LIGO-G07XXXX-00-Z
Glitch Rejection Capabilities of a
Coherent Burst Detection Algorithm
Maria Principe, Innocenzo M. Pinto
TWG, University of Sannio @ Benevento, INFN and LSC
The Waves Group
4th ILIAS-GW Annual General Meeting
Universität Tübingen, October 8-9 2007
Outlook
 Sought signals vs local disturbances:
GW bursts, glitches and atoms
 Simplest coherent network algorithm
 Rationale and model
 Conclusions and future work
4th ILIAS-GW Annual General Meeting
Universität Tubingen, October 8,9 2007
Sought Signals: GWB
(Stolen from Katsavounidis, LIGO-G-070033-00-Z)
4th ILIAS-GW Annual General Meeting
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Sought Signals: GWBs
 Poorly modeled or unmodeled transient signals:
 Sine-Gaussians and Gaussians also probed
 Gross Features:
 Time duration: 1-100 ms typical
 Center frequency: 50 Hz up to few kHz
 Expected strenght ~ 3.6 10-22 Hz-1/2 ( SNR~ 10 )
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Triggered or Untriggered
GWB Searches
 TRIGGERED SEARCH
Targets events which produce EM or neutrino
signatures (e.g. supernovae, gamma-ray bursts).
These signatures provide independent estimates
of time of occurrence and source position.
A small subset of the data stream must be sieved.
 UNTRIGGERED (“BLIND”) SEARCH
No information available as to time of occurrence,
and direction of arrival (DOA), both to be estimated
from data. All available data must be sieved.
4th ILIAS-GW Annual General Meeting
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Glitches
 Non-GWB transients (glitches) show up
in several IFO channels
 Glitches in each channel tend to cluster
in TF plane [Mukherjee, LIGO P070051-00-Z ]
 Strategies to identify/reject some of them
make use of knowledge about the coupling
of instrumental channels with the main detector output. [Ajith, ArXiv:0705.1111]
 Glitches observed in data (DARM_ERR) seem to fall
into a few simple categories (e.g., SG, RD)
[Saulson, LIGO G-070548-00-Z]
 Glitches occur in each detector as Poisson processes
with a characteristic rate λ
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Atoms (1/2)
 Both GW and noisy bursts can be modeled as atoms
(Gabor, Rihaczek) in the TF plane.
 Atoms are transient signals with “almost” compact timefrequency support. Can be characterized by fewest
moments, e.g., time-frequency barycenter (t0, f0) and
spreads (σt ,σf) [P. Flandrin, Time-Frequency/ Time-Scale
Methods, Academic Press,1999]
 The atom’s shape, as well as the ranges of its moments
and the related probability distributions, can be inferred
from theoretical and/or experimental evidences.
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Atoms (2/2)
 A simplest choice for the atoms, for both GW and spurious
noise bursts is perhaps the Sine-Gaussian (SG)
h  t   h0 sin  2 f0 t  t0   e
t  Q
2 f0
 t t0   t2
2
 f  f0 Q
 Spurious glitches can be statistically characterized in terms of the
distributions (priors) of their relevant parameters
Q, f0 , t0 , h0
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Network Operation
 A single detector cannot discriminate a GW burst from
a transient (instrumental) glitch
 Need to operate an ensemble of GW detector
 How many ? How oriented ?
 Two network data analysis strategies developed
 incoherent (e.g. coincidence; experience from bar-detectors)
 coherent (e.g. Gursel-Tinto technique)
 Key benefits:
 Reject spurious glitches;
 Identify direction of arrival (blind search).
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Coherent Network Analysis
 Exploits the redundancy of the network:
 only 2 unknown quantities, h+(t) and h×(t), while D ≥2
detector outputs (over-determined problem, redundant
network)
 Network redundancy is crucial to estimate the DOA, and
to reject spurious transient signals
 Expected to achieve better performance compared to
incoherent analysis [Arnaud et al, PRD 68 (2003) 102005];
 Improved performance paid in terms of heavier computational load.
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Rationale of this Work
Abundant Literature exists about coherent algorithms
performance for DOA retrieval and signal detection.
Only a few papers discuss in quantitative terms the
capabilities of coherent algorithm in rejecting spurious
glitches [Chatterjee et al., LIGO-P060009-01-E];
We propose a simple approach to quantify such capabilities, for the special case of the LH-LL-V network, and
the possibly simplest coherent algorithm, proposed by
Rakhmanov and Klimenko [CQG 22 (2005) S1311];
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(M)-RK Statistic(s)
 Output of i-th detector
Vi t   i  s    Fi   s  h  t   Fi  s  h t   ni t  ,  i   s  


