Estimation of spatially varying parameters with application to hyperbolic SPDEs

TL;DR

This paper develops a method for estimating spatially varying parameters in hyperbolic SPDEs using Metropolis-Hastings, showing improved estimates over joint estimation methods in signal processing applications.

Contribution

It introduces a Monte Carlo-based approach with Metropolis-Hastings for spatial parameter estimation in hyperbolic SPDEs, enhancing accuracy over traditional joint estimation.

Findings

Metropolis-Hastings yields better parameter estimates.

Application to advection and wave equations demonstrates effectiveness.

Method outperforms joint estimation in accuracy.

Abstract

Parameter estimation is a growing area of interest in statistical signal processing. Some parameters in real-life applications vary in space as opposed to those that are static. Most common methods in estimating parameters involve solving an optimization problem where the cost function is assembled variously; for example, maximum likelihood and maximum a posteriori methods. However, these methods do not have exact solutions to most real-life problems. It is for this reason that Monte Carlo methods are preferred. In this paper, we treat the estimation of parameters which vary with space. We use Metropolis-Hastings algorithm as a selection criteria for the maximum filter likelihood. Comparisons are made with the use of joint estimation of both the spatially varying parameters and the state. We illustrate the procedures employed in this paper by means of two hyperbolic SPDEs: the advection and the wave equation. The Metropolis-Hastings procedure registers better estimates.