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2005 Unbinned Point Source Analysis UpdatePowerPoint Presentation

2005 Unbinned Point Source Analysis Update

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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

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

- Gaussian likelihood

Review -- Methods Compare methods at fixed points in the sky

- Test hypothesis of no signal with likelihood ratio:
- Compare likelihood ratio to distribution obtained in trials randomized in RA to compute significance

- 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 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?

- 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

- No Poisson Statistics

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

- Past AMANDA point source searches
- 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 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)

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

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

- 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

- Addition of energy estimator to likelihood function
- The future:
- Analyze 2000-2005(6) (possibly 1997-2006)

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