compiled:
25 October
Inference on white dwarf binary systems using the
first round Mock LISA Data Challenges data sets
Alexander Stroeer1,2, John Veitch1, Christian Röver3,
Ed Bloomer4, James Clark4, Nelson Christensen5,
Martin Hendry4, Chris Messenger4, Renate Meyer3,
Matthew Pitkin4, Jennifer Toher4, Richard Umstätter3,
Alberto Vecchio1,2 and Graham Woan4
1 School of Physics & Astronomy, University of Birmingham, Birmingham, UK
2 Department of Physics & Astronomy, Northwestern University, Evanston, IL, USA
3 Department of Statistics, The University of Auckland, Auckland, New Zealand
4 Department of Physics & Astronomy, University of Glasgow, Glasgow, UK
5 Physics & Astronomy, Carleton College, Northfield, MN, USA
Abstract. We report on the analysis of selected single source data sets from the
first round of the Mock LISA Data Challenges (MLDC) for white dwarf binaries. We
implemented an end-to-end pipeline consisting of a grid-based coherent pre-processing
unit for signal detection, and an automatic Markov Chain Monte Carlo post-processing
unit for signal evaluation. We demonstrate that signal detection with our coherent
approach is secure and accurate, and is increased in accuracy and supplemented
with additional information on the signal parameters by our Markov Chain Monte
Carlo approach. We also demonstrate that the Markov Chain Monte Carlo routine is
additionally able to determine accurately the noise level in the frequency window of
interest.
PACS numbers: 04.80.Nn, 02.70.Uu.
Submitted to: Classical and Quantum Gravity
1. Introduction
The data obtained from LISA [1] will contain a large number of white dwarf binary
systems across the whole observational window [2]. At frequencies below ∼ 3 mHz the
sources are so abundant that they produce a stochastic foreground whose intensity
dominates the instrumental noise [3]. The closer (and louder) sources will still be
sufficiently bright to be individually resolvable. Above ∼ 3 mHz the sources become
sufficiently sparse in parameter space (and in particular in the frequency domain) that
the detectable sources become individually resolvable. The identification of white dwarfs
in the LISA data set represents one of the most interesting analysis problems posed by
the mission: the total number of signals in the data set is unknown, the effective noise
http://arxiv.org/abs/0704.0048v2
WD MLDC1 2
level affecting the measurements is not easily estimated from the data streams, and
there is a large number of overlapping sources to the limit of confusion.
Bayesian inference provides a clear framework to tackle such a problem [4, 5, 6].
Some of us have carried out exploratory studies and “proof of concept” analyses
on simplified problems that have demonstrated that Bayesian techniques do indeed
show good potential for LISA applications [11, 10, 12]. Similarly other authors have
successfully implemented techniques using Bayesian inference [18, 17, 16]. In this paper
we present the first results of an end-to-end analysis pipeline developed in the context of
the Mock LISA Data Challenges that has evolved from our earlier work. This pipeline
is applied to the simplest single-source challenge data sets 1.1.1a and 1.1.1b and all the
results presented here are obtained after the release of the key files. In a companion
paper [19], we present results that we have obtained for the analysis of the data sets
containing gravitational radiation from a massive-black-hole binary inspiral. Our group
submitted an entry for the MLDC analysing the blind data set 1.1.1c [13, 14]: however
that result suffered from the fact that the pipeline was not complete, the analysis code
was inefficient and we encountered hardware problems with the Beowulf cluster used to
perform the analysis.
The results that we present here are obtained with a two-stage end-to-end analysis
pipeline: (i) we first process the data set with a grid-based coherent algorithm to identify
candidate signals; (ii) we then follow up the candidate signals with a Markov Chain
Monte Carlo code to obtain probability density function on the model parameters. Our
method differs from other MCMC methods that have been proposed and applied to
the MLDC data in the context of white dwarf binaries [18, 17, 16]: the MCMC is not
used to search, but only in the final stage of the analysis to produce posterior density
functions of the model parameters. The noise spectral level is included as one of the
unknown parameters and is estimated together with the parameters of the gravitational
wave source(s).
2. Analysis method
In this section we describe the two stage approach that we have adopted for the analysis.
The signal produced by a white dwarf binary system is modelled as monochromatic in
the source reference frame, following the conventions adopted in the first MLDC [7, 8, 9].
It is described by 7 parameters: ecliptic latitude ϑe and longitude ϕe, inclination ι and
polarisation angle Ψ, frequency at a reference time f0 and corresponding overall phase
Φ0 and amplitude A.
The data distributed for the MLDC are the three TDI v1.5 Michelson observables
X , Y and Z ‡. From those we construct the two orthogonal TDI outputs
A = (2X − Y − Z)/3 (1)
E = (Z − Y )/
3 (2)
‡ In our MCMC analysis we use the data set produced using the LISA Simulator.
WD MLDC1 3
by diagonalizing the noise covariance matrix following the procedure presented in [23].
The noise affecting the channels A and E is uncorrelated and described by the one-sided
noise spectral density Sn(f). We model the LISA response function in the low frequency
limit in order to improve the computational efficiency of our analysis.
2.1. First stage: Grid based search
The first stage of the pipeline consists of a fast search of the data for the best
matched filter based on the well-known F -statistic algorithm, first developed for triaxial
pulsar signals in the context of ground-based observations [20]. This exploits the Fast
Fourier Transform to perform matching in the frequency domain to templates which are
generated at an array of fixed points in the parameter space.
