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[10.02] Structure in Photon Maps and Time Series: A New Approach to Bayesian Blocks

J. D. Scargle (NASA Ames Research Center), J. P. Norris (NASA Goddard Space Flight Center)

The Bayesian Blocks algorithm finds the most probable piecewise constant ("blocky") representation for time series in the form of binned, time-tagged, or time-to-spill photon counting data. In (Scargle, 1998, ApJ 504, 405) the number of blocks was determined in an ad hoc iterative procedure. Another approach maximizes the posterior -- after marginalizing all parameters except the number of blocks -- computed with Markov Chain Monte Carlo methods.

A new, better algorithm starts with the Voronoi tessellation of the individual events in an arbitrary dimensioned data space. (This generalization allows solution of problems such as detection of clusters in high dimensional parameter spaces, and identification of structures in images.) In successive steps, these many cells are merged to form fewer, larger ones. The decision to merge two cells or keep them apart is based on comparison of the corresponding posterior probabilities. Let P(N,V) be the posterior for a Poisson model of a volume of size V containing N events, a function easily calculated explicitly. Then cells j and k are merged if

P( N_j + N_k, V_j + V_k ) > P( N_j, V_j ) P( N_k, V_k )

and kept separate otherwise. When this criterion favors the merging of no further cells computation halts. Local structures ("shots") in the variability of Cygnus X-1 and RXTE 1118+480 were detected in this way, using time-tagged photon data from the USA X-ray Telescope. Since no time bins are invoked, the full sub-millisecond time resolution of the USA instrument is maintained. The method contains no parameters other than those defining prior probability distributions, and therefor yields objective structure estimates. For image data, the cells need not be restricted to be simply connected, e.g. in order to treat background regions surrounding sources. Partly funded by the NASA Applied Information Systems Research Program.

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The author(s) of this abstract have provided an email address for comments about the abstract: jeffrey@sunshine.arc.nasa.gov