Geometry of Goodness-of-Fit Testing in High Dimensional Low Sample Size Modelling

We introduce a new approach to goodness-of-fit testing in the high dimensional, sparse extended multinomial context. The paper takes a computational information geometric approach, extending classical higher order asymptotic theory. We show why the Wald – equivalently, the Pearson X2 and score statistics – are unworkable in this context, but that the deviance has a simple, accurate and tractable sampling distribution even for moderate sample sizes. Issues of uniformity of asymptotic approximations across model space are discussed. A variety of important applications and extensions are noted

Recent advances in directional statistics

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed.Comment: 61 page

The Information Geometry of Sparse Goodness-of-Fit Testing

This paper takes an information-geometric approach to the challenging issue of goodness-of-fit testing in the high dimensional, low sample size context where—potentially—boundary effects dominate. The main contributions of this paper are threefold: first, we present and prove two new theorems on the behaviour of commonly used test statistics in this context; second, we investigate—in the novel environment of the extended multinomial model—the links between information geometry-based divergences and standard goodness-of-fit statistics, allowing us to formalise relationships which have been missing in the literature; finally, we use simulation studies to validate and illustrate our theoretical results and to explore currently open research questions about the way that discretisation effects can dominate sampling distributions near the boundary. Novelly accommodating these discretisation effects contrasts sharply with the essentially continuous approach of skewness and other corrections flowing from standard higher-order asymptotic analysis

X-ray spectral modelling of the AGN obscuring region in the CDFS: Bayesian model selection and catalogue

AGN are known to have complex X-ray spectra that depend on both the properties of the accreting SMBH (e.g. mass, accretion rate) and the distribution of obscuring material in its vicinity ("torus"). Often however, simple and even unphysical models are adopted to represent the X-ray spectra of AGN. In the case of blank field surveys in particular, this should have an impact on e.g. the determination of the AGN luminosity function, the inferred accretion history of the Universe and also on our understanding of the relation between AGN and their host galaxies. We develop a Bayesian framework for model comparison and parameter estimation of X-ray spectra. We take into account uncertainties associated with X-ray data and photometric redshifts. We also demonstrate how Bayesian model comparison can be used to select among ten different physically motivated X-ray spectral models the one that provides a better representation of the observations. Despite the use of low-count spectra, our methodology is able to draw strong inferences on the geometry of the torus. For a sample of 350 AGN in the 4 Ms Chandra Deep Field South field, our analysis identifies four components needed to represent the diversity of the observed X-ray spectra: (abridged). Simpler models are ruled out with decisive evidence in favour of a geometrically extended structure with significant Compton scattering. Regarding the geometry of the obscurer, there is strong evidence against both a completely closed or entirely open toroidal geometry, in favour of an intermediate case. The additional Compton reflection required by data over that predicted by toroidal geometry models, may be a sign of a density gradient in the torus or reflection off the accretion disk. Finally, we release a catalogue with estimated parameters such as the accretion luminosity in the 2-10 keV band and the column density, $N_{H}$, of the obscurer.Comment: 28 pages, 18 figures, catalogue available from https://www.mpe.mpg.de/~jbuchner/agn_torus/analysis/cdfs4Ms_cat/, software available from https://github.com/JohannesBuchner/BX

Computational statistics using the Bayesian Inference Engine

This paper introduces the Bayesian Inference Engine (BIE), a general parallel, optimised software package for parameter inference and model selection. This package is motivated by the analysis needs of modern astronomical surveys and the need to organise and reuse expensive derived data. The BIE is the first platform for computational statistics designed explicitly to enable Bayesian update and model comparison for astronomical problems. Bayesian update is based on the representation of high-dimensional posterior distributions using metric-ball-tree based kernel density estimation. Among its algorithmic offerings, the BIE emphasises hybrid tempered MCMC schemes that robustly sample multimodal posterior distributions in high-dimensional parameter spaces. Moreover, the BIE is implements a full persistence or serialisation system that stores the full byte-level image of the running inference and previously characterised posterior distributions for later use. Two new algorithms to compute the marginal likelihood from the posterior distribution, developed for and implemented in the BIE, enable model comparison for complex models and data sets. Finally, the BIE was designed to be a collaborative platform for applying Bayesian methodology to astronomy. It includes an extensible object-oriented and easily extended framework that implements every aspect of the Bayesian inference. By providing a variety of statistical algorithms for all phases of the inference problem, a scientist may explore a variety of approaches with a single model and data implementation. Additional technical details and download details are available from http://www.astro.umass.edu/bie. The BIE is distributed under the GNU GPL.Comment: Resubmitted version. Additional technical details and download details are available from http://www.astro.umass.edu/bie. The BIE is distributed under the GNU GP

Simulation of ultrasonic lamb wave generation, propagation and detection for an air coupled robotic scanner

A computer simulator, to facilitate the design and assessment of a reconfigurable, air-coupled ultrasonic scanner is described and evaluated. The specific scanning system comprises a team of remote sensing agents, in the form of miniature robotic platforms that can reposition non-contact Lamb wave transducers over a plate type of structure, for the purpose of non-destructive evaluation (NDE). The overall objective is to implement reconfigurable array scanning, where transmission and reception are facilitated by different sensing agents which can be organised in a variety of pulse-echo and pitch-catch configurations, with guided waves used to generate data in the form of 2-D and 3-D images. The ability to reconfigure the scanner adaptively requires an understanding of the ultrasonic wave generation, its propagation and interaction with potential defects and boundaries. Transducer behaviour has been simulated using a linear systems approximation, with wave propagation in the structure modelled using the local interaction simulation approach (LISA). Integration of the linear systems and LISA approaches are validated for use in Lamb wave scanning by comparison with both analytic techniques and more computationally intensive commercial finite element/difference codes. Starting with fundamental dispersion data, the paper goes on to describe the simulation of wave propagation and the subsequent interaction with artificial defects and plate boundaries, before presenting a theoretical image obtained from a team of sensing agents based on the current generation of sensors and instrumentation