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Burst detection method
in wavelet domain
(WaveBurst)
S.Klimenko, G.Mitselmakher
University of Florida
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


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Wavelets
Time-Frequency analysis
Coincidence
Statistical approach
Summary
S.Klimenko, December 2003, GWDAW
Wavelet basis

basis {Y(t)} :
 bank of template waveforms
 Y0 -mother wavelet
 a=2 – stationary wavelet
Fourier
Yjk  a Y0 (a t  k )
j/2
Haar
local
orthogonal
not smooth
not
local
local,
smooth,
not
orthogonal
Mexican
hat
Marr
j
Daubechies
local
orthogonal
smooth
wavelet - natural basis for bursts
fewer functions are used for signal approximation – closer to match filter
S.Klimenko, December 2003, GWDAW
Wavelet Transform
decomposition in basis {Y(t)}
critically sampled
DWT
d0
DfxDt=0.5
d1
LP
dyadic
d2
a
a. wavelet transform tree
linear
HP
d1
d3
d4
d0
d2
a
b. wavelet transform binary tree
time-scale(frequency) spectrograms
S.Klimenko, December 2003, GWDAW
TF resolution
d0
d1
d2

depend on what nodes are selected for analysis
 dyadic – wavelet functions
 constant
 variable
wavelet packet – linear combination
of wavelet functions
 multi-resolution  select significant pixels
searching over all nodes and “combine” them into
clusters.
S.Klimenko, December 2003, GWDAW
Choice of Wavelet
sg850Hz
wavelet resolution: 64 Hz X 1/128 sec
Symlet
Daubechies
Biorthogonal
t=1 ms
Wavelet “time-scale” plane
t=100 ms
S.Klimenko, December 2003, GWDAW
burst analysis method
detection of excess power in wavelet domain



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.
use rank statistics
 calculated for each wavelet scale
 robust
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
Analysis pipeline
channel 1
wavelet transform,
data conditioning,
rank statistics
bp
channel 2
wavelet transform,
data conditioning
rank statistics
“coincidence”
IFO1 cluster
generation
channel 3,…
bp
wavelet transform,
data conditioning
rank statistics
“coincidence”
IFO2 cluster
generation
bp
IFO3 cluster
generation
“coincidence”
bp selection of loudest (black) pixels
(black pixel probability P~10% - 1.64 GN rms)
S.Klimenko, December 2003, GWDAW
Coincidence
accept
reject
no pixels
or
L<threshold

Given local occupancy P(t,f) in each channel, after coincidence the
black pixel occupancy is
2
PC (t , f )  P (t , f )

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.
S.Klimenko, December 2003, GWDAW
Cluster Analysis (independent for each IFO)
cluster  T-F plot area with high occupancy
cluster halo
cluster core
positive
negative
S.Klimenko, December 2003, GWDAW
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]
Statistical Approach


statistics of pixels & clusters (triggers)
parametric
 Gaussian noise
 pixels are statistically independent

non-parametric
 pixels are statistically independent
 based on rank statistics:
data: {xi}: |xk1| < | xk2| < … < |xkn|
rank: {Ri}:
n
n-1
1
yi   ( Ri )  u(xi )
 – some function
u – sign function
example: Van der Waerden transform, RG(0,1)
S.Klimenko, December 2003, GWDAW
non-parametric pixel statistics

calculate pixel likelihood from its rank:
 Ri 
yi   ln 
  u ( xi )
 nP 


Derived from rank statistics  non-parametric
likelihood pdf - exponential
percentile probability
xi
S.Klimenko, December 2003, GWDAW
Ri
nP
statistics of filter noise (non-parametric)

non-parametric cluster likelihood
 Ri 
Yk   i 0 ln 

 nP 
k

sum of k (statistically independent) pixels has gamma
distribution
k 1 Yk
Yk e
pdf (Yk ) 
(k )
P=10%
single pixel likelihood
y
S.Klimenko, December 2003, GWDAW
statistics of filter noise (parametric)
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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
y
x 2  x 2p
2
  (1  x
,
Gaussian noise
)
 2 1
p
pdf ( y )  e y ,
P=10%
k  0 yi
xp=1.64
k
y
S.Klimenko, December 2003, GWDAW
cluster confidence

cluster confidence: C = -ln(survival probability)


1
C (Yk )   ln  ( k )  x k 1e x dx 
Yk


parametric C
pdf(C) is exponential regardless of k.
parametric C

S2 inj
non-parametric C
S.Klimenko, December 2003, GWDAW
S2 inj
non-parametric C
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
S.Klimenko, December 2003, GWDAW