PHYSTAT2003, SLAC, Stanford, California, September 8-11, 2003
Bayesian Separation of Independent Sources in Astrophysical
Radiation Maps Using MCMC
E.E. Kuruoğlu and P.M. Comparetti
Istituto di Scienza e Tecnologie dell’Informazione, CNR, via G. Moruzzi 1, 56124 Pisa, Italy.
In this work, we present a novel approach to the recovery of independent sources of radiations in sky maps.
The work is motivated by the need to resolve CMB and other specific sources of radiation from a mixture of
sources in the observations that are to be made by the Planck satellite after its launch in 2007. In particular,
we present a numerical approach for Bayesian estimation namely Markov Chain Monte Carlo (MCMC) that
exploits available prior information about radiation sources and hence differs from most of other work in the
literature which are generally blind. MCMC provides large flexibility in modelling the problem and avoids
analytical difficulties by resorting to numerical techniques. Results demonstrate the success and the flexibility
of the approach.
1. INTRODUCTION
ESA’s Planck satellite, which is to be launched
in 2007, will provide 9 all-sky maps ranging in frequency from 30 GHz to 900 GHz, and in angular
resolution from 30 to 4.5 arcminutes. Celestial microwave radiation is generated by various astronomical sources, and the measured signals are superimpositions of the source signals, corrupted by measurement
noise. Source signals include the cosmic microwave
background (CMB), the thermal Galactic dust radiation, the synchrotron radiation (caused by the interaction of the electrons with magnetic field of the
galaxy) and the free-free radiation (due to the thermal
bremstrahlung from hot electrons when accelarated by
ions in the interstellar gas); among which CMB is of
paramount importance since it is a relic radiation remaining from the first instant light was able to travel
in the universe and therefore contains the picture of
the very early universe. In addition, the measurement
of the anisotropies in the CMB will place fundamental
constraints on models for the evolution of large scale
structure in the universe. Each of the other source signals is also of interest in cosmology and astrophysics.
Our goal is to reconstruct these signals.
We implement a Markov Chain Monte Carlo
(MCMC) algorithm to perform Bayesian source separation, with application to the separation of signals of
different origin in sky radiation maps. The problem
is formulated as the separation of an instantaneous
linear mixing
x = As + n
who implemented the FastICA algorithm and its noisy
version which had limited success in the presence of
significant noise. A source model was introduced in
Kuruoglu et al. [2003] implementing the Independent
Factor Analysis (IFA) technique which also included
the noise in the mixing model. Despite this added flexibility, IFA uses a fixed source model which lacks freedom in modelling source model parameters and moreover could not deal with non-stationary noise which is
the case in our problem. In this work, we aim at overcoming these limitations by providing a full Bayesian
analysis equipped with statistical priors for the source
parameters and utilizing the prior information about
the mixing as well.
Other related work include Snoussi et al. [2001]
which uses an Expectation-Maximization (EM) algorithm for obtaining the mixing parameters to obtain
a locally optimal maximum a posteriori estimate for
the mixing matrix and other model parameters, then
solves the problem of estimating the source signals
given the parameter estimates, while in our method
all unknowns, including the source signals, are estimated jointly, providing a globally optimal minimum
mean squared error (MMSE) estimate. Miskin and
MacKay [2000] use a variational Bayesian approach
which uses several approximations to avoid analytical difficulties. MCMC approach, on the other hand,
does not make any analytical approximations and bypasses the analytical difficulties by using a Bayesian
sampling strategy.
(1)
where s contains the astrophysical sources, x houses
the observations over various frequency channels and
A is the mixing matrix where each row contains the
frequency response of each source at a certain frequency channel.
The problem has been dealt with using other methods by several researchers in the literature including Baccigalupi et al. [2000] and Maino et al. [2002]
Since the MCMC methods provide samples from the
full posterior distribution, one can easily infer other
functions of the parameters and their uncertainties.
