Using Data Augmentation and Stochastic Differential Equations in Spatio Temporal Modeling

One of the biggest challenges in spatiotemporal modeling is indeed how to manage the large amount of missing information. Data augmentation techniques are frequently used to infer about missing values, unobserved or latent processes, approximation of continuous time processes that are discretely observed.

The literature treating the inference when modeling using stochastic differential equations (SDE) that are partially observed has been growing in recent years. Many attempts have been made to tackle this problem, from very different perspectives. The goal of this thesis is not a comparison of the different methods. The focus is, instead, on Bayesian inference for the SDE in a spatial context, using a data augmentation approach. While other methods can be less computationally intensive or more accurate in some cases, the main advantage of the Bayesian approach based on model augmentation is the general scope of applicability. In Chapter 2 we propose some methods to model space time data as noisy realizations of an underlying system of nonlinear SDEs. The parameters of this system are realizations of spatially correlated Gaussian processes. Models that are formulated in this fashion are complex and present several challenges in their estimation. Standard methods degenerate when the the level of refinement in the discretization gets larger. The innovation algorithm overcomes such problems. We present an extension of the innvoation scheme for the case of high-dimensional parameter spaces. Our algorithm, although presented in spatial SDE examples, can be actually applied in any general multivariate SDE setting.

In Chapter 3 we discuss additional insights regarding SDE with a spatial interpretation: spatial dependence is enforced through the driving Brownian motion.

In Chapter 4 we discuss some possible refinement on the SDE parameter estimation. Such refinements, that involve second order SDE approximations, have actually a more general scope than spatiotemporal modeling and can be applied in a variety of settings.

In the last chapter we propose some methodology ideas for fitting space-time models to data that are collected in a wireless sensor network when suppression and failure in transmission are considered. In this case also we make use of data augmentation techniques but in conjunction with linear constraints on the missing values.