Transcript Seminar at ICRR (2005.01.19)

Network analysis and statistical issues
Lucio Baggio
An introductive seminar to ICRR’s GW group
Topics of this presentation
Gravitational wave bursts networks
From the single detector to a worldwide network
IGEC (International GW Collaboration)
Long-term search with four detectors; directional search and statistical issues
Setting confidence intervals
From raw data to probability statements; likelihood/Byesian vs frequentist methods
False discovery probability
Multiple tests and large surveys change the overall confidence of the first detection
Miscellaneous topics
The LIGO-AURIGA white paper on joint data analysis;problems with non-aligned
or different detectors; coherent data analysis.
Network analysis is
unavodable, as far as
background estimation is
concerned
Gravitational wave burst events
For fast (~1÷10ms) gw signals the
impulse response of the optimal filter
for the signal amplitude is an
exponentially damped oscillation
Even at a very low amplitude the
signals from astrophysical sources
are expected to be rare.
A candidate event in the gravitational
wave channel is any single extreme
value in a more or less constant time
window.
Background events come from the
extreme distribution for an (almost)
Gaussian stochastic process
The background in practice (1)
Amplitude distribution of events
AURIGA, Jun 12-21 1997
simulation (gaussian)
vetoed (2 test)
L. Baggio et al.
 2 testing of optimal filters for
gravitational wave signals: an
experimental implementation.
Phys. Rev. D, 61:102001–9, 2000
The background in practice (2)
Amplitude distribution of events
AURIGA Nov. 13-14, 2004
cumulative event rate above threshold
false alarm rate [hour-1]
after vetoing
epoch vetoes (50% of time)
vetoed glitches
Remaining events after vetoing
The background in practice (3)
Cumulative power distribution of events
TAMA Nov. 13-14, 2004
from the presentation at The 9th Gravitational Wave Data Analysis Workshop
(December 15-18, 2004, Annecy, France)
10
10
10
10
10
10
0
DT8
–1
–2
–3
–4
–5
Gaussian noise
Rate [events/sec]
10
DT9 (before veto)
DT9
DT6
–6
10
0
10
1
2
3
4
10
10
10
10
Event Power Threshold P
( th)
5
10
6
The background in practice (4)
Environmental Monitoring
• Try to eliminate locally all possible false signals
• Detectors for many possible sources (seismic, acoustic, electromagnetic, muon)
• Also trend (slowly-varying) information (tilts, temperature, weather)
• Matched filter techniques for `known' signals  this can only decrease
background (no confidece for not matched signal) but not increase the (unknown)
confidence for remaining signals.
Two good reasons for multiple detector analysis
1.
the rate of background candidates can be estimated reliably
2.
the background rate of the network can be less than that of the single detector
Non-coeherent methods
coincidences among detectors (also non-GW: e.g., optical, g-ray , X-ray, neutrino)
Coeherent methods
Correlations
Maximum likelihood (e.g.: weighted average)
8
M-fold coincidence search
A coincidence is defined as a multiple detection on many detectors of triggers with
estimated time of arrival so close that there is a common overlap between their time
windows tw. The latter are defined by the estimated timing error.
The ideal “off-source” measure of the background cannot be truly performed (no way to
shield the detector). The surrogate solution consists in computing coincidence search
after proper delays dtk (greater than the timing errors) have been applied to event series.
Then, the coincidences due to real signals disappear, and only background coincidences
are left.
M-fold coincidence search (2)
M
 b (t )  C(t )    ( k ) (t )
The expected coincidence rate is given by:
C(t) depends on the choice of the the time error boxes:
vary with detector
equal and constant
C(t)  M  2tw 
false
alarm
rate
-1
[yr ]
M 1
M
C(t )   2t
k 1 h  k
10
0.1
0.01
1E-3
1E-4
1E-5
1E-6
2E-21
vary with event
Monte Carlo
(by shifted times
resampled statistics)
From IGEC 1997-2000: example
of predicted mean false alarm
rates. Notice the dramatic
improvement when adding a
third detector: the occurrence of
a 3-fold coincidence would be
interpreted inevitably as a
