Iterative and Geometric Methods for State Estimation in Non-linear Models

Abstract

Many problems in science and engineering involve estimating a dynamic signal from indirect measurements subject to noise, where points can either evolve in continuous time or in discrete time. These problems are often formalised as inference in probabilistic state-space models, which are also frequently assumed to be Markovian. For inferring the value of the signal at a particular point in time, methods of inference can be divided into different classes, namely prediction, filtering (tracking), and smoothing. In prediction, only past measurements of the signal are used to infer its present value, whereas in filtering both past and present measurements are used, and in smoothing past, present, and future measurements are used. Prediction is useful in situations where decisions need to be made contingent on a future value of the signal before future measurements are made. On the other hand, filtering is useful when the signal needs to be inferred as the measurements arrive, that is on-line. Lastly, smoothing is the preferred choice when none of the aforementioned constraints are present as it allows the use of the entire sequence of measurements to infer the signal. In this thesis, the filtering and smoothing problems and their applications are examined. In particular, iterative Gaussian filters and smoothers are developed for both inferring continuous and discrete time signals. Furthermore, it is shown that methods for inference in state-space models can be applied to the field of probabilistic numerics. More specifically, estimating the solutions to ordinary differential equations can be formulated as inference in a probabilistic state-space model, hence the solutions can be inferred using either Gaussian filtering methods or sequential Monte Carlo. Another theme of this thesis is the exploitation of geometry - in a broad sense. Firstly, the geometry of probability densities, namely information geometry, is exploited to approximately infer the signal in filtering. Geometry is also exploited in terms of the geometry of the state-space, the space where the signal takes its values. That is, for tracking a time-varying unit vector, a continuous-time dynamic model is posed that respects the geometry of the unit sphere. Subsequently, a filtering algorithm is developed based on the von Mises-Fisher distribution for inference in this model. The method is demonstrated to have applications in tracking the local gravity and magnetic field vectors using a smartphone. Lastly, the geometry of L_2 spaces is used to approximate a stochastic differential equation with an ordinary differential equation with random coefficients. On this basis filtering and smoothing algorithms are developed.