Statistical problems in network data analysis: burst searches by narrowband detectors L.Baggio and G.A.Prodi ICRR TokyoUniv.Trento and INFN IGEC time coincidence search is taking advantage of “a priori” information template search: matched filters optimized for short and rare transient gw with flat Fourier transform over the detector frequency band many trials at once: - different detector configurations (9 pairs + 7 triples + 2 four-fold) - many target thresholds on the searched gw amplitude (  30) - directional / non directional searches narrowband detectors & same directional sensitivity Cons: probing a smaller volume of the signal parameter space Pros: simpler problem GravStat 2005

… IGEC cont`d data selection and time coincidence search: - control of false dismissal probability - balance between efficiency of detection and background fluctuations background noise estimation - high statistics: 10 3 time lags for detector pairs 10 4 – 10 5 detector triples - goodness of fit tests with background model (Poisson) blind analysis (“good will”): - tuning of procedures on time shifted data by looking at all the observation time (no playground) … what if evidence for a claim would appear ? “GW candidates will be given special attention …” - IGEC-2 agreed on a blind data exchange (secret time lag) GravStat 2005

Poisson statistics For each couple of detectors and amplitude selection, the resampled statistics allows to test Poisson hypothesis for accidental coincidences.   Example: EX-NA background (one-tail  2 p-level 0.71) As for all two-fold combinations a fairly big number of tests are performed, the overall agreement of the histogram of p-levels with uniform distribution says the last word on the goodness-of-the-fit. verified GravStat 2005

A few basics: confidence belts and coverage experimental data physical unknown GravStat 2005

I  can be chosen arbitrarily within this “horizontal” constraint Feldman & Cousins (1998) and variations (Giunti 1999, Roe & Woodroofe 1999,...) Freedom of choice of confidence belt Fixed frequentistic coverage GravStat 2005

Plot of the likelihood integral vs. minimum (conservative) coverage min  C(  ), with background counts N b = Confidence intervals from likelihood integral I fixed, solve for : Compute the coverage Let Poisson pdf: Likelihood: GravStat 2005

Example: Poisson background N b = % 95% 85% N Likelihood integral

Plot of the likelihood integral vs. minimum (conservative) coverage min  C(  ), with background counts N b = Confidence intervals from likelihood integral I fixed, solve for : Compute the coverage Let Poisson pdf: Likelihood: GravStat 2005

Multiple configurations/selection/grouping within IGEC analysis GravStat 2005

Resampling statistics of accidental claims event time series coverage“claims” (0.555) [1] (0.326) [1] expected found Easy to set up a blind search GravStat 2005

Keep track of the number of trials (and their correlation) ! IGEC-1 final results consist of a few sets of tens of Confidence Intervals with min{C}=95%  the “false positives” would hide true discoveries requiring more than  5 two- sided C.I. to reach 0.1% confidence for rejecting H 0 the procedure was good for Upper Limits, but NOT optimized for discoveries  Need to decrease the “false alarm probability” (type I error) GravStat 2005

Freedom of choice of confidence belt Fine tune of the false alarm probability

Example: confidence belt from likelihood integral Poisson background N b = 7.0 Min{C}=95% 1 - C(N  ) P{false alarm} < 0.1% P{false alarm} < 5 % GravStat 2005

What false alarm threshold should be used to claim evidence for rejecting the null H 0 ? GravStat 2005 control the overall false detection probability: Familywise Error Rate <  requires single C.I. with P{false alarm} <  /m Pro: rare mistakes Con: high detection inefficiency control the mean False Discovery Rate: R = total number of reported discoveries F + = actual number of false positives Benjamini & Hochberg (JRSS-B (1995) 57: ) Miller et. al. (A J 122: Dec 2001;

Typically, the measured values of p are biased toward 0. signal The p-values are uniformly distributed in [0,1] if the assumed hypothesis is true Usually, the alternative hypothesis is not known. However, the presence of a signal would contribute to bias the p-values distribution. p-level 1 background pdf FDR control

Sketch of Benjamini & Hochberg FDR control procedure choose your desired bound q on ; OK if p-values are independent or positively correlated compute p-values {p 1, p 2, … p m } for a set of tests, and sort them in creasing order; p-value m determine the threshold T= p k by finding the index k such that p j <(q/m) j for every j>k; reject H 0 q T counts in case NO signal is present (H 0 is true), the procedure is equivalent to the control of the FamilyWise Error Rate at confidence < q

Open questions  check the fluctuations of the random variable FDR with respect to the mean.  check how the expected uniform distribution of p-values for the null H 0 can be biased (systematics, …)  would the colleagues agree that overcoming the threshold chosen to control FDR means & requires reporting a rejection of the null hypothesis ? To me rejection of the null is a claim for an excess correlation in the observatory at the true time, not taken into account in the measured noise background at different time lags. It could NOT be gws, but a paper reporting the H 0 rejection is worthwhile and due. GravStat 2005