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Network analysis and statistical issues

Network analysis and statistical issues. Lucio Baggio An introductive seminar to ICRR’s GW group. Topics of this presentation. Gravitational wave bursts networks. From the single detector to a worldwide network. IGEC (International GW Collaboration).

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Network analysis and statistical issues

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  1. Network analysis and statistical issues Lucio Baggio An introductive seminar to ICRR’s GW group

  2. Topics of this presentation Gravitational wave bursts networks From the single detector to a worldwide network IGEC (International GW Collaboration) Long-term search with four detectors; directional search and statistical issues Setting confidence intervals From raw data to probability statements; likelihood/Byesian vs frequentist methods False discovery probability Multiple tests and large surveys change the overall confidence of the first detection Miscellaneous topics The LIGO-AURIGA white paper on joint data analysis;problems with non-aligned or different detectors; coherent data analysis.

  3. Network analysis is unavodable, as far as background estimation is concerned

  4. Gravitational wave burst events For fast (~1÷10ms) gw signals the impulse response of the optimal filter for the signal amplitude is an exponentially damped oscillation Even at a very low amplitude the signals from astrophysical sources are expected to be rare. A candidate event in the gravitational wave channel is any single extreme value in a more or less constant time window. Background events come from the extreme distribution for an (almost) Gaussian stochastic process

  5. The background in practice (1) Amplitude distribution of events AURIGA, Jun 12-21 1997 simulation (gaussian) vetoed (2 test) L. Baggio et al. 2 testing of optimal filters for gravitational wave signals: an experimental implementation. Phys. Rev. D, 61:102001–9, 2000

  6. cumulative event rate above threshold false alarm rate [hour-1] after vetoing epoch vetoes (50% of time) vetoed glitches Remaining events after vetoing The background in practice (2) Amplitude distribution of events AURIGA Nov. 13-14, 2004

  7. DT8 DT9 (before veto) DT9 DT6 The background in practice (3) Cumulative power distribution of events TAMA Nov. 13-14, 2004 from the presentation at The 9th Gravitational Wave Data Analysis Workshop (December 15-18, 2004, Annecy, France)

  8. Environmental Monitoring • Try to eliminate locally all possible false signals • Detectors for many possible sources (seismic, acoustic, electromagnetic, muon) • Also trend (slowly-varying) information (tilts, temperature, weather) • Matched filter techniques for `known' signals  this can only decrease background (no confidece for not matched signal) but not increase the (unknown) confidence for remaining signals. The background in practice (4) • Two good reasons for multiple detector analysis • the rate of background candidates can be estimated reliably • the background rate of the network can be less than that of the single detector Non-coeherent methods coincidences among detectors (also non-GW: e.g., optical, g-ray , X-ray, neutrino) Coeherent methods Correlations Maximum likelihood (e.g.: weighted average)

  9. M-fold coincidence search A coincidence is defined as a multiple detection on many detectors of triggers with estimated time of arrival so close that there is a common overlap between their time windows tw. The latter are defined by the estimated timing error. The ideal “off-source” measure of the background cannot be truly performed (no way to shield the detector). The surrogate solution consists in computing coincidence search after proper delays dtk (greater than the timing errors) have been applied to event series. Then, the coincidences due to real signals disappear, and only background coincidences are left.

  10. M-fold coincidence search (2) The expected coincidence rate is given by: C(t) depends on the choice of the the time error boxes: vary with event vary with detector equal and constant Monte Carlo (by shifted times resampled statistics) From IGEC 1997-2000: example of predicted mean false alarm rates. Notice the dramatic improvement when adding a third detector: the occurrence of a 3-fold coincidence would be interpreted inevitably as a gravitational wave signal. In practice, when no signal is detected in coincidence, the upper limit is determined by the total observation time

  11. International networks of GW detectors Interferometers Operative: GEO600 – (Germany/UK) LIGO Hanford 2km – (USA) LIGO Hanford 4km – (USA) LIGO Livingstone 4km – (USA) TAMA300 – (Japan) Upcoming: VIRGO – (Italy/France) CLIO – (Japan) Resonant bars ALLEGRO – (USA) AURIGA – (Italy) EXPLORER – (CERN, Geneva) NAUTILUS – (Italy) EXPLORER GEO600 Virgo, AURIGA, NAUTLUS LIGO TAMA300 CLIO100

  12. International networks of GW detectors 1969 -- Argonne National Laboratory and at the University of Maryland J. Weber, Phys. Rev. Lett. 22, 1320–1324 (1969) 1973-1974 – Phys. Rev. D 14, 893-906 (1976) 15 years of worldwide networks 1989 – 2 bars, 3 months E. Amaldi et al., Astron. Astrophys. 216, 325 (1989). 1991 – 2 bars, 120 days P. Astone et al., Phys. Rev. D59, 122001 (1999). 1995-1996 – 2 detectors, 6 months P. Astone et al., Astropart. Phys. 10, 83 (1999). 1989 – 2 interferometers, 2 days D. Nicholson et al., Phys. Lett. A 218, 175 (1996). 1997-2000 – 2, 3, 4 resonant detectors, resp. 2 years, 6 months, 1 month P. Astone et al., Phys. Rev. D68, 022001 (2003). 2001 – 2 detectors, 11 days TAMA300-LISM collaboration (2004) Phys. Rev. D70, 042003 (2004) 2001 – 2 detectors, 90 days P. Astone et al., Class. Quant. Grav 19, 5449 (2002). 2002 – 3 detectors, 17 days LIGO collaboration B. Abbott et al.,Phys. Rev. D69, 102001 (2004) GW detected? If NOT, why?