The data from an individual detector in the frequency domain d̃(f) is supposed to
contain a signal plus Gaussian noise, d̃(f) = h̃(f) + ñ(f). We define the logarithmic
likelihood as a measure of match, as given by logL ≈ (d̃|h̃)− 1
(h̃|h̃) with (·|·) denoting
the scalar product as defined in [20]. A single signal in the F -statistic algorithm is
re-parameterised as a linear function of four orthogonal variables, and the frequency
f0. The detection statistic is based on four parameters AF , BF , CF and DF , found by
integrating over the response functions a(t) and b(t) to the two polarisation states of
the gravitational wave signal [20],
∫ Tobs
a(t)2dt (3)
b(t)2dt (4)
a(t)b(t)dt, (5)
DF = AFBF − C2F (6)
Tobs denotes the total observed time for the data set being analysed. The optimal
detection statistic 2F , which is pre-maximised over the nuisance parameters h0, ι, φ0
and ψ is
2F = 8
Sn(f)Tobs
BF |Fa|2 + AF |Fb|2 − 2CF ×R(FaFb)
. (7)
Fa and Fb are the demodulated Fourier transforms of the data,
∫ Tobs
d(t)a(t)e−iΦ(t)dt; Fb =
∫ Tobs
d(t)b(t)e−iΦ(t)dt, (8)
Φ(t) is the phase of the gravitational wave signal, as is described in [22].
As the LISA array moves in space, the frequency f0 is affected by Doppler
modulations. This modulation changes with differing position of the source in the sky,
implying the need to recalculate the modulations and thus a(t) and b(t) for each sky
position that is tested - a significant factor in the performance of this approach. The
differing modulation structure however also allows us to estimate the location of the
WD MLDC1 4
source in the sky by maximising the 2F value. The resolution possible on the sky with
this method is not as good as from a full Bayesian posterior probability calculation as
performed in the parameter estimation stage, as shown in an example for Challenge
1.1.1a in figure 1. Nevertheless, since this statistic can be computed fairly quickly it
serves as a useful way of finding initial values to feed into the MCMC routine, as adopted
within the pipeline. The resolution achievable on the sky increases with frequency, which
implies that the mismatch between filter and signal falls off more rapidly at higher
frequencies, requiring a greater number of templates to cover the sky. Therefore for
challenge 1.1.1b at f ≈ 3mHz a sky grid of size 5,752 points was used, in comparison
with 765 points for challenge 1.1.1a at f ≈ 1mHz.
The F -statistic search was implemented using the LIGO “Lalapps” suite of software
[24], in which the pulsar search code was modified by Reinhard Prix and John Whelan
to use the LISA response function for the TDI variables X , Y , and Z [21]. These input
data streams were given in the form of Short Fourier Transforms, each of length one day,
created from the MLDC1 challenge data. For each challenge the full specified range of
frequencies was searched for the signal as it would be in a blind search. The code was
run on a single CPU and executed in a few hours, with the run-time increasing at higher
frequency due to the higher resolution of sky and frequency grid that had to be used.
The candidate chosen to pass to the MCMC stage was simply that which triggered the
highest value of 2F .
2.2. Second stage: Markov Chain Monte Carlo follow-up
According to Bayes’ theorem, the posterior probability, p(m̃|d̃) of a model m̃ given the
data d̃ depends on the prior distribution p(m̃), containing the information known before
the analysis, the likelihood L(d̃|m̃) of the model and a normalisation factor p(d̃)
p(m̃|d̃) = L(d̃|m̃)p(m̃)
p(d̃)
The posterior probability density function shows the joint probability density of given
values of parameters of the model m̃, conditional on the data d̃.
We implemented Bayes’ theorem using data in the form of TDI variables A and E
and modelled our template according to the Long Wavelength Approximation directly
in the Fourier domain [25] to gain computational speed. The logarithmic likelihood
L(d̃|m̃) in this stage explicitly included its dependence on the one-sided noise spectral
density Sn(f)
logL(d̃|m̃) = const. −
log Sn(f) − (d̃− h̃|d̃− h̃), (10)
shown here for either A or E, with the combined likelihood as sum of the individual
likelihoods. We restricted our analysis to a sufficiently narrow frequency window in
order to be able to approximate the noise spectral density as constant, Sn(f) = S0. This
window was set as the interval in frequency that contains at least 98% of the power of our
WD MLDC1 5
Ecliptic Longitude
2F as a function of sky position, at a frequency 0.001063 Hz
1 2 3 4 5 6
Figure 1. The variation of 2F values for the search for unknown signal 1.1.1a, as a
function of sky position, parameterised by ecliptic latitude β and longitude λ. The
distribution is multi-modal and non-Gaussian, and has a poor resolution in comparison
with that can be achieved with the MCMC and a Bayesian likelihood, but by finding the
maximum it serves well as a starting point for the more refined parameter estimation
below.
model m̃, with the interval’s upper and lower limits given by f±(2/year)(5+2πf0AU/c)
[25]. S0 is therefore an additional parameter to be inferred within the model m̃ in Eq. 10.