The great flexibility of the sampling approach allows
us to make appropriate modelling choices for our problem. We can therefore relax the assumption that noise
is stationary, and work under the realistic assumption
that antenna noise is Gaussian but non-stationary,
with a different but known variance at each pixel.
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PHYSTAT2003, SLAC, Stanford, California, September 8-11, 2003
2. MARKOV CHAIN MONTE CARLO
For a detailed description of the Markov Chain
Monte Carlo approach the reader is referred to Gelman et al. [1995].
In this work, we tried various sampling schemes including Metropolis-Hastings but here we present results with Gibbs sampling which we preferred because
of computational convenience. In this case, the samples are generated using the conditional posterior as
the sampling kernel. For assessing convergence, we
generated several parallel Metropolis chains and followed both inter and intra-sequence variances. We
made decisions on convergence looking at the ratio of
the marginal posterior variance of the estimand which
is a weighted average of inter and intra-sequence variances to the intra-sequence variance (Gelman et al.
[1995]).
We assume a Gaussian mixture model for the
sources. In that sense, the source model is similar to
the one in Kuruoglu et al. [2003]; however, unlike that
work here the model parameters are assigned prior distributions. In particular, the means of the Gaussian
components are assigned Gaussian priors, the variances inverse Gamma distributions and the index parameter which chooses between Gaussian components
is assigned a Dirichlet prior. The priors are chosen to
be conjugate to simplify the computations.
We also exploited the prior knowledge about the
mixing matrix, the elements of which are related
through equations of black body radiation as detailed
in Baccigalupi et al. [2000]. This information reduces
the unknowns in the matrix. The remaining elements
are assigned non-informative priors.
3. SIMULATION STUDIES
To test the performance of our approach, we use
synthetic but realistic radiation maps of CMB generated using Seljak and Zaldarriaga [1996], and synchrotron and galactic dust extrapolated from the low
resolution images obtained during the COBE mission.
We use the 30 GHz, 70GHz and 143 GHz channels.
The original images are mixed using the matrix


1.26 37.0 0.13
(2)
A =  1.14 2.91 0.55  .
0.78 0.34 1.79
Noise with an average SNRs of 12.56µK (at 30GHz),
11.307µK (at 70GHz) and 1.839µK (at 143GHz) was
added to the mixtures.
The original images, the mixtures and the MMSE
estimates obtained by the Gibbs-MCMC technique
are given in Figure 1 where for each pixel, its marginal
posterior mean value is shown. As seen from the figure, the method has been very successful in separating
Figure 1: top row: original images of CMB, synchrotron
and galactic dust, middle row: artificial mixtures over
143GHz, 70GHz and 30 GHz channels with additive
noise, bottom row: estimates using MCMC
Table I Comparison of performances of separation by
MCMC and FastICA by SIR of the estimates.
CMB Synchrotron Dust
MCMC 19.5
18.2
27.6
FastICA 4.5
11.4
10.5
the signals. The success of the method in the presence
of noise and its clear superiority to FastICA (Hyvarinen and Oja [1997]) can be seen in Table I where respective signal-to-interference ratios (SIR) are given.
4. CONCLUSIONS
In this short contribution, we presented a numerical Bayesian approach, namely MCMC with a Gibbs
sampling scheme, for the separation of the mixtures of
astrophysical sources in radiation maps. The method
provides a very flexible source model and includes
noise in the mixing model in contrast with some previous work. It also exploits available prior information about the sources and the mixing and obtains a
posterior which gives one the freedom of inference for
various statistical variables of interest. The simulation results demonstrate the success of the method.
Currently, we are exploring different estimators and
are testing our method over different regions of the
sky which show varied characteristics to provide comparisons with other existing techniques.
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PHYSTAT2003, SLAC, Stanford, California, September 8-11, 2003
Acknowledgments
Authors would like to thank the Planck teams in
Trieste and Bologna for providing the images.
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