Alanna Connors

STATISTICS, HANDLE WITH CARE: DETECTING MULTIPLE MODEL COMPONENTS WITH THE LIKELIHOOD RATIO TEST

The likelihood ratio test (LRT) and the related F-test, popularized in astrophysics by Eadie and coworkers in 1971, Bevington in 1969, Lampton, Margon, & Bowyer, in 1976, Cash in 1979, and Avni in 1978, do not (even asymptotically) adhere to their nominal χ2 and F-distributions in many statistical tests common in astrophysics, thereby casting many marginal line or source detections and nondetections into doubt. Although the above authors illustrate the many legitimate uses of these statistics, in some important cases it can be impossible to compute the correct false positive rate. For example, it has become common practice to use the LRT or the F-test to detect a line in a spectral model or a source above background despite the lack of certain required regularity conditions. (These applications were not originally suggested by Cash or by Bevington.) In these and other settings that involve testing a hypothesis that is on the boundary of the parameter space, contrary to common practice, the nominal χ2 distribution for the LRT or the F-distribution for the F-test should not be used. In this paper, we characterize an important class of problems in which the LRT and the F-test fail and illustrate this nonstandard behavior. We briefly sketch several possible acceptable alternatives, focusing on Bayesian posterior predictive probability values. We present this method in some detail since it is a simple, robust, and intuitive approach. This alternative method is illustrated using the gamma-ray burst of 1997 May 8 (GRB 970508) to investigate the presence of an Fe K emission line during the initial phase of the observation. There are many legitimate uses of the LRT and the F-test in astrophysics, and even when these tests are inappropriate, there remain several statistical alternatives (e.g., judicious use of error bars and Bayes factors). Nevertheless, there are numerous cases of the inappropriate use of the LRT and similar tests in the literature, bringing substantive scientific results into question.

Instrument description and performance of the Imaging Gamma-Ray Telescope COMPTEL aboard the Compton Gamma-Ray Observatory

The imaging Compton telescope COMPTEL is one of the four instruments on board the Compton Gamma-Ray Observatory (GRO), which was launched on 1991 April 5 by the space shuttle Atlantis into an Earth orbit of 450 km altitude. COMPTEL is exploring the 1-30 MeV energy range with an angular resolution (1σ) between 1° and 2° within a large field of view of about 1 steradian. Its energy resolution (8.8% FWHM at 1.27 MeV) makes it a powerful gamma-ray line spectrometer. Its effective area (for on-axis incidence) varies between 10 and 50 cm 2 depending on energy and event selections. Within a 14 day observation period COMPTEL is able to detect sources which are about 20 times weaker than the Crab. The measurement principle of COMPTEL also allows the measurements of solar neutrons

Observations of GRB 990123 by the Compton gamma ray observatory

GRB 990123 was the first burst from which simultaneous optical, X-ray, and gamma-ray emission was detected; its afterglow has been followed by an extensive set of radio, optical, and X-ray observations. We have studied the gamma-ray burst itself as observed by the Compton Gamma Ray Observatory detectors. We find that gamma-ray fluxes are not correlated with the simultaneous optical observations and that the gamma-ray spectra cannot be extrapolated simply to the optical fluxes. The burst is well fitted by the standard four-parameter GRB function, with the exception that excess emission compared with this function is observed below ~15 keV during some time intervals. The burst is characterized by the typical hard-to-soft and hardness-intensity correlation spectral evolution patterns. The energy of the peak of the νfν spectrum, Ep, reaches an unusually high value during the first intensity spike, 1470 ± 110 keV, and then falls to ~300 keV during the tail of the burst. The high-energy spectrum above ~1 MeV is consistent with a power law with a photon index of about -3. By fluence, GRB 990123 is brighter than all but 0.4% of the GRBs observed with BATSE, clearly placing it on the - power-law portion of the intensity distribution. However, the redshift measured for the afterglow is inconsistent with the Euclidean interpretation of the - power law. Using the redshift value of ≥1.61 and assuming isotropic emission, the gamma-ray energy exceeds 1054 ergs.

