(710c) Nonlinear State Estimation Using Parent and Nested Particles In Sequential Monte Carlo

State estimation is a fundamental element of engineering operations because it provides the knowledge for implementing process monitoring, control, optimization and safety related tasks. Many engineering systems tend to be nonlinear in nature and the uncertainties may be characterized by non-Gaussian pdfs. The recursive Bayesian solution to estimation is available for half a century, but faithful implementation is restricted to linear Gaussian systems. 

Recently there has been a surge of interest in solving the nonlinear non-Gaussian conditional mean estimation, following the introduction of sequential Monte Carlo approach or particle filters. It is possible to design fast and accurate particle filters for nonlinear state estimation problems in engineering, tracking and guidance, meteorology and oceanography. There has been a flurry of recent research activity in chemical engineering regarding nonlinear and constrained estimation using the Monte Carlo approach with various simplifications such as Gaussianity and statistical linearization. Optimistic projections for the spread of particle filters in industrial practice are based on the availability of cheap computing resources. While computational cost is a crucial factor, the Monte Carlo approach can also gain in performance by focusing on a multiscale view of the state space. The effects of computationally expedient approximations can be limited to localized state space at a smaller scale while significantly alleviating the effects on the global state space of the filter.

The particle filter is a generic sequential Monte Carlo implementation of the optimal Bayesian filter. It approximately retrieves the properties of the a posteriori density by using a set of samples (particles) from the density. However, it is generally not feasible to compute the a posteriori density in closed-form, such that samples may be drawn from it. The particle filter is a combination of importance sampling and resampling techniques in lieu of sampling the conditional density. The importance sampling technique provides a means to sample a set of relevant support points (particles) from a chosen importance density that is amenable to sampling. The support points are assigned appropriate weights that reflect their relative importance such that the particles and associated weights provide an approximate discrete representation of the unknown a posteriori density.

The importance density is clearly a crucial filter design choice. Any probability measure with support including the support of the a posteriori density may be chosen as the importance density. The optimal importance density minimizes the variance of the weights conditioned on states and measurements. However, the optimal importance density is generally not available, which leads to suboptimal choices. The transition density is the most frequently used suboptimal importance density because it is readily available. Although easy to use, the transition density can be a poor choice in cases where there is little overlap between the importance density and the likelihood function. This can be particularly problematic when highly accurate measurements yield narrow and peaked likelihood functions. Furthermore, the transition density has no knowledge of the latest measurement information, as a result of which many of the the support points may not be naturally located in the high likelihood region suggested by the latest measurement. In practice, one needs to resort to simplifications such as linearization of nonlinear transformation and/or fixed shape Gaussian approximation to estimate the support of practically useful importance densities. Both approximations can be detrimental in revealing the importance support when applied for approximation on a global scale, i.e. over all of the  appropriate state space. These effects may be mitigated by searching for multiple importance densities supported on state space that is localized in scale instead of a single importance density on a global scale.

It is proposed here that nonlinear transformation of the conditional density and Bayesian update with measurement is achieved by localized importance sampling, performed over multiple subsets of state space (local scale) that results in sampling the support of the desired density (global scale). The proposed approach uses nested particle sets related to the parent particles of the filter. The purpose of the nested child particles is to generate local (nested) importance densities with a knowledge of the measurement information as well as the local behavior of nonlinearity in state space. These local scale importance densities in turn generate the parent particles as support points for the determination of the global importance weights. The forward propagations of the nests in time are representative of the local behavior of the system nonlinearity in state space and the corresponding local transition density. It is conceivable that the propagated child particles may be corrected with the measurement information using various forms of approximate nonlinear filters, including the particle filter itself. The nesting approach broadly subsumes previously known local linearization extended Kalman particle filter and the the unscented particle filter as well as lay out a framework for the inclusion of other nonlinear filters in the particle filter approach.

This paper investigates the formulation of local importance densities as Gaussian approximations derived from the nested particles. More generally, it is also proposed here that kernel density estimation may be used to formulate more accurate and appropriate importance densities by avoiding the Gaussian approximation. Unlike the existing approaches of local linearized particle filters, the proposed nesting approach allows for two-way interaction between the nested particles (localized in state space) and the parent  particles (over global state space of the filter). The gains in performance are demonstrated with a benchmark nonlinear estimation problem from the literature. Preliminary results indicate that the estimation error can be reduced by orders of magnitude with small increase in computational cost due to additional particles in the nests.