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Network analysis and statistical issues
Lucio Baggio
An introductive seminar to ICRR’s GW group
Topics of this presentation
Setting confidence intervals
False discovery probability
Gravitational wave bursts networksFrom the single detector to a worldwide network
IGEC (International GW Collaboration)Long-term search with four detectors; directional search and statistical issues
From raw data to probability statements; likelihood/Byesian vs frequentist methods
Multiple tests and large surveys change the overall confidence of the first detection
Miscellaneous topicsThe 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
Amplitude distribution of eventsAURIGA, Jun 12-21 1997
The background in practice (1)
vetoed (2 test)
simulation (gaussian)
L. Baggio et al.
2 testing of optimal filters for gravitational wave signals: an experimental implementation.
Phys. Rev. D, 61:102001–9, 2000
Amplitude distribution of eventsAURIGA Nov. 13-14, 2004
Remaining events after vetoingvetoed glitches
epoch vetoes (50% of time)
cumulative event rate above thresholdfalse alarm rate [hour-1]
after vetoing
The background in practice (2)
Cumulative power distribution of eventsTAMA Nov. 13-14, 2004from the presentation at The 9th Gravitational Wave Data Analysis Workshop (December 15-18, 2004, Annecy, France)
The background in practice (3)
100 101 102 103 104 105 106
10–6
10–5
10–4
10–3
10–2
10–1
100
Event Power Threshold (Pth)
Rat
e [
even
ts/s
ec]
Gau
ssian n
oise
DT9
DT8
DT6
DT9 (before veto)
8
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.
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)
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
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.
(3)2wt
(2)t (3)t
(1)t
(1)2wt
(2)2wt 1st detector 2nd detector 3rd detector
coincidence! timing error box
(2)t (3)t
(1)t 1st detector 2nd detector 3rd detector
dt2
dt3
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)
( )
1
( )( ) ( )M
kb
k
t tC tThe expected coincidence rate is given by:
1( ) 2
M
wC t M t
( )
1
( ) 2M
hw
k h k
C t t
C(t) depends on the choice of the the time error boxes:
equal and constant vary with detector vary with event
Monte Carlo(by shifted times
resampled statistics)
2E-21 1E-201E-6
1E-5
1E-4
1E-3
0.01
0.1
1
10
AL-AU AL-AU-NA
falsealarm rate
[yr-1]
common search threshold [Hz-1]
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.
In practice, when no signal is detected in coincidence, the upper limit is determined by the total observation time
International networks of GW detectorsInterferometers
Operative:
GEO600 – (Germany/UK)
LIGO Hanford 2km – (USA)
LIGO Hanford 4km – (USA)
LIGO Livingstone 4km – (USA)
TAMA300 – (Japan)
Upcoming:
VIRGO – (Italy/France)
CLIO – (Japan)
Resonant bars
ALLEGRO – (USA)
AURIGA – (Italy)
EXPLORER – (CERN, Geneva)
NAUTILUS – (Italy)
LIGOGEO600 Virgo, AURIGA, NAUTLUS
TAMA300CLIO100
EXPLORER
International networks of GW detectors
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 monthsP. Astone et al., Astropart. Phys. 10, 83 (1999).
1989 – 2 interferometers, 2 daysD. 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 daysTAMA300-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 daysLIGO collaborationB. Abbott et al., Phys. Rev. D 69, 102001 (2004)
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)
GW detected?If NOT, why?
The International Gravitational Event
Collaboration
http://igec.lnl.infn.it
LSU group: ALLEGRO (LSU) http://gravity.phys.lsu.eduLouisiana State University, Baton Rouge - Louisiana
AURIGA group: AURIGA (INFN-LNL) http://www.auriga.lnl.infn.itINFN of Padova, Trento, Ferrara, Firenze, LNLUniversities of Padova, Trento, Ferrara, FirenzeIFN- CNR, Trento – Italia
NIOBE group: NIOBE (UWA) http://www.gravity.pd.uwa.edu.auUniversity of Western Australia, Perth, Australia
ROG group: EXPLORER (CERN) http://www.roma1.infn.it/rog/rogmain.htmlNAUTILUS (INFN-LNF)
INFN of Roma and LNFUniversities of Roma, L’AquilaCNR IFSI and IESS, Roma - Italia
The International Gravitational Event Collaboration
The IGEC protocol
Detector
DATA ACQUISITION
raw data
evt
> start_anl
evt
evt
evt
evt
Time of arrival
Amplitude SNR
DATA ANALYSIS
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
fraction of time in monthly bins
exchange threshold
21 16 10 Hz 21 13 6 10 Hz
21 13 10 Hz
ALLEGRO
AURIGA
NAUTILUS
EXPLORER
NIOBE
Fourier amplitude of burst gw
0 0( ) ( )h Ht tt
arrival time
The exchanged data
0
1
2
3
4
5
6
7
8
9
10
0 6 12 18 24 30 36 42 48 54 60
time (hours)
gaps
minimum detectable amplitude
(aka exchange threshold)
events amplitude and time of arrival
ampl
itude
(H
z-1·1
0-21)
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 !
