Pulsar Detection and Parameter Estimation with MCMC - Six

Parameters

Nelson ChristensenPhysics and Astronomy

Carleton College

GWDAWDecember 2003

Collaborators

Rejean Dupuis, John Veitch, Graham Woan Department of Physics and AstronomyGlasgow University

Renate Meyer, Richard UmstaetterDepartment of StatisticsUniversity of Auckland

Markov Chain Monte Carlo

● Computational Bayesian Technique● Metropolis-Hastings Routine● Estimate Parameters and Generate Summary

Statistics (PDFs, cross correlation, etc)● 6 Unknown Parameters (so far): h

0, f,

df/dt● Initial Application SN1987a: Location known but

other parameters not

Starting Point

The Starting point for the MCMC is the same likelihood and priors used in the LSC time domain (bayesian) search

Remember Rejean's talk

Time domain method (Rejean)

• The phase evolution can be removed by heterodyning to dc.– Heterodyne (multiply by e-i (t)) calibrated time domain data from detectors.– This process reduces a potential GW signal h(t) to a slow varying

complex signal y(t) which reflects the beam pattern of the interferometer.– By means of averaging and filtering, we calculate an estimate of this

signal y(t) every 1 minute which we call Bk.

The Bk’s are our data which we compare with the model

y t =14Fƒ t h0 1ƒ cos 2 i e i 2? 0 i

2F X žtŸh0 cos i e i2 ? 0

Bayesian analysis (Rejean)

A Bayesian approach is used to determine the joint posterior distribution of the probability of the unknown parameters,

a=(h0f,df/dt), via the likelihood:

p˜a# Bk p

˜a p Bk #

˜a

Posterior Prior Likelihood

Model

p Bk #˜a exp

k

Bk y t k ;˜a

2 ?k2

2

¿

Bk’s are processed data Noise estimate

Metropolis Hasting Routine

Generate a chain of parameter values

PDFs generated from the distribution of chain values

Quasi-Random (somewhat intelligent) walk through parameter space

4 or 5 parameter search is straightforward h

0, , f

6th parameter df/dt causes problems

Distributions for f and df/dt are VERY narrow Hard to find

Reparametrization New Variables

fstart

=f + (1/2)(df/dt)Tstart

fend

=f + (1/2)(df/dt)Tend

S2 Injected Signals: Analysis using a Metropolis-Hastings Markov chain

Monte Carlo Routine

Here we present results for analysis of signals injected during S2.

Injected Signal Parameters

Signal 2:Unknown Parameters of Injected Signal:

h0 = 20.0 (± calibration errors, in some units) = 0.0 = 0.0 cos()=0.0

df/dt = -5.0e-9

Rejean put in an offset in a f to make it non-zero. f = 6.17e-4

H1 signal 524 minutes of data, 458 minutes for H2, and 394 minutes for L1.

One Parameter Hiccup

Parameter of the phase : Injected with wrong phase (-), and with factor of 2 for gravity wave this resulted in an effective value of

= -/4

H1 Parameter Estimation Result

H1 Parameter Estimation Result

H1 Parameter Estimation Result

Mean h0 1.99e+01 psi -1.62e-02 phase -7.495e-01 cosiota 4.20e-02 df 6.19e-04 dfdot -5.09e-09

Quantiles for each variable: 2.5% h0 1.55e+01 psi -1.17e-01phase -1.40e+00cosiota -9.12e-02df 6.06e-04 dfdot -5.66e-09

97.5%h0 2.41e+01psi 8.37e-02phase -2.96e-02cosiota 1.82e-01df 6.31e-04dfdot -4.48e-09

H2 Parameter Estimation Result

H2 Parameter Estimation Result

H2 Parameter Estimation Result

Mean h0 1.71e+01 psi -1.63e-02 phase -7.09e-01 cosiota 4.01e-02 df 6.19e-04 dfdot -5.09e-09

Quantiles for each variable: 2.5% h0 1.13e+01 psi -1.90e-01phase -1.48e+00cosiota -2.02e-01df 5.95e-04 dfdot -5.83e-09

97.5%h0 2.34e+01psi 1.31e-01phase 5.00e-01cosiota 2.69e-01df 6.34e-04dfdot -3.94e-09

Tested Code on Synthesized Data

f 0.007 Hzdf/dt -2.5e-10d2f/dt2 0.0h0 “varied” 0.4 1.0 0.5RA 1.23DEC 0.32110 days worth of data

Noise k = 1.0

Estimate of h0

Estimate of df

Estimate of df/dt

Goal of our Work

SN1987a Location known, but all other parameters unknown.

Heterodyne 1/60 Hz bandwidth: 5 Hz search with 300 processes.

Also useful for radio observed pulsar: work in concert with time domain search.

John Veitch (Glasgow) is presently running code on E10 and S3 injections.

Delayed Rejection

ABSTRACT

Presented is a Markov chain Monte Carlo technique for finding a laser interferometer detected pulsar signal and estimating 6 unknown parameters, including the pulsar frequency and frequency derivative. The technique used is called Delayed Rejection in Reversible Jump Metropolis-Hastings. This method will be explained, as well as noting how a simple extension to multiple computer processors will allow for a wider frequency band search. The goal of this research is to search for a possible (but radio quiet) source at a know location; for example, SN1987A. The code has been successfully demonstrated with synthesized data, and with LIGO S2 injected signals. The results of the code on these artificial signals will be presented.

The Details of the Technique

This talk will describe the details of how the 6 parameter MCMC search works. However, I have not written this yet. The results of the technique

using S2 injected signals are presented here for LSC review. Results on synthesized and ficticious data are not included here (but will eventually make it

into the talk).