We implemented an automatic Random Walk Metropolis sampler (Stroeer &
Vecchio 2007, in. prep.) to sample from the posterior probability density function in
form of a Markov chain. Metropolis sampling eliminates the need to explicitly calculate
the normalisation constant in Bayes’ theorem, and the evolving Markov chain gives
easy access to joint as well as marginalised posterior density distribution. The sampler
was started from the parameter set which triggered the highest value of 2F in our grid
based coherent run of the analysis (see former section). The automated function of the
Metropolis sampling is achieved by controlling the sampling step-size with adaptive
acceptance probability techniques [26]. The sampler therefore does not depend on
assumptions about the signal in the data set in order to perform successfully and reliably;
it develops a suitable algorithm and approach by itself based on the properties of the
likelihood as found on the fly, in the initial steps of the sampler. The length of our
Markov chain was pre-set to 106, with the initial 104 chain states discarded as the
“burn-in” phase of our sampler. The runtime for one data analysis run is 5 hours on a
single 2 GHz CPU on the Tsunami cluster of the University of Birmingham.
WD MLDC1 6
Figure 2. The marginalised posterior probability density functions of the eight
unknown parameters – the seven parameters that describe the signal and the noise
spectral density S0 – for the the challenge data set 1.1.1a. The vertical black solid
line denotes the true value of the parameter (for the polarisation angle the true value
modulo π/2), and the grey dashed line the initial value for the MCMC analysis as
determined by the template of the first-stage that produces the maximum value of the
F -statistic. In the case of the noise spectral density the first stage of the analysis does
not provide an estimate; the true value of this parameter is taken to be the value of
the instrumental noise spectrum used to generate the data set and provided in [9].
WD MLDC1 7
Figure 3. The marginalised posterior probability density functions of the eight
unknown parameters for the the challenge data set 1.1.1b. Labels are as in Figure
3. Results
We found that the most promising candidate signal from the F -statistic search already
matched the true embedded signal to high accuracy, particularly in frequency and
sky location. Our MCMC sampler, as a post-processing unit, thus only needed 1000
iterations to burn in and to establish a reliable sampling from the posterior. The
marginalised posteriors are shown in Figs. 2 and 3. We found, as seen in latter
figures, that the MCMC sampler further refined the initial guesses from the F -statistic,
as measured by the absolute difference between the true value of a given parameter
and the median of the marginalised posterior recovered for that parameter. Table 1
WD MLDC1 8
Table 1. Details about the results from Challenge 1.1.1a and Challenge 1.1.1b. S0,
the constant one-sided noise spectral density within our narrow frequency window, is
compared to the true one sided noise spectral density at the true frequency of the
signal, Ψ is given modulo π/2. Int90 denotes the minimum interval to include 90% of
MCMC states for given parameter, ∆mode denotes the absolute difference between the
true value of a signal parameter and the mode of its recovered posterior; ∆median and
∆mean denote the equivalent absolute difference for median and mean of the posterior
respectively; σ denotes the sampled standard deviation of the posterior as derived
from the median. We further quote the signal-to-noise ratio (SNR) for a template
using the true values of the source and the recovered values of the data analysis run,
as derived from the median of the individual posterior distributions, and the correlation
C between these two templates.
Int90 ∆mode ∆median ∆mean σ
Challenge 1.1.1a
10−41Hz−1
(3.53257, 4.72639) -0.42084 -0.440278 -0.452456 0.36704
ϑe/ rad (0.958409, 1.03165) -0.0147383 -0.0149381 -0.0148725 0.0222861
ϕe/ rad (5.05376, 5.13528) -0.00550139 -0.00569547 -0.00579889 0.0247886
Ψ/ rad (1.32475, 0.500553) 0.1768 0.1823 0.1902 0.1908
ι/ rad (0.097761, 1.0008) -0.0459747 0.190001 0.23459 0.295211
A/10−22 (1.61976, 2.67967) 0.664371 0.358844 0.298978 0.368524
f0/ mHz (1.06273, 1.06273) -1.19664e-06 -1.22207e-06 -1.22259e-06 1.04422e-06
Φ0/ rad (3.10668, 5.808) -0.164989 0.00998525 0.229659 0.829146
SNR true = 51.024497 recovered = 50.648600
C true vs. recovered = 0.99689
Challenge 1.1.1b
10−41Hz−1
(0.876833, 1.38959) -0.0679571 -0.0906557 -0.0996144 0.16017
ϑe/ rad (-0.121611, 0.0116916) -0.0343353 -0.151185 -0.150328 0.0406552
ϕe/ rad (4.60969, 4.63537) 0.00265723 0.00305564 0.00302203 0.00779893
Ψ/ rad (0.246328, 0.362409) 0.0301541 0.0311747 0.0311268 0.0353938
ι/ rad (1.22036, 1.33338) -0.0430412 -0.040458 -0.0394818 0.0348383
A/10−22 (0.45001, 0.542454) -0.016442 -0.0151921 -0.0149907 0.0281154
f0/ mHz (3.00036, 3.00036) 3.1221e-07 2.49289e-07 2.42807e-07 8.18111e-07
Φ0/ rad (5.83869, 6.19411) 0.137219 0.119301 0.119921 0.502384
SNR true = 36.587444 recovered = 37.368806
C true vs. recovered = 0.97897
shows details of the statistics of recovered posterior distributions. We highlight that
the majority of the true values of the parameters are within one standard deviation
of the median of the posterior, with a small percentage within two sampled standard
deviations. In addition, every true value of a parameter of the signal is within the
minimum interval of the posterior to cover 90% of all MCMC state values. Recovered
signal-to-noise ratios are measured as SNR = (s|h)/
(h|h), and the match C =
(htrue|hmed)/
(htrue|htrue) (hmed|hmed) between a template constructed from the true
values and a template from the median values of the individual posterior distributions,
yielding a correlation that is always higher than 0.97. Noise levels are determined
accurately and within 1 to 1.5 sampled standard deviations. Nevertheless we note that
WD MLDC1 9
our run on Challenge 1.1.1a shows a lower match and higher differences between true
value and recovered value of parameters as compared to the run on Challenge 1.1.1b. It
also exhibits tailing posterior distributions in inclination and amplitude, although the
SNR of Challenge 1.1.1a is twice the value of Challenge 1.1.1b.
