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David Esch, David A. van Dyk (Harvard University), Vinay L. Kashyap, Aneta Siemiginowska (Harvard-Smithsonian Center for Astrophysics), Alanna Connors (Wellesley College)
In this paper we attack the outstanding problem of the analysis of spatial-spectral data by developing models for low count high resolution images that allow the energy spectrum to vary across the image. We begin with simple elliptic images (e.g., for a dispersed point source, a globular cluster, or a cluster of galaxies) that are modeled with bivariate Gaussian or Lorentzian distributions. The point spread function is approximated by an analytic form based on conditional bivariate normal distributions with means and variance-covariance matrices (parameterized via a eigen decomposition) that depend on the location of the incoming photon on the detector. Although an approximation to the actual point spread function, this solution will greatly accelerate the computationally expensive algorithms that we use for Bayesian model fitting.
We incorporate spectral information using the model developed in van Dyk et al. (1999, this publication) conditional on the location of the photon on the detector. The shape of the continuum and the absorption rate in this conditional distribution are allowed to vary smoothly across the image. This is accomplished via a Generalized Liner Model for the Poisson photon intensity parameters in each spectral cross spatial cell. The Generalized Linear Model parameterizes the log intensity parameters as a linear function of some transformation of the associated Energy (e.g., as in a power law) and location on the detector. The Poisson nature of the model easily allows for the low count high resolution data of the several new generation space-based telescopes which are expected over the next ten years. The model also allows for in-space and telescope absorption of photons and background emissions in a manner analogous to van Dyk et al. (1999). This highly structured hierarchical model can be fit in the Bayesian paradigm using Markov Chain Monte Carlo methods (e.g., the Gibbs sampler and the Metropolis-Hastings algorithm).
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