Transcript LSC talk on August 22, 2002 - DCC

How optimal are wavelet TF methods?
S.Klimenko





Introduction
Time-Frequency analysis
Comparison with optimal filters
Example with BH-BH merger
Summary
S.Klimenko, August 2003, LSC @ Hannover
LIGO-G030455-00-Z
Introduction

Match filter – optimal detection of signal of known form m(t) (M(w))
 
S 2
N


 M (w ) 2
dω
P
(
w
)
n


1
2
,
(Wainstein, Zubakov)
Many GW waveforms (like mergers, SN,..) are not well known, therefore
other search filters are required.

Excess power filters:
 band-pass filter
e 

(Flanagan, Hughes: gr-qc/9701039v2 1997)
SNRBP 1 / 2
SNRMF

1
2f
f –filter bandwidth
 - signal duration
e for BH-BH mergers ~ 0.2-0.5
 Excess Power: (Anderson et al., PRD, V63, 042003)

What is e for wavelet time-frequency methods (like WaveBurst ETG)?
S.Klimenko, August 2003, LSC @ Hannover
LIGO-G030455-00-Z
Time-Frequency Transform

TF decomposition in a basis of (preferably orthonormal) waveforms
{Yt} - “bank of templates”
wavelet - natural basis for bursts
Fourier
Symlet 58
time-frequency spectrograms
S.Klimenko, August 2003, LSC @ Hannover
LIGO-G030455-00-Z
Symlet 58 packet (4,7)
Time-Frequency Analysis

Analysis steps:
 Select “black” pixels by setting threshold xp on pixels amplitude
The threshold xp defines black pixel probability p
 cluster reconstruction – construct an “event” out of elementary pixels

 Set second threshold(s) on cluster strength
Match filter, if burst matches one of the basis functions (template)
 NS 2opt  x 2

– noise rms per pixel
x=w/ – wavelet amplitude / 
If basis is not optimal for a burst, its energy will be spread over some
area of the TF plot
 
S 2
N TF


 k2
2
x
2
i
e
k – noise rms per k pixels
S.Klimenko, August 2003, LSC @ Hannover
LIGO-G030455-00-Z
 xi 
x2  k
2

1
k
statistics of filter noise


assume that detector noise is white, gaussian
after black pixel selection (|x|>xp) gaussian tails
y

x 2  x 2p
2
pdf ( y )  e
,
y
,
  1  x

 2 1
p
sum of k (statistically independent) pixels has gamma
distribution
yk 
1
2
 x
k
2
i
x
2
p

k 1  y k
yk e
pdf ( yk ) 
(k )
S.Klimenko, August 2003, LSC @ Hannover
p=1%
LIGO-G030455-00-Z
y
z-domain

cluster confidence: z = -ln(survival probability)


1
z( yk )   ln  ( k )  x k 1e x dx 
yk



noise pdf(z) is exponential regardless of k.

control false alarm rate with set of thresholds zt(k) on cluster strength
in z-domain
 zt ( k )
f alarm  e
fk
cluster rates

k

“canonical” threshold set
zt (k )  z0  ln( k )
f alarm  p  f sampling  e
data rate
S.Klimenko, August 2003, LSC @ Hannover
LIGO-G030455-00-Z
 z0
effective distance to source



given a source h(t), the filter response in z-domain is different
depending on how good is approximation of h(t) with the
basis functions{Yt}
d1 - distance to “optimal” source (k=1)
dk – distance to “non-optimal” source with the same
z-response
effectiveness:
e  d k / d1
p=10%
same significance
& false alarm rate as
for MF
S.Klimenko, August 2003, LSC @ Hannover
k=20
LIGO-G030455-00-Z
k=5
k=1
effective distance(snr,k)
e vs SNR
k=3
k=5
k=15
k=30
e vs cluster size
 
S 2
N
S.Klimenko, August 2003, LSC @ Hannover
red – snr=20
blk - snr=25
blue- snr=30
LIGO-G030455-00-Z
cluster size

select transforms that produce more compact clusters
resolution, properties of wavelet filters, orthogonality
octave resolution:
good if fc ~ 1/
=10ms
=1ms~1/850Hz
=100ms
variable resolution
=1ms
=10ms
=100ms
sg850Hz
t resolution: 1/128sec
S.Klimenko, August 2003, LSC @ Hannover
const resolution:
more robust
to cover larger parameter space
the analysis could be conducted
at several different resolutions
LIGO-G030455-00-Z
response to “templates” {Yt}
h(t+dt)=Yi(t), 0<dt<t, t – time resolution of the {Yt} grid
time resolution 1/128 sec
frequency

optimal
“quasi-optimal”
time
black – Symlet (14,6) template
red - SG850, t=10ms


0.8Etot
Average cluster size of ~5 at optimal resolution.
Doesn’t make sense to look for 1-pixel clusters
S.Klimenko, August 2003, LSC @ Hannover
LIGO-G030455-00-Z
0.9Etot
BH-BH mergers

BH-BH mergers
start frequency:
duration:
bandwidth:

(Flanagan, Hughes: gr-qc/9701039v2 1997)
f start  
0.02
M
  205Hz  
  50M  5ms  20MM
f ~
o


20 M
0.13



f qnr  M  1300 Hz  M
BH-BH simulation
(J.Baker et al, astro-ph/0202469v1)
S.Klimenko, August 2003, LSC @ Hannover
20 M o
M
LIGO-G030455-00-Z
o

response to simulated BH-BH mergers
octave resolution
time resolution 1/128 sec
80
40 20
10
need even better resolution
for 10-20Mo black holes


quasi-optimal
k=5
resolution should be >=10ms
If proven by theory, that for BH-BH mergers
fmerger ~1/ ,
it allows a priori selection of a “quasi-optimal” basis
S.Klimenko, August 2003, LSC @ Hannover
LIGO-G030455-00-Z
e for BH-BH mergers
EP filter
k>1,p=10%
p=10%
p=1%
1 -band-pass
5
e~0.7-0.8

ways to increase e
 higher black pixel probability
 ignore small clusters (k=1,2), which contribute most to false alarm
rate and use lower threshold for larger clusters.
S.Klimenko, August 2003, LSC @ Hannover
LIGO-G030455-00-Z
Summary
• wavelet and match filter are compared by using a simple
approximation of the wavelet filter noise.
• filter performance depends on how optimal is the wavelet
resolution with respect to detected gravity waves.
• filter performance could be improved by increasing the
black pixel probability and by ignoring small (k=1,2)
clusters
• expected
efficiency for BH-BH mergers with respect to
match filter: 0.7-0.8
S.Klimenko, August 2003, LSC @ Hannover
LIGO-G030455-00-Z