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  • Research Article
  • Open Access

Optimal Nonparametric Covariance Function Estimation for Any Family of Nonstationary Random Processes

  • 1Email author and
  • 1
EURASIP Journal on Advances in Signal Processing20112011:140797

  • Received: 28 June 2010
  • Accepted: 29 December 2010
  • Published:


A covariance function estimate of a zero-mean nonstationary random process in discrete time is accomplished from one observed realization by weighting observations with a kernel function. Several kernel functions have been proposed in the literature. In this paper, we prove that the mean square error (MSE) optimal kernel function for any parameterized family of random processes can be computed as the solution to a system of linear equations. Even though the resulting kernel is optimized for members of the chosen family, it seems to be robust in the sense that it is often close to optimal for many other random processes as well. We also investigate a few examples of families, including a family of locally stationary processes, nonstationary AR-processes, and chirp processes, and their respective MSE optimal kernel functions.


  • Covariance
  • Information Technology
  • Linear Equation
  • Mean Square Error
  • Kernel Function

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Authors’ Affiliations

Division of Mathematical Statistics, Centre for Mathematical Sciences, Lund University, 221 00 Lund, Sweden


© Johan Sandberg and Maria Hansson-Sandsten. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.