M. Principe, GWDAW-10, 16th December 2005, Brownsville, Texas Modeling the Performance of Networks...

M. Principe, GWDAW-10, 16th December 2005, Brownsville, Texas

Modeling the Performance of Networks of Gravitational-Wave Detectors in Bursts

Search

Maria Principe1, Patrick Sutton2

1University of Sannio, Benevento, Italy2LIGO-CALTECH


GW Bursts Search GWBs are generated by systems such as core-collapse

supernovae, black-hole mergers and gamma-ray bursters

Poor theoretical knowledge of the source and of the resulting GW signal

Multiple-detector search is required, but» Different noise spectra» Different alignment» Different algorithms» Non-Gaussian, non-stationary data

It’s not obvious how to use the different detectors optimally

We wrote a Network Simulator for helping the tuning of GWB searches

Carlo Principe

GWBs are expected to be produced from astrophysical sources such as core-collapse supernovae, black-hole mergers and gamma-ray bursts. But theorethical knowledge of the source and of the resulting GWsignal is quite limited. Multiple detetctor search is required to improve our confidence to claim a GW burst. But such a cooperative analysis presents a number of difficulties because of differences in noide spectra, alignments and algorithm and it's not obvious how to use optimally the different detectors. We built a software Network Simulator for helping the tuning of analyses.

maria

Since it is very difficult to distinguish between noise bursts and GW signals in any one detector, it will be essential to use multiple detector statistics for actual signal detection to reject false alarms


Multiple-Detector Searches

GW search codes typically have a single threshold (e.g. on SNR or significance) which is varied after trigger generation to tune the analysis.

Multi-detector GWB searches can be tuned according Neyman-Pearson criterion

Achieve maximum probability of detection while not allowing the probability of false alarm to exceed a certain value

max{ },D FP such that P

ligo

The NP cr states that we should construct our decision rule in order to have maximim probability of detection while not allowing the probability of false alarm to exceed a certain value alfa.The maximization is over all decision rules.So in our case we should choose the best threshold set, which allow for the best detection probability while keeping FAR below specified threshold

ligo

This threshold is used to perform a selection of trigger list: all triggers with SNR below specified threshold are nelected and remaining triggers are possible candidate GWB. Obviously higher thresholds result in lower FAR in but poorer detection efficiency for low amplitude GW signal, and lower thresholds allow weak signlas to be seen but they increase false event rate.


Target of the work Develop a software tool in Matlab to find the optimal

tuning of analyses in actual network GWB search» Input:

– Lists of triggers from raw data and from injected signals

» Output:– Optimal trigger threshold for each detector to satisfy N-P criterion– Network efficiency– Predicted false alarm rate

» Available on CVS archive

Such a tool could be also useful» to simulate the behavior of GW detectors in trigger-based searches for

GW bursts (GWBs)» to estimate sensitivity to populations of signals other than those directly

tested in the search» to estimate the effect of uncertainties in the properties of the individual

detectors (calibration,..)

ligo

estimate sensitivity for simulated signals, not yet detected, but that are likely to happen

ligo

that means for a single detector

ligo

Goal of my project is developing a software network simulator, which must be a quantitative model for GW detectors network, to find the best tuning of network analyses, which means to find experimantally the best power threshold set to satisfy NP-cr. It can be also useful to simulate..., to quantify the effects of uncertainties or possible errors in the description od detectors, such as uncertainty in calibration,...


The idea Single-IFO Event Generation:

» Get trigger from ETG (Q-pipeline, Excess Power, TFClusters, …)» Select trigger threshold for each IFO

Single-Detector False Alarm Rate:» Estimate for selected threshold

Network False Alarm Rate:» Estimate after

– Time Coincidence test in all IFOs.– Frequency, amplitude comparisons.

