| The first part of this thesis is focused on the measure of spatio-temporal variation of pathogen variants, and the analysis of a set of incomplete multinomial data. Data on pathogen variants often suffer from some data problems including (i) low cell counts, (ii) incomplete classification, and (iii) unseen variants. For incomplete multinomial data with disjoint sources of missingness, we show that the posterior distribution admits an independent sampling scheme. Consequently we can derive closed-form solutions for the posterior distribution of the number of categories and the Bayes factor for assessing spatial heterogeneity. We extend this idea to develop an efficient block Gibbs sampler for general incomplete multinomial data having possibly overlapping sources of missingness.;The second part of the thesis is focused on nonlinear state-space modeling for estimating the SIR model as the first step to quantifying cross-immunity. By discretizing the differential equation defining the SIR model with variable birth rate via the Runge-Kutta method, we develop a new nonlinear state-space model. In principle, the method of conditional least squares (CLS) may be applied to estimate a nonlinear state-space model. However, the method of CLS requires computing the 1-step ahead predictor which is generally intractable for a nonlinear state-space model. We study a new algorithm known as the Unscented Kalman filter (UKF) that was earlier proposed in the engineering literature for approximately computing the 1-step ahead predictor and its prediction variance. We study the large sample properties of the approximate CLS estimator obtained by replacing the 1-step ahead predictors in the objective function by their UKF approximations. We derive some results characterizing the error rate of the UKF approximation. These new results suggest that under suitable regularity conditions, the approximate CLS method via UKF enjoys the same asymptotic properties of the exact CLS estimator. The theoretical results are illustrated by some simulation studies and a real application of fitting an SIR model to a large-scale epidemiological monitoring dataset. |