< 5% false dismissal for k =4.5 (Tchebyceff inequality)
strongly dependent on SNR !
2 2ij i jt k
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
relat
ive c
ount
s
10-5
10-4
10-3
10-2
10-1
1
relat
ive c
ount
s
10-5
10-4
10-3
10-2
10-1
1
NIOBENIOBEAMP/THR1 10
NAUTILUSNAUTILUSAMP/THR1 10
AURIGAAURIGAAMP/THR1 10
ALLEGROALLEGROAMP/THR1 10
EXPLOREREXPLORERAMP/THR1 10
normalized to each detector threshold for trigger search
typical trigger search thresholds:SNR 3 ALLEGRO, NIOBESNR 5 AURIGA, EXPLORER, NAUTILUS The amplitude range is much wider than expected extreme distribution: non modeled outliers dominate at high SNR
time
amplitude
time
amplitude
time
amplitude
time
amplitude
A
False alarm reduction by amplitude selection
Corollary:
Selected events have naturally consistent amplitudes
With a small increase of minimum amplitude, the false alarm rate drops dramatically.
0
1
2
3
4
5
6
7
8
9
10
0 6 12 18 24 30 36 42 48 54 60
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 6 12 18 24 30 36 42 48 54 60
amplitude directional sensitivity
2sin GC2sin GC
0
1
2
3
4
5
6
7
8
9
10
0 6 12 18 24 30 36 42 48 54 60
time (hours)
ampl
itude
(H
z-1·1
0-21)
time (hours)
Sensitivity modulation for directional searcham
plitu
de (
Hz-1
·10-2
1)
0
1
2
3
4
5
6
7
8
9
10
0 6 12 18 24 30 36 42 48 54 60
A small digression: different antenna patterns and the relevance of signal
polarization
• In order to reconstruct the wave amplitude h, any amplitude has to be divided by
Introduction
• At any given time, the antenna pattern is:
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 )),(2cos(),(),,( AF
),,( F
• 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 cross-correlation search
F1F2 is an “extension” of the “AND” logic of IGEC 2-fold coincidence
This has been extensively used by IGEC: first step is a data selection obtained by putting a threshold F-1 on each detector
For linearly polarized signal, does not vary with time.The product of antenna pattern as a function of is given by:
)cos()4cos(
)2cos()2cos(
)()(
21212
121
2121
21
AA
AA
FF
)()( 21 FF
)()( 21 FF
The relative phase 1-2 between detectors affects the sensitivity of the pair.
Linearly polarized signals
AURIGA -TAMA sky coverage: (1) linearly polarized signal
)cos()4cos()()( 21212
12121 AAFF
AURIGA2
TAMA2
21
F
22
F
02
21
AURIGA x TAMA 21 FF
If:
the signal is circularly polarized:
Amplitude h(t) is varying on timescales longer than 1/f0
Then:
The measured amplitude is simply h(t), therefore it depends only on the magnitude of the antenna patterns. In case of two detectors:
The effect of relative phase 1-2 is limited to a spurious time shift t which adds to the light-speed delay of propagation:
(Gursel and Tinto, Phys Rev D 40, 12 (1989) )
Circularly polarized signals
0
21
2 ft
F
F
tf 02
22
22
12
12
21 FFFFAA
0f
h
)2sin(
)2cos()(
thh
h
04
1
ft
)cos()4cos()()( 21212
12121 AAFF
AURIGA2
TAMA2
2221 11 FFA
2222 22 FFA
AURIGA -TAMA sky coverage: (2) circularly polarized signal
AURIGA x TAMA 21 AA
AURIGA x TAMA 21AA AURIGA x TAMA 21 FF
AURIGA -TAMA sky coverage
Linearly polarized signalCircularly polarized signal
IGEC (continued)
0
1
2
3
4
5
6
7
8
9
10
0 6 12 18 24 30 36 42 48 54 60time (hours)
Data selection at work
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.
