Hierarchical Bayesian Random Sets with Applications to Growth Models

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

Knowing if a medical treatment is effective or anticipating the path of a severe storm is of great importance to the public in terms of potential loss of life and property. Such experiments or phenomena can be described by defining appropriate random sets, which can be done is several ways. A "hitting function" approach involves calculation of the probabilities that the set intersects a given class of test sets, e.g., discs. The major difficulty is that a likelihood cannot be specified in a simple way in order to provide inference on model parameters, while the choice of test set can potentially alter the parameter estimates. In addition, random sets can often be viewed as marked point processes (MPP), where the marks define the characteristics of a well known geometric object, e.g., the radius of a disc centered at an event from the point process. However elegant this view might be, it does not eliminate the aforementioned problems in obtaining a likelihood for the random set and conducting a statistical analysis. In this proposal, the investigator considers an approach that models a point in the random set and not the entire set directly. In this way, random sets are viewed as if they were created by an underlying process that helps realize the observed data, and thus the observed data are modeled hierarchically, given the underlying random process. The hierarchy can be described as [Data|Process]x[Process|Parameters]x[Parameters]. Thus, an alternative approach, such as a MMP, serves as the second stage in this hierarchical formulation. This approach is more general with its major advantage being that it easily facilitates inference on model parameters. The investigator studies several models for random sets based on the general setting of a Boolean model, as well as, models to capture the evolution of the random object over time. The inclusion of relevant covariate information is also considered. The investigator efficiently models a random set via a multistage hierarchical Bayesian framework. Several novel growth models are proposed, as well as the corresponding Bayesian formulation that provides inference and prediction. The models are applied to modeling forestry (random tree objects) as well as storm cell development as obtained from weather radar over time (storm cell evolution). Every experiment or phenomenon can be described by defining appropriate objects that describe its main characteristics. Thus, understanding the characteristics of an object and its evolution is needed in almost every scientific discipline. From observing tumor growth to assess the effectiveness of a medical treatment, studying the movement of storm or tornado systems based on radar images, to the investigation of an epidemic as it spreads through a populated area, defining and studying growth models has become an increasingly critical research topic. Such research is of great importance to the public in terms of potential loss of life and property, since knowing, for example, the path of a tornado or a hurricane can then be used to provide ample warning to the population of the area, in an effort to minimize the effects of these disasters. The proposed methodology not only helps to better understand the evolution of objects in the four dimensional world we live in, but also provides a measure of uncertainty in the forecasts, whereas, traditional deterministic models require perfect knowledge of the phenomenon of interest, which is typically an unrealistic assumption, and thus are often inadequate to describe the phenomenon. The investigator's major concern is the development and study of statistical models that capture the evolution of such random objects.