Estimation of Resolution and reduced Likelihood

analysis-

Kai SchattoUniversity of Mainz

IceCube Colaboration Meeting, Berlin 21. September 09

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Contents

Part 1: Status of Cramer-Rao modul

Part 2: Studies on reduced likelihood

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Cramer-Rao-Bound Cramér-Rao inequality expresses lower bound for

resolution of individual track

Cramér-Rao inequality:

Fisher information matrix:

p=propability density function(in our case Pandel function)

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Cramer-Rao-Bound

Fisher information matrix:

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Cramer-Rao-Bound

Fisher information matrix:

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Cramer-Rao-Bound

Fisher information matrix:

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Cramer-Rao-Bound

Fisher information matrix:

precalculated calculated on the run

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Cut efficiency

Cutting the geometric mean of azimuth and zenith estimation

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Cut efficiency

Cutting the geometric mean of azimuth and zenith estimation

50 %

6 %

3 %

1 %

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Real resol vs estimator

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New: Depth-dependent scatter length

we use the Pandel function as pdf, which depends on lambda:

before a fixed lambda value of 33m was used, which is not accurate due to in ice dust layers for instance

now the Cramer-Rao estimation is calculated with the lambda values from:

„Ackermann, M., et al. (2006), Optical properties of deep glacial ice at the South Pole, J. Geophys. Res., 111, D13203, doi:10.1029/2005JD006687“

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New: Depth-dependent scatter length

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New: Convoluted Pandel using gauss convoluted pandel from:

Implementation of a Gauss convoluted Pandel PDFfor track reconstruction in Neutrino Telescopes,N. van Eijndhoven, O. Fadiran and G. Japaridze

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New: Convoluted Pandel using gauss convoluted pandel from:

Implementation of a Gauss convoluted Pandel PDFfor track reconstruction in Neutrino Telescopes,N. van Eijndhoven, O. Fadiran and G. Japaridze

t/ns

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New: Convoluted Pandel

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Still fast to compute using a precalculated table On an Intel Xeon 2.5GHz quad core

0.91ms/event a paraboloid calculation takes 178ms no need for high level reco

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Part 2: Studies on reduced likelihood

Part 1: Status of Cramer-Rao modul

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llh!=Chi^2

definition of reduced likelihood

but likelihood differs from Chi^2

get further information in Studies on reduced likelihood, Lutz Köpke

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reduced Likelihood vs NChannel

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Nchannel independent rLlh

Fit to find rlogl without NChdependency

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Nchannel independent rLlh

Fit to find rlogl without NChdependency

Reduced Llh with (NCh-0.8)

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Nchannel independent rLlh

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Cut comparison

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Correlation

correlation factor for background = 0.1

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Conclusion & status

Cramer Rao module: Cramér-Rao module currently in testing & review stage fast computation and practical for early cuts not correlating too strong with rLlh cuts integration into offline software soon

rLlh:

rLlh cuts are hidden channel cuts other name for that what‘s currently called „reduced Likelihood“ -> effective Likelihood ??? reduced likelihood similar to Chi^2 with (N-0.8)

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Thanks for listening!