ANALYSIS OF ENERGY SPECTRA WITH LOW PHOTON COUNTS VIA BAYESIAN POSTERIOR SIMULATION

Over the past 10 years Bayesian methods have rapidly grown more popular in many scientific disciplines as several computationally intensive statistical algorithms have become feasible with increased computer power. In this paper we begin with a general description of the Bayesian paradigm for statistical inference and the various state-of-the-art model-fitting techniques that we employ (e.g., the Gibbs sampler and the Metropolis-Hastings algorithm). These algorithms are very flexible and can be used to fit models that account for the highly hierarchical structure inherent in the collection of high-quality spectra and thus can keep pace with the accelerating progress of new space telescope designs. The methods we develop, which will soon be available in the Chandra Interactive Analysis of Observations (CIAO) software, explicitly model photon arrivals as a Poisson process and thus have no difficulty with high-resolution low-count X-ray and γ-ray data. We expect these methods to be useful not only for the recently launched Chandra X-Ray Observatory and XMM but also for new generation telescopes such as Constellation X, GLAST, etc. In the context of two examples (quasar S5 0014+813 and hybrid-chromosphere supergiant star α TrA), we illustrate a new highly structured model and how Bayesian posterior sampling can be used to compute estimates, error bars, and credible intervals for the various model parameters. Application of our method to the high-energy tail of the ASCA spectrum of α TrA confirms that even at a quiescent state, the coronal plasma on this hybrid-chromosphere star is indeed at high temperatures (>10 MK) that normally characterize flaring plasma on the Sun. We are also able to constrain the coronal metallicity and find that although it is subject to large uncertainties, it is consistent with the photospheric measurements.

On Computing Upper Limits to Source Intensities

A common problem in astrophysics is determining how bright a source could be and still not be detected in an observation. Despite the simplicity with which the problem can be stated, the solution involves complicated statistical issues that require careful analysis. In contrast to the more familiar confidence bound, this concept has never been formally analyzed, leading to a great variety of often ad hoc solutions. Here we formulate and describe the problem in a self-consistent manner. Detection significance is usually defined by the acceptable proportion of false positives (background fluctuations that are claimed as detections, or Type I error), and we invoke the complementary concept of false negatives (real sources that go undetected, or Type II error), based on the statistical power of a test, to compute an upper limit to the detectable source intensity. To determine the minimum intensity that a source must have for it to be detected, we first define a detection threshold and then compute the probabilities of detecting sources of various intensities at the given threshold. The intensity that corresponds to the specified Type II error probability defines that minimum intensity and is identified as the upper limit. Thus, an upper limit is a characteristic of the detection procedure rather than the strength of any particular source. It should not be confused with confidence intervals or other estimates of source intensity. This is particularly important given the large number of catalogs that are being generated from increasingly sensitive surveys. We discuss, with examples, the differences between these upper limits and confidence bounds. Both measures are useful quantities that should be reported in order to extract the most science from catalogs, though they answer different statistical questions: an upper bound describes an inference range on the source intensity, while an upper limit calibrates the detection process. We provide a recipe for computing upper limits that applies to all detection algorithms.

Gamma-Ray-Burst Spectral Shapes from 2 keV to 500 MeV

We present three gamma-ray-burst spectra for bright bursts over very wide energy ranges. These were created from BATSE, COMPTEL, and OSSE data. The three spectra are for GRB 910503 (from 20 keV to 300 MeV), GRB 910601 (from 28 keV to 10.5 MeV), and GRB 910814 (from 103 keV to 500 MeV). A composite spectrum of 19 bright bursts is presented from 41 keV to 1.9 MeV (with no weak lines visible) for use in calculating average redshift corrections in cosmological models. Expanding fireball models with shocked synchrotron emission are predicted to have low-energy spectral slope (νFν ∝ να) that asymptotically approaches α = 4/3 such that α should never exceed 4/3. This prediction is tested with more than 100 bright bursts with BATSE and Ginga data. Over 90% of the bursts have spectral slopes in agreement with this prediction. For only one burst (GRB 870303, which has reported cyclotron lines) can a strong case be made that the slope violates the model limit, and then only from 2-5 keV.

AN IMAGE RESTORATION TECHNIQUE WITH ERROR ESTIMATES

Image restoration including deconvolution techniques offers a powerful tool to improve resolution in images and to extract information on the multiscale structure stored in astronomical observations. We present a new method for statistical deconvolution, which we call expectation through Markov Chain Monte Carlo (EMC2). This method is designed to remedy several shortfalls of currently used deconvolution and restoration techniques for Poisson data. We use a wavelet-like multiscale representation of the true image to achieve smoothing at all scales of resolution simultaneously, thus capturing detailed features in the image at the same time as larger scale extended features. Thus, this method smooths the image, while maintaining the ability to effectively reconstruct point sources and sharp features in the image. We use a principled, fully Bayesian model-based analysis, which produces extensive information about the uncertainty in the fitted smooth image, allowing assessment of the errors in the resulting reconstruction. Our method also includes automatic fitting of the multiscale smoothing parameters. We show several examples of application of EMC2 to both simulated data and a real astronomical X-ray source.

