1. A statistical test for point source searches - Aart Heijboer - AWG - Cern june 2002 A statistical test for point source searches Aart Heijboer contents: Motivation Hypothesis testing reminder Likelihood ratio test Calculation of Likelihood formulas ingredients Event generators test results Conclusions and plans under construction! Results shown only serve as illustration!

2. A statistical test for point source searches - Aart Heijboer - AWG - Cern june 2002 Motivation Suppose we use a binned method and find a candidate bin; we would like to know a bit more about these events: What is the energy? Are the events located together within the bin? is the angular separation compatible with the measured muon energy? not unimportant 2 o x 2 o bin less signal like more signal like what is the energy of this event? Try to develop a method that uses all information (no information loss by binning or clustering)

3. A statistical test for point source searches - Aart Heijboer - AWG - Cern june 2002 probability density Hypothesis testing - reminder Given the data, which is more likely? H 0 : only atmospheric neutrinos are present or H 1 : in addition to the background there is some signal How to decide between the two: Choose a parameter that is a function of the data (data), the 'test statistic'. The distribution of l (data) should be sensitive to whether the data was 'caused' by H 0 or H 1. Define the a region where is unlikely if H 0 is true: Reject H 0 if is in this region rejection region acceptance region H0H0 H1H1

4. A statistical test for point source searches - Aart Heijboer - AWG - Cern june 2002 Hypothesis testing reminder. probability density 1-level of significance 1-power rejection region acceptance region H0H0 H1H1 two important parameters of a test: level of significance (aka size or confidence level)  1- the probability of rejecting H 0, when it H 0 is true. the power   the probability of accepting H 0 when H 1 is true at a fixed level of significance, the power is related to the sensitivity for 'detecting' H 1. for a given H 1 a large power results in the rejection of H0 at high confidence level.

5. A statistical test for point source searches - Aart Heijboer - AWG - Cern june 2002 How to choose the test statistic? We are free to any function of the data as test statistic ! examples for H 1 = 'there is a point source' number events in direction bin how to define bin-size? number of events in a cluster how to define clustering algorithm? minimum of the difference in direction of all pairs of events If H1 is completely specified (no unknown parameters): there exists a recipe for the best possible test statistic! If H1 has free parameters: there is a recipe that usually performs very well.

6. A statistical test for point source searches - Aart Heijboer - AWG - Cern june 2002 The likelihood ratio test statistic If H 1 is completely specified, for example H1 = 'there is a point source of neutrinos at (l,b) =399,-4 with flux 1.10 -4 E -2 GeV -1 m -2 s -1 ', Then the most powerful test is the Neyman-Pearson test (likelihood ratio test): Choose:  log of the ratio of the probabilities of the data (x) under H 1 and H 0.... so we have to calculate the probability of the observed data for a given flux. If H 1 has free parameters: for example H1 = 'there is a point source with a power law spectrum somewhere in the sky', Then is usual to choose the unknown parameters so that they maximise the probability of the data with these parameters, do a likelihood ratio test this is the 'maximum likelihood ratio' test. H1 :  bg  sig H0  bg H1 :  bg  sig H0  bg unknown: has free parameters

7. A statistical test for point source searches - Aart Heijboer - AWG - Cern june 2002 likelihood of all data for flux  Rate of this kind of event Calculation of the likelihood reconstructed muon direction and energy likelihood of event i Likelihood is related to the predicted event rate: Depends only on the data total number of predicted events We know how to calculate event rates except for this factor

8. A statistical test for point source searches - Aart Heijboer - AWG - Cern june 2002 Calculating the likelihood final approximation: split energy and direction dependence & parameterize the direction-term as function of reconstructed energy This is the probability of measuring muon direction when the true neutrino direction is for an event with a reconstructed energy. It is called the point spread function (PSF)

9. A statistical test for point source searches - Aart Heijboer - AWG - Cern june 2002 Calculating the likelihood evaluate at reconstructed coordinates. Integrate over coordinates (l,b) The flux is given by atmospheric background and a point source: peaked in a few degrees weakly dependent on l,b

10. A statistical test for point source searches - Aart Heijboer - AWG - Cern june 2002 Calculating the likelihood: summary prob. of event i background-rate at reconstructed coordinates and energy Point spread function rate of signal events at reconstructed muon energy H1 :  bg  sig H0  bg simply set the signal flux to 0

11. A statistical test for point source searches - Aart Heijboer - AWG - Cern june 2002 Ingredients: The point spread function Distribution of angle between true neutrino and rec. muon for different bins in muon energy (will use rec. muon energy in future). Parameterised with Landau.

