Multilevel Bayesian framework for modeling the production, propagation and detection of ultra-high energy cosmic rays

Ultra-high energy cosmic rays (UHECRs) are atomic nuclei with energies over ten million times energies accessible to human-made particle accelerators. Evidence suggests that they originate from relatively nearby extragalactic sources, but the nature of the sources is unknown. We develop a multilevel Bayesian framework for assessing association of UHECRs and candidate source populations, and Markov chain Monte Carlo algorithms for estimating model parameters and comparing models by computing, via Chib's method, marginal likelihoods and Bayes factors. We demonstrate the framework by analyzing measurements of 69 UHECRs observed by the Pierre Auger Observatory (PAO) from 2004-2009, using a volume-complete catalog of 17 local active galactic nuclei (AGN) out to 15 megaparsecs as candidate sources. An early portion of the data ("period 1," with 14 events) was used by PAO to set an energy cut maximizing the anisotropy in period 1; the 69 measurements include this "tuned" subset, and subsequent "untuned" events with energies above the same cutoff. Also, measurement errors are approximately summarized. These factors are problematic for independent analyses of PAO data. Within the context of "standard candle" source models (i.e., with a common isotropic emission rate), and considering only the 55 untuned events, there is no significant evidence favoring association of UHECRs with local AGN vs. an isotropic background. The highest-probability associations are with the two nearest, adjacent AGN, Centaurus A and NGC 4945. If the association model is adopted, the fraction of UHECRs that may be associated is likely nonzero but is well below 50%. Our framework enables estimation of the angular scale for deflection of cosmic rays by cosmic magnetic fields; relatively modest scales of $\approx\!3^{\circ}$ to $30^{\circ}$ are favored. Models that assign a large fraction of UHECRs to a single nearby source (e.g., Centaurus A) are ruled out unless very large deflection scales are specified a priori, and even then they are disfavored. However, including the period 1 data alters the conclusions significantly, and a simulation study supports the idea that the period 1 data are anomalous, presumably due to the tuning. Accurate and optimal analysis of future data will likely require more complete disclosure of the data.Comment: Published in at http://dx.doi.org/10.1214/13-AOAS654 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

On The Modeling Of Multiple Functional Outcomes With Spatially Heterogeneous Shape Characteristics

This dissertation presents an approach for analyzing functional data with multiple outcomes that exhibits spatially heterogeneous shape characteristics. An example of data of this type that motivated this study is a data from a diffusion tensor imaging (DTI) study of neuronal tract in multiple sclerosis (MS) patients. DTI is an imaging technique for measuring the diffusion of water that can be used to detect abnormalities in brain tissue. DTI tractography can be summarized by 3 functional outcomes, measuring the diffusion in different directions. One of the main and most common difficulties in functional data analysis is the large number of parameters to be estimated. This is especially challenging when multiple functional outcomes are considered. To accommodate this problem, a copula approach is adopted so that the marginal distribution and the dependence structure are estimated independently. In addition to fast computation, the two-step approach also allows flexibility in the specification of the distribution of the data as the marginal distribution and copula distribution can be specified separately. The first part of this dissertation presents an estimation algorithm using the copula approach. The marginal distribution parameters are estimated using methodology based on maximum likelihood and penalized splines. In the estimation for the dependence structure, the Karhunen-Loeve expansion and an EM algorithm are used to significantly reduce the dimension of the problem. This allows the dependence within the same outcome and across different outcomes to be captured even in the case of many functional outcomes. The second part of this dissertation demonstrates the application of the methodology to the DTI study. The goal is to identify the locations where the abnormalities occur and also explain the characteristics of the abnormalities in MS patients. The difference in the marginal distributions and structure dependence in the MS group from the healthy control group is then used to develop a method for predicting case status for patients. The last part of the dissertation explores the DTI study in longitudinal setting. A larger dataset that contains DTI data from multiple visits is studied. We adopted a multilevel approach to investigate how the DTI tractography in MS patients varies over time