gravitational wave signal.
AL-AU
AL-AU-NA
1
( h)
w
k 1
1E-20
-1
common search threshold [Hz ]
In practice, when no signal is
detected in coincidence, the
upper limit is determined by the
total observation time
International networks of GW detectors
Interferometers
Resonant bars
Operative:
ALLEGRO – (USA)
GEO600 – (Germany/UK)
AURIGA – (Italy)
LIGO Hanford 2km – (USA)
EXPLORER – (CERN, Geneva)
LIGO Hanford 4km – (USA)
NAUTILUS – (Italy)
LIGO Livingstone 4km – (USA)
EXPLORER
GEO600
TAMA300 – (Japan)
Virgo, AURIGA, NAUTLUS
LIGO
TAMA300
CLIO100
Upcoming:
VIRGO – (Italy/France)
CLIO – (Japan)
International networks of GW detectors
1969 -- Argonne National Laboratory and at the University of Maryland
J. Weber, Phys. Rev. Lett. 22, 1320–1324 (1969)
1973-1974 – Phys. Rev. D 14, 893-906 (1976)
15 years of worldwide networks
1989 – 2 bars, 3 months
E. Amaldi et al., Astron. Astrophys. 216, 325 (1989).
1991 – 2 bars, 120 days
P. Astone et al., Phys. Rev. D 59, 122001 (1999).
1995-1996 – 2 detectors, 6 months
P. Astone et al., Astropart. Phys. 10, 83 (1999).
1989 – 2 interferometers, 2 days
D. Nicholson et al., Phys. Lett. A 218, 175 (1996).
1997-2000 – 2, 3, 4 resonant detectors, resp. 2 years, 6 months, 1 month
P. Astone et al., Phys. Rev. D 68, 022001 (2003).
2001 – 2 detectors, 11 days
TAMA300-LISM collaboration (2004)
Phys. Rev. D 70, 042003 (2004)
2001 – 2 detectors, 90 days
P. Astone et al., Class. Quant. Grav 19, 5449 (2002).
2002 – 3 detectors, 17 days
LIGO collaboration
B. Abbott et al., Phys. Rev. D 69, 102001 (2004)
GW detected?
If NOT, why?
The International
Gravitational Event
Collaboration
The International Gravitational Event Collaboration
http://igec.lnl.infn.it
LSU group:
ALLEGRO (LSU)
http://gravity.phys.lsu.edu
Louisiana State University, Baton Rouge - Louisiana
AURIGA group: AURIGA (INFN-LNL)
http://www.auriga.lnl.infn.it
INFN of Padova, Trento, Ferrara, Firenze, LNL
Universities of Padova, Trento, Ferrara, Firenze
IFN- CNR, Trento – Italia
ROG group:
EXPLORER (CERN)
NAUTILUS (INFN-LNF)
INFN of Roma and LNF
Universities of Roma, L’Aquila
CNR IFSI and IESS, Roma - Italia
http://www.roma1.infn.it/rog/rogmain.html
NIOBE group: NIOBE (UWA)
http://www.gravity.pd.uwa.edu.au
University of Western Australia, Perth, Australia
The IGEC protocol
The source of IGEC data are different
data analysis applied to individual
detector outputs.
The IGEC members are only asked to
follow a few general guidelines in order
to characterize in a consistent way the
parameters of the candidate events and
the detector status at any time.
Further
data
conditioning
and
background estimation are performed in
a coordinated way
Exchanged periods of observation 1997-2000
ALLEGRO
AURIGA
NAUTILUS
EXPLORER
NIOBE
fraction of time in monthly bins
21
exchange threshold
1
 6  10 Hz
3  6  1021 Hz 1
 3  1021 Hz 1
Fourier amplitude of burst gw
h(t )  H 0   (t  t0 )
arrival time
amplitude (Hz-1·10-21)
The exchanged data
gaps
events amplitude
and time of arrival
10
9
8
7
6
5
4
3
2
1
0
0
6
12
18
24
30
36
42
48
54
60
time (hours)
minimum detectable amplitude
(aka exchange threshold)
M-fold coincidence search (revised)
A coincidence is defined when for all 0<i,j<M t i – t j< tij~0.1 sec
Coincidence windows tij depend on timing error, which is
 non-gaussian at low SNR !
tij  k  i2   2j 
< 5% false dismissal for k =4.5
(Tchebyceff inequality)
 strongly dependent on SNR !
To make things even worse, we would like the sequence of event times to be
described by a (possibly non-homogeneous) Poisson point series, which
means rare and independent triggers, but this was not the case.
Timing error uncertainty (AURIGA, for -like bursts )
Auto- and cross-correlation of time series (clustering)
 Auto-correlation of time of arrival on timescales ~100s
 No cross-correlation
AL = ALLEGRO
AU = AURIGA
EX = EXPLORER
NA = NAUTILUS
NI = NIOBE
x-axis: seconds
y-axis: counts
Amplitude distributions of exchanged events
normalized to each detector threshold for trigger search
1
1
-1
relative counts
10
-2
-2
10
10
-3
-3
10
relative counts
-1
10
10
-4
-4
10
10
-5
-5
10
10
1
10 AMP/THR
ALLEGRO
1
10 AMP/THR
AURIGA
1
10 AMP/THR
EXPLORER
1
10 AMP/THR
NAUTILUS
1
10 AMP/THR
NIOBE

