Spectral and cross-spectral analysis of uneven time series with the smoothed Lomb–Scargle periodogram and Monte Carlo evaluation of statistical significance

Many spectral analysis techniques have been designed assuming sequences taken with a constant sampling interval. However, there are empirical time series in the geosciences (sediment cores, fossil abundance data, isotope analysis, …) that do not follow regular sampling because of missing data, gapped data, random sampling or incomplete sequences, among other reasons. In general, interpolating an uneven series in order to obtain a succession with a constant sampling interval alters the spectral content of the series. In such cases it is preferable to follow an approach that works with the uneven data directly, avoiding the need for an explicit interpolation step. The Lomb–Scargle periodogram is a popular choice in such circumstances, as there are programs available in the public domain for its computation. One new computer program for spectral analysis improves the standard Lomb–Scargle periodogram approach in two ways: (1) It explicitly adjusts the statistical significance to any bias introduced by variance reduction smoothing, and (2) it uses a permutation test to evaluate confidence levels, which is better suited than parametric methods when neighbouring frequencies are highly correlated.Another novel program for cross-spectral analysis offers the advantage of estimating the Lomb–Scargle cross-periodogram of two uneven time series defined on the same interval, and it evaluates the confidence levels of the estimated cross-spectra by a non-parametric computer intensive permutation test. Thus, the cross-spectrum, the squared coherence spectrum, the phase spectrum, and the Monte Carlo statistical significance of the cross-spectrum and the squared-coherence spectrum can be obtained.Both of the programs are written in ANSI Fortran 77, in view of its simplicity and compatibility. The program code is of public domain, provided on the website of the journal (http://www.iamg.org/index.php/publisher/articleview/frmArticleID/112/).Different examples (with simulated and real data) are described in this paper to corroborate the methodology and the implementation of these two new programs.

► The task is spectral and cross-spectral analysis of uneven time series.
► The Lomb–Scargle method is extended to cross-spectral analysis.
► Uncertainty is estimated by an intensive Monte Carlo method.
► The methods are illustrated with synthetic and real data.