Time series, periodograms, and significance

The geophysical literature shows a wide and conflicting usage of methods employed to extract meaningful information on coherent oscillations from measurements. This makes it difficult, if not impossible, to relate the findings reported by different authors, Therefore, we have undertaken a critical investigation of the tests and methodology used for determining the presence of statistically significant coherent oscillations in periodograms derived from time series. Statistical significance tests are only valid when performed on the independent frequencies present in a measurement. Both the number of possible independent frequencies in a periodogram and the significance tests are determined by the number of degrees of freedom, which is the number of true independent measurements, present in the time series, rather than the number of sample points in the measurement. The number of degrees of freedom is an intrinsic property of the data, and it must be determined from the serial coherence of the time series. As part of this investigation, a detailed study has been performed which clearly illustrates the deleterious effects that the apparently innocent and commonly used processes of filtering, de-trending, and tapering of data have on periodogram analysis and the consequent difficulties in the interpretation of the statistical significance thus derived. For the sake of clarity, a specific example of actual field measurements containing unevenly-spaced measurements, gaps, etc., as well as synthetic examples, have been used to illustrate the periodogram approach, and pitfalls, leading to the (statistical) significance tests for the presence of coherent oscillations. Among the insights of this investigation are: (1) the concept of a time series being (statistically) band limited by its own serial coherence and thus having a critical sampling rate which defines one of the necessary requirements for the proper statistical design of an experiment; (2) the design of a critical test for the maximum number of significant frequencies which can be used to describe a time series, while retaining intact the variance of the test sample; (3) a demonstration of the unnecessary difficulties that manipulation of the data brings into the statistical significance interpretation of said data; and (4) the resolution and correction of the apparent discrepancy in significance results obtained by the use of the conventional Lomb-Scargle significance test, when compared with the long-standing Schuster-Walker and Fisher tests.