Bayesian modeling of source confusion in LISA data

Nelson Christensen, Sarah Vigeland

Carleton College

Richard Umstätter, Renate Meyer

University of Auckland

Graham Woan, Vimal Simha, John Veitch,

Martin Hendry

University of Glasgow

Source Confusion for LISA

LISA sensitivity range 10-5 - 0.1 Hz

Most abundant source will be close-by White Dwarf Binaries: 0.1 mHz to 3 mHz

Source confusion below 1 mHz

Resolvable sources above 5 mHz

1 mHz to 5 mHz band presents a data analysis challenge

50 000 to 100 000 potential signals

Markov chain Monte Carlo

Developed a toy problem - N sinusoidal signals within noise

Apply a numerical Bayesian technique, a Metropolis-Hastings algorithm

Estimate parameters associated with each sinusoid, signal number N and noise level s

Bayesian – Occam's Razor preference for smaller N

Extraction of Parameters for Signal

We assume a simple signal that consists of N superposed sinusoidal signals:

We fit this model to the data:

Likelihood and Bayes

Simple uniform priors are chosen for the amplitude

parameters, and frequency. The MCMC provides an

estimate of the (unnormalized) posterior density:

20 signals (10 shown). Output chains from the MCMC

MCMC takes

a quasi-

intelligent

walk through

the parameter

space.

Histograms of

the post-

convergence

chains provide

estimates of

the posterior

PDFs

Reversible Jump MCMC

Total number of signals, N, is unknown

Need to apply a non-standard MCMC

Different N, different model

Create or annihilate signals

Split and merge signals

Also estimate noise level s

Splitting and Merging Signals

Merge: Two randomly selected signals i≠j with parameter vectors

Si=[Ai(1),Ai(2),fi] and Sj=[Aj(1),Aj(2),fj] are merged into a new signal with

parameter vector Si*=[Ai(1)+ Aj(1), Ai(2)+ Aj(1), (fi+fj)/2]

Split: For a randomly chosen signal i with parameter vector

Si=[Ai(1),Ai(2),fi] a proposal u=[uA(1), uA(2), uf] is sampled from a

distribution qs and split up into signals with parameter vectors

Si*=[Ai(1)/2, Ai(2)/2, fi] + u and Sj*=[Aj(1)/2, Aj(2)/2, fj] - u

Creating and Annihilating Signals

Annihilation: A randomly selected chosen signal i with

Si=[Ai(1),Ai(2),fi] is simply killed.

Creation: A vector u=[uA(1), uA(2), uf] is sampled from

a distribution qc and the new signal with

parameter vector Si* = u is added.

Accept or Reject

New parameters and new models are accepted or

rejected based on Metropolis-Hastings acceptance

methods (also using delayed rejection).

Example with N=20 signals, s=2.2, T=2 000s of data sampled at

fs=1 Hz, and max amplitude for a signal of A=1.0

MCMC estimates

18 signals

MCMC generated

power spectral

density better

than classical

periodogram

Missed 2 signals

f=0.23 Hz and 0.47Hz

Example with N=100 signals, s=0.75, T=10 000s of data sampled at

fs=1 Hz, and max amplitude for a signal of A=0.5

MCMC generated

power spectral

density is

again better

than classical

periodogram

We can be more

sure about the real

values of our

parameters if there

are fewer

combinations

that plausibly fit the

data.

MCMC estimates 97 signals

MCMC Good at Resolving Signals with Similar Frequencies

5 injected signals, T = 500s, noise level s=3.6, sampled at 1 Hz

n	A1	A2	frequency

1	2	2	0.2

2	-2	-2	0.201

3	-1	1	0.4

4	1	-1	0.403

5	0	-1	0.15

Not clear that there

are five signals

here.

4 of 5 signals

found in output

small signal at

f=0.15 missed

Example of densities

Estimates of posterior PDFs

A1	A2	frequency

2	2	0.2

-2	-2	0.201

Example of densities

Estimates of posterior PDFs

A1	A2	frequency

-1	1	0.4

1	-1	0.403

Example of densities

Estimates of posterior PDFs

A1	A2	frequency

-1	1	0.4

1	-1	0.403

What Next

Get good numbers on scalability: computation time as N and T increase. Appears roughly linear so far with N and with T.

Realistic Signals. Start coupling this work with the LISA simulator

2004-12-05T11:33:58