Hierarchical models and spatio-temporal processes in data analysis

Abstract

Spatio-temporal processes evolve in space and time. As a consequence data from such processes, usually possess both temporal and spatial dependence which have to be reflected by fitted stochastic models. The existing models may fail to provide a proper fit to many Spatio-temporal data, especially when the dataset possesses features that contradict the assumption of the normality. In this thesis, we present the extension of the existing hierarchical models that are capable of capturing the complicated structures in both large-dimensional and small datasets. A non-linear general Spatio-temporal model is developed by extending Serfling's model. The model is used in the derivation of the locally weighted scatterplot smoothing (LOESS) predictor for both the spatial interpolation and the temporal prediction of spatio-temporal processes. Theoretical findings are applied to real data of an outbreak of influenza in Southern Germany. The study shows that, the LOESS outperforms the classical linear predictor in terms of providing smaller values for the performance measure computed by cross-validation. We then, extend the random effects model, by introducing the correlation coefficient between random effects in its definition. Bayesian inference procedures for the model parameters are obtained when objective priors, are employed for the model parameters. We consider the problem of Bayesian estimation of heterogeneity parameter in the generalized random effects model and compare the obtained theoretical results with the existing approaches from frequentists and Bayesian statistics. The findings are used for consensus building in meta-analyses of measurement results for the Newtonian constant of gravitation: We, therefore, recommend, the use of our new models, to overcome the dependence problem in data arising from spatio-temporal processes and data with small observations.