Muon reconstruction and optimal eventclassification in AMANDA
Gary C. Hill
University of Wisconsin- Madisonfor the AMANDA Collaboration
INTERNATIONAL WORKSHOP ON
Ultra High Energy Neutrino Telescopes
Chiba University, Chiba, Japan July 29, 2003

Muon reconstruction and optimal event classification in AMANDA – p.1/29

Outline
Maximum likelihood reconstruction in the presenceof light scattering in ice
Optimal event classification - our job ofreconstruction isn’t done until we’ve assigned themost probable origin (background, signal) to anobserved event
Zenith weighted “Bayesian” event
reconstruction
Optimal classification with modern machinelearning methods
Try to give a unifying theme to the problem ofreconstruction and event classification

Muon reconstruction and optimal event classification in AMANDA – p.2/29

AMANDA-II Location

Muon reconstruction and optimal event classification in AMANDA – p.3/29

AMANDA-II Experiment
19 strings
677 Optical Modules (OM)
200 meters diameter
500 meters tallcompleted in 1999
1997-99 AMANDA-B10
10 strings, 300 OM

Muon reconstruction and optimal event classification in AMANDA – p.4/29

Experimentalevent

Muon reconstruction and optimal event classification in AMANDA – p.5/29

Potential (muon) event origins?
ProtonProton

Atmospheric
Muon

AtmosphericMuon Bundle

Muon
Detector
Atmospheric
Neutrino

Consider three typesof hypothetical origin
to which we will try toassign an event

Upgoing
Muons

Downgoing muons
Upgoing
atmospheric
neutrinos

EarthProton

Atmosphere

Extraterrestrial
Neutrino

Upgoing extraterrestrial neutrinos

Muon reconstruction and optimal event classification in AMANDA – p.6/29

Reconstruction principle
Cherenkov photons
are detected byPMTs
tracks are reconstructed by maximumlikelihood
method of photon
arrival times

Muon reconstruction and optimal event classification in AMANDA – p.7/29

Muon track reconstruction
Cherenkov photons from the muons are recordedby the array optical modules
each module records photon arrival times andamplitudes
















an event is described by a vector of times andamplitudes of all the hits :













Wish to fit a track hypothesis :Muon reconstruction and optimal event classification in AMANDA – p.8/29

Fitting a muon track hypothesis tothe event information




andwe need the



To connect the eventthe track hypothesis
likelihood function

Muon reconstruction and optimal event classification in AMANDA – p.9/29

Muon track Cherenkov cone
geometry-axis

PMT
η

OM
rip

θc

µdx
t 0 , r0, E 0

Cherenkov lightθcGiven a track hypothesis we can calculate the expected
photon arrival times
from an unscattered
Cherenkov cone

Muon reconstruction and optimal event classification in AMANDA – p.10/29

Likelihood reconstruction in theabsence of scattering#$ &%'(. %)

#,0/.

Essentially12'( "!

*-,+ !

Straightforward form of)Expected photon arrival times derived fromCherenkov geometry smeared with Gaussian PMT
jitterfit

This method insufficient in ice with scattering
Need to use a likelihood with full photon
propagation information

Muon reconstruction and optimal event classification in AMANDA – p.11/29

Detemining the PMT time residuals
Time residual is the delay in photon arrival timeafter the expected “direct” Cherenkov arrival time
Full photon propagation simulation (e.g. PTD(Albrecht Karle), Photonics (Ped Miocinović)) used
to tabulate residuals as a function of possible
muon tracks
These tables can be used as the reconstruction
likelihood

Muon reconstruction and optimal event classification in AMANDA – p.12/29

Analytic form - the Pandel function

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9Dirk Pandel (diploma student at DESY-Zeuthen inmid-90’s) solved the propagation equations of lightin the presence of absorption and scattering andfound an analytic form for the time residuals

Muon reconstruction and optimal event classification in AMANDA – p.13/29

Fitting the Pandel function freeparameters
The free parameters are fitted to make the Pandel
form of the residual distributions match the fullphoton simulation
Gives an analytic form that can be used in thereconstruction algorithm
Need to add PMT jitter - old method was a simple
patching of a Gaussian with the Pandel
Recently an analytic form of the convolution of thePandel with a Gaussian was found (GeorgeJaparidze)

Muon reconstruction and optimal event classification in AMANDA – p.14/29

1010−1−2

Delay prob / ns

Delay prob / ns

Time residuals - full simulation andPandel fit

d = 8m1010
1010−4−5d = 71m−31010−3−4100200

400
time delay / ns

−6−70500

1000
1500
time delay / ns

Pandel fit (red) to photon tables (black)

Muon reconstruction and optimal event classification in AMANDA – p.15/29

Reconstructing an event!






Our likelihood function

is given by the track geometry and the time residualfunction (tabulated photon simulation or Pandelfunction)
use a minimisation algorithm (Nelder-Meadsimplex, Powell’s gradient descent, Minuit) to fit thetrack parameters

Muon reconstruction and optimal event classification in AMANDA – p.16/29

Reconstruction performance

zenith angle
Pandel function andphoton tables yield
similar results

space angle

Muon reconstruction and optimal event classification in AMANDA – p.17/29

; ` a\\?

PPTPP\\` bc; b` aac-

how is thiscounted for?

