4th ILIAS-GW Annual General Meeting Universität Tübingen, October 8-9 2007

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 Tubingen, 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 Universität Tubingen, October 8,9 2007

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 )

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

TRIGGERED SEARCHTargets 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.

Triggered or UntriggeredGWB Searches

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 couplingof instrumental channels with the main det-ector 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

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

Glitches

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

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 time-frequency 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.

A simplest choice for the atoms, for both GW and spurious noise bursts is perhaps the Sine-Gaussian (SG)

Spurious glitches can be statistic-ally characterized in terms of thedistributions (priors) of their rele-vant parameters Q, f0 , t0 , h0

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

2 20

0 0 0sin 2 tt th t h f t t e

Atoms (2/2)

02t Q f 0f f Q

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

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).

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

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 com-putational load.

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

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 capa-bilities, 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];

1 11 1

2 2 2 2

3 33 3

F FV nh

V F F nh

V nF F

GW Polarization waveforms at Earth’s center

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

(M)-RK Statistic(s)

Output of i-th detector

i i s i s i s iV t F h t F h t n t

rank-2 antenna response matrix

1 ˆi s i sr k

c

Antenna Patterns

In matrix form

,

The matrix is also rank-2

,

1 1 1

2 2 2

3 3 3

V F F

V F F

V F F

1 1 2 2 3 3 0AV t A V t A V t

1 2 3 3 2

2 3 1 1 3

3 1 2 2 1

A F F F F

A F F F F

A F F F F

1( ) ( )i i j j k k iW A A V A V V t (no noise)

Wi can be used as a noisy template

,

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

The Ai (Ωs)

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

Detection Statistics

Define the noisy-template based correlations:

A suitable function of the Ci, must be formed to be used as a detection statistic. Several choices possible.

R&K proposed

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

for which the statistical properties can still be obtained in analytic form

max1,...,

max ii D

C C

1

DoptLC i i

i

C a C

, (s known, fixed)

,

,i i iC V W , i =1,2,…,D

1 0

1 0

| |arg max

| |

opt optLC LC

i s opt optLC LC

C H C Ha

stdev C H stdev C H

Explicit expression of Ci

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:

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

Statistical Distribution of Ci

2i s i k i s

k

V t E f 2,i i iC N

j li j l

i i

A At n t n t

A A

1

0

, , , , ,sN

j li i j i l k k k s

ki i

A AC V V V V s v s t v t t k f

A A

2 22 2,i s j i l i nA A A A Q i(Ωs)

22 2 2 2 2 2 2 2, , 1i s i k n i n i s i s n i i n s

k

V t N E f Q Q N AWGN power equal in any detector

Noise term in the template

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

Statistical Distribution of Ci :

H1 hypothesis

2 22 2

i s i s rss i s rss sF h F h f

2/ 2/rss s k

k

h t h t

22 2 21i s i s n i i n sQ Q N ∞Ai → 0

independent on Ai

Choosing Ai Wi(t) as a template

00

ii s i s i s

AA

22 2 2 2 2 2 2 2 2

0ii s i i s j l n i s j l n s

AA A A E f A A N

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

Statistical Distribution of Ci

H0 hypothesis (AWGN only)

(0)

2(0) 4

( ) 0

( ) (1 ( ))

i s

i s n s i sN Q

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.

(1) 2i s i k i s

k

V t E f 22(1) 2 2 2 2 2 2 2

, , 1i s i k n i n i s i s n i i n sk

V t N E f Q Q N

H0

(AWGN)

H1

(GWburst)

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

K-R ROCs (AWGN only)

1 ( )1 ( ) MFd hP erfc erfc d

22( )

2,MF s rss

hn

f hd

Deflection of perfect MF actingon GW waveform

Would be one for the perfect MF acting on the GW waveform

( ) 2( ) ( | |) ( ) ( )MFs h i s s sd F Q N Z

1

( ) 2( ) (1 ) 1 ( | |)1

MFis i s h

i

QZ Q N d F

Q

1/ 2( ) | | ( )s sF Z

1/ 22 2 2 2| | cos 2 sin 2F F F

2 22 ,

.

rss rss rssh h h

pol angle

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

The 2 function

(+), 10

H+, dmf=20

(), 20

(+), 20

(), 10

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

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 para-meters. The resulting distribution will be different from the AWGN-only 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.

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

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 sim-plifying working assumptions that in each detector

P(at least a glitch) P(one glitch occurs)P(glitch and GWB)

H0 (AWGN+glitches) hyp.

0

Glitches SG-atoms,

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.

2 20

0 0 0sin 2 tt th t h f t t e

Working Assumptions

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

These quantities must be averaged out over random (exponential) inter-arrival times between events and over (uniform) SG parameters.

Denote as the averaged quantities.

H0 (AWGN+glitches) hyp.Marginal Distribution of Ci

(0 ) 2

2(0 ) 2 2 2

2 2 4

( ; , ) ( ; , )

( ) (0; ) ( / ) (0; ) ( / ) (0; )

2 ( / ) ( ; , ) 1

(

j li ij ij i j il il i l

i i

i s n i ii i j i jj j l i ll l

j l i jl jl j l n i s

ij

A At t P glitch

A A

P glitch Q A A A A

P glitch A A A t P glitch Q N

; , ) ( ; ), ( ; )ij i j i i j jt t t t t

(0 ) (0 ),i i

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

K-R ROCs (AWGN+glitches)

1 ( )' 1 ' ' ( ) ' MFd hP erfc erfc d

2( )

2,MF s rss

hn

f hd

Deflection of perfect MF actingon GW waveform

Would be one for the perfect MF acting on the GW waveform

(0 )

(0)

(0 )

(1)

'( ) ( )

'( ) ( ) 1

is s

i

is s

i

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

ROC (AWGN+glitches): Cmax

0 0.02 0.04 0.06 0.08 0.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PFA

PD

Cmax

hrss2 =0.0381

hrss2 =0.0244

hrss2 =0.0137

hrss2 =0.0061

hrss2 =0.0015

5.88

4.70

3.53

2.351.17

PRELIMINARY RESULTS

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

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 (chirplets [Sutton, GWDAW 10, UTB, 2005])

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

2 1 0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

x

0( | )PDF x H 1( | )PDF x H

x > , H1 acceptedx < , H1 rejected

0( | )prob x H 1( | )prob x H

Detection/Decision

For low , should be > E(x|H0) + stdev(x|H0) For low , should be

E(x|H1) > + stdev(x|H1)Detector performance depends on ratio

1 0

0 1

( | ) ( | )

( | ) ( | )

E x H E x Hd

stdev x H stdev x H

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

NP strategy : assign false alarm probability; deduce from 1stMequation .

2

00

0

11

1

1/ 2 / 2

( | )[ | ]

( | )

( | )[ | ] 1

( | )

( ) (2 ) t

z

E x HPDF x H dx erfc

stdev x H

E x HPDF x H dx erfc

stdev x H

erfc z e dt

NP-Strategy

4th ILIAS-GW Annual General Meeting Universität Tubingen, October 8,9 2007

ROCs

For given signal and noise, plot the

curve {1-(), ()}, known as the

Receiver Operating Characteristic

00

0

11

1

( | )[ | ]

( | )

( | )[ | ] 1

( | )

E x HPDF x H dx erfc

stdev x H

E x HPDF x H dx erfc

stdev x H

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0.7

0.75

0.8

0.85

0.9

0.95

1

1-

Each point on the curve corresponds to a different i.e. a different decision-rule.One can prove that =slope