  13. The International Gravitational Event Collaboration

  14. The International Gravitational Event Collaboration http://igec.lnl.infn.it LSU group: ALLEGRO (LSU)http://gravity.phys.lsu.edu Louisiana State University, Baton Rouge - Louisiana AURIGA group: AURIGA (INFN-LNL) http://www.auriga.lnl.infn.it INFN of Padova, Trento, Ferrara, Firenze, LNL Universities of Padova, Trento, Ferrara, Firenze IFN- CNR, Trento – Italia ROG group: EXPLORER (CERN) http://www.roma1.infn.it/rog/rogmain.html NAUTILUS (INFN-LNF) INFN of Roma and LNF Universities of Roma, L’Aquila CNR IFSI and IESS, Roma - Italia NIOBE group: NIOBE (UWA) http://www.gravity.pd.uwa.edu.au University of Western Australia, Perth, Australia

  15. The IGEC protocol The source of IGEC data are different data analysis applied to individual detector outputs. The IGEC members are only asked to follow a few general guidelines in order to characterize in a consistent way the parameters of the candidate events and the detector status at any time. Further data conditioning and background estimation are performed in a coordinated way

  16. Fourier amplitude of burst gw exchange threshold arrival time Exchanged periods of observation 1997-2000 ALLEGRO AURIGA NAUTILUS EXPLORER NIOBE fraction of time in monthly bins

  17. The exchanged data gaps events amplitude and time of arrival amplitude (Hz-1·10-21) time (hours) minimum detectable amplitude (aka exchange threshold)

  18. M-fold coincidence search (revised) Ønon-gaussian at low SNR !  < 5% false dismissal for k =4.5 (Tchebyceff inequality) Østrongly dependent on SNR ! A coincidence is defined when for all 0<i,j<M t i – t j< tij~0.1 sec Coincidence windows tij depend on timing error, which is To make things even worse, we would like the sequence of event times to be described by a (possibly non-homogeneous) Poisson point series, which means rare and independent triggers, but this was not the case.

  19. Timing error uncertainty (AURIGA, for -like bursts)

  20. Auto- and cross-correlation of time series (clustering)  Auto-correlation of time of arrival on timescales ~100s  No cross-correlation AL = ALLEGRO AU = AURIGA EX = EXPLORER NA = NAUTILUS NI = NIOBE x-axis: seconds y-axis: counts

  21. Amplitude distributions of exchanged events normalized to each detector threshold for trigger search ·typical trigger search thresholds: SNR 3ALLEGRO, NIOBE SNR 5 AURIGA, EXPLORER, NAUTILUS The amplitude range is much wider than expected extreme distribution: non modeled outliers dominate at high SNR

  22. False alarm reduction by amplitude selection With a small increase of minimum amplitude, the false alarm rate drops dramatically. Corollary: Selected events have naturally consistent amplitudes

  23. Sensitivity modulation for directional search amplitude directional sensitivity amplitude (Hz-1·10-21) time (hours) amplitude (Hz-1·10-21) time (hours)

  24. A small digression: different antenna patterns and the relevance of signal polarization

  25. Introduction • At any given time, the antenna pattern is: • it is a sinusoidal function of polarization, i.e. any gravitational wave detector is a linear polarizer • it depends on declination  and right ascension  through the magnitude A and the phase  • In order to reconstruct the wave amplitude h, any amplitude has to be divided by • This has been extensively used by IGEC: first step is a data selection obtained by putting a threshold  F-1on each detector • We will characterize the directional sensitivity of a detector pair by theproduct of their antenna patterns F1 and F2 • F1F2 is inversely proportional to the square of wave amplitude h2 in a cross-correlation search • F1F2 is an “extension” of the “AND” logic of IGEC 2-fold coincidence

  26. d1-d2 = p/2 d1-d2 = p/4 d1-d2 = 0 Linearly polarized signals For linearly polarized signal,  does not vary with time. The product of antenna pattern as a function of  is given by: The relative phase1-2 between detectors affects the sensitivity of the pair.