typical trigger search thresholds:
SNR 3 ALLEGRO, NIOBE
SNR 5 AURIGA, EXPLORER, NAUTILUS
The amplitude range is much wider than expected extreme
distribution: non modeled outliers dominate at high SNR
False alarm reduction by amplitude selection
With a small increase of minimum amplitude, the false alarm rate drops dramatically.
amplitude
time
Corollary:
Selected events have naturally consistent amplitudes
amplitude (Hz-1·10-21)
Sensitivity modulation for directional search
10
9
8
7
6
5
4
3
2
1
0
0
6
12
18
24
30
36
42
48
54
60
amplitude (Hz-1·10-21)
time (hours)
1.0
10
0.9
9
amplitude
directional
sensitivity
0.8
8
7
0.7
6
0.6
sin 2 GC
0.5
5
sin 2 GC
0.4
4
0.3
3
2
0.2
1
0.1
0
0.0
0
6
12
18
24
30
36
42
48
54
60
time (hours)
A small digression:
different antenna patterns
and the relevance of signal
polarization
Introduction
• At any given time, the antenna pattern is:
F ( ,  , )  A( ,  ) cos( 2   ( ,  ))
 it is a sinusoidal function of polarization , i.e. any gravitational wave
detector is a linear polarizer
 it depends on declination  and right ascension  through the
magnitude A and the phase 
•
In order to reconstruct the wave amplitude h, any amplitude has to be
divided by
F ( ,  , )
This has been extensively used by IGEC: first step is a data selection obtained by
putting a threshold  F-1 on each detector
•
We will characterize the directional sensitivity of a detector pair by the product of
their antenna patterns F1 and F2
 F1F2 is inversely proportional to the square of wave amplitude h2 in a crosscorrelation search
 F1F2 is an “extension” of the “AND” logic of IGEC 2-fold coincidence
Linearly polarized signals
For linearly polarized signal,  does not vary with time.
The product of antenna pattern as a function of  is given by:
F1 ( ) F2 ( ) 
 A1 A2 cos( 2   1 ) cos( 2   2 ) 
 A1 A2  12 cos( 4   1   2 )  cos( 1   2 )
The relative phase 1-2 between detectors affects the sensitivity of the pair.
F1 ( )  F2 ( )
F1 ( )  F2 ( )
22 p/2
0
11
p/4
AURIGA -TAMA sky coverage: (1) linearly polarized signal
F1
AURIGA2
1   2
2

p 2
F1 ( ) F2 ( )  A1 A2 
TAMA2
1
2
cos( 4  1   2 )  cos(1   2 )
F2
2

AURIGA x TAMA
0
F1  F2

Circularly polarized signals
If:

f0
the signal is circularly polarized:
 h 
 cos( 2 ) 
   h(t ) 

h
sin(
2

)


 

  2pf 0  t
h

F
Amplitude h(t) is varying on timescales longer than 1/f0
F
Then:

The measured amplitude is simply h(t), therefore it depends only on the magnitude of
the antenna patterns. In case of two detectors:
A1 A2  F 1  F 1  F 2  F
2
2
2
2
2

