Accuracy of Discrete Markov Approximation in the Problems of Estimation of Random Field Characteristics

TL;DR

This paper investigates how well Markov fields can approximate the covariance matrices of random fields, enabling efficient computational algorithms for various estimation problems, with accuracy depending on the connectivity parameter m.

Contribution

It introduces a method for approximating arbitrary random fields with Markov fields using covariance matrices, and evaluates the approximation accuracy through computer simulations.

Findings

Approximation accuracy improves with increasing connectivity coefficient m.

Even small m values can provide sufficient accuracy for many problems.

The approach balances estimation accuracy and computational complexity.

Abstract

The covariance matrix of measurements of Markov random fields (processes) has useful properties that allow to develop effective computational algorithms for many problems in the study of Markov fields on the basis of field observations (parametric identification problems, filtering problems, interpolation problems and others). Therefore, approximation of arbitrary random fields by Markov fields is of great interest, as it gives an opportunity to use computationally efficient algorithms of Markov fields analysis to study them. The paper deals with approximation of the covariance matrix of the field being observed with the help of covariance matrix of a multiply connected (m-connected) Markov field. Using computer simulation, the accuracy of such replacements at different values of the connectivity coefficient m for the problem of parametric identification of deterministic component of the field has been studied. Various models of deterministic polynomial component of the field and a number of covariance functions that are often used as a mathematical model of real random noise and measurements noise have been reviewed. It has been shown that for many problems such approximation, even at small connectivity m values of approximating Markov field, provides necessary accuracy. This allows to achieve good compromise between accuracy of the estimates and complexity of calculating them.