  27. AURIGA x TAMA AURIGA2 AURIGA -TAMA sky coverage: (1) linearly polarized signal TAMA2

  28. If: • the signal is circularly polarized: • Amplitude h(t) is varying on timescales longer than 1/f0 Then: • The measured amplitude is simply h(t), therefore it depends only on the magnitude of the antenna patterns. In case of two detectors: • The effect of relative phase 1-2 is limited to a spurious time shifttwhich adds to the light-speed delay of propagation: • (Gursel and Tinto, Phys Rev D 40, 12 (1989) ) Circularly polarized signals y

  29. AURIGA x TAMA AURIGA -TAMA sky coverage: (2) circularly polarized signal AURIGA2 TAMA2

  30. AURIGA x TAMA AURIGA -TAMA sky coverage Linearly polarized signal Circularly polarized signal AURIGA x TAMA

  31. IGEC (continued)

  32. Data selection at work Duty time is shortened at each detector in order to have efficiency at least 50% A major false alarm reduction is achieved by excluding low amplitude events. amplitude (Hz-1·10-21) time (hours)

  33. amplitude of burst gw Duty cycle cut: single detectors total time when exchange threshold has been lower than gw amplitude

  34. Duty cycle cut: network (1) Galactic Center coverage

  35. Duty cycle cut: network (2) searchthreshold6 10 -21/Hz searchthreshold3 10 -21/Hz

  36. time coincidence constraint • The Tchebyscheff inequality provides a robust (with respect to timing error statistics) and general method to limit conservatively the false dismissal false alarms  k • amplitude consistency check: gw generates events with correlated amplitudes • testing (same as above) fraction of found gw coincidences fluctuations of accidental background  When optimizing the (partial) efficiency of detection versus false alarms, we are lead to maximize the ratio A coincidence can be missed because of… False dismissal probability • data conditioning. • The common search thresholdHt guarantees that no gw signal in the selected data are lost because of poor network setup. • …however the efficiency of detection is still undetermined (depends on distribution of signal amplitude, direction, polarization) Best choice for 1997-2000 data: false dismissal in time coincidence less than 5%  30% no amplitude consistency test

  37. Resampling statistics by time shifts amplitude (Hz-1·10-21) time (hours) We can approximately resample the stochastic process by time shift. The in the resampled data the gw sources are off, along with any correlated noise Ergodicity holds at least up to timescales of the order of one hour. The samples are independent as long as the shift is longer than the maximum time window for coincidence search (few seconds)

  38. Example: EX-NA background (one-tail 2 p-level 0.71)  verified Poisson statistics  For each couple of detectors and amplitude selection, the resampled statistics allows to test Poisson hypothesis for accidental coincidences. 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. 

  39. Setting (frequentist) confidence intervals

  40. Unified vs flip-flop approach (1) experimental data physical results hypothesis testing (CL) null upper limit x mup(CL) estimation (with error bars) claim Flip-flop method m(x)± kCLsm

  41. Unified vs flip-flop approach (2) experimental data physical results estimation (with confidence interval) confidence belt x mmin(CL) < m <mmax(CL) Unified approach

  42. Setting confidence intervals IGEC approach is Frequentist in that it computes the confidence level or coverage as theprobability that the confidence interval contains the true value Unified in that it prescribes how to set a confidence interval automatically leading to a gw detection claim or an upper limit however, different from F&C References G.J.Feldman and R.D.Cousins, Phys. Rev. D57 (1998) 3873 B. Roe and M. Woodroofe, Phys. Rev. D 63 (2001) 013009 F. Porter, Nucl. Inst. Meth. A368 (1986), http://www.cithep.caltech.edu/~fcp/statistics/ Particle Data Group: http://pdg.lbl.gov/2002/statrpp.pdf

  43. A few basics: confidence belts and coverage physical unknown experimental data

  44. For each outcome x one should be able to determine a confidence interval Ix For each possible m, the measures which lead to a confidence interval consistent with the true value have probability C(m), i.e. 1-C(m) is thefalse dismissal probability A few basics (2) physical unknown confidence interval coverage experimental data

  45. Feldman & Cousins (1998) and variations (Giunti 1999, Roe & Woodroofe 1999, ...) Roe & Woodroofe [2000]: a Bayesian inspired frequentistic approach Im can be chosen arbitrarily within this “horizontal” constraint Ix can be chosen arbitrarily within this “vertical” constraint Fixed frequentistic coverage Freedom of choice of confidence belt Maximization of “likelyhood” Fine tune of the false discovery probability Non-unified approaches Other requirements...

  46. In general the bounds obtained as a solution to these equations have a coverage (or confidence level) different from “CL” Confidence level, likelyhood, maybe probability? The term “CL” is often found associated with equations like likelihood integral likelihood ratio relative to the maximum likelihood ratio (hipothesis testing)

  47. Let • Poisson pdf: • Likelihood: • I fixed, solve for : • Compute the coverage Confidence intervals from likelihood integral

  48. 99% 95% Likelihood integral 85% N Example: Poisson background Nb = 7.0

  49. Dependence of the coverage from the background likelihood integral = 0.90 Nb=0.01-0.1-1.0-3.0-7.0-10

  50. From likelihood integral to coverage Plot of the likelihood integral vs. minimum (conservative) coverage minN C(N ), for sample values of the background counts Nb, spanning the range Nb=0.01-10

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