1. S.Klimenko, December 2003, GWDAW Burst detection method in wavelet domain (WaveBurst) S.Klimenko, G.Mitselmakher University of Florida l Wavelets l Time-Frequency analysis l Coincidence l Statistical approach l Summary

2. S.Klimenko, December 2003, GWDAW Wavelet basis Daubechies l basis  t   bank of template waveforms   0 - mother wavelet  a=2 – stationary wavelet Fourier wavelet - natural basis for bursts fewer functions are used for signal approximation – closer to match filter not local Haar local orthogonal not smooth local, smooth, not orthogonal Marr Mexican hat local orthogonal smooth

3. S.Klimenko, December 2003, GWDAW Wavelet Transform decomposition in basis {  (t)} d4d4 d3d3 d2d2 d1d1 d0d0 a a. wavelet transform tree b. wavelet transform binary tree d0d0 d1d1 d2d2 a dyadic linear time-scale(frequency) spectrograms critically sampled DWT  fx  t=0.5 LP HP

4. S.Klimenko, December 2003, GWDAW TF resolution d0d0 d1d1 d2d2 l depend on what nodes are selected for analysis  dyadic – wavelet functions  constant  variable  multi-resolution  select significant pixels searching over all nodes and “combine” them into clusters. wavelet packet – linear combination of wavelet functions

5. S.Klimenko, December 2003, GWDAW Choice of Wavelet Wavelet “time-scale” plane wavelet resolution: 64 Hz X 1/128 sec Symlet Daubechies Biorthogonal  =1 ms  =100 ms sg850Hz

6. S.Klimenko, December 2003, GWDAW burst analysis method detection of excess power in wavelet domain l use wavelets  flexible tiling of the TF-plane by using wavelet packets  variety of basis waveforms for bursts approximation  low spectral leakage  wavelets in DMT, LAL, LDAS: Haar, Daubechies, Symlet, Biorthogonal, Meyers. l use rank statistics  calculated for each wavelet scale  robust l use local T-F coincidence rules  works for 2 and more interferometers  coincidence at pixel level applied before triggers are produced

7. S.Klimenko, December 2003, GWDAW “coincidence” Analysis pipeline bp  selection of loudest (black) pixels (black pixel probability P ~10% - 1.64 GN rms) wavelet transform, data conditioning, rank statistics channel 1 IFO1 cluster generation bp wavelet transform, data conditioning rank statistics channel 2 IFO2 cluster generation bp “coincidence” wavelet transform, data conditioning rank statistics channel 3,… IFO3 cluster generation bp “coincidence”

8. S.Klimenko, December 2003, GWDAW Coincidence accept l Given local occupancy P(t,f) in each channel, after coincidence the black pixel occupancy is for example if P=10%, average occupancy after coincidence is 1% l can use various coincidence policies  allows customization of the pipeline for specific burst searches. reject no pixels or L <threshold

9. S.Klimenko, December 2003, GWDAW Cluster Analysis (independent for each IFO) Cluster Parameters size – number of pixels in the core volume – total number of pixels density – size/volume amplitude – maximum amplitude power - wavelet amplitude/noise rms energy - power x size asymmetry – (#positive - #negative)/size confidence – cluster confidence neighbors – total number of neighbors frequency - core minimal frequency [Hz] band - frequency band of the core [Hz] time - GPS time of the core beginning duration - core duration in time [sec] cluster core positive negative cluster halo cluster  T-F plot area with high occupancy

10. S.Klimenko, December 2003, GWDAW Statistical Approach l statistics of pixels & clusters (triggers) l parametric  Gaussian noise  pixels are statistically independent l non-parametric  pixels are statistically independent  based on rank statistics:  – some function u – sign function data: { x i }: |x k1 | < | x k2 | < … < |x kn | rank: { R i }: n n-1 1 example: Van der Waerden transform, R  G(0,1)

11. S.Klimenko, December 2003, GWDAW non-parametric pixel statistics l calculate pixel likelihood from its rank: l Derived from rank statistics  non-parametric l likelihood pdf - exponential xixi percentile probability

12. S.Klimenko, December 2003, GWDAW statistics of filter noise (non-parametric) l non-parametric cluster likelihood l sum of k (statistically independent) pixels has gamma distribution P=10% y single pixel likelihood

13. S.Klimenko, December 2003, GWDAW statistics of filter noise (parametric) P =10% x p =1.64 y Gaussian noise l x: assume that detector noise is gaussian l y: after black pixel selection (| x |> x p )  gaussian tails Y k : sum of k independent pixels distributed as  k

14. S.Klimenko, December 2003, GWDAW cluster confidence l cluster confidence: C = -ln (survival probability) l pdf(C) is exponential regardless of k. S2 inj non-parametric C parametric C S2 inj non-parametric C parametric C

15. S.Klimenko, December 2003, GWDAW Summary A wavelet time-frequency method for detection of un- modeled bursts of GW radiation is presented  Allows different scale resolutions and wide choice of template waveforms.  Uses non-parametric statistics  robust operation with non-gaussian detector noise  simple tuning, predictable false alarm rates  Works for multiple interferometers  TF coincidence at pixel level  low black pixel threshold