Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Optimal Nonparametric Covariance Function Estimation for Any Family of Nonstationary Random Processes | Hindawi Publishing Corporation EURASIP Journal on Advances in Signal Processing Volume 2011 Article ID 140797 7 pages doi 2011 140797 Research Article Optimal Nonparametric Covariance Function Estimation for Any Family of Nonstationary Random Processes Johan Sandberg EURASIP Member and Maria Hansson-Sandsten EURASIP Member Division of Mathematical Statistics Centre for Mathematical Sciences Lund University 221 00 Lund Sweden Correspondence should be addressed to Johan Sandberg sandberg@ Received 28 June 2010 Revised 15 November 2010 Accepted 29 December 2010 Academic Editor Antonio Napolitano Copyright 2011 J. Sandberg and M. Hansson-Sandsten. 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. 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. 1. Introduction In several applications including statistical time-frequency analysis 1-4 the covariance function of a nonstationary random process has to be estimated from one single observed realization. We assume that the complex-valued process which we denote by x t t e Z is in .