The effect of relative phase 1-2 is limited to a spurious time shift t which adds to the
light-speed delay of propagation:
t 
1   2
2p f 0
t 
1
4 f0
(Gursel and Tinto, Phys Rev D 40, 12 (1989) )
AURIGA -TAMA sky coverage: (2) circularly polarized signal
A1  F1  F1
2
AURIGA2
2
2
F1 ( ) F2 ( )  A1 A2  12 cos( 4  1   2 )  cos(1   2 )
AURIGA x TAMA
TAMA2
A2  F2  F2
2
2
2
A1  A2
AURIGA -TAMA sky coverage
Circularly polarized signal
AURIGA x TAMA
A1 A2
Linearly polarized signal
AURIGA x TAMA
F1  F2

IGEC (continued)
Data selection at work
amplitude (Hz-1·10-21)
Duty time is shortened at each
detector in order to have efficiency
at least 50%
A major false alarm reduction is
achieved by excluding low
amplitude events.
10
9
8
7
6
5
4
3
2
1
0
0
6
12
18
24
30
36
42
48
54
60
time (hours)
Duty cycle cut: single detectors
total time when exchange threshold has been lower than gw amplitude
amplitude of burst gw
Duty cycle cut: network
Center
Galactic center coverage:Galactic
1997-2000
10000
single
3-fold
4yr limit
days
1000
100
coverage
DETECTORS TIME FRACTION
(days)
of 4 yr
search threshold 6 E-21 /Hz
1 or more
61%
894
2 or more
27%
397
3 or more
5%
70
4
0.5%
7.2
search threshold 3 E-21 /Hz
1 or more
25%
359
2 or more
5%
70
3 or more
0.2%
3
4
-
10
1
1.E-21
2-fold
4-fold
(1)
1.E-20
search threshold (Hz -1)
Duty cycle cut: network
(2)
duty cycle percentage
search threshold 6  10 -21/Hz
100%
1
75%
2
50%
3
25%
0%
01-Jan-97
4
31-Dec-97
30-Dec-98
29-Dec-99
duty cycle percentage
search threshold 3  10 -21/Hz
100%
1
75%
2
50%
3
25%
0%
01-Jan-97
4
31-Dec-97
30-Dec-98
29-Dec-99
False dismissal probability
A coincidence can be missed because of…
• data conditioning.
The common search threshold Ht guarantees that no gw signal in the
selected data are lost because of poor network setup.
…however the efficiency of detection is still undetermined (depends
on distribution of signal amplitude, direction, polarization)
• time coincidence constraint
The Tchebyscheff inequality provides a robust (with respect to timing
error statistics) and general method to limit conservatively the false dismissal
1
ti  t j  k  i2   2j  false dismissal  2
k
false alarms  k
• amplitude consistency check: gw generates events with correlated amplitudes
testing Ai  Aj  A (same as above)
 When optimizing the (partial)
efficiency of detection versus false
fraction of found gw coincidences
alarms, we are lead to maximize the ratio
fluctuations of accidental background
Best choice for 1997-2000 data:
false dismissal in time coincidence less than 5%  30%
no amplitude consistency test
amplitude (Hz-1·10-21)
Resampling statistics by time shifts
10
9
8
7
6
5
4
3
2
1
0
0
6
12
18
24
30
36
42
48
54
60
time (hours)
We can approximately resample the stochastic process by time shift.
The in the resampled data the gw sources are off, along with any
correlated noise
Ergodicity holds at least up to timescales of the order of one hour.
The samples are independent as long as the shift is longer than the
maximum time window for coincidence search (few seconds)
 Poisson statistics verified
 For
each couple of
detectors and amplitude
selection, the resampled
statistics allows to test
Poisson hypothesis for
accidental coincidences.
Example: EX-NA background
(one-tail 2 p-level 0.71)

