Abstract
The state-space formulation for time-dependent models has been long used invarious applications in science and engineering. While the classical Kalman filter
(KF) provides optimal posterior estimation under linear Gaussian models, filtering
in nonlinear and non-Gaussian environments remains challenging.
Based on the Monte Carlo approximation, the classical particle filter (PF) can provide
more precise estimation under nonlinear non-Gaussian models. However, it suffers from
particle degeneracy. Drawing from optimal transport theory, the stochastic map filter
(SMF) accommodates a solution to this problem, but its performance is influenced by
the limited flexibility of nonlinear map parameterisation. To account for these issues,
a hybrid particle-stochastic map filter (PSMF) is first proposed in this thesis, where
the two parts of the split likelihood are assimilated by the PF and SMF, respectively.
Systematic resampling and smoothing are employed to alleviate the particle degeneracy
caused by the PF. Furthermore, two PSMF variants based on the linear and nonlinear
maps (PSMF-L and PSMF-NL) are proposed, and their filtering performance is compared
with various benchmark filters under different nonlinear non-Gaussian models.
Although achieving accurate filtering results, the particle-based filters require expensive computations because of the large number of samples involved. Instead, robust
Kalman filters (RKFs) provide efficient solutions for the linear models with heavy-tailed
noise, by adopting the recursive estimation framework of the KF. To exploit the stochastic
characteristics of the noise, the use of heavy-tailed distributions which can fit various
practical noises constitutes a viable solution. Hence, this thesis also introduces a novel
RKF framework, RKF-SGαS, where the signal noise is assumed to be Gaussian and the
heavy-tailed measurement noise is modelled by the sub-Gaussian α-stable (SGαS) distribution. The corresponding joint posterior distribution of the state vector and auxiliary
random variables is estimated by the variational Bayesian (VB) approach. Four different
minimum mean square error (MMSE) estimators of the scale function are presented.
Besides, the RKF-SGαS is compared with the state-of-the-art RKFs under three kinds of
heavy-tailed measurement noises, and the simulation results demonstrate its estimation
accuracy and efficiency.
One notable limitation of the proposed RKF-SGαS is its reliance on precise model
parameters, and substantial model errors can potentially impede its filtering performance. Therefore, this thesis also introduces a data-driven RKF method, referred to as
RKFnet, which combines the conventional RKF framework with a deep learning technique. An unsupervised scheduled sampling technique (USS) is proposed to improve the
i
stability of the training process. Furthermore, the advantages of the proposed RKFnet
are quantified with respect to various traditional RKFs.
Date of Award | 8 Dec 2023 |
---|---|
Original language | English |
Awarding Institution |
|
Supervisor | Alin Achim (Supervisor) |