Sequential Monte Carlo methods for data assimilation in strongly nonlinear dynamics.

Data assimilation is the process of estimating the state of dynamic systems (linear or nonlinear, Gaussian or non-Gaussian) as accurately as possible from noisy observational data. Although the Three Dimensional Variational (3D-VAR) methods, Four Dimensional Variational (4D-VAR) methods and Ensemble Kalman filter (EnKF) methods are widely used and effective for linear and Gaussian dynamics, new methods of data assimilation are required for the general situation, that is, nonlinear non-Gaussian dynamics. General Bayesian recursive estimation theory is reviewed in this thesis. The Bayesian estimation approach provides a rather general and powerful framework for handling nonlinear, non-Gaussian, as well as linear, Gaussian estimation problems. Despite a general solution to the nonlinear estimation problem, there is no closed-form solution in the general case. Therefore, approximate techniques have to be employed. In this thesis, the sequential Monte Carlo (SMC) methods, commonly referred to as the particle filter, is presented to tackle non-linear, non-Gaussian estimation problems. In this thesis, we use the SMC methods only for the nonlinear state estimation problem, however, it can also be used for the nonlinear parameter estimation problem. In order to demonstrate the new methods in the general nonlinear non-Gaussian case, we compare Sequential Monte Carlo (SMC) methods with the Ensemble Kalman Filter (EnKF) by performing data assimilation in nonlinear and non-Gaussian dynamic systems. The models used in this study are referred to as state-space models. The Lorenz 1963 and 1966 models serve as test beds for examining the properties of these assimilation methods when used in highly nonlinear dynamics. The application of Sequential Monte Carlo methods to different fixed parameters in dynamic models is considered. Four different scenarios in the Lorenz 1063 [sic] model and three different scenarios in the Lorenz 1996 model are designed in this study for both the SMC methods and EnKF method with different filter sizThe original print copy of this thesis may be available here: http://wizard.unbc.ca/record=b160051

Inference for Differential Equation Models using Relaxation via Dynamical Systems

Statistical regression models whose mean functions are represented by ordinary differential equations (ODEs) can be used to describe phenomenons dynamical in nature, which are abundant in areas such as biology, climatology and genetics. The estimation of parameters of ODE based models is essential for understanding its dynamics, but the lack of an analytical solution of the ODE makes the parameter estimation challenging. The aim of this paper is to propose a general and fast framework of statistical inference for ODE based models by relaxation of the underlying ODE system. Relaxation is achieved by a properly chosen numerical procedure, such as the Runge-Kutta, and by introducing additive Gaussian noises with small variances. Consequently, filtering methods can be applied to obtain the posterior distribution of the parameters in the Bayesian framework. The main advantage of the proposed method is computation speed. In a simulation study, the proposed method was at least 14 times faster than the other methods. Theoretical results which guarantee the convergence of the posterior of the approximated dynamical system to the posterior of true model are presented. Explicit expressions are given that relate the order and the mesh size of the Runge-Kutta procedure to the rate of convergence of the approximated posterior as a function of sample size

A new approach to particle based smoothed marginal MAP

We present here a new method of finding the MAP state estimator from the weighted particles representation of marginal smoother distribution. This is in contrast to the usual practice, where the particle with the highest weight is selected as the MAP, although the latter is not necessarily the most probable state estimate. The method developed here uses only particles with corresponding filtering and smoothing weights. We apply this estimator for finding the unknown initial state of a dynamical system and addressing the parameter estimation problem