MULTI-FREQUENTIAL PERIODOGRAM ANALYSIS AND THE DETECTION OF PERIODIC COMPONENTS IN TIME SERIES
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Résumé
Abstract The spectral analysis of Gaussian linear time-series processes is usually based on uni-frequential tools because the spectral density functions of degree 2 and higher are identically zero and there is no polyspectrum in this case. In finite samples, such an approach does not allow the resolution of closely adjacent spectral lines, except by using autoregressive models of excessively high order in the method of maximum entropy. In this article, multi-frequential periodograms designed for the analysis of discrete and mixed spectra are defined and studied for their properties in finite samples. For a given vector of frequencies ω, the sum of squares of the corresponding trigonometric regression model fitted to a time series by unweighted least squares defines the multi-frequential periodogram statistic IM(ω). When ω is unknown, it follows from the properties of nonlinear models whose parameters separate (i.e., the frequencies and the cosine and sine coefficients here) that the least-squares estimator of frequencies is obtained by maximizing I M(ω). The first-order, second-order and distribution properties of I M(ω) are established theoretically in finite samples, and are compared with those of Schuster's uni-frequential periodogram statistic. In the multi-frequential periodogram analysis, the least-squares estimator of frequencies is proved to be theoretically unbiased in finite samples if the number of periodic components of the time series is correctly estimated. Here, this number is estimated at the end of a stepwise procedure based on pseudo-Flikelihood ratio tests. Simulations are used to compare the stepwise procedure involving I M(ω) with a stepwise procedure using Schuster's periodogram, to study an approximation of the asymptotic theory for the frequency estimators in finite samples in relation to the proximity and signal-to-noise ratio of the periodic components, and to assess the robustness of I M(ω) against autocorrelation in the analysis of mixed spectra. Overall, the results show an improvement of the new method over the classical approach when spectral lines are adjacent. Finally, three examples with real data illustrate specific aspects of the method, and extensions (i.e., unequally spaced observations, trend modeling, replicated time series, periodogram matrices) are outlined. Keywords: Trigonometric regressionNonlinear models whose parameters separateLeast-squares estimation of frequenciesFinite-sample properties of periodogram statistics and frequency estimatorsAnalysis of discrete and mixed spectra ACKNOWLEDGMENTS This research work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds pour la Formation de Chercheurs et l'Aide à la Recherche du Québec (FCAR). This work benefited from discussions with Dr. Guy Gérard (who passed over on August 12, 1997) and Dr. José Paris at the early stage of development of the method. I am gratetul to Dr. Pierre Legendre for having provided technical support for further developments and to Philippe Casgrain for his valuable contribution to the PerioMod software. I am indebted to Dr. David Brillinger for the enlightening discussions that we had on a draft of this article when I was a Visiting Professor in the Department of Statistics at UC Berkeley in the summer of 1999. Special thanks go to Dr. Maurice Priestley for having provided me references that were missing in a previous version of the manuscript. I thank the Associate Editor for comments and suggestions that help improve the presentation. This article is dedicated to defunct Prof. Guy Gérard.
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