The analysis of big data is of great interest today, and this comes with challenges of improving precision and efficiency in estimation and prediction. We study binary data with covariates from numerous small areas, where direct estimation is not reliable, and there is a need to borrow strength from the ensemble. This is generally done using Bayesian logistic regression, but because there are numerous small areas, the exact computation for the logistic regression model becomes challenging. Therefore, we develop an integrated multivariate normal approximation (IMNA) method for binary data with covariates within the Bayesian paradigm, and this procedure is assisted by the empirical logistic transform. Our main goal is to provide the theory of IMNA and to show that it is many times faster than the exact logistic regression method with almost the same accuracy. We apply the IMNA method to the health status binary data (excellent health or otherwise) from the Nepal Living Standards Survey with more than 60,000 households (small areas). We estimate the proportion of Nepalese in excellent health condition for each household. For these data IMNA gives estimates of the household proportions as precise as those from the logistic regression model and it is more than fifty times faster (20 seconds versus 1,066 seconds), and clearly this gain is transferable to bigger data problems.

Mapping of mortality rates has been a valuable public health tool. We describe novel Bayesian methods for constructing maps which do not depend on a post stratification of the estimated rates. We also construct posterior modal maps rather than posterior mean maps. Our methods are illustrated using mortality data from chronic obstructive pulmonary diseases (COPD) in the continental United States. Poisson regression models have attracted much attention in the scientific community for their superiority in modeling rare events (including mortality counts from COPD). Christiansen and Morris (JASA 1997) described a hierarchical Bayesian model for heterogeneous Poisson counts under the exchangeability assumption. We extend this model to include latent classes (groups of similar Poisson rates unknown to an investigator). Also, it is standard practice to construct maps using quantiles (e.g., quintiles) of the estimated mortality rates. For example, based on quintiles, the mortality rates are cut into 5 equal size groups, each containing $20\%$ of the data, and a different color is applied to each of them on the map. A potential problem is that, this method assumes an equal number of data in each group, but this is often not the case. The latent class model produces a method to construct maps without using quantiles, providing a more natural representation of the colors. Typically, for rare events, the posterior densities of the rates are skewed, making the posterior mean map inappropriate and inaccurate. Thus, although it is standard practice to present the posterior mean maps, we also develop a method to provide the joint posterior modal map (i.e., the map with the highest posterior probability over the ensemble). For the COPD data, collected 1988-1992 over 798 health service areas, we use Markov chain Monte Carlo methods to fit the model, and an output analysis is used to construct the new maps.

In high range-resolution (HRR) radar systems, the returns from a single target may fall in multiple adjacent range bins which individually vary in amplitude. A target following this representation is commonly referred to as an extended target and results in more information about the target. However, extracting this information from the radar returns is challenging due to several complexities. These complexities include the single dimensional nature of the radar measurements, complexities associated with the scattering of electromagnetic waves, and complex environments in which radar systems are required to operate. There are several applications of HRR radar systems which extract target information with varying levels of success. A commonly used application is that of imaging referred to as synthetic aperture radar (SAR) and inverse SAR (ISAR) imaging. These techniques combine multiple single dimension measurements in order to obtain a single two dimensional image. These techniques rely on rotational motion between the target and the radar occurring during the collection of the single dimension measurements. In the case of ISAR, the radar is stationary while motion is induced by the target. There are several difficulties associated with the unknown motion of the target when standard Doppler processing techniques are used to synthesise ISAR images. In this dissertation, a non-standard Dop-pler approach, based on Bayesian inference techniques, was considered to address the difficulties. The target and observations were modelled with a non-linear state space model. Several different Bayesian techniques were implemented to infer the hidden states of the model, which coincide with the unknown characteristics of the target. A simulation platform was designed in order to analyse the performance of the implemented techniques. The implemented techniques were capable of successfully tracking a randomly generated target in a controlled environment. The influence of varying several parameters, related to the characteristics of the target and the implemented techniques, was explored. Finally, a comparison was made between standard Doppler processing and the Bayesian methods proposed. / Dissertation (MEng)--University of Pretoria, 2013. / gm2014 / Electrical, Electronic and Computer Engineering / unrestricted