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

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

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

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

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

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

S.Klimenko, December 2003, GWDAW “coincidence” Analysis pipeline bp  selection of loudest (black) pixels (black pixel probability P ~10% 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”

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

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

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)

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

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

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

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

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