A New Algorithm for Circulant Rational Covariance Extension and Applications to Finite-interval Smoothing

TL;DR

This paper introduces a novel algorithm for circulant rational covariance extension that leverages nonlinear Yule-Walker equations, offering a robust and conceptually clear approach to finite-interval smoothing problems.

Contribution

It proposes a new iterative algorithm based on nonlinear Yule-Walker equations that converges to the solution of a variational problem, simplifying finite-interval smoothing.

Findings

Algorithm converges to the unique variational solution.

Provides a simpler alternative to Riccati-based smoothing methods.

Demonstrates successful application to finite-interval smoothing.

Abstract

The partial stochastic realization of periodic processes from finite covariance data has recently been solved by Lindquist and Picci based on convex optimization of a generalized entropy functional. The meaning and the role of this criterion have an unclear origin. In this paper we propose a solution based on a nonlinear generalization of the classical Yule-Walker type equations and on a new iterative algorithm which is shown to converge to the same (unique) solution of the variational problem. This provides a conceptual link to the variational principles and at the same time yields a robust algorithm which can for example be successfully applied to finite-interval smoothing problems providing a simpler procedure if compared with the classical Riccati-based calculations.