4. Conclusions
We have presented a new approach to LISA data analysis in the form of an end-to-end
pipeline. We first detected and identified candidate signals in the LISA data stream with
a grid-based coherent algorithm, and then post-processed the most promising candidate
signals with an automatic Markov Chain Monte Carlo code to obtain probability
densities for the model’s parameters. We demonstrated successful identification and
post-processing of the signals from the double white dwarf single source MLDC data
sets 1.1.1a and 1.1.1b. Furthermore, the automatic Markov Chain Monte Carlo code
successfully identified the noise level within a small frequency window of interest in
these data sets. We note that a parallel approach to the data analysis of binary inspiral
signals is being developed by Röver et al, with a Markov Chain Monte Carlo method
that can successfully post-process a candidate signal generated from the true parameters
of the signal. Signal detection in a pre-processing stage is currently being tested within
parallel tempered MCMC methods and/or time-frequency analyses [19].
We identify two prominent and promising features of our pipeline: its ability to
determine good initial conditions for the MCMC and its ability to run the MCMC
automatically. As we have demonstrated in this paper, the width of the marginalised
posterior density for the frequency parameter is extremely narrow. It is therefore vital
that the initial estimate of the frequency is within this region, as the almost flat structure
of the posterior PDF outside this region gives little to no information on the location of
the peak. The chances of finding the mode through a random sampling are decreased
further still with a larger prior range for the parameter. Adding an F -statistic search as
the first stage in the pipeline solves this problem, since the frequency and position in the
sky are recovered very accurately, to within the limits of the posterior probability region
of interest, before the MCMC performs post-processing and parameter estimation. The
automatic feature of the MCMC ensures a successful post-processing for the other
astrophysical parameters that may have been located outside the posterior region of
interest by the F -statistic approach, as in the case for the amplitude of Challenge
1.1.1a. Convergence is aided by the ability of our code to increase or decrease sampling
step-sizes according to its experience of the sampling quality of the posterior during the
burn-in phase.
We are working on an extension of the pipeline as shown in this document to
successfully tackle multi-source data sets, required for the second round of the MLDC.
Current work includes the exploration of our grid-based coherent search on such data
streams in order to automatically identify the most promising individual candidate
signals, and the implementation of an automatic Reversible Jump Markov Chain Monte
WD MLDC1 10
Carlo routine (e.g. as already demonstrated in [10]) to find the trans-dimensional
probability density functions of the parameters of an unknown total number of signals.
We highlight that the noise level determination presented here already serves as a key
ingredient to round 2, where the simulation of a galactic white dwarf binary population
introduces additional confusion noise levels from unresolvable sources.
Acknowledgements
Nelson Christensen’s work was supported by the National Science Foundation grant
PHY-0553422 and the Fulbright Scholar Program. Alberto Vecchio’s work was partially
supported by the Packard Foundation and the National Science Foundation. The
University of Auckland group was supported by the Royal Society of New Zealand
Marsden Fund Grant UOA-204.
References
[1] Bender B L et al 1998 LISA Pre-Phase A Report; Second Edition, MPQ 233
[2] Nelemans G, Yungelson L R and Portegies Zwart S F 2001 Astron. and Astrophys. 375 890
[3] Farmer A J and Phinney E S 2003 Mon. Not. R. Astron. Soc 346 1197
[4] Jaynes E T Probability theory: The logic of science 2003 Cambridge University Press
[5] Gregory P C Bayesian logical data analysis for the physical sciences 2005 Cambridge University
Press
[6] Gelman A, Carlin J B, Stern H, and Rubin D B Bayesian data analysis 1997 Chapman & Hall
CRC Boca Raton
[7] Arnaud K A et al 2006 AIP Conf. Proc. 873 619 Preprint gr-qc/0609105
[8] Arnaud K A et al 2006 AIP Conf. Proc. 873 625 Preprint gr-qc/0609106
[9] Mock LISA Data Challenge Task Force, “Document for Challenge 1,”
svn.sourceforge.net/viewvc/lisatools/Docs/challenge1.pdf.
