Fast sequential parameter inference for dynamic state space models

Abstract

Many problems in science require estimation and inference on systems that generate data over time. Such systems, quite common in statistical signal processing, time series analysis and econometrics, can be stated in a state-space form. Estimation is made on the state of the state-space model, using a sequence of noisy measurements made on the system. This difficult problem of estimating the parameters in real time, has generated a lot of interest in the statistical community, especially since the latter half of the last century. One area that is particularly important is the estimation of parameters which do not evolve over time. The parameters in the dynamic state-space model generally have a nonGaussian posterior distribution and holds a nonlinear relationship with the data. Sequential inference of these static parameters requires novel statistical techniques. Addressing the challenges of such a problem provides the focus for the research contributions presented in this thesis. A functional approximation update of the posterior distribution of the parameters is developed. The approximate posterior is explored at a sufficient number of points on a grid which is computed at good evaluation points. The grid is re-assessed at each time point for addition/reduction of grid points. Bayes Law and the structure of the state-space model are used to sequentially update the posterior density of the model parameters as new observations arrive. These approximations rely on already existing state estimation techniques such as the Kalman filter and its nonlinear extensions, as well as integrated nested Laplace approximation. However, the method is quite general and can be used for any existing state estimation algorithm. This new methodology of sequential updating makes the calculation of posterior both fast and accurate, while it can be applied to a wide class of models existing in literature. The above method is applied to three different state-space models namely, linear model with Gaussian errors, nonlinear model and model with non-Gaussian errors, and comparison with some other existing methods has been discussed.