| The Kalman filter in various forms has been around for a long time and is the standard analytic approach to state estimation when the state space model is linear and Gaussian or approximately linear and Gaussian (Jazwinski 1970). I discuss the Kalman filter in detail in this thesis to provide a standard with which to compare other approaches to state estimation.; Over the last 10 years or so there has been a growing interest in numerical approaches to state estimation. This approach, the particle filter (Arulampalam 2002), (Kitagawa 1996), has the advantage that it can be applied to a wider range of state space models than the Kalman filter, including nonlinear models with non-Gaussian noise processes. I discuss both the SIS/SIR and the Metropolis-Hastings approach to particle filtering in this thesis, with a special emphasis on the latter.; When formulating a state space model, the variance of the noise processes as well as other unknown parameters must be estimated either from observation or from prior knowledge. Some methods of estimating the variance of the noise processes of a state space model are developed in this thesis. These methods are then applied to real data drawn from the field of ocean optics for which a linear/Gaussian state space model has been formulated. The results of these methods are shown to be comparable to the MLE provided by the Kalman filter. |
