Modeling and analyzing phenomena, particularly interactions amongst moving particles

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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.