State-space models have proven invaluable in the analysis of dynamic data, specifically time series data. They provide a natural and interpretable framework to learn about and describe dynamic processes. State-space models also provide a flexible framework for embedding prescriptive, mathematical models in ways that account for multiple sources of uncertainty. When considered within the more general directed graphical model formalism, state-space models can be reimagined and extended into arenas beyond the reach of traditional state-space models. In three papers, we consider various applications of and extensions to state-space models. All papers stem from collaborations with Los Alamos National Laboratory.

In the field of space weather forecasting, many modeling approaches have been developed in the last 25 years. These approaches attempt to make sense of the dynamic and not-well-understood relationships between electron flux intensities and relevant covariates. Many of these forecasting models possess inherent limitations because they are static in nature and thus are constrained to customized and narrow time windows. In Chapter 2, we discuss these limitations and present an alternate approach to space weather forecasting utilizing dynamic linear models (DLMs). Benefits of dynamic modeling when compared to static modeling are discussed and ground work is laid for future dynamic forecasting endeavors in space weather. This work was published in the journal Space Weather under research article number 10.1002/2014SW001057 in June 2014.

Multiscale modeling involves decomposing or explicitly modeling processes that arise at multiple scales. In Chapter 3, we extend the DLM into the multiscale arena with the presentation of the multiscale dynamic linear model (MSDLM). We present the MSDLM within the directed graphical model formalism. In so doing, we provide the necessary background to consider multiscale modeling generally. The MSDLM is a multiscale time series model that is interpretable, flexible, and coherently combines multiple scales of information in a principled, unified framework. Estimation and sampling procedures are presented. We illustrate the efficacy of the MSDLM by revisiting the problem of space weather forecasting discussed in Chapter 2.

Forecasting seasonal influenza in the U.S. is challenging and consequential. It is challenging because there is uncertainty in the form of the disease transmission process, the process is only partially observed, and those observations are noisy. It is consequential because influenza poses serious risks to both national security and public health. In Chapter 4, we propose a non-Gaussian, nonlinear state-space model that embeds a compartmental model (i.e., a set of nonlinear, ordinary differential equations) into the state equations. The state-space framework provides valuable flexibility to the deterministic disease transmission process while simultaneously allowing and accounting for uncertainties in the parameters, the process, and the measurement mechanism. Prior specification is discussed in detail. Forecasting metrics are proposed and compared with competing models.