As
for
all
two-fold
combinations a fairly big
number of tests are
performed, the overall
agreement
of
the
histogram of p-levels with
uniform distribution says
the last word on the
goodness-of-the-fit.
Setting (frequentist)
confidence intervals
Unified vs flip-flop approach (1)
experimental
data
x
physical
results
hypothesis
testing
(CL)
null
claim
upper limit
mup(CL)
estimation
(with error bars)
Flip-flop method
m(x)  kCL m
Unified vs flip-flop approach (2)
experimental
data
x
Unified approach
physical
results
confidence
belt
estimation
(with confidence
interval)
mmin(CL)  m mmax(CL)
Setting confidence intervals
IGEC approach is
Frequentist in that it computes the confidence level or coverage as the
probability that the confidence interval contains the true value
Unified in that it prescribes how to set a confidence interval
automatically leading to a gw detection claim or an upper limit
however, different from F&C
References
G.J.Feldman and R.D.Cousins, Phys. Rev. D 57 (1998) 3873
B. Roe and M. Woodroofe, Phys. Rev. D 63 (2001) 013009
F. Porter, Nucl. Inst. Meth. A368 (1986), http://www.cithep.caltech.edu/~fcp/statistics/
Particle Data Group: http://pdg.lbl.gov/2002/statrpp.pdf
physical unknown
A few basics: confidence belts and coverage
1
coverage
0
m
x
x
x
experimental data
A few basics (2)
For each outcome x one should be able to determine a confidence interval Ix
For each possible m, the measures x  I m which lead to a confidence interval consistent
with the true value have probability C(m), i.e. 1-C(m) is the false dismissal probability
physical
unknown
coverage
C(m) 

x|mI x
pdf ( x ; m)
confidence
interval I x
m
Im
x
experimental data
Freedom of choice of confidence belt
Fixed frequentistic coverage
Maximization of “likelyhood”
Feldman & Cousins (1998) and variations (Giunti 1999, Roe & Woodroofe 1999, ...)
Roe &
[2000]:
a Bayesian probability
inspired frequentistic approach
Fine tune
ofWoodroofe
the false
discovery
C(m) CL
Non-unified approaches
(m ; x)dm

Other requirements...
Ix
Im can be chosen
arbitrarily
constraint
 within
" CL " this
 “horizontal”
C(m )

(m ; x)dm
usually
Ix can be chosen arbitrarily within this “vertical” constraint
GWfanatic
enthusiastic
skeptical
decision threshold
1
coverage
0
Confidence level, likelyhood,
maybe probability?
The term “CL” is often found associated with equations like

(m ; x)dm

(m ; x)dm
Ix
 " CL "  C(m)
usually
(m limit ; x)
 " CL "  C(m)
usually
max (m ; x)
likelihood integral
likelihood ratio relative
to the maximum
m
(m1 ; x)
 " CL "  C(m)
usually
(m2 ; x)
likelihood ratio
(hipothesis testing)
In general the bounds obtained as a solution to these equations have a
coverage (or confidence level) different from “CL”
Confidence intervals from likelihood integral
• Let Nc  Nb  N
 Nb  Tobs
e   Nb  N  
Nc
• Poisson pdf: f ( Nc ; N ) 
 Nb  N  
Nc !
• Likelihood:
(N ; Nc )  f (Nc ; N )
 (Ninf ; Nc )  (Nsup ; Nc )

1 N
• I fixed, solve for 0  Ninf  Nsup: 

sup


I

(
N
;
N
)
dN
(N ; Nc )dN
  0

c


N

inf
• Compute the coverage C(N ) 

