Random Processes: Data Analysis and Theory

随机过程数据分析已成为当代科学的一个重要主题。受到野生生物和林业问题的启发，时空点过程数据和平面上曲线轨迹的统计处理技术以及随机过程数据的统计方法得到了发展。特别是概率学家已经发现了关于向量值随机微分方程的各种理论结果，但它们的统计和数据分析性质还没有完全发展起来。采用随机微分方程来描述保护区动物的运动轨迹。这些方程的不同之处在于包含了解释的路径，有时还有时间滞后。动物的运动受到高栅栏的限制这一事实也影响了分析。在工作中，需要对随机模型进行近似。本文将研究大样本近似的精度。对野火数据采用了点处理、平滑和时间序列分析等工具。随着火灾季节的进行和未来几年，未来风险和可能损失的预测器正在开发中。所讨论的两个具体问题具有广泛的影响和社会重要性。问题是野火的风险评估，以及人类在动物保护区活动对动物行为的影响的调查。在后一种情况下的问题是，人类在多大程度上可以利用野生动物和人类共享栖息地。从设计好的实验中可以获得数据，这些实验包括人类在保护区步行、骑马、骑自行车和乘坐全地形车辆。这两个问题都是使用来自美国农业部林业局的数据集来研究的。这项工作具有特别广泛的影响，因为有可能减少火灾对人类的威胁，并保持自然资源的价值。对风险作为时间和空间的函数进行预测，将有助于有效地配置消防资源。这项工作的智力价值包括，在这些应用的推动下，出现了各种有趣的分析问题，这些问题将在研究中得到解决。数据集很大，因此预计在理解这些方法的适用性方面将取得真正的进展。

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.