[10] Stroeer A, Gair J and Vecchio A 2006 Automatic Bayesian inference for LISA data analysis
strategies Preprint gr-qc/0609010
[11] Umstätter R, Christensen N, Hendry M, Meyer R, Simha V, Veitch J, Vigeland S and Woan G
2005 Phys Rev D 72 022001
[12] Wickham E D L, Stroeer A and Vecchio A 2006 Class Quantum Grav 23 819
[13] Bloomer E et al Report on MLDC1 available at http://astrogravs.nasa.gov/docs/mldc/round1/entries.html
[14] Arnaud K A et al 2007 Preprint gr-qc/0701139
[15] Arnaud K A et al 2007 Preprint gr-qc/0701170
[16] Crowder, J., and Cornish, N. J. 2007 Phys. Rev. D 75 043008
[17] Crowder J, Cornish N J and Reddinger J L 2006 Phys. Rev. D 73 063011
[18] Cornish N J and Crowder J 2005 Phys. Rev. D 72 043005
[19] Röver C et al in this volume
[20] Jaranowski P, Królak A and Schutz B F 1998 Phys. Rev. D 58 063001
[21] Prix R and Whelan J 2006 Technical note
[22] Brady P R, Creighton T, Cutler C and Schutz B F 1997 Phys. Rev. D 57 2101
[23] Prince T A, Tinto M, Larson S L and Armstrong J W 2002 Phys. Rev. D 66 122002
[24] LAL Home Page: http://www.lsc-group.phys.uwm.edu/daswg/projects/lal.html
[25] Cornish N J, Larson S L 2003 Phys. rev. D 67 103001
[26] Atchade Y F, Rosenthal J S 2005 Bernoulli 11 815-828
http://arxiv.org/abs/gr-qc/0609105
http://arxiv.org/abs/gr-qc/0609106
http://arxiv.org/abs/gr-qc/0609010
http://astrogravs.nasa.gov/docs/mldc/round1/entries.html
http://arxiv.org/abs/gr-qc/0701139
http://arxiv.org/abs/gr-qc/0701170
http://www.lsc-group.phys.uwm.edu/daswg/projects/lal.html
	Introduction
	Analysis method
	First stage: Grid based search
	Second stage: Markov Chain Monte Carlo follow-up
	Results
	Conclusions

  We report on the analysis of selected single source data sets from the first
round of the Mock LISA Data Challenges (MLDC) for white dwarf binaries. We
implemented an end-to-end pipeline consisting of a grid-based coherent
pre-processing unit for signal detection, and an automatic Markov Chain Monte
Carlo post-processing unit for signal evaluation. We demonstrate that signal
detection with our coherent approach is secure and accurate, and is increased
in accuracy and supplemented with additional information on the signal
parameters by our Markov Chain Monte Carlo approach. We also demonstrate that
the Markov Chain Monte Carlo routine is additionally able to determine
accurately the noise level in the frequency window of interest.

Introduction
The data obtained from LISA [1] will contain a large number of white dwarf binary
systems across the whole observational window [2]. At frequencies below ∼ 3 mHz the
sources are so abundant that they produce a stochastic foreground whose intensity
dominates the instrumental noise [3]. The closer (and louder) sources will still be
sufficiently bright to be individually resolvable. Above ∼ 3 mHz the sources become
sufficiently sparse in parameter space (and in particular in the frequency domain) that
the detectable sources become individually resolvable. The identification of white dwarfs
in the LISA data set represents one of the most interesting analysis problems posed by
the mission: the total number of signals in the data set is unknown, the effective noise
http://arxiv.org/abs/0704.0048v2
WD MLDC1 2
level affecting the measurements is not easily estimated from the data streams, and
there is a large number of overlapping sources to the limit of confusion.
Bayesian inference provides a clear framework to tackle such a problem [4, 5, 6].
Some of us have carried out exploratory studies and “proof of concept” analyses
on simplified problems that have demonstrated that Bayesian techniques do indeed
show good potential for LISA applications [11, 10, 12]. Similarly other authors have
successfully implemented techniques using Bayesian inference [18, 17, 16]. In this paper
we present the first results of an end-to-end analysis pipeline developed in the context of
the Mock LISA Data Challenges that has evolved from our earlier work. This pipeline
is applied to the simplest single-source challenge data sets 1.1.1a and 1.1.1b and all the
results presented here are obtained after the release of the key files. In a companion
paper [19], we present results that we have obtained for the analysis of the data sets
containing gravitational radiation from a massive-black-hole binary inspiral. Our group
submitted an entry for the MLDC analysing the blind data set 1.1.1c [13, 14]: however
that result suffered from the fact that the pipeline was not complete, the analysis code
was inefficient and we encountered hardware problems with the Beowulf cluster used to
perform the analysis.
The results that we present here are obtained with a two-stage end-to-end analysis
pipeline: (i) we first process the data set with a grid-based coherent algorithm to identify
candidate signals; (ii) we then follow up the candidate signals with a Markov Chain
Monte Carlo code to obtain probability density function on the model parameters. Our
method differs from other MCMC methods that have been proposed and applied to
the MLDC data in the context of white dwarf binaries [18, 17, 16]: the MCMC is not
used to search, but only in the final stage of the analysis to produce posterior density
functions of the model parameters. The noise spectral level is included as one of the
unknown parameters and is estimated together with the parameters of the gravitational
wave source(s).
2. Analysis method
In this section we describe the two stage approach that we have adopted for the analysis.
The signal produced by a white dwarf binary system is modelled as monochromatic in
the source reference frame, following the conventions adopted in the first MLDC [7, 8, 9].
It is described by 7 parameters: ecliptic latitude ϑe and longitude ϕe, inclination ι and
polarisation angle Ψ, frequency at a reference time f0 and corresponding overall phase
Φ0 and amplitude A.
