Statistical Properties for Coherence Estimators From Evolutionary Spectra

Abstract

Abstract-Evolutionary spectra were developed by Priestley to extend spectral analysis to some nonstationary time series, in particularly semistationary processes, of which the ubiquitous uniformly modulated processes are a subclass. Coherence is well-defined for bivariate semistationary processes and can be estimated from such processes. We consider Priestley's estimator for the evolutionary spectral density matrix, and show that its elements can be written as weighted multitaper estimators with calculable weights and tapers. Under Gaussianity an approximating Wishartdistribution model follows for the spectral matrix, valid for all frequencies except small computable intervals near zero and Nyquist. Moreover, the critically important degrees of freedom are known. Consequently the statistical distribution of the coherence is given by Goodman's distribution and the raw coherence estimate can be accurately debiased. Theoretical results are verified using a model for wind fluctuations: simulations give excellent agreement between the mean debiased coherence estimates and true coherence, and between the proposed and empirical distributions of coherence.

Estimation of the Evolutionary Spectra With Application to Stationarity Test

“…2. The blue circle line is the relationship between N and the corresponding optimal K evaluated according to (18) and the red dot line is the relationship between N and the corresponding optimal K evaluated according to (19). relationship between the N and the corresponding optimal K is shown in Fig.…”

“…2. Note that (19) is derived in [23] and the optimal K scales roughly as N −4/5 . For each N , the K that minimize the MSE do not vary much in both cases.…”

“…1. The blue circle line corresponds to the MSE in (18) with respect to the optimal K for each N and the red dot line corresponds to the MSE in (19) with respect to the optimal K for each N . Fig.…”

“…There are a few other related works in the literature. In [19], the authors expressed Priestley's two step approach in the form of the multitaper formulation, where the number of tapers becomes the number of neighbors of the targeted time t for the smoothing step. However, in this work we have removed the smoothing step and apply the multitaper method over the same segment of process, i.e., with the same targeted time t and frequency w. With additional smoothness assumptions on the underlying spectra, the authors in [20] have investigated the statistical properties of the spectra estimate.…”

Estimating multiresolution dependency graphs within the stationary wavelet framework

“…The traditional family of complex exponential functions {e iωt } used in spectral analysis is not expressive enough to deal with non-stationarity [2]. To deal with this issue, a theory of evolutionary spectra was proposed by Priestly [2] (also see [5], [16]) whereby the exponential family is replaced by the more general class {φ t (ω)}, in particular he considers families of the form φ t (ω) = C t (ω)e iωt , where C t (ω) allows one to localise the basis about point t.…”