Date of Award
Engineering and Applied Science
Li, X. Rong
Markov processes have been widely studied and used for modeling problems. A Markov process has two main components (i.e., an evolution law and an initial distribution). Markov processes are not suitable for modeling some problems, for example, the problem of predicting a trajectory with a known destination. Such a problem has three main components: an origin, an evolution law, and a destination. The conditionally Markov (CM) process is a powerful mathematical tool for generalizing the Markov process. One class of CM processes, called $CM_L$, fits the above components of trajectories with a destination. The CM process combines the Markov property and conditioning. The CM process has various classes that are more general and powerful than the Markov process, are useful for modeling various problems, and possess many Markov-like attractive properties.
Reciprocal processes were introduced in connection to a problem in quantum mechanics and have been studied for years. But the existing viewpoint for studying reciprocal processes is not revealing and may lead to complicated results which are not necessarily easy to apply.
We define and study various classes of Gaussian CM sequences, obtain their models and characterizations, study their relationships, demonstrate their applications, and provide general guidelines for applying Gaussian CM sequences. We develop various results about Gaussian CM sequences to provide a foundation and tools for general application of Gaussian CM sequences including trajectory modeling and prediction.
We initiate the CM viewpoint to study reciprocal processes, demonstrate its significance, obtain simple and easy to apply results for Gaussian reciprocal sequences, and recommend studying reciprocal processes from the CM viewpoint. For example, we present a relationship between CM and reciprocal processes that provides a foundation for studying reciprocal processes from the CM viewpoint. Then, we obtain a model for nonsingular Gaussian reciprocal sequences with white dynamic noise, which is easy to apply. Also, this model is extended to the case of singular sequences and its application is demonstrated. A model for singular sequences has not been possible for years based on the existing viewpoint for studying reciprocal processes. This demonstrates the significance of studying reciprocal processes from the CM viewpoint.
Rezaie, Reza, "Gaussian Conditionally Markov Sequences: Theory with Application" (2019). University of New Orleans Theses and Dissertations. 2679.