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2005 Unbinned Point Source Analysis Update. Jim Braun IceCube Fall 2006 Collaboration Meeting. d. Nch = 20. Nch = 24. Nch = 26. a. Case 1: N bin = 3. d. Nch = 28. Nch = 60. Nch = 102. a. Case 2: N bin = 3. Review -- Inefficiency of Binned Methods. Unused information Event loss

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2005 unbinned point source analysis update

2005 Unbinned Point Source Analysis Update

Jim Braun

IceCube Fall 2006 Collaboration Meeting


Review inefficiency of binned methods

d

Nch = 20

Nch = 24

Nch = 26

a

Case 1: Nbin = 3

d

Nch = 28

Nch = 60

Nch = 102

a

Case 2: Nbin = 3

Review -- Inefficiency of Binned Methods

  • Unused information

    • Event loss

    • Distribution of events within bin

    • Track resolution

    • Event energy

  • Optimization

    • Bin sizes optimized to set the lowest flux limit are not optimal for 5s discovery

  • Unbinned search methods should be better in every way

    • Except work needed to implement them


Review methods

x1

x2

Review -- Methods

  • Comparison of two likelihood approaches with standard binned approach

    • Gaussian likelihood

      • Assume signal distributed according to 2D gaussian determined from MC

    • Paraboloid likelihood

      • Space angle error estimated on event-by-event basis

    • The signal + uniform background hypothesis contains an unknown number of signal events out of Nband total events in declination band around source. Minimize -Log likelihood to find best number of signal events


Review methods1
Review -- Methods

  • Test hypothesis of no signal with likelihood ratio:

  • Compare likelihood ratio to distribution obtained in trials randomized in RA to compute significance

  • Compare methods at fixed points in the sky

    • Simulate signal point source events with neutrino MC in fixed declination bands

    • Choose 1000 random background events from neutrino MC

    • Apply 2005 filter and 2000-2004 point source quality cuts

    • For binned search, optimize bin radius to minimize m90(Nbkgd)/Ns


  • Detection probability

    Detection Probability d=22.5oa=180o, 1000 Background Events

    Likelihood

    Binned

    (Cone)

    5s

    3s

    Detection Probability

    • Gaussian and paraboloid methods perform similarly

      • Paraboloid resolution quality cut applied to simulation, paraboloid method may improve with looser cut

    • Clear 15%-20% decrease in number of events needed to achieve a given significance and detection probability compared to binned method

    • More to gain for hard spectra

      • Use energy information in likelihood formulation


    What if there is no signal
    What if there is no Signal?

    • In the absence of signal, how do limits (sensitivity) of unbinned searches compare with binned?

    • Sensitivity of binned searches:

      • Calculate Nbkgd for optimal search bin at selected zenith angles

      • Look up m90(Nbkgd) from Feldman-Cousins Poisson tables

      • Sensitivity = m90(Nbkgd) * F / Ns(F)

    • Unbinned searches

      • No Poisson Statistics

        • No ‘number’ of observed events

      • Need to create analysis-specific Feldman-Cousins confidence tables


    Feldman cousins tables
    Feldman-Cousins Tables

    • Given an observation of observable o, we would like to place limits on some physical parameter m

      • Past AMANDA point source searches

        • Observable o = number of events in the search bin

        • Parameter m = neutrino flux from a source in direction of search bin

    • We can calculate P(o|m)

      • For a search bin with N events and B expected background, P(o|m) is Poisson probability of N events given mean (m + B)

    • For each m, integrate probability until desired coverage is reached (typically 90%)

      • Order by P(o|m)/P(o|mbest) to determine which values of the observable are included in acceptance region

    • This ‘confidence belt’ in o-m space contains 90% of total probability

      • In 90% of observations of observable o, the true value of m will lie in the confidence belt.

      • 90% upper and lower confidence limits given observable o correspond to confidence belt maximum and minimum values of m


    Feldman cousins tables1
    Feldman-Cousins Tables

    • Construction of confidence belts for likelihood searches

      • m = Poisson mean number of true events, corresponding to flux

      • o = ANY observable

        • Choose Till’s significance estimate as the observable

    • Need table of P(z|m) on a fine grid of m

      • Choose number of signal events (N) from Poisson distribution with mean m

      • Calculate significance estimate and repeat ~10k times

      • Significance estimate distribution yields P(z|m)


    Feldman cousins tables2

    P(z|m) d=22.5, 1000 Background Events

    FC 90% Conf. Band d=22.5, 1000 Bkgd Events

    Feldman-Cousins Tables

    • Easier in practice:

    • Can simulate sources with Nt events and weight by Poisson probability of Nt for a given m

    • Confidence belts constructed by integrating probability for each m to 90%

    • Average upper limitcalculable using confidence band and z distribution for m = 0


    Sensitivity comparison

    Gaussian LH

    Paraboloid LH

    q

    Sensitivity Comparison

    • Compare sensitivity of likelihood methods to sensitivity of binned cone search at three zenith angles

    • 22%-24% better sensitivity at d=22.5o , similar to gain in detection probability

    • Again, more to gain for hard spectra with energy information in likelihood function

      • If Nch is cut parameter, then for E-2 fluxes limits should be better than with optimal Nch cut


    Roadmap to unblinding
    Roadmap to Unblinding

    • Significant work yet to be done to unblind 2005!

      • Addition of energy estimator to likelihood function

        • May be as simple as Nch

      • 2005 neutrino sample selection

        • Cuts intended to maximize neutrino efficiency

    • The future:

      • Analyze 2000-2005(6) (possibly 1997-2006)



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