Particle Filters for High Dimensional Spatial Systems

Abstract

The objective of this work is to develop new filtering methodologies that allow state-space models to be applied to high dimensional spatial systems with fewer and less restrictive assumptions than the currently practical methods. Reducing the assumptions increases the range of systems that the state-space framework can be applied to and therefore the range of systems for which the uncertainty in estimates can be quantified and statements about the risk of particular outcomes made. The particle filter was developed to meet this objective because restrictive assumptions are fundamental to the alternative methods. Two barriers to applying particle filters to high dimension spatial systems were identified. The first barrier is the lack of a flexible and practically applicable high dimensional noise distribution for the evolution equation in the case of non-negative states. The second barrier is the tendency of the Monte Carlo ensemble approximating the state distribution updated by observations to collapse down to a single point. The first barrier is overcome by defining the evolution equation noise distribution using very flexible meta-elliptical distributions. The second barrier is overcome by using a particle smoother across a sequence of spatial locations to generate the Monte Carlo ensemble. Because this location-domain particle smoother only considers one location at a time, the dimensionality of the sampling problem is reduced and a diverse ensemble can be generated. The location-domain particle smoother requires that the evolution noise distribution be defined using a meta-elliptical distribution and that the observation errors at different locations are independent. If the system has spatial resolution that is 'too fine' and there are 'too many' observed locations then the number of distinct particles can fall below an acceptable level at the beginning of the location sequence. A second method for overcoming ensemble collapse is proposed for these systems. In the second method a particle smoother is used to generate separate samples from the marginal state distributions at each location. The marginal samples are combined into a single sample from the joint state distribution spanning all of the locations using a copula. This second method requires that the state distribution is meta-elliptical and that the observation errors at different locations are independent. The assumptions required by the proposed methods are fewer and vastly less restrictive than the assumptions required by currently practical methods. The statistical properties of the new methods are explored in a simulation study and found to out-perform a standard particle filter and the popular ensemble Kalman filter when the Kalman assumptions are violated. A demonstration of the new methods using a real example is also provided.