The data distributed for the MLDC are the three TDI v1.5 Michelson observables
X , Y and Z ‡. From those we construct the two orthogonal TDI outputs
A = (2X − Y − Z)/3 (1)
E = (Z − Y )/
3 (2)
‡ In our MCMC analysis we use the data set produced using the LISA Simulator.
WD MLDC1 3
by diagonalizing the noise covariance matrix following the procedure presented in [23].
The noise affecting the channels A and E is uncorrelated and described by the one-sided
noise spectral density Sn(f). We model the LISA response function in the low frequency
limit in order to improve the computational efficiency of our analysis.
2.1. First stage: Grid based search
The first stage of the pipeline consists of a fast search of the data for the best
matched filter based on the well-known F -statistic algorithm, first developed for triaxial
pulsar signals in the context of ground-based observations [20]. This exploits the Fast
Fourier Transform to perform matching in the frequency domain to templates which are
generated at an array of fixed points in the parameter space.
The data from an individual detector in the frequency domain d̃(f) is supposed to
contain a signal plus Gaussian noise, d̃(f) = h̃(f) + ñ(f). We define the logarithmic
likelihood as a measure of match, as given by logL ≈ (d̃|h̃)− 1
(h̃|h̃) with (·|·) denoting
the scalar product as defined in [20]. A single signal in the F -statistic algorithm is
re-parameterised as a linear function of four orthogonal variables, and the frequency
f0. The detection statistic is based on four parameters AF , BF , CF and DF , found by
integrating over the response functions a(t) and b(t) to the two polarisation states of
the gravitational wave signal [20],
∫ Tobs
a(t)2dt (3)
b(t)2dt (4)
a(t)b(t)dt, (5)
DF = AFBF − C2F (6)
Tobs denotes the total observed time for the data set being analysed. The optimal
detection statistic 2F , which is pre-maximised over the nuisance parameters h0, ι, φ0
and ψ is
2F = 8
Sn(f)Tobs
BF |Fa|2 + AF |Fb|2 − 2CF ×R(FaFb)
. (7)
Fa and Fb are the demodulated Fourier transforms of the data,
∫ Tobs
d(t)a(t)e−iΦ(t)dt; Fb =
∫ Tobs
d(t)b(t)e−iΦ(t)dt, (8)
Φ(t) is the phase of the gravitational wave signal, as is described in [22].
As the LISA array moves in space, the frequency f0 is affected by Doppler
modulations. This modulation changes with differing position of the source in the sky,
implying the need to recalculate the modulations and thus a(t) and b(t) for each sky
position that is tested - a significant factor in the performance of this approach. The
differing modulation structure however also allows us to estimate the location of the
WD MLDC1 4
source in the sky by maximising the 2F value. The resolution possible on the sky with
this method is not as good as from a full Bayesian posterior probability calculation as
performed in the parameter estimation stage, as shown in an example for Challenge
1.1.1a in figure 1. Nevertheless, since this statistic can be computed fairly quickly it
serves as a useful way of finding initial values to feed into the MCMC routine, as adopted
within the pipeline. The resolution achievable on the sky increases with frequency, which
implies that the mismatch between filter and signal falls off more rapidly at higher
frequencies, requiring a greater number of templates to cover the sky. Therefore for
challenge 1.1.1b at f ≈ 3mHz a sky grid of size 5,752 points was used, in comparison
with 765 points for challenge 1.1.1a at f ≈ 1mHz.
The F -statistic search was implemented using the LIGO “Lalapps” suite of software
[24], in which the pulsar search code was modified by Reinhard Prix and John Whelan
to use the LISA response function for the TDI variables X , Y , and Z [21]. These input
data streams were given in the form of Short Fourier Transforms, each of length one day,
created from the MLDC1 challenge data. For each challenge the full specified range of
frequencies was searched for the signal as it would be in a blind search. The code was
run on a single CPU and executed in a few hours, with the run-time increasing at higher
frequency due to the higher resolution of sky and frequency grid that had to be used.
The candidate chosen to pass to the MCMC stage was simply that which triggered the
highest value of 2F .
2.2. Second stage: Markov Chain Monte Carlo follow-up
According to Bayes’ theorem, the posterior probability, p(m̃|d̃) of a model m̃ given the
data d̃ depends on the prior distribution p(m̃), containing the information known before
the analysis, the likelihood L(d̃|m̃) of the model and a normalisation factor p(d̃)
p(m̃|d̃) = L(d̃|m̃)p(m̃)
p(d̃)
The posterior probability density function shows the joint probability density of given
values of parameters of the model m̃, conditional on the data d̃.