Nc|Ninf  N  Nsup
f (Nc ; N )  I
Example: Poisson background Nb = 7.0
10
10
9
9
99.9%
8
8
7
99%
7
95%
6
6
N 5
50%
50%
5
95%
4
99%
4
99.9%
3
3
2
2
1
1
0
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
coincidence counts Nc
99%
Nc  Nb  N 
95%
Likelihood
integral
85%
N
Dependence of the coverage
from the background
likelihood integral = 0.90
Nb=0.01-0.1-1.0-3.0-7.0-10
From likelihood integral to coverage
Plot of the likelihood integral vs. minimum (conservative) coverage minN C(N ),
for sample values of the background counts Nb, spanning the range Nb=0.01-10
IGEC results (and what we
learned from experience)
Setting confidence intervals on IGEC results
GOAL: estimate the number (rate) of gw detected with amplitude  Ht
Example: confidence interval with coverage  95%
“upper limit”: true value outside
with coverage  95%
18
16
14
12
Ngw
10
8
6
Ht
4
2
0
1.0
10.0
search threshold [10-21/Hz]
100.0
Uninterpreted upper limits
…on RATE of BURST GW from the GALACTIC CENTER DIRECTION
whose measured amplitude is greater than the search threshold
no model is assumed for the sources, apart from being a random time series
1,000
100
0.60
0.80
rate
(year –1)
0.90
0.95
10
1
1E-21
1E-20
search threshold
(Hz -1 )
1E-19
true rate value is under the curves with a probability = coverage
ensured
minimum
coverage
Upper limits after amplitude selection
18
16
systematic search on thresholds
many trials !
14
12
Ngw
10
8
all upper limits but one:
6
4
2
0
1.0
10.0
100.0
search threshold [10-21/Hz]
overall false alarm probability
33%
at least one detection in case
NO GW are in the data
NULL HYPOTHESIS WELL IN
AGREEMENT WITH THE
OBSERVATIONS
Multiple configurations/selection/grouping
within IGEC analysis
Resampling statistics of accidental claims
event time series
500
counts
400
300
200
100
0
expected
found
coverage
“claims”
0.90
0.866 (0.555) [1]
0.95
0.404 (0.326) [1]
0
1
2
3
4
5
numer of false alarms
Resampling  blind analysis!
False discovery rate:
setting the probability of
false claim of detection
Why FDR?
When should I care of multiple test procedures?.
•
All sky surveys: many source directions and polarizations are tried
•
Template banks
•
Wide-open-eyes searches: many analysis pipelines are tried altogether,
with different amplitude thresholds, signal durations, and so on
•
Periodic updates of results: every new science run is a chance for a
“discovery”. “Maybe next one is the good one”.
•
Many graphical representations or aggregations of the data: “If I change
the binning, maybe the signal shows up better…
Preliminary (1) : hypothesis testing
False discoveries
(false positives)
Null (Ho) True
Background (noise)
Alternative True
signal
Null Retained
(can’t reject)
Reject Null =
Accept Alternative
Total
U
B
Type I Error α = εb
mo
Type II Error β = 1- εs
T
S
m1
R = S+B
m-R
m
inefficiency
Detected
signals
(true positives)
Reported
signal
candidates
Preliminary (2): p-level
Assume you have a model for the noise that affects the measure x.
You derive a test statistics t(x) from x.
F(t) is the distribution of t when x is sampled from noise only (off-signal).
The p-level associated with t(x) is the value of the distribution of t in t(x):
p = F(t) = P(t>t(x))
•
•
Example: 2 test  p is the “one-tail” 2 probability associated with n
counts (assuming d degrees of freedom)
The distribution of p is always linearly raising in case of agreement of
the noise with the model P(p)=p  dP/dp = 1
Usually, the alternative hypothesis is not known.
However, for our purposes it is pdf
sufficient assuming that the
signal can be distinguished 1
from the noise, i.e. dP/dp  1.
Typically, the measured values of
p are biased toward 0.
signal
background
p-level
Usual multiple testing procedures
For each hypothesis test, the condition {p<  reject null} leads to false
positives with a probability 
In case of multiple tests (need not to be the same test statistics, nor the same
tested null hypothesis), let p={p1, p2, … pm} be the set of p-levels. m is the trial
factor.
We select “discoveries” using a threshold T(p): {pj<T(p) reject null}.
• Uncorrected testing: T(p)= 
–The probability that at least one rejection is wrong is
P(B>0) = 1 – (1- )m ~ m
hence false discovery is guaranteed for m large enough
• Fixed total 1st type errors (Bonferroni): T(p)= /m
–Controls familywise error rate in the most stringent manner:
P(B>0) = 
–This makes mistakes rare…
–… but in the end efficiency (2nd type errors) becomes negligible!!
Controlling false discovery fraction
We desire to control (=bound) the ratio of false discoveries over the
total number of claims: B/R = B/(B+S)  q.
The level T(p) is then chosen accordingly.
Let us make a simple case when signals are easily separable (e.g. high SNR)
pdf

