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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

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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
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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

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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

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Signal Detection and Analysis


Matched filtering
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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
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MCMC Approach

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LALInference Burst


Used MCMC-based nested sampling to explore parameter space
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Uses a Sine-Gaussian template to model waveform Calculates evidence Z directly Parameter posteriors can be obtained by resampling MCMC points

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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
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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

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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
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Low-latency sky localization

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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

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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

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

Courtesy of R. Essick

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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

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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
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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?
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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
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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

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Make search coherent among detectors: enables amplitude consistency

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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:
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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

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Prior on strain


For narrow-band signals with sources uniform in v ol um e :
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Energy flux: Marginalize over energy and distance:



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

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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
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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
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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



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Preliminary Results ("Rectangular")


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

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Preliminary Results ("Rectangular")


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

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Preliminary Results ("Unmodeled")


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

Overfitted data!
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Conclusions and Future Outlook


LIB
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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
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Low-latency pipeline
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