12. A statistical test for point source searches - Aart Heijboer - AWG - Cern june 2002 Ingredients: The point spread function II dl (deg) db (deg) Now as function of difference in angular coordinates

13. A statistical test for point source searches - Aart Heijboer - AWG - Cern june 2002 Ingredients: neutrino effective area & P(E  |E ) Used for calculating predicted event rates as function of the neutrino flux OFF-TOPPIC: this table is now implented in CALCRATE to give event rate vs muon energy

14. A statistical test for point source searches - Aart Heijboer - AWG - Cern june 2002 Events & event generators Bartol+RQPM flux for atmospheric neutrinos (= highest flux) Need realistic sample (can not use event weighting for this!) have selected a number of events from production with P(select)]weight. can make program available if you want. Written point source mode for GENHEN generates events for specified declination available in CVS (GENHEN v5r1) Same event selection as atmospheric neutrinos To get a distribution of the test statistic, we need to simulate many full ANTARES experiments (i.e. many years of data taking). For each of them the test statistic must be calculated.  use same sample of events but mix the detection times (event mixing). This can also be done with the real data! atmospheric neutrinos points source neutrinos nb: no atmospheric muons. (hopefully negligible w.r.t atm. neutrinos)

15. A statistical test for point source searches - Aart Heijboer - AWG - Cern june 2002 Clustering The search for sources can be restricted to regions of the sky where events are present, therefore search for clusters evaluate S+B/S likelihood ratio for each cluster cluster with largest likelihood ratio is source candidate clustering algorithm is simple: for each event { find all events within  degrees }  is e.g. 1.5 degrees NB: clustering only serves to speed up the computation (not waste time on areas of the sky where there are no events) The search method is not a 'clustering method' one year of background events

16. A statistical test for point source searches - Aart Heijboer - AWG - Cern june 2002 Fitting source parameters We are looking for point sources with a power law spectrum: flux contains 4 unknown variables source position (2 parameters) spectral index flux 'normalisation' In the test statistic, the Maximum Likelihood is needed: for each cluster, these parameters are fitted (MINUIT) to the data additional advantage: good (best) estimator of the source-position and spectrum. tested by making clusters of 3 or 6 events at known position

17. A statistical test for point source searches - Aart Heijboer - AWG - Cern june 2002 Fitting source parameters: test results true values conclusion: this works (but need to check spectral index)

18. A statistical test for point source searches - Aart Heijboer - AWG - Cern june 2002 Test results Simulated 4 x 1000 x 1 year data-taking periods mix atmospheric event-times and azimuth angles add N signal events from weighted sample of point source events at (ra,dec) = ( 0, -0.5) find test statistic: find all candidate clusters & calculate test stat. test stat. of all data is largest test stat. of all clusters N = Poisson( Rate(  ) )  0 :background only experiments (H 0 )  1 10 -3 E -2 GeV -1 m -2 s -1 (~3 sig. events/year)  10 -3 E -2 GeV -1 m -2 s -1 (~6 sig. events/year)  10 -3 E -2 GeV -1 m -2 s -1 (~12 sig. events/year)

19. A statistical test for point source searches - Aart Heijboer - AWG - Cern june 2002 test results preliminary! ' good' separation between background and signal better separation if there are more signal events

20. A statistical test for point source searches - Aart Heijboer - AWG - Cern june 2002 test results: fitted source position  0 :background only experiments (H 0 )  1 10 -3 E -2 GeV -1 m -2 s -1 (~3 sig. events/year)  1 10 -3 E -2 GeV -1 m -2 s -1  10 -3 E -2 GeV -1 m -2 s -1 cut at 1.4 better resolution if there are more events.

21. A statistical test for point source searches - Aart Heijboer - AWG - Cern june 2002 Conclusions A statistical test was described using the maximum likelihood ratio as test statistic The test 'automatically' takes into account likelihood of measured muon energy for the source spectrum variation of angular resolution point spread function with energy. Work done on event generators to get realistic samples of signal and atmospheric background. First results look sensible: the method seems to be working! Lot of work to be done start using reconstructed muon energy in stead of true. check that atmospheric muons are negligable. check for errors by comparing with binned method. calculate sensitivity / exclusion power Additional ideas for improvement loosen selection cut speed up clustering algorithm the end