Single-Detector Efficiency:» Compute efficiency to optimally oriented sources for chosen trigger threshold

Network Efficiency:» Measure based on known single-detector efficiencies

Best trigger threshold set, satisfying N-P criterion:» Threshold set with best network efficiency and FAR below desired value

ligo

Event Trigger generator

ligo

Coincidence tests allow network FAR to reduce

ligo

it's hard to compare frequency and amplitude for detectors with different noise curve sensitivity and not-aligned. For LIGO-TAMA search was choosen as frequency range of analysis the one where all the interferometers have approximately the same efficiency.Otherwise nw efficiency would be affected by the least sensitive detector. Amplitude coparisons are complicated for not-aligned detector, because they are sensitive to different combinations of hte 2 polarization coponents og a GW.So, a simple amplitude comparison can only be applied to aligned detectors.

maria

The idea at the base of this project is outlined here.First step is collecting trigger lists generated from the different algorithms in detectors and choose a power threshold to perform a selection of triggers. Afterwards, single-det FAR is estimated and nw FAR, after temporal coincidence test, frequency and ampitude comparison.Next step is estimating detector efficiency for chosen threshold. Subsequentially, nw efficiency is computed based on the known single-detector efficiency curves. After having developed functions to do all of this, a master function was done to find the best threshold set to satisfy NP cr.

Carlo Principe

Only for rho sets satisfying our condition on the nw FAR.maria11/12/2005Calculations of false rate and efficiency are designed to follow the way real searches are done.


Test: LIGO-Virgo Simulated Data

We demonstrate the Network Simulator using Q-pipeline (S. Chatterji) triggers from the LIGO-Virgo project 1B simulated data

» With burst injections» Gaussian noise at LIGO/Virgo design sensitivity

Injected simulated signals:» linearly polarized Gaussian-modulated sinusoids

» fixed amplitude hrss = 5 x 10-22 Hz-1/2

» Q = 2π fc = 15

» central frequency fc = 235 Hz

» Duration = 10 ms

0

2

0 0( ) sin 2

( ) 0

t t

rssh t h f t t e

h t

x

ligo

This analysis requires to choose a target population, inculding waveform and the distribution of sources over the sky.This family was selected. These signals (tot 16800) were added to the data stream before passing through TFCluster or Excess Power algorithm.

ligo

hrss is the root sum square amplitude of the plus polarization and it is found to be a convenient measure of the signal strength

ligo

Data used are those of S2 run...To estimate sensitivity of each detector and of the nw simulated GWB are added to the data stream from individual detectors before passing thorough ETG algorithm (TFCluster or Exess Power) and new data are re-analyzed in the same manner as is done in the actual GW search.

maria

Q = fc/sigmaf


Detector False Alarm Rate

Estimate single-detector time FAR

» Based on trigger list and total observation time

» Background noise is modeled as a Poisson process

8 10 12 14 16 18 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

rho

far

(Hz)

H1H1

Rate of background noise events occurring above selected threshold

obs

e

T

NR ˆ

ligo

shourov9/21/2005Avoid confusing "false alarm rate" with "false alarm probability".false_rate = false_probability * measurement_ratemeasurement rate is the number of measurements per second, which is usually difficult to predict. ForTFClusters, it depends on the statistical independence of time-frequency tiles.

ligo

Detector FAR is estimated in order to estimate nw FAR. It is estimated using trigger list and knowing total obser time. Poisson, so the best estiamator for the parameter of the process is the number of false events divided by tot obs tm. In the table are shown estimated FAR for LIGO detectors using run 15 data with no injections and no threshold.

maria

For these data far for each detector has the same trend vs rho. We are dealing with simulated data, data with no glitches or non-stationarity. So, all of the detectors have the same noise properties (except for small differences in the shape of the Virgo noise, which should be irrelevant for Q pipeline). Of course, for real detector noise the false rates will be different.


Network False Alarm Rate Actual GWB searches require a candidate GWB to be observed

simultaneously by all detectors, to minimize the probability of falsely claiming a GW detection. Also often require similar measured frequency and/or amplitude.