B B
T ( p) 

m0 m
S
B
m0
q
cumulative
counts
R

B mT ( p )
FDR  q  
R
R
p
m

T ( p) q

R
m
B
S

T ( p)
p
Benjamini & Hochberg FDR control procedure
Among the procedures that accomplish this task, one simple recipe was proposed
by Benjamini & Hochberg (JRSS-B (1995) 57:289-300)
• compute p-values {p1, p2, … pm} for a set of tests, and sort them in creasing order;
• choose your desired FDR q (don’t ask too much!);
• determine the threshold T(p)= pk by finding the index k such that pj<(q/m) j/c(m)
for every j>k;
• define c(m)=1 if p-values are independent or positively correlated; otherwise
c(m)=Sumj(1/j)
m
q/c(m)
reject H0

T ( p)
p
LIGO – AURIGA:
coincidence vs correlation
LIGO-AURIGA MoU
A working group for the joint burst search in LIGO and AURIGA has been
formed, with the purpose to:
» develop methodologies for bar/interferometer searches, to be tested on real data
» time coincidence, triggered based search on a 2-week coincidence period (Dec
24, 2003 – Jan 9, 2004)
» explore coherent methods
‘best’ single-sided PSD
Simulations and methodological
studies are in progress.
White paper on joint analysis
Two methods will be explored in parallel:
Method 1:
• IGEC style, but with a new definition of consistent amplitude estimator in
order to face the radically different spectral densities of the two kind of
detectors (interferometers and bars).
• To fully exploit IGEC philosophy, as the detectors are not parallel,
polarization effects should be taken into account (multiple trials on
polarization grid).
Method 2:
• No assumptions are made on direction or waveform.
• A CorrPower search (see poster) is applied to the LIGO interferometers
around the time of the AURIGA triggers.
• Efficiency for classes of waveforms and source population is performed
through Monte Carlo simulation, LIGO-style (see talks by Zweizig, Yakushin,
Klimenko).
• The accidental rate (background) is obtained with unphysical time-shifts
between data streams.
Summary of non-directional “IGEC style” coincidence search
 Detectors: PARALLEL, BARS
 Shh: SIMILAR FREQUENCY RANGE
 Search: NON DIRECTIONAL
 Template: BURST = (t)
HS
The search coincidence is performed in a
subset of the data such that:
 the efficiency is at least 50% above the
threshold (HS)
 significant false alarm reduction is
accomplished
The number of detectors in coincidence
considered is self-adapting
detector 1
detector 2
AND
AND
detector 2
AND
detector 3
This strategy can be made directional
Cross-correlation search (naïve)
Threshold crossing
after correlation
detector 1
detector 1 * detector 2
detector 2
 x  x    T
1
j 1,n
j
Detectors: PARALLEL
Shh: SAME FREQUENCY RANGE NEEDED
Search: NON DIRECTIONAL
Template: NO
Selection based on data quality can be
implemented before cross-correlating.
T
w(1) w( 2 )




The efficiency is to be determined a posteriori
using Monte Carlo.
2
j
The information which is usually included in
cross-correlation takes into account statistical
properties of the data streams but not
geometrical ones, as those related to antenna
patterns.
Comparison between “IGEC style” and cross-correlation
IGEC style search was designed for
template searches. The template guarantees
that it is possible to have consistent estimators
of signal amplitude and arrival time.
A bank of templates may be required to cover
different class of signals. Anyway in burst
search we don’t know how well the template fits
the signal
A template-less IGEC style search can be
Some more work is needed to extend
IGEC in case of template-less search
among (spectrally) different detectors.
Hint: the amplitude estimators should
have spectral weights common to all
detectors, to be consistent without a
template. The trade-off will be between
between efficiency loss and network gain
(sky coverage and false alarm rate)
easily implemented in case of detectors with
equal detector bandwidth. In fact it is possible to
define a consistent amplitude estimator.
(Karhunen-Loeve, power…)
Template
search
Cross-correlation among identical
detectors is the most used method to cope
with lack of templates.
1
Sh  Sh
Cross-correlation in general is not
efficient with non-overlapping frequency
bandwidths, even for wide band signals.
Sh
1
2 
 k Sh
2 
Template-less
search