ACCOUNTING FOR CALIBRATION UNCERTAINTIES IN X-RAY ANALYSIS: EFFECTIVE AREAS IN SPECTRAL FITTING

While considerable advance has been made to account for statistical uncertainties in astronomical analyses, systematic instrumental uncertainties have been generally ignored. This can be crucial to a proper interpretation of analysis results because instrumental calibration uncertainty is a form of systematic uncertainty. Ignoring it can underestimate error bars and introduce bias into the fitted values of model parameters. Accounting for such uncertainties currently requires extensive case-specific simulations if using existing analysis packages. Here, we present general statistical methods that incorporate calibration uncertainties into spectral analysis of high-energy data. We first present a method based on multiple imputation that can be applied with any fitting method, but is necessarily approximate. We then describe a more exact Bayesian approach that works in conjunction with a Markov chain Monte Carlo based fitting. We explore methods for improving computational efficiency, and in particular detail a method of summarizing calibration uncertainties with a principal component analysis of samples of plausible calibration files. This method is implemented using recently codified Chandra effective area uncertainties for low-resolution spectral analysis and is verified using both simulated and actual Chandra data. Our procedure for incorporating effective area uncertainty is easily generalized to other types of calibration uncertainties.

A Maximum Entropy Map of the 511 keV Positron Annihilation Line Emission Distribution Near the Galactic Center

We have applied the maximum entropy method to a large data set based on Compton Gamma Ray Observatory (CGRO)/OSSE, Solar Maximum Mission, Transient Gamma-Ray Spectrometer, Gamma-Ray Imaging Spectrometer, HEXAGONE, and FIGARO observational results of the Galactic center 511 keV radiation in order to produce a map of the 511 keV line emission. This map suggests two components: a central bulge with a flux of 5.7 ± 0.29 × 10-4 photons cm-2 s-1 and a Galactic plane (GP) component with a flux of 2.2 ± 0.4 × 10-4 photons cm-2 s-1. The central bulge is located at l = -047 ± 024, b = 01 ± 018, and FWHM = 528 ± 048. We note that the position of the GP component coincides with a strong hot spot in the COMPTEL map of 1.8 MeV26Al line emission. A comparison between CGRO/OSSE and other instruments with larger field of view suggests an extended diffuse 511 keV emission. An interesting hot spot at l = -4°, b = 7° with a flux of ~2 × 10-4 photons cm-2 s-1 is shown on our map. A bootstrap test indicates that the significance level of this feature is 3.5 σ.

The X-Ray Characteristics of a Classical Gamma-Ray Burst and Its Afterglow

The serendipitous observation of GRB 780506 by coaligned γ-ray (HEAO 1 A-4 0.02-6 MeV) and X-ray (HEAO 1 A-2 2-60 keV) instruments during a 6 hr pointing at a blank section of the sky gave us unprecedented high signal-to-noise ratio X-ray spectra and light curves of a γ-ray burst and its afterglow. First, the spectra: although the time resolution was only 10.24 s, it was possible to derive unambiguous position constraints and a reliable instrument response. We therefore could see two breaks in the initial spectrum, one consistent with a peak in νFν of ~45 keV and one below 4 keV, consistent with strong absorption. The event exhibited dramatic spectral variability, hardening as it rose to each peak, and softening as it fell, similar to behavior reported above 100 keV from other bursts. The initial strong turnover below a few keV evolved into a slight excess as the burst progressed. The spectral shape varied widely outside low-energy limits prescribed by current relativistic shock models. At no time was there evidence for any lines, either in emission or absorption. Overall, ~20% of the total energy was radiated in nonthermal 2-30 keV X-rays. Second, the light curve: the X-ray proportional counters were able to monitor the flux for an hour before and some hours after burst onset. Two minutes after it ended, HEAO 1 A-2 detected a faint resurgence of 2-10 keV flux, rising to a peak ~7 minutes after burst onset, followed by irregular emission with a best-fit decay time of ½ hr. The position of the source of the extended flux was consistent with that of the burst and was probably not from a serendipitous transient. We estimated that the entire afterglow radiated between 3% and 30% of the >1 keV energy radiated during the burst. In this paper, we present the 2-300 keV HEAO 1 A-2 and A-4 light curves and spectra from GRB 780506 and its apparent ~1 keV afterglow.