Network Simulator can estimate the false rate after any of these coincidence tests:

» time coincidence » frequency coincidence (optional)» amplitude coincidence (optional, applied only to user-specified detectors)

Expected network FAR is given by

NTFAR: Rate at which background noise events occur simultaneously in all detectors

NFFAP: Probability for background noise events to occur in frequency coincidence in all detectors

NAFAP: Probability for background noise events (that are frequency coincident) to occur with approximately the same amplitude in specified detectors

nwhnwfnwtnw PPRR ___ˆˆˆˆ

ligo

Nw FAR is given by the product of these three quantities, to which we refer as nw time FAR, nw frequency FAR and nw amplitude FAR

ligo

Succesive step was estimating NwFAR. To minimize the probability of falsely claiming a GW detection... Triggers passing time coincidence test are required to be in frequency coincidence and further outliving triggers are required to be in amplitude coincidence. Amplitudute comparison is made hard beacuse of differences in the alignements of the detcteors. Not aligned detectors are sensitive to different combinations of the 2 polarization component of the GW signal, so a trivial ampl comparison is possible only for aligned detectors.NwFAR is then given by the product of these 3 quantities. the first one is... and I will refereto it with NTFAR,..The product rule is valid in the hypothesis tha time, frequency and amplitude coincidences are indipendent from each other and this is supposed to be true

maria

Calculations of false rate and efficiency are designed to follow the way real searches are done.


Time coincidenceRt_nw - NTFAR Time Coincidence test:

» a, wtij user-specified quantities

» A set of event triggers is defined to be in coincidence if each pair is in coincidence

The expected network background rate for a set of N detectors with rates Ri is

estimated by Monte Carlo method [1]

ti j ij i jt t w a t t

ti - peak time of the event i

wtij - coincidence window

for the pair (i, j)

Δti - duration of the event i

Ndet= 2

Ndet= 3

[1] L. Baggio et al. 2002 Classical and Quantum Gravity 19

det

1_

ˆˆN

ii

obs

fnwt R

T

VR

Tobs

Tobs

ligo

Ideally Wt should be as short as possible, to minimize the rate of accidental coincidences, while still being long enough that all simuated signals detected are in coinicdence

Carlo Principe

The set of all possible pairings between two triggers in a 2 detectors configuration can be represented with a lattice in a 2 dimensional plane, obtained by the tensor product of the original single detector event lists. In the simplified case when every trigger of one detector has the same time window, coincidence search can be seen as a particular selection of events in a stripe (fiducial volume) along the bisector. A time-shifted search means counting triggers pairs that fall inside a new traslated stripe.In a similar way, in a three-detector configuration, the geometrical rapresentation of the fiducial volume is a tube obtained when developing the 3-dimensal error box along the bisector.

maria

estimate now the first quantity in that product. Its values depends on the test performed on triggers.2 events are defined to be in tm coincidence if this condition is satisfied, where..4 safetyProbability that a background noise event can occur in the all detectors in time coincidencewt takes into account for the light travel time between the detectorsIn practice 10-20 ms longer than the light travel timeUser can specify different wtij for each pair of detectorsSecond term can be considered as an allowance for the uncertainty in the determination of the peak timeMention the figures are taken by baggio et al. paper

ligo

because it allows for the estimated peak time of coincident triggers to be farther apart if the trigers are long compared to Wt


Frequency and Amplitude Coincidence

Pf_nw – NFFAP

Frequency Coincidence condition:

» a, wfij user-specified quantities

Compute fraction of noise events which satisfy coincidence condition by Monte Carlo

Ph_nw – NAFAP

Amplitude Coincidence condition:

» whij user-specified quantities

» Usually applied only to aligned detectors

Compute fraction of noise events (frequency coincident) which satisfy coincidence condition by Monte Carlo

fi j ij i jf f w a f f

fi - central frequency of the event i

wfij - coincidence window

for the pair (i, j)

Δfi - frequency bandwidth of the event i

hi j ijH H w

H = log(h)

h = observed amplitude

whij - coincidence window

for the pair (i, j)

ligo

Concerning the NFFAR, estimating background noise distribution over cf and bw is required first, in order to estimate NFFAR thorug Monte Cralo.The frequency coincidence test is quite similar to that performed in time.the rule to define multiple events to be in coincidence is the same. performing a Monte Carlo integration ovre frquency and bandwidth, the obtained result is..That is the probability that background noise events occuring in all detectros satisfy coincidence condition.