We implemented Bayes’ theorem using data in the form of TDI variables A and E
and modelled our template according to the Long Wavelength Approximation directly
in the Fourier domain [25] to gain computational speed. The logarithmic likelihood
L(d̃|m̃) in this stage explicitly included its dependence on the one-sided noise spectral
density Sn(f)
logL(d̃|m̃) = const. −
log Sn(f) − (d̃− h̃|d̃− h̃), (10)
shown here for either A or E, with the combined likelihood as sum of the individual
likelihoods. We restricted our analysis to a sufficiently narrow frequency window in
order to be able to approximate the noise spectral density as constant, Sn(f) = S0. This
window was set as the interval in frequency that contains at least 98% of the power of our
WD MLDC1 5
Ecliptic Longitude
2F as a function of sky position, at a frequency 0.001063 Hz
1 2 3 4 5 6
Figure 1. The variation of 2F values for the search for unknown signal 1.1.1a, as a
function of sky position, parameterised by ecliptic latitude β and longitude λ. The
distribution is multi-modal and non-Gaussian, and has a poor resolution in comparison
with that can be achieved with the MCMC and a Bayesian likelihood, but by finding the
maximum it serves well as a starting point for the more refined parameter estimation
below.
model m̃, with the interval’s upper and lower limits given by f±(2/year)(5+2πf0AU/c)
[25]. S0 is therefore an additional parameter to be inferred within the model m̃ in Eq. 10.
We implemented an automatic Random Walk Metropolis sampler (Stroeer &
Vecchio 2007, in. prep.) to sample from the posterior probability density function in
form of a Markov chain. Metropolis sampling eliminates the need to explicitly calculate
the normalisation constant in Bayes’ theorem, and the evolving Markov chain gives
easy access to joint as well as marginalised posterior density distribution. The sampler
was started from the parameter set which triggered the highest value of 2F in our grid
based coherent run of the analysis (see former section). The automated function of the
Metropolis sampling is achieved by controlling the sampling step-size with adaptive
acceptance probability techniques [26]. The sampler therefore does not depend on
assumptions about the signal in the data set in order to perform successfully and reliably;
it develops a suitable algorithm and approach by itself based on the properties of the
likelihood as found on the fly, in the initial steps of the sampler. The length of our
Markov chain was pre-set to 106, with the initial 104 chain states discarded as the
“burn-in” phase of our sampler. The runtime for one data analysis run is 5 hours on a
single 2 GHz CPU on the Tsunami cluster of the University of Birmingham.
WD MLDC1 6
Figure 2. The marginalised posterior probability density functions of the eight
unknown parameters – the seven parameters that describe the signal and the noise
spectral density S0 – for the the challenge data set 1.1.1a. The vertical black solid
line denotes the true value of the parameter (for the polarisation angle the true value
modulo π/2), and the grey dashed line the initial value for the MCMC analysis as
determined by the template of the first-stage that produces the maximum value of the
F -statistic. In the case of the noise spectral density the first stage of the analysis does
not provide an estimate; the true value of this parameter is taken to be the value of
the instrumental noise spectrum used to generate the data set and provided in [9].
WD MLDC1 7
Figure 3. The marginalised posterior probability density functions of the eight
unknown parameters for the the challenge data set 1.1.1b. Labels are as in Figure
3. Results
We found that the most promising candidate signal from the F -statistic search already
matched the true embedded signal to high accuracy, particularly in frequency and
sky location. Our MCMC sampler, as a post-processing unit, thus only needed 1000
iterations to burn in and to establish a reliable sampling from the posterior. The
marginalised posteriors are shown in Figs. 2 and 3. We found, as seen in latter
figures, that the MCMC sampler further refined the initial guesses from the F -statistic,
as measured by the absolute difference between the true value of a given parameter
and the median of the marginalised posterior recovered for that parameter. Table 1
WD MLDC1 8
Table 1. Details about the results from Challenge 1.1.1a and Challenge 1.1.1b. S0,
the constant one-sided noise spectral density within our narrow frequency window, is
compared to the true one sided noise spectral density at the true frequency of the
signal, Ψ is given modulo π/2. Int90 denotes the minimum interval to include 90% of
MCMC states for given parameter, ∆mode denotes the absolute difference between the
true value of a signal parameter and the mode of its recovered posterior; ∆median and
∆mean denote the equivalent absolute difference for median and mean of the posterior
respectively; σ denotes the sampled standard deviation of the posterior as derived
from the median. We further quote the signal-to-noise ratio (SNR) for a template
using the true values of the source and the recovered values of the data analysis run,
as derived from the median of the individual posterior distributions, and the correlation
C between these two templates.
