Nonlinear bayesian filtering with applications to estimation and navigation

In principle, general approaches to optimal nonlinear filtering can be described in a unified way from the recursive Bayesian approach. The central idea to this recur- sive Bayesian estimation is to determine the probability density function of the state vector of the nonlinear systems conditioned on the available measurements. However, the optimal exact solution to this Bayesian filtering problem is intractable since it requires an infinite dimensional process. For practical nonlinear filtering applications approximate solutions are required. Recently efficient and accurate approximate non- linear filters as alternatives to the extended Kalman filter are proposed for recursive nonlinear estimation of the states and parameters of dynamical systems. First, as sampling-based nonlinear filters, the sigma point filters, the unscented Kalman fil- ter and the divided difference filter are investigated. Secondly, a direct numerical nonlinear filter is introduced where the state conditional probability density is calcu- lated by applying fast numerical solvers to the Fokker-Planck equation in continuous- discrete system models. As simulation-based nonlinear filters, a universally effective algorithm, called the sequential Monte Carlo filter, that recursively utilizes a set of weighted samples to approximate the distributions of the state variables or param- eters, is investigated for dealing with nonlinear and non-Gaussian systems. Recentparticle filtering algorithms, which are developed independently in various engineer- ing fields, are investigated in a unified way. Furthermore, a new type of particle filter is proposed by integrating the divided difference filter with a particle filtering framework, leading to the divided difference particle filter. Sub-optimality of the ap- proximate nonlinear filters due to unknown system uncertainties can be compensated by using an adaptive filtering method that estimates both the state and system error statistics. For accurate identification of the time-varying parameters of dynamic sys- tems, new adaptive nonlinear filters that integrate the presented nonlinear filtering algorithms with noise estimation algorithms are derived. For qualitative and quantitative performance analysis among the proposed non- linear filters, systematic methods for measuring the nonlinearities, biasness, and op- timality of the proposed nonlinear filters are introduced. The proposed nonlinear optimal and sub-optimal filtering algorithms with applications to spacecraft orbit es- timation and autonomous navigation are investigated. Simulation results indicate that the advantages of the proposed nonlinear filters make these attractive alterna- tives to the extended Kalman filter

An Information-Theoretic Analysis of Discrete-Time Control and Filtering Limitations by the I-MMSE Relationships

Fundamental limitations or performance trade-offs/limits are important properties and constraints of control and filtering systems. Among various trade-off metrics, total information rate, which characterizes the sensitivity trade-offs and average performance of control and filtering systems, is conventionally studied by using the (differential) entropy rate and Kolmogorov-Bode formula. In this paper, by extending the famous I-MMSE (mutual information -- minimum mean-square error) relationship to the discrete-time additive white Gaussian channels with and without feedback, a new paradigm is introduced to estimate and analyze total information rate as a control and filtering trade-off metric. Under this framework, we enrich the trade-off properties of total information rate for a variety of discrete-time control and filtering systems, e.g., LTI, LTV, and nonlinear, and also provide an alternative approach to investigate total information rate via optimal estimation.Comment: Neng Wan and Dapeng Li contributed equally to this pape

A Simulation Approach to Optimal Stopping Under Partial Information

We study the numerical solution of nonlinear partially observed optimal stopping problems. The system state is taken to be a multi-dimensional diffusion and drives the drift of the observation process, which is another multi-dimensional diffusion with correlated noise. Such models where the controller is not fully aware of her environment are of interest in applied probability and financial mathematics. We propose a new approximate numerical algorithm based on the particle filtering and regression Monte Carlo methods. The algorithm maintains a continuous state-space and yields an integrated approach to the filtering and control sub-problems. Our approach is entirely simulation-based and therefore allows for a robust implementation with respect to model specification. We carry out the error analysis of our scheme and illustrate with several computational examples. An extension to discretely observed stochastic volatility models is also considered

State and Parameter Estimation of Partially Observed Linear Ordinary Differential Equations with Deterministic Optimal Control

Ordinary Differential Equations are a simple but powerful framework for modeling complex systems. Parameter estimation from times series can be done by Nonlinear Least Squares (or other classical approaches), but this can give unsatisfactory results because the inverse problem can be ill-posed, even when the differential equation is linear. Following recent approaches that use approximate solutions of the ODE model, we propose a new method that converts parameter estimation into an optimal control problem: our objective is to determine a control and a parameter that are as close as possible to the data. We derive then a criterion that makes a balance between discrepancy with data and with the model, and we minimize it by using optimization in functions spaces: our approach is related to the so-called Deterministic Kalman Filtering, but different from the usual statistical Kalman filtering. e show the root-$n$ consistency and asymptotic normality of the estimators for the parameter and for the states. Experiments in a toy model and in a real case shows that our approach is generally more accurate and more reliable than Nonlinear Least Squares and Generalized Smoothing, even in misspecified cases.Comment: 45 pages, 1 figur

