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Bayesian approaches to the detection and analysis of unmodeled gravitational wave signals
Talk by Ryan Lynch
MIT LIGO Lab October 21 2014

Collaborators: Salvatore Vitale, Reed Essick

1

Background on LIGO

Gravitational Waves

Caused by disturbances to a stable spacetime manifold Expected to propagate at speed of light Expected to have 2 independent polarizations Sources

http://www.johnstonsarchive.net/relativity/pictures.html

Unmodeled bursts Binary Coalescence Periodic Sources Stochastic Background

http://mucholderthen.tumblr.com/post/55970198303/gravitational-waves-patterns-in-space-time-a

3

LIGO-VIRGO Interferometer Network

Observatories are large Michelson interferometers (to 0th order) LIGO: Two observatories with 4 km arms in US (Hanford, Washington & Livingston, Louisiana) Virgo: One observatory with 3km arms (Cascina, Italy)

http://phys.columbia.edu/~millis/1900/readings/LIGO.pdf http://www.ligo.caltech.edu/LIGO_web/PR/scripts/draw_lg.html

4

Signal Detection and Analysis

Matched filtering

Assume data is of form:

h(t) is gravitational wave signal n(t) is detector noise

Define noise-weighted inner product
where

Assuming stationary Gaussian noise, can write likelihood (under hypothesis H) as
5

MCMC Approach

6

LALInference Burst

Used MCMC-based nested sampling to explore parameter space

Uses a Sine-Gaussian template to model waveform Calculates evidence Z directly Parameter posteriors can be obtained by resampling MCMC points

7

Nested Sampling (Skilling 2004)

Skilling 2004

R e p a ra m e t r iz e e v i d e n c e

with

Scatter "live points" through initial Monte Carlo (i.e., draw from prior) C a lc u la t e li k e l ih o o d a t e a c h l iv e p o in t By drawing from prior, replace lowest likelihood live point with new live point

N e w p o in t o f h ig h e r lik e lih o o d : f o r m s n e s t e d li k e l ih o o d c o n t o u r s
8

Algorithm calculates Z while converging to max likelihood

Parameter estimation and posteriors

If live points are recorded, have access to the posterior through
Q parametrizes number of waveform cycles

f0 is oscillation frequency in t-domin, central frequency in fdomain

Courtesy of R. Essick

9

Bayes Factors and Signal Detection

Write Bayes' theorem as: Taking the "odds ratio" of two hypotheses we find the important quantity to be the Bayes factor:
where

Signal vs. Gaussian Noise: Coherent Signal vs. Incoherent Glitches:
Useful because coherent prior is more sharply peaked in parameter space 10

Signals vs. False Alarms
Signals

Ln B

C,I

Glitches

Ln B

h,0
11

Low-latency sky localization

12

Single Detector Sensitivity
Most sensitive when source is overhead "Null" spots when source is in plane of detector

http://docuserv.ligo.caltech.edu/docs/public/T/T970101-B.pdf

Effective polarization to which single detector is sensitive

Single detectors have very poor sky localization capabilities

13

Network Sensitivity

For 2-detector (HL) case:
Max Sensitivity Eigenvalue Motivation for priors! Min Sensitivity Eigenvalue

Max eigenvalue similar to sensitivity of single detector

Min eigenvalue much smaller in magnitude across most of sky

14

Performing Sky Localization

Time-of-arrival measurements give rings on sky for each pair of detectors

Courtesy of R. Essick

15

Performing Sky Localization

Triangulation:
2 Detectors (HL)

Ring-like localization for 2 detectors

3 Detectors (HLV)

Intersection of rings -> point-like localization for 3 detectors

Courtesy of R. Essick

Priors and amplitude consistency can provide modulation along timing rings

16

Burst Sky Localization

For BNS CBC events

http://arxiv.org/abs/1404.5623

Some gravitational wave signals expected to have EM counterparts and afterglows Challenge: Need accurate localization with low latency

Searching over entire sky is computationally expensive Sky location posteriors tend to be fragmented and nonlocalized How to model burst signals and search over parameter space?
17

Low-latency Bayesian Approach

Can do full parameter estimation and sky localization follow-up with LIB on timescales of hours Our goal: design a low-latency, all-sky sky localization pipeline

Allow for varying degree of signal strain ( h(f) ) modeling

Marginalize over all strain amplitudes through Gaussian integration This requires expansion of prior in terms of Gaussians

Make search coherent among detectors: enables amplitude consistency

18

Ratio of Gaussian noise realizations with and without signal present

Likelihood

Beta signifies the detector, i,j signify the polarization

Define Likelihood ratio as:

d ( f ) i s t h e d e t e c te d d a t a hj (f) is gravitational wave strain (j th polarization) Fx,+(, ) are antennae patterns t0 is signal's central time S(f) is noise PSD
Defined for each sky pixel

Expansion reveals useful a quantity to be:

Sensitivity matrix:
19

Strain Model

Model strain as independent "rectangular" functions over specified frequency intervals

hi(f) = ai for f1 < f < f2 , else 0
intervals intervals

In limit N In limit N signal

1, we get a "rectangular" template N
freq bins

, we have a completely unmodeled

20

Prior on strain

For narrow-band signals with sources uniform in v ol um e :

Energy flux: Marginalize over energy and distance:

Find best fits of coefficients for Gaussian expansion:
Blue is hrss-4 Red is Gaussian model

21

Final Formulation
Not necessarily true!

Assume Marginalize over each hj(f) by performing Gaussian integral:

Determinant term

Can marginalize over t0 using discrete fast Fourier transforms Dilemmas:

In "unmodeled" limit: determinant term acts as Occam factor that penalizes us for overfitting the data In single "rectangular" limit: don't want to include frequency bins without si g n a l
22

Model selection

Integrate over sky position to get a Bayes factor for signal vs. noise Can use prior to set h(f) to zero at any frequency bin

Creates "window" where signal is allowed to live Reduces number of parameters

The proper thing to do would be to marginalize over a grid of "window" models In favor of computational speed, currently just maximize over a set of "windows" designed to converge on true signal

23

Preliminary Results ("Rectangular")

Threshold events:
2-detector (HL) network 3-detector (HLV) network

24

Preliminary Results ("Rectangular")

Loud events
2-detector (HL) network 3-detector (HLV) network

25

Preliminary Results ("Unmodeled")

Threshold events:
2-detector (HL) network 3-detector (HLV) network

Overfitted data!
26

Conclusions and Future Outlook

LIB

Already proven as parameter estimation tool (arXiv:1409.2435) Detection pipeline using Bayes factor cuts looks promising, statistical study in the works Should understand trade-offs with number of live points (latency, accuracy of evidence, accuracy of posterior) Preliminary results appear to be consistent with LIB with latencies of ~ (30 minutes) / (# CPUs) Can implement better priors, strictly speaking Need to perform statistical tests to optimize model selection and compare to LIB results
27

Low-latency pipeline