Multi-Frequential Periodogram Analysis and the Detection of Periodic Components in Time Series

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 !, 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(!). When ! is unknown, it follows from the properties of nonlinear models whose parameters separate (i.e., the frequencies and the cosine and ORDER REPRINTS sine coefficients here) that the least-squares estimator of frequencies is obtained by maximizing I M (!). The first-order, second-order and distribution properties of I M (!) 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-F likelihood ratio tests. Simulations are used to compare the stepwise procedure involving I M (!) 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 (!) 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.Key Words: Trigonometric regression; Nonlinear models whose parameters separate; Least-squares estimation of frequencies; Finite-sample properties of periodogram statistics and frequency estimators; Analysis of discrete and mixed spectra 1064 DUTILLEUL

Multifrequential periodogram analysis of earthquake occurrence: An alternative approach to the Schuster spectrum, with two examples in central California

“…In Tables and , 5 to 10 period estimates are listed (from left to right), following the order in which the significant periodic components were detected in the MFPA stepwise procedure. This procedure, which includes a likelihood‐ratio F ‐test at each step, was assessed in a simulation study where the number of periodic components in the time series was known and has been shown to be more accurate in the estimation of this number than the stepwise procedure based on the Schuster periodogram [ Dutilleul , , ]. Period estimates close to one another (e.g., 13.1 and 11.7 months for CSAF‐PKD, original, M ≥ 2.5 in Table ) suggest a pseudoperiodic signal with an intermediate period (e.g., about 12 months) and fluctuations with varying amplitude.…”

“…The presence of several pseudoperiodic signals in a time series rapidly increases the number of frequency components found to be statistically significant in the MFPA stepwise procedure, because they are required to reproduce the signals and their characteristics. There is no statistical bias in the estimation because the MFPA is multifrequential and its statistic is modified to be exact between the Fourier frequencies [ Dutilleul , , ].…”

“…The fact that the Schuster periodogram may show one peak when there are actually two periodicities close to one another has been documented in simulation studies where the “true” periodic components of time series were known. Unlike the Schuster periodogram, the MFPA can resolve periodic components of 12 and 15 time units for series lengths of 50 and more [ Dutilleul , , ].…”

“…When K = 1 and the frequency ω 1 corresponds to an integer number of cycles over the time series, i.e., it is a Fourier frequency, then I M ( ω 1 ) = I C ( ω 1 ). Mathematical, statistical, and computational details about the method of MFPA can be found in Dutilleul []. In summary, (i) under general conditions, the statistic I M ( ω ) follows a chi‐square distribution with 2 K degrees of freedom; (ii) a stepwise procedure, which includes a likelihood‐ratio F ‐test at each step to assess the increase in the explanation of the variability of the time series by the addition of one more frequency in ω , eventually provides an estimate of the number of periodic components in the time series; and (iii) the maximization of I M ( ω ), performed for a given K , can be effective whether the stochastic process underlying the observed time series is autocorrelated or not.…”

“…Here, we explore seismicity catalogs spanning 12 to 26 years in two areas of central California, looking for periodicities in earthquake occurrence at periods theoretically ranging from 2 months (i.e., the shortest period at the monthly scale) to half of the time span (i.e., 6 to 13 years). To do this, we use the Schuster test and spectrum as well as the Schuster periodogram analysis and its multifrequential extension and generalization by Dutilleul [], called “multifrequential periodogram analysis” (MFPA) [ Dutilleul , ], and compare results. We consider both the complete and declustered versions of the event catalogs.…”