Statistical problems in network data analysis:burst searches by narrowband detectors
L.Baggio and G.A.ProdiICRR Tokyo Univ.Trento and INFN
IGEC time coincidence search is taking advantage of “a priori” information
• template search: matched filters optimized for short and rare transient gw with flat Fourier transform over the detector frequency band
• many trials at once: - different detector configurations (9 pairs + 7 triples + 2 four-fold)- many target thresholds on the searched gw amplitude (30)- directional / non directional searches
narrowband detectors & same directional sensitivityCons: probing a smaller volume of the signal parameter spacePros: simpler problem
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… IGEC cont`d
• data selection and time coincidence search: - control of false dismissal probability- balance between efficiency of detection and background fluctuations
• background noise estimation - high statistics: 103 time lags for detector pairs
104 – 105 detector triples- goodness of fit tests with background model (Poisson)
• blind analysis (“good will”):- tuning of procedures on time shifted data by looking at all the
observation time (no playground)
… what if evidence for a claim would appear ?“GW candidates will be given special attention …”
- IGEC-2 agreed on a blind data exchange (secret time lag)
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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.
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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
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( )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 coverage
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Plot of the likelihood integral vs. minimum (conservative) coverage minC(), with background counts Nb=0.01-10
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:
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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
Plot of the likelihood integral vs. minimum (conservative) coverage minC(), with background counts Nb=0.01-10
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:
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Multiple configurations/selection/grouping within IGEC analysis
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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
Easy to set up a blind search
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Keep track of the number of trials (and their correlation) !
IGEC-1 final results consist of a few sets of tens of Confidence Intervals with min{C}=95%
the “false positives” would hide true discoveries requiring more than 5 two-sided C.I. to reach 0.1% confidence for rejecting H0
the procedure was good for Upper Limits, but NOT optimized for discoveries
Need to decrease the “false alarm probability” (type I error)
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Freedom of choice of confidence belt
Fine tune of the false alarm probability
0
GW enthusiastic
fanatic skeptical
c bN N N
Example: confidence belt from likelihood integralPoisson background Nb = 7.0
Min{C}=95%
1 - C(N )
P{false alarm} < 0.1%
P{false alarm} < 5 %
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What false alarm threshold should be used to claim evidence for rejecting the null H0?
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• control the overall false detection probability: Familywise Error Rate < requires single C.I. with P{false alarm} < /mPro: rare mistakesCon: high detection inefficiency
• control the mean False Discovery Rate: R = total number of reported discoveriesF+ = actual number of false positives
Benjamini & Hochberg (JRSS-B (1995) 57:289-300)Miller et. al. (A J 122: 3492-3505 Dec 2001; http://arxiv.org/abs/astro-ph/0107034)
Fq
R
Typically, the measured values of p are biased toward 0. signal
The p-values are uniformly distributed in [0,1] if the assumed hypothesis is true
Usually, the alternative hypothesis is not known. However, the presence of a signal
would contribute to bias the p-values distribution.
p-level
1
background
FDR control
Sketch of Benjamini & Hochberg FDR control procedure
• choose your desired bound q on <FDR>;
• OK if p-values are independent or positively correlated
• compute p-values {p1, p2, … pm} for a set of tests, and sort them in creasing order;
p-value
m
• determine the threshold T= pk by finding the index k such that pj<(q/m) j for every j>k;
reject H0 q
T
counts
• in case NO signal is present (H0 is true), the procedure is equivalent to the control of the FamilyWise Error Rate at confidence < q
Open questions
check the fluctuations of the random variable FDR with respect to the mean.
check how the expected uniform distribution of p-values for the null H0 can be biased (systematics, …)
would the colleagues agree that overcoming the threshold chosen to control FDR means & requires reporting a rejection of the null hypothesis ?
To me rejection of the null is a claim for an excess correlation in the observatory at the true time, not taken into account in the measured noise background at different time lags. It could NOT be gws, but a paper reporting the H0 rejection is worthwhile and due.
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