Optimal Nonparametric Covariance Function Estimation for Any Family of Nonstationary Random Processes
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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, [email protected] 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 ∈ Z}, is in discrete time and / Tn = {1, . . . , n}. has finite support: x(t) = 0 for all t ∈ Most often, the mean of the process is assumed to be known or already estimated and, hereby, we can, without loss of generality, assume that the mean of the process is zero. An estimate of the covariance function defined and denoted by rx (s, t) = E[x(s)x(t)∗ ] is then accomplished by a weighted average of observations of x(s + k)x(t + k)∗ with different weights for different k, [5, 6], where ∗ denotes complex conjugate. Presumably, the weights, also known as the kernel function, are allowed to vary with the time-lag τ = s − t. We denote and define this estimator by Rx;H : Tn2 → C: Rx;H (s, t) =
1 H(k, s − t)x(s + k)x(t + k)∗ , (1) |Ks−t | k∈K s−t
where Kτ is the set {−n + 1 + |τ |, . . . , n − 1 − |τ |}, and H is a kernel function which belongs to the set H = {H : Kτ × T = {−n + 1, . . . , n − 1} → C} of all possible kernel functions, and
where we denote the cardinality of a set S by |S|. Some care has to be taken in order for this estimate to be nonnegative definite, but as this problem has appropriate solutions [7], we will not discuss it further. Naturally, one wishes to ch
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