Modelling spatial trajectories
The study of trajectories has been basic to science for many centuries. One can mention the motion of the planets, the meanderings of animals and the routes of ships. More recently there has been considerable modeling and statistical analysis of biological and ecological processes of moving particles. The models may be motivated formally by difference and dif- ferential equations and by potential functions. Initially, following Liebnitz and Newton, such models were described by deterministic differential equations, but variability around observed paths has led to the introduction of random variables and to the development of stochastic calculi. The results obtained from the fitting of such models are highly useful. They may be employed for: simple description, summary, comparison, simulation, prediction, model appraisal, bootstrapping, and also employed for the estimation of derived quantities of inter- est. The potential function approach, to be presented in Section 1.3.4, will be found to have the advantage that an equation of motion is set down quite directly and that explanatories, including attractors, repellors, and time-varying fields may be included conveniently.