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Overlapping Astronomical Sources: Utilizing Spectral Information
David Jones Advisor: Xiao-Li Meng Collaborators: Vinay Kashyap (CfA) and David van Dyk (Imperial College) CHASC Astrostatistics Group

April 15, 2014

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Introduction

X-ray telescope data:
spatial co ordinates of photon detections photon energy

Instrument error: diffraction in the telescope means recorded photon positions are spread out according to the point spread function (PSF)

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Introduction

PSFs overlap for sources near each other Aim: inference for number of sources and their intensities, positions and spectral distributions Key points: (i) obtain posterior distribution of number of sources, (ii) use spectral information

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Basic Model and Notation

(xi , yi ) = spatial coordinates of photon i k = # sources µj = centre of source j si = latent variable indicating which source photon i is from nj = n=1 1{si =j } = # photons detected from component j {0, 1, . . . , k } i (xi , yi )|si = j , µj , k (n0 , n1 , . . . , nk )|w , k (w0 , w1 , . . . , wk )|k µj |k PSFj centred at µj for i = 1, . . . , n Mult(n; (w0 , w1 , . . . , wk )) Dirichlet(, , . . . , ) Uniform over the image j = 1, 2, . . . , k

k Pois() Component with label 0 is background and its "PSF" is uniform over the image (so its "centre" is irrelevant) Reasonably insensitive to , the prior mean number of sources

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3rd Dimension: Spectral Data

We can distinguish the background from the sources better if we jointly model spatial and spectral information: ei |si = j , j ,
j



Gamma(j , j ) for j {1, . . . , k } Uniform to some maximum Gamma(a , b ) Gamma(a , b )

e i |s i = 0 j
j

Using a (correctly) "informative" prior on si and si versus a diffuse prior made very little difference to results.

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Computation: RJMCMC

Similar to Richardson & Green 1997 Knowledge of the PSF makes things much easier Insensitive to the prior k Pois() e.g. posterior when k = 10:

(a) = 1

(b) = 10

Figure: Average p osterior probabilities of each value of k across ten datasets

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Simulation Study: Example

100 datasets simulated for each configuration Analysis with and without energy data Summarize posterior of k by posterior probability of two sources
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Simulation Study: PSF (King 1962)

King density has Cauchy tails Gaussian PSF leads to over-fitting in real data

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Simulation Study: Data Generation
Bright source: n1 Pois(m1 = 1000) Dim source: n2 Pois(m2 = 1000/r ) where r = 1, 2, 10, 50 gives the relative intensity 'Source region': the region defined by PSF density greater than 10% of the maximum (essentially a circle with radius 1) d = the probability a photon from a source falls within its own region

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Simulation Study: Data Generation

Background per 'source region': Pois(bdm2 ) where relative background b = 0.001, 0.01, 0.1, 1 Overall background n0 Pois image area bdm source region area
2

Separation: the distance between the sources. Values: 0.5, 1, 1.5, 2

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Simulation Study: Data Generation
Note: units should be pulse invariant (PI) channel

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Median Posterior Probability at k=2: No Energy

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Median Posterior Probability at k=2: Energy

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Median SE of Dim Source Posterior Mean Position: No Energy

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Median SE of Dim Source Posterior Mean Position: Energy

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Strong Background Posterior Mean Positions: No Energy

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Strong Background Posterior Mean Positions: Energy

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XMM Data

Additional question: how do the spectral distributions of the sources compare?

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Parameter Inference

Table: Posterior parameter estimation for FK Aqr and FL Aqr (using sp ectral data)
µ11 120.973 0.001 0.000 µ12 124.873 0.001 0.000 µ21 121.396 0.002 0.000 µ22 127.326 0.002 0.000 w1 0.732 0.001 0.001 w2 0.189 0.001 0.003 wb 0.079 0.000 0.004 1 3.195 0.008 0.002 2 3.121 0.014 0.004 1 0.005 0.000 0.002 2 0.005 0.000 0.005

Mean SD SD/Mean

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Componentwise posterior spectral distributions

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Posteriors of source spectral parameters

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Chandra Data

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Gamma Mixture Spectral Model

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Chandra k Results

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Locations: 90% credible regions

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Chandra data spectral model contribution

1. Spectral model gives some constraints on the spectral distributions helping us to infer source properties more precisely
Posterior standard deviations are smaller with the spectral model Without it some fainter sources are o ccasionally not found

2. It also offers some robustness to chance or systematic variations in the PSF and background
Background is more easily distinguished from sources Sp ectral mo del plays a large role in the likeliho od so a strange shap ed source will be unlikely to b e split unless the spectral data also supports two sources

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Summary and extensions

Coherent method for dealing with overlapping sources that uses spectral as well as spatial information Flexibility to include other phenomenon Temporal model? Flares and other activity change the intensity and spectral distribution of sources over time Approximation to full method could be desirable in some cases

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S. Richardson, P. J. Green On Bayesian analysis of mixtures with an unknown number of components (with discussion), J. R. Statist. Soc. B, 59, 731792, 1997; corrigendum, 60 (1998), 661. I. King, The structure of star clusters. I. An empirical density law, The Astronomical Journal, 67 (1962), 471. C. M. Bishop, N. M. Nasrabadi, Pattern recognition and machine learning, Vol. 1. New York: springer, 2006. A. P. Dempster, N. M. Laird, D. B. Rubin. Maximum likelihood from incomplete data via the EM algorithm, Journal of the Royal Statistical Society, Series B (Methodological) (1977): 1-38. S. P. Brooks, A. Gelman, General Methods for Monitoring Convergence of Iterative Simulations, Journal of Computational and Graphical Statistics, Vol. 7, No. 4. (Dec., 1998), pp. 434-455.

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XMM data spectral distribution

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Four models

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