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Detection: Overlapping Sources
David Jones
Harvard University Statistics Department

November 12, 2013

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Introduction Model Example Simulation study Chandra data XMM data Summary and discussion

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Introduction

X-ray data: coordinates of photon detections, photon energy 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 of number of sources, (ii) use spectral information

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Basic Model and Notation

yij k µi ni

= spatial coordinates of photon j from source i = # sources (components) = centre of source i = # photons detected from source i yij |µi , ni , k (n0 , n1 , . . . , nk )|w , k (w0 , w1 , . . . , wk )|k µi |k PSF centred at µi j = 1, . . . , ni , i = 0, . . . , k Mult(n; (w0 , w1 , . . . , wk )) Dirichlet(, , . . . , ) Uniform over the image i = 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: eij |i , e0
i



Gamma(i , i ) for i = 1, . . . , k and j = 1, . . . , ni Uniform to some maximum for j = 1, . . . , n0 Gamma(a , b ) Gamma(a , b )

j

i i

Using a (correctly) "informative" prior on i and i 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 easier Insensitive to the prior on k e.g. posterior when k = 10 and = 3: Posterior of number of sources (k) 8 9 10 11 12 0.058 0.141 0.222 0.220 0.157 0.019 0.022 0.029 0.027 0.021

Mean SD

7 0.029 0.018

13 0.082 0.014

Used posterior probabilities given by 10 chains

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Example

Region occupied by the three sources (2 SD) is about 28% of the area and contains about 41% of the observations Within this sources region around 48% is background Positions (-2, 0), (0, 1), (1.5, 0) with intensities 50, 100, 150 respectively

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Posterior of k

Mean over 10 chains of the posterior probabilities (range indicated) When the spectral data is ignored we do not find the faintest source

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Parameter Inference

Truth Sp ectral data ignored Mean SD MSE SD/Mean Sp ectral data included Mean SD MSE SD/Mean

µ11 -2

µ12 0

µ21 0

µ22 1

µ31 1.5

µ32 0

w1 0.038

w2 0.077

w3 0.115

wb 0.769

3

0.5

-1.266 0.069 0.543

0.839 0.125 0.718

0.401 0.067 0.165

0.549 0.068 0.207

1.798 0.030 0.090

-0.054 0.046 0.005

0.049 0.002 0.050

0.067 0.002 0.027

0.086 0.003 0.032

0.798 0.001 0.001

NA NA NA NA

NA NA NA NA

-1.790 0.037 0.045

-0.101 0.064 0.014

-0.234 0.033 0.056

1.042 0.026 0.002

1.584 0.019 0.007

-0.044 0.022 0.002

0.040 0.001 0.036

0.077 0.001 0.018

0.115 0.002 0.014

0.768 0.000 0.000

2.827 0.013 0.030 0.004

0.459 0.003 0.002 0.006

The effects are less pronounced when the sources are more easily distinguished from the background

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Allocation of Photons
Table: Allocation breakdown: (a) ignoring sp ectral data

Source (intensity) Background (10/sq) Left (50) Middle (100) Right (150)

No. Photons 1015 38 97 152

Allo cation Breakdown Background Left Middle 0.876 0.035 0.040 0.798 0.121 0.067 0.502 0.168 0.189 0.481 0.043 0.159

Right 0.049 0.014 0.141 0.317

Table: Allocation breakdown: (b) using sp ectral data

Source (intensity) Background (10/sq) Left (50) Middle (100) Right (150)

No. Photons 1015 38 97 152

Allo cation Breakdown Background Left Middle 0.894 0.024 0.038 0.531 0.278 0.165 0.293 0.122 0.346 0.305 0.028 0.141

Right 0.045 0.026 0.239 0.526

Background is more easily distinguished from the sources when we include the spectral data
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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(1000) Dim source: n2 Pois(1000/r ) where r = 1, 2, 10, 50 gives the relative intensity Background per 'source region': n0 Pois(bd 1000/r ) where relative background b = 0.001, 0.01, 0.1, 1. Here d = 0.52 is the proportion of photons from a source within the region defined by density greater than 10% of the maximum (essentially a circle with radius 1)

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Simulation Study: Data Generation

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Simulation Study: Example

50 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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Posterior Probability at k=2: No Energy

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Posterior Probability at k=2: Energy

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Average MSE of Positions: No Energy

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Average MSE of Positions: Energy

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Chandra Data

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Chandra k Results

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Locations

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XMM Data

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

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k posterior

Mean over 10 chains of the posterior probabilities (range indicated) Spectral information focuses posterior on 2 sources

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Parameter Inference

Table: Parameter estimation for FK Aqr and FL Aqr (using spectral data)
Mean SD SD/Mean µ11 120.988 0.001 0.000 µ12 124.891 0.002 0.000 µ21 121.366 0.016 0.000 µ22 127.376 0.027 0.000 w1 0.808 0.001 0.001 w2 0.182 0.001 0.005 wb 0.009 0.000 0.011 3.182 0.000 0.000 0.005 0.000 0.000

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Componentwise posterior spectral distributions

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Posteriors of source spectral parameters

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Summary

Coherent method for dealing with overlapping sources that uses spectral as well as spatial information Flexibility to include other phenomenon How to combine Chandra datsets? Other models/computation possible Approximation to full method could be desirable

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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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