Topics in particle filtering and smoothing

Particle filtering/smoothing is a relatively new promising class of algorithms\ud to deal with the estimation problems in nonlinear and/or non-\ud Gaussian systems. Currently, this is a very active area of research and\ud there are many issues that are not either properly addressed or are still\ud open.\ud One of the key issues in particle filtering is a suitable choice of the\ud importance function. The optimal importance function which includes the\ud information from the most recent observation, is difficult to obtain in most\ud practical situations. In this thesis, we present a new Gaussian approximation\ud to this optimal importance function using the moment matching\ud method and compare it with some other recently proposed importance\ud functions.\ud In particle filtering/smoothing, the posterior is represented as a weighted\ud particle cloud. We develop a new algorithm for extracting the smoothed\ud marginal maximum a posteriori (MAP) estimate from the available particle\ud cloud of the marginal smoother, generated using either the forwardbackward\ud smoother or the two filter smoother. The smoothed marginal\ud MAP estimator is then applied to estimate the unknown initial state of a\ud dynamic system.\ud There are many approaches to deal with the unknown static system\ud parameters within particle filtering/smoothing set up. One common approach\ud is to model the parameters as a part of the state vector. This is\ud followed by adding artificial process noises to this model and then estimate\ud the parameters along with the other state variables. Although this\ud approach may work well in (certain) practical situations, the added process\ud noises may result in a unnecessary loss of accuracy of the estimated\ud parameters. Here we propose some new particle filtering/smoothing based\ud algorithms, where we avoid any effect of the artificial dynamics on the\ud estimate of the parameters

On forward-backward SDE approaches to continuous-time minimum variance estimation

The work of Kalman and Bucy has established a duality between filtering and optimal estimation in the context of time-continuous linear systems. This duality has recently been extended to time-continuous nonlinear systems in terms of an optimization problem constrained by a backward stochastic partial differential equation. Here we revisit this problem from the perspective of appropriate forward-backward stochastic differential equations. This approach sheds new light on the estimation problem and provides a unifying perspective. It is also demonstrated that certain formulations of the estimation problem lead to deterministic formulations similar to the linear Gaussian case as originally investigated by Kalman and Bucy. Finally, optimal control of partially observed diffusion processes is discussed as an application of the newly proposed estimators

Essays on nonlinear filtering with applications in finance

In this dissertation, I discuss the nonlinear filtering problem and its applications in finance. In the first chapter, I present a new filtering approach for nonlinear and non- Gaussian state space models. This approach builds on the well-established Kalman filter, featuring a state-dependent least-square linearization of the nonlinear function and a Gaussian-mixture approximation to the error distribution, and it applies the quasi-Monte Carlo method for numerical integration during the computation. The theoretical analysis shows that when the model is Gaussian, the filtering distribution based on the proposed approach can capture the true first two moments of the state variable; when the model is non-Gaussian, the filtering distribution can always be represented by a Gaussian mixture. This study also provides an analysis of the stability of this new filtering approach. In addition, I propose two methods to estimate the unknown parameters of the model. The first is an off-line likelihood-based method, and the second is an on-line dual estimation method. I also establish the consistency of the proposed quasi-maximum likelihood estimator under general conditions. To illustrate the proposed approach, I discuss several numerical examples using simulated data and compare my approach with other existing methods. I find that the proposed approach can outperform these methods in terms of speed and accuracy. The second chapter examines pairs trading using a general state space model framework. I model the spread between the prices of two assets as an unobservable state variable. I show how to use the filtered spread to carry out statistical arbitrage. I also propose a new trading strategy and present a Monte Carlo-based approach to select the optimal trading rule. The third chapter, coauthored with Li Chen, presents a new approximation scheme for the price and exercise policy of American options. The scheme is based on Hermite polynomial expansions of the transition density of the underlying asset dynamics and the early exercise premium representation of the American option price. The proposed approximations of the price and optimal exercise boundary are shown to be convergent to the true ones

New advances in H∞ control and filtering for nonlinear systems

The main objective of this special issue is to summarise recent advances in H∞ control and filtering for nonlinear systems, including time-delay, hybrid and stochastic systems. The published papers provide new ideas and approaches, clearly indicating the advances made in problem statements, methodologies or applications with respect to the existing results. The special issue also includes papers focusing on advanced and non-traditional methods and presenting considerable novelties in theoretical background or experimental setup. Some papers present applications to newly emerging fields, such as network-based control and estimation