Polynomial-chaos-based Bayesian approach for state and parameter estimations

Two new recursive approaches have been developed to provide accurate estimates for posterior moments of both parameters and system states while making use of the generalized polynomial-chaos framework for uncertainty propagation. The main idea of the generalized polynomial-chaos method is to expand random state and input parameter variables involved in a stochastic differential/difference equation in a polynomial expansion. These polynomials are associated with the prior probability density function for the input parameters. Later, Galerkin projection is used to obtain a deterministic system of equations for the expansion coefficients. The first proposed approach provides means to update prior expansion coefficients by constraining the polynomial-chaos expansion to satisfy a specified number of posterior moment constraints derived from Bayes's rule. The second proposed approach makes use of the minimum variance formulation to update generalized polynomial-chaos coefficients. The main advantage of the proposed methods is that they not only provide a point estimate for the states and parameters, but they also provide the associated uncertainty estimates along these point estimates. Numerical experiments involving four benchmark problems are considered to illustrate the properties of the proposed methods.