Latent state estimation in a class of nonlinear systems

Abstract

The problem of estimating latent or unobserved states of a dynamical system from observed data is studied in this thesis. Approximate filtering methods for discrete time series for a class of nonlinear systems are considered, which, in turn, require sampling from a partially specified discrete distribution. A new algorithm is proposed to sample from partially specified discrete distribution, where the specification is in terms of the first few moments of the distribution. This algorithm generates deterministic sigma points and corresponding probability weights, which match exactly a specified mean vector, a specified covariance matrix, the average of specified marginal skewness and the average of specified marginal kurtosis. Both the deterministic particles and the probability weights are given in closed form and no numerical optimization is required. This algorithm is then used in approximate Bayesian filtering for generation of particles and the associated probability weights which propagate higher order moment information about latent states. This method is extended to generate random sigma points (or particles) and corresponding probability weights that match the same moments. The algorithm is also shown to be useful in scenario generation for financial optimization. For a variety of important distributions, the proposed moment-matching algorithm for generating particles is shown to lead to approximation which is very close to maximum entropy approximation. In a separate, but related contribution to the field of nonlinear state estimation, a closed-form linear minimum variance filter is derived for the systems with stochastic parameter uncertainties. The expressions for eigenvalues of the perturbed filter are derived for comparison with eigenvalues of the unperturbed Kalman filter. Moment-matching approximation is proposed for the nonlinear systems with multiplicative stochastic noise.