Modeling of Stationary Periodic Time Series by ARMA Representations

TL;DR

This paper surveys recent advances in modeling stationary periodic time series using ARMA representations, focusing on the covariance extension problem, dual optimization solutions, and applications to image processing.

Contribution

It provides a comprehensive overview of the rational covariance extension problem, dual convex optimization solutions, and extensions to multivariate cases with practical applications.

Findings

Solution of the covariance extension problem via dual convex optimization

Reformulation using circulant matrices and connections to reciprocal processes

Extension of theory to multivariate time series and application to image processing

Abstract

This is a survey of some recent results on the rational circulant covariance extension problem: Given a partial sequence $(c_0,c_1,\dots,c_n)$ of covariance lags $c_k=\mathbb{E}\{y(t+k)\overline{y(t)}\}$ emanating from a stationary periodic process $\{y(t)\}$ with period $2N>2n$, find all possible rational spectral functions of $\{y(t)\}$ of degree at most $2n$ or, equivalently, all bilateral and unilateral ARMA models of order at most $n$, having this partial covariance sequence. Each representation is obtained as the solution of a pair of dual convex optimization problems. This theory is then reformulated in terms of circulant matrices and the connections to reciprocal processes and the covariance selection problem is explained. Next it is shown how the theory can be extended to the multivariate case. Finally, an application to image processing is presented.