V1   F1 F1 
 n1 
In matrix form
V    F  F    h    n 
Polarization
2
2
2 
  2
 GW
h
Antenna
Patterns


 n 

V3waveforms
  F3 F3at
 3

 Earth’s
center
V F  F  
1
1
1
r i  kˆ   s 
c
rank-2 antenna response
matrix
1



V
F
F
 The  2 2 2  matrix is also rank-2 AV
1 1  t   A2V2  t   AV
3 3 t   0
V F  F  
 
 
3 3 3
A

F
F

F
F2
1
2
3
3
(no noise)
Wi   Ai1 ( AjV j  AkVk )  Vi (t ) ,
, Wi can be used as a noisy template
A2  F3 F1  F1 F3
A3  F1 F2  F2 F1
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The Ai (Ωs)
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Detection Statistics
Define the noisy-template based correlations:
Ci  Vi ,Wi
, i =1,2,…,D
A suitable function of the Ci, must be formed to be used as a
detection statistic. Several choices possible.
 R&K proposed
Cmax  max Ci
i 1,..., D
, (s known, fixed)
This is not the best one (does not exploit all the information collected)
 A better choice is a linear combination of the Ci maximizing the
deflection
opt
opt
D





C
|
H


C
LC
1
LC | H 0 
opt


CLC  ai Ci , ai  s   arg max
opt
opt
stdev CLC
| H1   stdev CLC
| H 0 
i 1

for which the statistical properties can still be obtained in analytic form
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Statistical Distribution of Ci
 Explicit expression of Ci
Aj
Al

Ci 
Vi ,V j 
Vi ,Vl ,
Ai
Ai
s ,v 
N s 1
 s t  v t  ,
k 0
k
k
tk  k f s
 In view of the large (>> 103) number of samples in the integration
window, the (extended) CLT applies, the Ci being sums of many
independent random variables:
Ci  N  i ,  i2 
Noise term in the
template
 2     V 2 t
i
s

i
k
 
2
n
i  s   Vi 2  tk   Ei  f s

2
 ,i
 
k
2
n

2
 ,i
AWGN
power equal
2
2 2
N s  Ei f sin
 Qidetector
  Qi  n N s
any
n 1
 
k
Aj
Al
 i  t    n j  t   nl  t 
Ai
Ai
2
2

   s    Aj Ai    Al Ai    n2


2
 ,i
Q i(Ωs)
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Statistical Distribution of Ci :
H1 hypothesis
2
2

 2

 2

i   s    Fi   s   hrss   Fi   s   hrss   f s


independent on Ai
 / 2
  s   i  s   1  Qi   Qi 
2
i
2
n

2 2
n
h 
 ts  h2/   tk 
rss
Ns
∞
Ai → 0
k
 Choosing Ai Wi(t) as a template
i  s   i  s  Ai  s   0
Ai 0

2
i
 s   A   s  A
A
0
2
i
2
i
2
j
i
 A   Ei f s   A  A
2
l
2
n
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2
j
2
l
 
2 2
n
Ns
Statistical Distribution of Ci
H0 hypothesis (AWGN only)
i(0) (s )  0
H0
(AWGN)
H1
(GWburst)
2
 (s )    n4 N s (1  Qi (s ))
i(1)  s   Vi 2  tk   Ei  f s
(0)
i

(1)
i
 s 
2
k
 Vi  tk    
2
2
n
2
 ,i
 
2
n
 N s  Ei f s 1  Qi   Qi 
2
 ,i
2
n

2 2
n
k
This is all we need to compute ROCs. ROC may be written
in such a way so as to highlight difference with “perfect”
matched filter.
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Ns
K-R ROCs (AWGN only)
Pd  1    erfc  erfc 1 ( )  d h( MF ) 
Would be one for the perfect MF
acting on the GW waveform
 d
( MF )
h
2
2
f s hrss
  2 ,
n
Deflection of
perfect MF acting
on GW waveform
( s )  (d h( MF ) | F |) 2 Qi ( s )  N s  Z ( s )
h  h
2
rss