ampl
itude
(H
z-1·1
0-21)
amplitude of burst gw
Duty cycle cut: single detectors
total time when exchange threshold has been lower than gw amplitude
1
10
100
1000
10000
1.E-21 1.E-20search threshold (Hz-1)
day
s
single 2-fold
3-fold 4-fold
4yr limit
Duty cycle cut: network (1)
Galactic Center coverage
DETECTORS TIME (days)
FRACTION of 4 yr
search threshold 6 E-21 /Hz
1 or more 894
61%
2 or more 397
27%
3 or more 70
5%
4 7.2
0.5%
search threshold 3 E-21 /Hz
1 or more 359
25%
2 or more 70
5%
3 or more 3
0.2%
4 -
-
Duty cycle cut: network (2)
0%
25%
50%
75%
100%
01-Jan-97 31-Dec-97 30-Dec-98 29-Dec-99
duty
cyc
le p
erce
ntag
e
1
2
3
4
0%
25%
50%
75%
100%
01-Jan-97 31-Dec-97 30-Dec-98 29-Dec-99
duty
cyc
le p
erce
ntag
e
1
2
3
4
search threshold 6 10 -21/Hz
search threshold 3 10 -21/Hz
False dismissal probability
• 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)
Best choice for 1997-2000 data:false dismissal in time coincidence less than 5% 30%no amplitude consistency test
• time coincidence constraintThe Tchebyscheff inequality provides a robust (with respect to timing
error statistics) and general method to limit conservatively the false dismissal
2 22
1i j i jt t k false dismissal
k
false alarms k
• amplitude consistency check: gw generates events with correlated amplitudes testing (same as above) i jA A A
A coincidence can be missed because of…
fraction of found gw coincidences
fluctuations of accidental background
When optimizing the (partial) efficiency of detection versus false alarms, we are lead to maximize the ratio
0
1
2
3
4
5
6
7
8
9
10
0 6 12 18 24 30 36 42 48 54 60
0
1
2
3
4
5
6
7
8
9
10
0 6 12 18 24 30 36 42 48 54 60
0
1
2
3
4
5
6
7
8
9
10
0 6 12 18 24 30 36 42 48 54 60
0
1
2
3
4
5
6
7
8
9
10
0 6 12 18 24 30 36 42 48 54 60
0
1
2
3
4
5
6
7
8
9
10
0 6 12 18 24 30 36 42 48 54 60time (hours)
Resampling statistics by time shiftsam
plitu
de (
Hz-1
·10-2
1)
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
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.
verified
Setting (frequentist) confidence intervals
Unified vs flip-flop approach (1)
experimental data
physical results
hypothesis testing (CL)
x upper limit
estimation (with error bars)
(x)kCL
up(CL)
Flip-flop method
null
claim
Unified vs flip-flop approach (2)
experimental data
physical results
confidence belt
xestimation (with confidence interval)
Unified approach
min(CL)max(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) 3873B. Roe and M. Woodroofe, Phys. Rev. D 63 (2001) 013009F. Porter, Nucl. Inst. Meth. A368 (1986), http://www.cithep.caltech.edu/~fcp/statistics/Particle Data Group: http://pdg.lbl.gov/2002/statrpp.pdf
A few basics: confidence belts and coverage
x
x
x
( ; )p d f x
0 1 coverage
0 1 coverage
0 1 coverage experimental data
phys
ical
unk
now
n
A few basics (2)
experimental data
physical unknown
confidence interval
x
coverage
|
( ) ( ; )xx I
C pdf xI
xI
For each outcome x one should be able to determine a confidence interval Ix
For each possible , the measures which lead to a confidence interval consistent with the true value have probability C(), i.e. 1-C() is the false dismissal probability
x I
( )C CL
I can be chosen arbitrarily within this “horizontal” constraint
Feldman & Cousins (1998) and variations (Giunti 1999, Roe & Woodroofe 1999, ...)
0 1 coverage
Freedom of choice of confidence belt
Fixed frequentistic coverageMaximization of “likelyhood”
( ; )
" " ( )( ; )
xI
usually
x d
CL Cx d
Ix can be chosen arbitrarily within this “vertical” constraint
Roe & Woodroofe [2000]: a Bayesian inspired frequentistic approachFine tune of the false discovery probability
0
GW enthusiastic
fanatic skeptical
Non-unified approaches
decision threshold
Other requirements...
Confidence level, likelyhood, maybe probability?
