Spatial-temporal nonlinear filtering based on hierarchical statistical models

A hierarchical statistical model is made up generically of a data model, a process model, and occasionally a prior model for all the unknown parameters. The process model, known as the state equations in the filtering literature, is where most of the scientist’s physical/chemical/biological knowledge about the problem is used. In the case of a dynamically changing configuration of objects moving through a spatial domain of interest, that knowledge is summarized through equations of motion with random perturbations. In this paper, our interest is in dynamically filtering noisy observations on these objects, where the state equations are nonlinear. Two recent methods of filtering, the Unscented Particle filter (UPF) and the Unscented Kalman filter, are presented and compared to the better known Extended Kalman filter. Other sources of nonlinearity arise when we wish to estimate nonlinear functions of the objects positions; it is here where the UPF shows its superiority, since optimal estimates and associated variances are straightforward to obtain. The longer computing time needed for the UPF is often not a big issue, with the ever faster processors that are available. This paper is a review of spatial-temporal nonlinear filtering, and we illustrate it in a Command and Control setting where the objects are highly mobile weapons, and the nonlinear function of object locations is a two-dimensional surface known as the danger-potential field.