A statistical study for estimating the instantaneous wave form of road traffic noise by three new estimation methods of generalized Kalman's filtering theory

Abstract

The problem of estimating the state of a stochastic environmental system from noisy observation is of central importance in the actual engineering field. Furthermore, from the technical point of view, there is very often a necessity of estimating especially any of statistical values like the higher-order moments, in the actual case where the observation process is a non-Gaussian process. In order to establish a unified method treating generally the above estimation problem under these actual needs, the so-called Bayesian point of view is first employed here. The chief purpose of this paper is how to extend the well-known results of Kalman and Bucy in the field of linear filtering and prediction problem by finding Bayes' theorem in a new form of unified series expansion, which is suitable for finding a recursive algorithm matched to our successive observation. Three different methods of extending Kalman's filter are newly derived in the unified forms of different type, for each of which recursive estimation formulas can be obtained in the form of a very compact and general solution for the so-called linear filtering (or prediction) problem. In addition, it is noticeable that the above new results include some generalization on the type of estimation processes, viz., those that could be generated by passing arbitrarily distributed input noise through an arbitrary nonlinear dynamical system (possibly time variant). Finally, the validity and the effectiveness of our theoretical results are experimentally clarified through several applications to actual urban noises in Hiroshima City.