The term “CL” is often found associated with equations like
( ; )
" "( ; )
xI
x d
CLx d
1
2
( ; )" "
( ; )x
CLx
limit( ; )
" "( ; )max
xCL
x
( )usually
C
( )usually
C
( )usually
C
In general the bounds obtained as a solution to these equations have a coverage (or confidence level) different from “CL”
likelihood integral
likelihood ratio relative to the maximum
likelihood ratio (hipothesis testing)
Confidence intervals from likelihood integral
• I fixed, solve for :
sup
inf
supinf
1
0
( ; ) ( ; )
( ; ) ( ; )
c c
N
c cN
N N N N
I N N dN N N dN supinf0 N N
• Compute the coverage
supinf|
( ) ( ; )c
cN N N N
C N f N N I
• Let
c b
b obs
N N N
N T
• Poisson pdf:
( ; )!
bc
N NN
c bc
ef N N N N
N
( ; ) ( ; )c cN N f N N• Likelihood:
c bN N N
Example: Poisson background Nb = 7.0
0
1
2
3
4
5
6
7
8
9
10
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
N
0
1
2
3
4
5
6
7
8
9
10
99%
99%
95%
95%
99.9%
99.9%
50%50%
99%
95%
85%
N
Likelihood integral
Dependence of the coverage from the background
Nb=0.01-0.1-1.0-3.0-7.0-10
likelihood integral = 0.90
From likelihood integral to coverage
Plot of the likelihood integral vs. minimum (conservative) coverage minC(), 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
Example: confidence interval with coverage 95%
0
2
4
6
8
10
12
14
16
18
1.0 10.0 100.0
search threshold [10-21/Hz]
Ngw
Ht
“upper limit”: true value outside with coverage 95%
GOAL: estimate the number (rate) of gw detected with amplitude Ht
Uninterpreted upper limits
1
10
100
1,000
1E-21 1E-20 1E-19
0.60
0.80
0.90
0.95
…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
ensured minimumcoverage
true rate value is under the curves with a probability = coverage
search threshold(Hz -1 )
rate(year –1)
Upper limits after amplitude selection
0
2
4
6
8
10
12
14
16
18
1.0 10.0 100.0
search threshold [10-21/Hz]
Ngw
systematic search on thresholdsmany trials !
all upper limits but one:
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
0
100
200
300
400
500
0 1 2 3 4 5
numer of false alarms
coun
ts
Resampling statistics of accidental claimsevent time series
coverage “claims”
0.90 0.866 (0.555) [1]
0.95 0.404 (0.326) [1]
expected found
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 testingFalse discoveries (false positives)
Detected signals
(true positives)
Reported signal candidates
inefficiency
Null Retained
(can’t reject)
RejectReject Null ==
AcceptAccept Alternative
Total
Null (Ho) True
Background (noise)
U B
Type I Error α = εb
mo
Alternative True signal
Type II Error β = 1- εs
T
S m1
m-R
R = S+B
m
Preliminary (2): p-levelAssume you have a model for the noise that affects the measure x.
However, for our purposes it is sufficient assuming that the signal can be distinguished from the noise, i.e. dP/dp 1. Typically, the measured values of p are biased toward 0.
signal
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)
Usually, the alternative hypothesis is not known.
p-level
1
background
• The distribution of p is always linearly raising in case of agreement of the noise with the model P(p)=p dP/dp = 1
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!!
p
S
m0
Let us make a simple case when signals are easily separable (e.g. high SNR)
Controlling false discovery fractionWe 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.
m
B
m
BpT
0
)(
B R
BqFDR
R
pmT )(
m
q
R
pT
)(
mq
)( pT p
B
S
cumulative counts
R
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)
• choose your desired FDR q (don’t ask too much!);
• define c(m)=1 if p-values are independent or positively correlated; otherwise c(m)=Sumj(1/j)
• compute p-values {p1, p2, … pm} for a set of tests, and sort them in creasing order;
p
m
• determine the threshold T(p)= pk by finding the index k such that pj<(q/m) j/c(m) for every j>k;
reject H0
)( pT
q/c(m)
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
detector 1
detector 2
detector 2
AND
AND
AND
detector 3
Detectors: PARALLEL, BARS Shh: SIMILAR FREQUENCY RANGE Search: NON DIRECTIONAL Template: BURST = (t)
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
This strategy can be made directional
HS
Cross-correlation search (naïve)
Detectors: PARALLEL Shh: SAME FREQUENCY RANGE NEEDED Search: NON DIRECTIONAL Template: NO
Selection based on data quality can be implemented before cross-correlating.
The efficiency is to be determined a posteriori using Monte Carlo.
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.
detector 1
detector 2
detector 1 * detector 2
Threshold crossing after correlation
Txxwwnj
jj ,1
21)2()1(
T
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 easily implemented in case of detectors with equal detector bandwidth. In fact it is possible to define a consistent amplitude estimator. (Karhunen-Loeve, power…)
Cross-correlation among identical detectors is the most used method to cope with lack of templates.
Cross-correlation in general is not efficient with non-overlapping frequency bandwidths, even for wide band signals.
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)
21hh SS
Templatesearch
Template-lesssearch
21hh SkS IG
EC
IGEC
cross-
corr
IGEC
cross-
corr
IGEC
