S.Klimenko, December 2003, GWDAW

Burst detection method in wavelet domain

(WaveBurst)

S.Klimenko, G.Mitselmakher

University of Florida Wavelets Time-Frequency analysis Coincidence Statistical approach Summary

S.Klimenko, December 2003, GWDAW

Wavelet basis

Daubechies

basis t bank of template waveforms 0 -mother wavelet a=2 – stationary wavelet

Fourier

wavelet - natural basis for burstsfewer functions are used for signal approximation – closer to match filter

ktaa jjjk 0

2/

notlocal

Haar localorthogonalnot smooth

local, smooth,

notorthogonal

MarrMexicanhat local

orthogonalsmooth

S.Klimenko, December 2003, GWDAW

Wavelet Transform decomposition in basis {(t)}

d4

d3

d2

d1

d0

aa. wavelet transform tree b. wavelet transform binary tree

d0

d1

d2

a

dyadic linear

time-scale(frequency) spectrograms

critically sampledDWT

fxt=0.5 LP HP

S.Klimenko, December 2003, GWDAW

TF resolution

d0

d1

d2

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 combinationof wavelet functions

S.Klimenko, December 2003, GWDAW

Choice of Wavelet

Wavelet “time-scale” plane

wavelet resolution: 64 Hz X 1/128 secSymlet Daubechies Biorthogonal

=1 ms

=100 ms

sg850Hz

S.Klimenko, December 2003, GWDAW

burst analysis methoddetection of excess power in wavelet domain

use waveletsflexible tiling of the TF-plane by using wavelet

packetsvariety of basis waveforms for bursts

approximation low spectral leakagewavelets in DMT, LAL, LDAS: Haar, Daubechies,

Symlet, Biorthogonal, Meyers. use rank statistics

calculated for each wavelet scale robust

use local T-F coincidence rulesworks for 2 and more interferometerscoincidence at pixel level applied before triggers

are produced

S.Klimenko, December 2003, GWDAW

“coincidence”

Analysis pipeline

bpselection 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”

S.Klimenko, December 2003, GWDAW

Coincidence

accept

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%

can use various coincidence policies allows customization of the pipeline for specific burst searches.

),(),( 2 ftPftPC

reject

no pixelsor

L<threshold

S.Klimenko, December 2003, GWDAW

Cluster Analysis (independent for each IFO)

Cluster Parameters

size – number of pixels in the corevolume – total number of pixelsdensity – size/volumeamplitude – maximum amplitudepower - wavelet amplitude/noise rmsenergy - power x sizeasymmetry – (#positive - #negative)/sizeconfidence – cluster confidenceneighbors – total number of neighborsfrequency - core minimal frequency [Hz]band - frequency band of the core

[Hz]time - GPS time of the core

beginningduration - core duration in time [sec]

cluster corepositive negative

cluster halo

cluster T-F plot area with high occupancy

S.Klimenko, December 2003, GWDAW

Statistical Approach statistics of pixels & clusters

(triggers) parametric

Gaussian noise pixels are statistically independent

non-parametric pixels are statistically independent based on rank statistics:

iii xuRy )( – some functionu – sign function

data: {xi}: |xk1| < | xk2| < … < |xkn|rank: {Ri}: n n-1 1

example: Van der Waerden transform, RG(0,1)

S.Klimenko, December 2003, GWDAW

non-parametric pixel statistics

calculate pixel likelihood from its rank:

Derived from rank statistics non-parametric

likelihood pdf - exponential

iii x

nPRy uln

nPRixi

percentile probability

S.Klimenko, December 2003, GWDAW

statistics of filter noise (non-parametric)

non-parametric cluster likelihood

sum of k (statistically independent) pixels has gamma distribution

)()(

1

keYYpdf

kYkk

k

k

ii

k nPRY

0ln

P=10%

y

single pixel likelihood

S.Klimenko, December 2003, GWDAW

statistics of filter noise (parametric)

,2

22

pxxy

,)( yeypdf

121 px

P=10%xp=1.64

y

Gaussian noise

x: assume that detector noise is gaussian y: after black pixel selection (|x|>xp)

gaussian tails Yk: sum of k independent pixels distributed

as k

k

ik y0

S.Klimenko, December 2003, GWDAW

cluster confidence

cluster confidence: C = -ln(survival probability)

pdf(C) is exponential regardless of k.

dxexYC

kY

xkkk

1)(

1ln)(

S2 inj

non-parametric C

para

met

ric C

S2 inj

non-parametric C

para

met

ric 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 interferometersTF coincidence at pixel levellow black pixel threshold