Bayesian multiscale deconvolution applied to gamma-ray spectroscopy

A common task in gamma-ray astronomy is to extract spectral information, such as model constraints and incident photon spectrum estimates, given the measured energy deposited in a detector and the detector response. This is the classic problem of spectral “deconvolution” or spectral inversion. The methods of forward folding (i.e., parameter fitting) and maximum entropy “deconvolution” (i.e., estimating independent input photon rates for each individual energy bin) have been used successfully for gamma-ray solar flares (e.g., Rank, 1997; Share and Murphy, 1995). These methods have worked well under certain conditions but there are situations were they don’t apply. These are: 1) when no reasonable model (e.g., fewer parameters than data bins) is yet known, for forward folding; 2) when one expects a mixture of broad and narrow features (e.g., solar flares), for the maximum entropy method; and 3) low count rates and low signal-to-noise, for both. Low count rates are a problem because these methods (as they have been implemented) assume Gaussian statistics but Poisson are applicable. Background subtraction techniques often lead to negative count rates. For Poisson data the Maximum Likelihood Estimator (MLE) with a Poisson likelihood is appropriate. Without a regularization term, trying to estimate the “true” individual input photon rates per bin can be an ill-posed problem, even without including both broad and narrow features in the spectrum (i.e., amultiscale approach). One way to implement this regularization is through the use of a suitable Bayesian prior. Nowak and Kolaczyk (1999) have developed a fast, robust, technique using a Bayesian multiscale framework that addresses these problems with added algorithmic advantages. We outline this new approach and demonstrate its use with time resolved solar flare gamma-ray spectroscopy

Evidence for a Galactic gamma ray halo

We present quantitative statistical evidence for a $\gamma$-ray emission halo surrounding the Galaxy. Maps of the emission are derived. EGRET data were analyzed in a wavelet-based non-parametric hypothesis testing framework, using a model of expected diffuse (Galactic + isotropic) emission as a null hypothesis. The results show a statistically significant large scale halo surrounding the center of the Milky Way as seen from Earth. The halo flux at high latitudes is somewhat smaller than the isotropic gamma-ray flux at the same energy, though of the same order (O(10^(-7)--10^(-6)) ph/cm^2/s/sr above 1 GeV).Comment: Final version accepted for publication in New Astronomy. Some additional results/discussion included, along with entirely revised figures. 19 pages, 15 figures, AASTeX. Better quality figs (PS and JPEG) are available at http://tigre.ucr.edu/halo/paper.htm

Spectra of a recent bright burst measured by CGRO-COMPTEL: GRB 990123

CGRO-COMPTEL measures gamma-ray burst positions, time-histories and spectra in the 0.1–30 MeV energy range, in both imaging “telescope” and single detector “burst spectroscopy” mode. GRB 990123, one of the most recent bright bursts seen by COMPTEL, was caught in the optical while the gamma-ray emission was ongoing. The burst spectral shape can be characterized by a peak in ν−Fν just below 1 MeV and a power-law tail above(photon index∼−2.4,) and flattening below. There is also spectral evolution by downward movement of the peak and/or softening of the power laws. We present light-curves, time resolved spectra and an image map for this burst

Bayesian Blocks, A New Method to Analyze Structure in Photon Counting Data

I describe a new time-domain algorithm for detecting localized structures (bursts), revealing pulse shapes, and generally characterizing intensity variations. The input is raw counting data, in any of three forms: time-tagged photon events (TTE), binned counts, or time-to-spill (TTS) data. The output is the most likely segmentation of the observation into time intervals during which the photon arrival rate is perceptibly constant -- i.e. has a fixed intensity without statistically significant variations. Since the analysis is based on Bayesian statistics, I call the resulting structures Bayesian Blocks. Unlike most, this method does not stipulate time bins -- instead the data themselves determine a piecewise constant representation. Therefore the analysis procedure itself does not impose a lower limit to the time scale on which variability can be detected. Locations, amplitudes, and rise and decay times of pulses within a time series can be estimated, independent of any pulse-shape model -- but only if they do not overlap too much, as deconvolution is not incorporated. The Bayesian Blocks method is demonstrated by analyzing pulse structure in BATSE $\gamma$-ray data. The MatLab scripts and sample data can be found on the WWW at: http://george.arc.nasa.gov/~scargle/papers.htmlComment: 42 pages, 2 figures; revision correcting mathematical errors; clarifications; removed Cyg X-1 sectio

Bayesian Exponential Random Graph Models with Nodal Random Effects

We extend the well-known and widely used Exponential Random Graph Model (ERGM) by including nodal random effects to compensate for heterogeneity in the nodes of a network. The Bayesian framework for ERGMs proposed by Caimo and Friel (2011) yields the basis of our modelling algorithm. A central question in network models is the question of model selection and following the Bayesian paradigm we focus on estimating Bayes factors. To do so we develop an approximate but feasible calculation of the Bayes factor which allows one to pursue model selection. Two data examples and a small simulation study illustrate our mixed model approach and the corresponding model selection.Comment: 23 pages, 9 figures, 3 table

Network inference - with confidence - from multivariate time series

Networks - collections of interacting elements or nodes - abound in the natural and manmade worlds. For many networks, complex spatiotemporal dynamics stem from patterns of physical interactions unknown to us. To infer these interactions, it is common to include edges between those nodes whose time series exhibit sufficient functional connectivity, typically defined as a measure of coupling exceeding a pre-determined threshold. However, when uncertainty exists in the original network measurements, uncertainty in the inferred network is likely, and hence a statistical propagation-of-error is needed. In this manuscript, we describe a principled and systematic procedure for the inference of functional connectivity networks from multivariate time series data. Our procedure yields as output both the inferred network and a quantification of uncertainty of the most fundamental interest: uncertainty in the number of edges. To illustrate this approach, we apply our procedure to simulated data and electrocorticogram data recorded from a human subject during an epileptic seizure. We demonstrate that the procedure is accurate and robust in both the determination of edges and the reporting of uncertainty associated with that determination.Comment: 12 pages, 7 figures (low resolution), submitte