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

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What if
is onlyslightly better than?

Also strong zenith
dependence ofV U]Y V U[ZX Y
Prior probability:

WVU

PP
Likelihood:

; b

R
R
R
R
R
RTPR PPRPTR
TRRT RTPPPRRRSBest upgoing hypothesis:

TPi.e.
more downgoing
muons passing
through the detector
,QPEventP TZRPrior probability:

Should we stillchooseoverWe know that^U]Y^U[Likelihood:

ZX YBest downgoing hypothesis:_^U

What is the most probable origin of
an observed event?

Muon reconstruction and optimal event classification in AMANDA – p.18/29

Joint conditional probability
distribution

e




d;

d;

; b; b

Most probable downgoing muon hypothesis is the onethat maximises joint probability distribution`d e`d ` bd;` e; bb

where, the flux of downgoing muons in
the vicinity of the detector.
Most probable upgoing hypothesis : maximise`d 

e

` b



where, the flux of upgoing muons in thevicinity of the detector (taken as uniform).

Muon reconstruction and optimal event classification in AMANDA – p.19/29

Zenith weighted reconstruction inpractice
Treat the downgoing muon prior as a simple
function of the zenith angle (polynomial fit tosimulated muon flux at the detector)
For each event, find the maximised downgoing
and upgoing likelihoods, then take the ratio.
Use this ratio as a cut parameter, optimised on
simulated downgoing and upgoing events
Rejection of mis-reconstructed atmospheric muonsimproved by a couple of orders of magnitude overconventional “all hypotheses are equal” method
Cuts are simplified (in principle, this is the only cut
we need)Muon reconstruction and optimal event classification in AMANDA – p.20/29

Bayesian statistics interpretation` ; ; ibb; ` ` i

j fk*

` bhgf*; bThe probabilities of observing an event due to up
and downward muons are found by integration overthe likelihood and priors

` bA; b

The ratio
is known as the Bayes’
discriminant and is the statistically most powerful
separator of classes of hypotheses

Muon reconstruction and optimal event classification in AMANDA – p.21/29

Are we evaluating the discriminant?

; nbb; n`  n; ; bbi; ` ` i` ` nj flmgf

We have approximated the Bayes’ discriminant ratio ofintegrals by the ratio of the maximum values of theintegrands :Muon reconstruction and optimal event classification in AMANDA – p.22/29

Don’t most physicists reject
Bayesian inference?
Absolutely yes when used incorrectly!

bo o 

is made fromo b

Usually taketo be uniform in some interval bpo b

After measuring , an inference onooB

Classic example is in upper limit calculations
where uniform priors are used to represent
subjective “degree-of-belief” about an unknown
physical quantity (e.g. the rate of a Poissonprocess , or the mass of a particle )

2

oo

2ob

o b



However
uniform does not yield same
uniform and bothinference as taking
choices of “metric” ( or ) are equally valid

Muon reconstruction and optimal event classification in AMANDA – p.23/29

What about our “Bayesian”
reconstruction?
Acid test - Advanced Statistics in Particle Physics
Workshop, Durham, 2002, Ty DeYoung with anaudience of the staunchest Bayesians andanti-Bayesians
Bayesians naturally said the technique was fine....

Muon reconstruction and optimal event classification in AMANDA – p.24/29

Bob Cousins for the frequentists....
Bayes theorem applies to all types of probability both subjective degree of belief (e.g. "I think themass of the Higgs is uniform in the interval 80-200
GeV") and to classical relative frequency
probabilities ("the distribution of cosmic rays
arriving at earth is uniform and follows a power law
energy spectrum");db

Our muon flux “prior” is a relative frequency
probability - it’s very easy to define- themuon flux is well measured, theoreticallycalculated and understood - not subjective at allMore explictly : the procedure is “Bayesian” only inthat Bayes’ theorem was used – definitely not“subjective” Bayesian!

Muon reconstruction and optimal event classification in AMANDA – p.25/29

NEVOD experiment developed thistechnique independently
When presenting this work in 2001 in Hamburg,
during discussion time A.A. Petruhkin from theNEVOD experiment explained how they did exactlythe same thing...

!w

tsutsvx5qr... and where able to separate an atmospheric
neutrino candidate from ato one backgroundof atmospheric muons in a tiny ()surface detector!

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Modern machine learningclassification
Machine learning - feed a routine a bunch oflabelled signal and background, build a model ofthe Bayes’ posterior for future classification of new
data
Neural networks are an example
Modern methods - Support Vector Machines,
Penalised Likelihood methods (Reproducing
Kernel Hilbert Space methods)

Muon reconstruction and optimal event classification in AMANDA – p.27/29

Penalised likelihood method
Build a model of the Bayes’ discriminant using
weighted sums of basis functions andregularisation methods to control the smoothnessof the solution9

2

Currently building a model of atmospheric andisotropic
neutrinos for our diffuse limit analysis(work in collaboration with UW Statistics)

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Conclusions
Reconstruction of muon tracks in a scattering
medium has been successful
Methods of optimally classifying events as signal
and background have been implemented (zenith
weighted reconstruction) and are underdevelopment (Penalised Likelihood Estimatemodel building)
These provide a unifying framework for thereconstruction and classification problem

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