 2
rss

  hrss

  pol. angle
( s ) | F | Z 1/ 2 ( s )
1


2

Qi

( MF )
2  
(d h | F |)  
, Z (s )  (1  Qi ) 1  N s
1  Qi





1/ 2
| F |  F cos 2  F sin 2 
2
2
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2
2
The 2 function
(+), 10
(+), 20
H+, dmf=20
(), 10
(), 20
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Glitches: a Recipe
 Assume a “network glitch set”, i.e., specify the presence,
firing-time, amplitude, center-frequency and t-f spread
parameters of the glitches (represented by a suitable atom) in
each detector.
 Compute the related distribution (first two moments) of the
detection statistic: this is a conditional distribution, corresponding to
the assumed “network glitch-pattern”;
Average out using the fiducial prior distributions of the glitch parameters. The resulting distribution will be different from the AWGNonly case (nonzero average and broader spread).
Use the resulting distribution for setting the detection threshold as
a function of the prescribed AWGN+glitch-mix false alarm rate.
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H0 (AWGN+glitches) hyp.
Working Assumptions
 The rate λ of Poisson process which models the occurrence of
glitches is assigned (e.g., in [0.1 , 0.5])
 We choose the analysis window T three times the maximum
duration of a bursts, i.e. T ~ 70 ms. Accordingly we make the simplifying working assumptions that in each detector
P(at least a glitch)  P(one glitch occurs)
P(glitch and GWB)  0
2
 t t0 
Glitches  SG-atoms, h  t   h0 sin  2 f 0  t  t0   e
f0 [Hz] ~ U( {700, 849, 1053, 1304, 1615, 2000})
t0 ~ Poisson(), h0 ~ U( [0, max]), Q = 8.9
“Loud” glitches (max : local SNR SNRmax) vetoed out.
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 t2
H0 (AWGN+glitches) hyp.
Marginal Distribution of Ci
 Aj

A
ij (tij ;i , j )  l il (til ;i ,l )  P 2  glitch 
Ai
 Ai

i(0  )   

 i(0  ) ( s )    n2 P  glitch  Qi ii (0;i )  ( Aj / Ai ) 2  jj (0; j )  ( Al / Ai ) 2 ll (0;l )  
2
2 P 2  glitch  ( Aj Al / Ai2 ) jl (t jl ; j , l )  1  P  glitch     n4Qi N s
ij (tij ;i ,  j )   (t  ti ;i ), (t  t j ; j )
 These quantities must be averaged out over random (exponential)
inter-arrival times between events and over (uniform) SG parameters.
Denote as
i(0  ) ,  i(0  ) the averaged quantities.
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K-R ROCs (AWGN+glitches)
P 'd  1   '  erfc   ' erfc 1 ( )   ' d h( MF ) 
Would be one for the perfect MF
acting on the GW waveform
d
( MF )
h

2
f s hrss

2
n
,
 i(0  )
 '( s )   ( s )  (0)
i
 i(0  ) 
 '( s )   ( s ) 1  (1) 
i 

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Deflection of
perfect MF acting
on GW waveform
ROC (AWGN+glitches): Cmax
C
max
1
5.88
0.9
0.8
h2rss =0.0244
0.7
h2rss =0.0137
4.70
0.6
PD
h2rss =0.0381
h2rss =0.0061
h2rss =0.0015
0.5
0.4
3.53
0.3
2.35
0.2
1.17
0.1
0
0
0.02
0.04
0.06
P
FA
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0.08
0.1
Conclusions
 All ingredients for assessing quantitatively the glitch
rejection capabilities of the LH-LL-V network have been
derived for the R-K coherent statistic. Extensive numerical
simulations for the triggered search case (known DOA,
and time of occurrence) in progress.
The more general case where the time and direction of
arrival are unknown and should be estimated can be also
formalized, and is under scrutiny.
Plans to use better atomic objects
GWDAW 10, UTB, 2005])
(chirplets [Sutton,
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Detection/Decision
1.0
PDF ( x | H 0 )
PDF ( x | H1 )
0.8
0.6
0.4   prob( x   | H1 )
  prob( x   | H 0 )
For low , should be
For low 0.2
, should be
 > E(x|H0) + stdev(x|H0)
Detector
performance depends on ratio
E(x|H1) >  + stdev(x|H
1)
E ( x | H1 )  E ( x | H 0 )
d

1
0
1
2
3
-2
stdev( x | H 0 )  stdev( x | H1 )
x >  , H1 accepted
x <  , H1 rejected 
0.0
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x
NP-Strategy



   E(x | H0 ) 
PDF [ x | H 0 ]dx  erfc 

stdev
(
x
|
H
)
0 


   E ( x | H1 ) 
   PDF [ x | H1 ]dx  1  erfc 

stdev
(
x
|
H
)

1 

erfc( z )  (2 )
1/ 2

e
t 2 / 2
dt
z
NP strategy : assign false alarm probability; deduce  from 1stMequation .
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ROCs




   E( x | H0 ) 
PDF [ x | H 0 ]dx  erfc 

stdev
(
x
|
H
)
0 


   E ( x | H1 ) 
PDF
[
x
|
H
]
dx

1

erfc


1

stdev
(
x
|
H
)

1 
For given signal and noise, plot the
curve {1-(), ()}, known as the
Receiver Operating Characteristic
1-
1
0.95
0.9
Each point on the curve
corresponds to a different 
i.e. a different decision-rule.
One can prove that =slope
0.85
0.8
0.75
0.7
0
0.05
0.1
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0.15
0.2
0.25
0.3
0.35
