Stochastic gradient systems: inference and applications

关于物体运动的数据在科学、工程和一般人类经验的许多领域已经变得很普遍。从牛顿时代起，机械运动就用微分方程解析地描述了。确定性微分方程的理论和应用已经发展到相应的随机微分方程的研究。在各种实际情况下，如何选择SDE的漂移函数一直是不清楚的。本研究将探讨使用势函数来提供漂移的公式。特别地，我们假定感兴趣的运动是由一个势函数控制的。漂移就是势函数的梯度。这种结构被称为随机梯度系统。作为实值的势函数比向量值的漂移函数更容易建模。估计的势函数可用于简单的描述、总结、比较、模拟、预测、模型评价、自举，并用于估计感兴趣的数量。这项工作将包括研究基于泛函随机微分方程的模型。这将允许在描述中包含时间历史。要研究的具体分析问题包括：观察时间间隔不均匀，开发模拟方法以显示变异性，允许模型评估和做出预测。在粒子运动研究中所研究的理论结构将应用于各种生物、生态和其他运动情况。特别是夏威夷僧海豹的GPS数据将被建模，有关海洋生物学家提出的问题将得到解决。这些数据很重要，因为僧海豹是美国最濒危的海洋哺乳动物。只剩下1300个。对象海豹迁徙路线的研究也将进行建模。这种动物是受1972年海洋哺乳动物法案保护的物种。继续，俄勒冈州斯塔基实验保护区的动物路径将包括在这项工作中。收集这些数据是为了研究野生动物、奶牛和人类共享的栖息地的管理。最后，将继续开展有关城市-野火界面野火风险模型的工作，将采用2003年圣地亚哥县火灾的数据进行建模。

Data on the motion of objects has become common in many fields of science, engineering, and general human experience. Since the time of Newton mechanical motion has been described analytically by differential equations. Deterministic differential equations theory and application have developed into corresponding study of stochastic differential equations (SDEs). In a variety of practical situations it has been unclear how to select the drift function of an SDE. This research will investigate the use of a potential function to provide a formula for drift. In particular it will be assumed that the motion of interest is governed by a potential function. The drift is then the potential function's gradient. This structure is referred to as a stochastic gradient system. Being real-valued a potential function is easier to model than a vector-valued drift function. An estimated potential function may be used for simple description, summary, comparison, simulation, prediction, model appraisal, bootstrapping, and employed for estimating quantities of interest. The work will include study of models based on functional stochastic differential equations. This will allow the inclusion of time history in the description. Specific analytic problems to be studied include: unequally spaced times of observation, development of simulation methods to display variability, allowing model appraisal and making predictions. The theoretical structure investigated in the particle motion research will be applied to a variety of biological, ecological and other motion situations. In particular Hawaiian monk seal GPS data will be modeled and questions asked by the concerned marine biologists addressed. These data are important because the monk seal is America's most endangered marine mammal. Only about 1300 remain. The study of migration routes of northern elephant seal will also be modeled. This animal is a protected species under the Marine Mammal Act of 1972. Continuing, paths of animals in the Starkey Experimental Reserve in Oregon will be included in the work. These data have been collected to study the management of habitats shared by wild animals, cows and people. Lastly, work will continue on risk models concerning wildfires at the urban-wildfire interface will be modeled employing data from the San Diego County fires of 2003.