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。受野生动物生物学和林业问题的启发，发展了时空点过程数据和平面曲线轨迹的统计处理技术，以及一般的随机过程数据的统计方法。特别是概率学家们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。用随机微分方程来描述保护区内动物的观测轨迹。这些方程的不同寻常之处在于包含了解释的路径，有时还会出现时间滞后。动物的运动受到高高的栅栏的限制，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。我们将研究这些近似的大样本精度。使用点过程、平滑和时间序列分析工具对野火数据进行处理。随着火灾季节的推进和未来几年的发展，正在制定关于未来风险和可能损失的预测指标。这些问题是对野火的风险估计，以及调查人类通过动物保护区对动物行为的影响。后一种情况下的一个问题是，野生动物和人类可以共享一个栖息地，人类的使用率达到什么水平。这些数据来自于在保护区内步行、骑马、骑自行车和全地形车辆的设计实验。这两个问题都是使用来自美国农业部林业局的数据集进行研究的。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。对风险随时间和空间变化的预测将允许有效地配置消防资源。这项工作的智力价值包括，在这些应用程序的激励下，产生了相当多有趣的分析问题，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.