Int90 ∆mode ∆median ∆mean σ
Challenge 1.1.1a
10−41Hz−1
(3.53257, 4.72639) -0.42084 -0.440278 -0.452456 0.36704
ϑe/ rad (0.958409, 1.03165) -0.0147383 -0.0149381 -0.0148725 0.0222861
ϕe/ rad (5.05376, 5.13528) -0.00550139 -0.00569547 -0.00579889 0.0247886
Ψ/ rad (1.32475, 0.500553) 0.1768 0.1823 0.1902 0.1908
ι/ rad (0.097761, 1.0008) -0.0459747 0.190001 0.23459 0.295211
A/10−22 (1.61976, 2.67967) 0.664371 0.358844 0.298978 0.368524
f0/ mHz (1.06273, 1.06273) -1.19664e-06 -1.22207e-06 -1.22259e-06 1.04422e-06
Φ0/ rad (3.10668, 5.808) -0.164989 0.00998525 0.229659 0.829146
SNR true = 51.024497 recovered = 50.648600
C true vs. recovered = 0.99689
Challenge 1.1.1b
10−41Hz−1
(0.876833, 1.38959) -0.0679571 -0.0906557 -0.0996144 0.16017
ϑe/ rad (-0.121611, 0.0116916) -0.0343353 -0.151185 -0.150328 0.0406552
ϕe/ rad (4.60969, 4.63537) 0.00265723 0.00305564 0.00302203 0.00779893
Ψ/ rad (0.246328, 0.362409) 0.0301541 0.0311747 0.0311268 0.0353938
ι/ rad (1.22036, 1.33338) -0.0430412 -0.040458 -0.0394818 0.0348383
A/10−22 (0.45001, 0.542454) -0.016442 -0.0151921 -0.0149907 0.0281154
f0/ mHz (3.00036, 3.00036) 3.1221e-07 2.49289e-07 2.42807e-07 8.18111e-07
Φ0/ rad (5.83869, 6.19411) 0.137219 0.119301 0.119921 0.502384
SNR true = 36.587444 recovered = 37.368806
C true vs. recovered = 0.97897
shows details of the statistics of recovered posterior distributions. We highlight that
the majority of the true values of the parameters are within one standard deviation
of the median of the posterior, with a small percentage within two sampled standard
deviations. In addition, every true value of a parameter of the signal is within the
minimum interval of the posterior to cover 90% of all MCMC state values. Recovered
signal-to-noise ratios are measured as SNR = (s|h)/
(h|h), and the match C =
(htrue|hmed)/
(htrue|htrue) (hmed|hmed) between a template constructed from the true
values and a template from the median values of the individual posterior distributions,
yielding a correlation that is always higher than 0.97. Noise levels are determined
accurately and within 1 to 1.5 sampled standard deviations. Nevertheless we note that
WD MLDC1 9
our run on Challenge 1.1.1a shows a lower match and higher differences between true
value and recovered value of parameters as compared to the run on Challenge 1.1.1b. It
also exhibits tailing posterior distributions in inclination and amplitude, although the
SNR of Challenge 1.1.1a is twice the value of Challenge 1.1.1b.
4. Conclusions
We have presented a new approach to LISA data analysis in the form of an end-to-end
pipeline. We first detected and identified candidate signals in the LISA data stream with
a grid-based coherent algorithm, and then post-processed the most promising candidate
signals with an automatic Markov Chain Monte Carlo code to obtain probability
densities for the model’s parameters. We demonstrated successful identification and
post-processing of the signals from the double white dwarf single source MLDC data
sets 1.1.1a and 1.1.1b. Furthermore, the automatic Markov Chain Monte Carlo code
successfully identified the noise level within a small frequency window of interest in
these data sets. We note that a parallel approach to the data analysis of binary inspiral
signals is being developed by Röver et al, with a Markov Chain Monte Carlo method
that can successfully post-process a candidate signal generated from the true parameters
of the signal. Signal detection in a pre-processing stage is currently being tested within
parallel tempered MCMC methods and/or time-frequency analyses [19].
We identify two prominent and promising features of our pipeline: its ability to
determine good initial conditions for the MCMC and its ability to run the MCMC
automatically. As we have demonstrated in this paper, the width of the marginalised
posterior density for the frequency parameter is extremely narrow. It is therefore vital
that the initial estimate of the frequency is within this region, as the almost flat structure
of the posterior PDF outside this region gives little to no information on the location of
the peak. The chances of finding the mode through a random sampling are decreased
further still with a larger prior range for the parameter. Adding an F -statistic search as
the first stage in the pipeline solves this problem, since the frequency and position in the
sky are recovered very accurately, to within the limits of the posterior probability region
of interest, before the MCMC performs post-processing and parameter estimation. The
automatic feature of the MCMC ensures a successful post-processing for the other
astrophysical parameters that may have been located outside the posterior region of
interest by the F -statistic approach, as in the case for the amplitude of Challenge
1.1.1a. Convergence is aided by the ability of our code to increase or decrease sampling
step-sizes according to its experience of the sampling quality of the posterior during the
burn-in phase.
We are working on an extension of the pipeline as shown in this document to
successfully tackle multi-source data sets, required for the second round of the MLDC.
Current work includes the exploration of our grid-based coherent search on such data
streams in order to automatically identify the most promising individual candidate
signals, and the implementation of an automatic Reversible Jump Markov Chain Monte
WD MLDC1 10
Carlo routine (e.g. as already demonstrated in [10]) to find the trans-dimensional
probability density functions of the parameters of an unknown total number of signals.
We highlight that the noise level determination presented here already serves as a key
ingredient to round 2, where the simulation of a galactic white dwarf binary population
introduces additional confusion noise levels from unresolvable sources.
Acknowledgements
Nelson Christensen’s work was supported by the National Science Foundation grant
PHY-0553422 and the Fulbright Scholar Program. Alberto Vecchio’s work was partially
supported by the Packard Foundation and the National Science Foundation. The
University of Auckland group was supported by the Royal Society of New Zealand
Marsden Fund Grant UOA-204.
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http://arxiv.org/abs/gr-qc/0609105
http://arxiv.org/abs/gr-qc/0609106
http://arxiv.org/abs/gr-qc/0609010
http://astrogravs.nasa.gov/docs/mldc/round1/entries.html
http://arxiv.org/abs/gr-qc/0701139
http://arxiv.org/abs/gr-qc/0701170
http://www.lsc-group.phys.uwm.edu/daswg/projects/lal.html
	Introduction
	Analysis method
	First stage: Grid based search
	Second stage: Markov Chain Monte Carlo follow-up
	Results
	Conclusions
