Nonlinear Filtering based on Log-homotopy Particle Flow : Methodological Clarification and Numerical Evaluation

The state estimation of dynamical systems based on measurements is an ubiquitous problem. This is relevant in applications like robotics, industrial manufacturing, computer vision, target tracking etc. Recursive Bayesian methodology can then be used to estimate the hidden states of a dynamical system. The procedure consists of two steps: a process update based on solving the equations modelling the state evolution, and a measurement update in which the prior knowledge about the system is improved based on the measurements. For most real world systems, both the evolution and the measurement models are nonlinear functions of the system states. Additionally, both models can also be perturbed by random noise sources, which could be non-Gaussian in their nature. Unlike linear Gaussian models, there does not exist any optimal estimation scheme for nonlinear/non-Gaussian scenarios. This thesis investigates a particular method for nonlinear and non-Gaussian data assimilation, termed as the log-homotopy based particle flow. Practical filters based on such flows have been known in the literature as Daum Huang filters (DHF), named after the developers. The key concept behind such filters is the gradual inclusion of measurements to counter a major drawback of single step update schemes like the particle filters i.e. namely the degeneracy. This could refer to a situation where the likelihood function has its probability mass well seperated from the prior density, and/or is peaked in comparison. Conventional sampling or grid based techniques do not perform well under such circumstances and in order to achieve a reasonable accuracy, could incur a high processing cost. DHF is a sampling based scheme, which provides a unique way to tackle this challenge thereby lowering the processing cost. This is achieved by dividing the single measurement update step into multiple sub steps, such that particles originating from their prior locations are graduated incrementally until they reach their final locations. The motion is controlled by a differential equation, which is numerically solved to yield the updated states. DH filters, even though not new in the literature, have not been fully explored in the detail yet. They lack the in-depth analysis that the other contemporary filters have gone through. Especially, the implementation details for the DHF are very application specific. In this work, we have pursued four main objectives. The first objective is the exploration of theoretical concepts behind DHF. Secondly, we build an understanding of the existing implementation framework and highlight its potential shortcomings. As a sub task to this, we carry out a detailed study of important factors that affect the performance of a DHF, and suggest possible improvements for each of those factors. The third objective is to use the improved implementation to derive new filtering algorithms. Finally, we have extended the DHF theory and derived new flow equations and filters to cater for more general scenarios. Improvements in the implementation architecture of a standard DHF is one of the key contributions of this thesis. The scope of the applicability of DHF is expanded by combining it with other schemes like the Sequential Markov chain Monte Carlo and the tensor decomposition based solution of the Fokker Planck equation, resulting in the development of new nonlinear filtering algorithms. The standard DHF, using improved implementation and the newly derived algorithms are tested in challenging simulated test scenarios. Detailed analysis have been carried out, together with the comparison against more established filtering schemes. Estimation error and the processing time are used as important performance parameters. We show that our new filtering algorithms exhibit marked performance improvements over the traditional schemes