
  either a muon or a cascade.
• Weights are numbers calculated from full
  phenomenology, i.e., photon arrival time
  distributions, given the kth estimate of the
  parameter set, .
• Taking the usual derivatives w.r.t. the
  parameters gives us a set of equations which are
  solved analytically to obtain new estimates of
  the parameters.
• Each iteration takes same amount of time as a
  F.G. method.
• Iterate a maximum of 10 times to obtain most
  likely decomposition.

9
EM - Application to Cascades
A complete model for an archetypical cascade

• Master Function for a cascade
• Taking derivatives w.r.t. directional, vertex and
  energy parameters decomposes master function into
  characteristic equations.
• No separate minimization algorithm required
• Perform C.O.G. fit.
• Solve planewave -gt
• Solve spherical wave characteristic equations -gt
• Solve Phit-Nohit charactistic equations -gt E
• Update weights (Pdirect)
• Goto (2)

10
Summary Conclusion
A new reconstruction paradigm exists

• Some References
• 1 Arthur Dempster, Nan Laird, and Donald Rubin.
  "Maximum likelihood from incomplete data via the
  EM algorithm". Journal of the Royal Statistical
  Society, Series B, 39(1)138, 1977
• 2 Hartley, H. (1958). Maximum likelihood
  estimation from incomplete data. Biometrics,
  14174194.
• 3 McLachlan, G. and Krishnan, T. (1997). The EM
  algorithm and extensions. Wiley series in
  probability and statistics. John Wiley Sons.
• 4 Minka, T. (1998). Expectation-Maximization as
  lower bound maximization. Tutorial published on
  the web at http//www-white.media.mit.edu/
  tpminka/papers/em.html.
• 5 Neal, R. and Hinton, G. (1998). A view of the
  EM algorithm that justifies incremental, sparse,
  and other variants. In Jordan, M., editor,
  Learning in Graphical Models. Kluwer Academic
  Press
• 6 Tanner, M. (1996). Tools for Statistical
  Inference. Springer Verlag, New York. Third
  Edition.

• New Muon likelihood function
• Direct hit proabilities
• Application of EM algorithm to Event
  Reconstruction
• Method searching for a working framework
• Code written in perl and OO-perl
• Could easily(?) be migrated to recoos, sieglinde
  or icetray.
• Testing needed
• Testing needed