Modeling and analyzing phenomena, particularly interactions amongst moving particles

考虑一个运动物体，其位置在已知时间是已知的。这是项目中要研究的数据/过程。它将被称为轨迹数据。将假设固有的可变性，从而使现象成为随机的。基本过程将假定满足随机微分方程(SDE)或泛函随机微分方程(FSDE)。在前一种情况下，假设物体在势场中运动，其梯度给出了SDE的漂移项，这将是方便的。当数据可用时，将为数据设置近似似然函数。这允许调用诸如估计、模拟和测试等推理过程。模型可以是参数模型，也可以是非参数模型。可能涉及一个或多个对象或粒子。在后一种情况下，可能会有相互作用。在存在解释变量的情况下，纳入解释变量可提高其适合性和可预测性。将开发该方法的实用和理论特性。如今，轨迹数据出现在许多地方，特别是自从全球定位系统(GPS)出现以来。在连续的时间估计集合或仅一个对象的个体的位置。时间可能是均匀分布的，也可能不是。它们对于不同的对象可能是不同的。这项工作的一个目标是为物体的轨迹建立数学和统计模型。这些模型可以用来检验有关运动的科学假设，也可以用来发现新奇事物。建模中可能会包含一些解释性变量，如水深测量。未来的走势可能只取决于最近的位置，也可能取决于更多过去的价值。模型需要反映这一点，并允许对假设进行检验。来自动物生物学、海洋生物学、天文学、生物物理学等领域的数据将被研究，以及不等间距数据的含义。

Consider a moving object whose locations are known at known times. This is the data/process to be studied in the project. It will be called trajectory data. Inherent variability will be assumed making the phenomenon random. The basic process will be assumed to satisfy a Stochastic Differential Equation (SDE) or a Functional Stochastic Differential Equation (FSDE). In the former case it will be convenient to assume that the object moves in a potential field whose gradient gives the drift term of the SDE. When data are available an approximate likelihood function will be set down for the data. This allows inference procedures such as estimation, simulation and testing to be invoked. The models may be parametric or nonparametric. One or several objects, or particles, may be involved. In the latter case there may be interactions. In the case that explanatory variables are present both fit and predictability may be improved by their inclusion. Practical and theoretical properties of the approach will be developed. Trajectory data appear in many places these days, particularly since the Global Positioning System (GPS) appeared. Locations of individuals of a collection or of just one object are estimated at a succession of times. The times may be equally spaced or not. They may be different for different objects. One object of this work is to develop mathematical and statistical models for the tracks of the objects. These models may be used to examine scientific hypotheses concerning movements as well as to discover novelties. There may be explanatory variables, such as bathymetry, to include in the modeling. The future movement may depend only on the most recent position or it may depend on more past values. The models need to reflect this and allow examination of the assumption. Data from fields including animal biology, marine biology, astronomy, biophysics will be studied as will the implications of unequally spaced data.