Random Processes: Data Analysis and Theory

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Random process data analysis has become a major theme of contemporary science. Motivated by problems from wildlife biology and forestry, techniques for the statistical handling of spatial-temporal point process data and of curved trajectories in the plane are developed as well as statistical methods for random process data generally. In particular probabilists have discovered various theoretical results concerning vector-valued stochastic differential equations, but their statistical and data analytic properties have not been totally developed. Stochastic differential equations are employed to describe the observed trajectories of the animals in the reserve. The equations are unusual in having paths of explanatories included and sometimes time lags. The fact that an animal's motion is bounded by a high fence also affects the analysis. In the work approximations to the random model are needed. The large sample accuracy of the approximations will be studied. The tools of point processes, smoothing and time series analysis are being employed with the wildfire data. Predictors of future risk and possible loss as the fire season proceeds and for future years are being developed.Two specific problems addressed are of broad impact and of societal importance. The problems are risk estimation for wildfires and the investigation of the effects that humans moving through an animal reserve have on the behavior of the animals. A question in the latter case is with what level of human usage can wild animals and humans share a habitat. There are data available from designed experiments with humans walking, riding horses, bicycling and on all-terrain vehicles traveling in the reserve. Both problems are studied using data sets come from the Forest Service, U. S. Department of Agriculture. This work is of particularly broad impact for there are the possibility of reducing human threat from fire, and of maintaining natural resource values. Predictions of the risk as a function of time and space will allow efficient placement of fire fighting resources.The intellectual merit of the work includes that quite a variety of interesting analytic problems arise, motivated by these applications, and these problems will be addressed in the research. The data sets are large so it is anticipated that real progress will be made on understanding the applicability of the methods.