Stochastic gradient systems: inference and applications

有关物体运动的数据在科学、工程和一般人类经验的许多领域中都很常见。自牛顿时代以来，机械运动一直用常微分方程式进行解析描述。确定性微分方程组的理论和应用已经发展到对随机微分方程组的相应研究。在各种实际情况下，如何选择SDE的漂移函数一直不清楚。这项研究将研究使用势函数来提供漂移的公式。具体地说，将假定感兴趣的运动受势函数支配。那么漂移就是势函数的梯度。这种结构被称为随机梯度系统。作为实值函数，势函数比矢量值漂移函数更容易建模。估计势函数可用于简单的描述、总结、比较、模拟、预测、模型评估、自举，并用于估计感兴趣的量。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括时间历史。要研究的具体分析问题包括：观察时间间隔不相等，开发模拟方法来显示可变性，允许进行模型评估和预测。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，将对夏威夷僧海豹GPS数据进行建模，并解决有关海洋生物学家提出的问题。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。目前只剩下约1300人。对北象海豹迁徙路线的研究也将被模型化。这种动物是受1972年《海洋哺乳动物法案》保护的物种。接下来，俄勒冈州斯塔基实验保护区的动物路径将被包括在工作中。收集这些数据是为了研究野生动物、牛和人共享的栖息地的管理。最后，关于城市野火的风险模型的工作将继续进行--野火界面将利用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.