Carlo Principe

The background noise distribution over amplitude is estimated taking into consideration only events being in frequency coincidence. In that way the estimated distribution is conditioned to the frequency coincidence. Multiplying P(h_coinc|f_coinc)*P(f_coinc) = P(h_coinc, f_coinc).So it's valuable to consider only events in frequency coincidence, because even if h_coinc is not independent on f_coinc, by that product I get anyway the joint probability.If I consider all the events, P(h_coinc)*P(f_coinc) could not be equal to the joint probability.


Single-Detector Efficiency

Compute time coincidences between triggers and injections

Tolerance for timing errors (~10 ms)

Use sigmoid fitting function (Blackburn & Chatterji)

Compute ‘optimally oriented’ efficiency curve, as function of hobs= hrss|F+|

(linearly polarized signals only)…as expected, efficiency gets worse increasing threshold

H1H1

ρ

ligo

After generation of trigger list from detectors and knowing parameters of injected signals, single-detector efficiency can be estimated at fixed SNR threshold. To do this is required to compute time coincidneces between triggers and analyzed injections, allowing for tolerance for timing errors, of the order of magnitude of 10 ms (10 or 20 ms).Curves obtained are optimally oriented curves, that are efficiency versus observalble amplitude, that is true injected amplitude times antenna resopnse factor. In this case only F+ factor because injected signals are supposed to have only h+ polarization component.Detector efficiency for different SNR thresholds is shown and as we expect, increasing the value, efficiency makes worse.

ligo

SNR is not meant as the usual concept. It's meant here as a general measure each ETG algorithm has its own measure of signal strenght. For TFCluster f.e. SNR^2 = sum_{f bins in-band} |h|^2/S(f).is the root square of sum over in-band frequencies of the ratio of estimated signal power to the background noise. This approximation sometimes breaks down, especially if the noise is fluctuating. F.e. sometimes miximum likelihood estimator, used in the case of event occurred to estimate how much of the power is due to the signal, gives this power zero and SNR is zero, consequently


Network Efficiency

cos(θ), φ and ψ dimensions are sampled uniformly or user can specify an arbitrary sampling map

Possibility to choose an arbitrary distribution (under process) Sigmoid fitting function turns out ok also for network efficiency curves

Solving by Monte Carlo

H1H2L1V1

H1-H2-L1-V1

[7 7 7 7]

[7 7 7 10]

2

10 0 0

( ) sin ( ( , , ) ) ( , ) ( )N

nw rss i obsi

E h E h p p

maria

Next step is computing network effeiciency basing on the known single-detector efficiencies.it basically averages nw efficiency over source angle and polarization angle.


Optimal tuning set

Fix a target network FAR

Choose a grid of trial threshold sets

For each set compute network FAR

For sets with FAR below target value compute network efficiency curve

Optimal threshold set is the set with 50% efficiency at the lowest signal amplitude

maria

Once we can compute nw efficiency and NFAR, a master function can be built to find best threshold set to satisfy NP cr. principal steps are:The efficiency level at which choose the lowest amplitude level can be specified by the user


LIGO-Virgo example

target FAR: 3.1688e-010 (1/century)

detector threshold (normalized energy) range: [9.5, 10], step : 0.1

Computed optimal set:

[9.7 9.7 9.7 9.7] Achieved network FAR:

3.1623e-010

H1H2L1V1all

maria

3.1688e-10 is one false event over 100 years.Looking at the distribution of energy in the no-injection triggers, it's easy to note that we can achieve such a network false rate for an energy threshold for each detector around 9.7.A former search was done making vary single detector threshold in that range


Future plans

Further improving and testing of optimal threshold set functions» Interpolation to find FAR=constant surface

» Clustering for non-Poisson events

Apply Network Simulator to real triggers sets» S2 Excess Power triggers

» Possibly tuning online Excess Power search


Acnowledgments

LIGO–CALTECH and Summer Undergraduate Research Fellowship program

Shourov Chatterji and the LIGO-Virgo joint working group for providing the Q-pipeline triggers we used to demonstrate the Network Simulator.

I.M. Pinto and Wavesgroup of University of Sannio


Thank you for your attention

Any questions?

M. Principe, GWDAW-10, 16th December 2005, Brownsville, Texas Modeling the Performance of Networks...

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