Abstract
Time series in the Earth Sciences are often characterized as self-affine long-range persistent, where the power spectral density, S, exhibits a power-law dependence on frequency, f, S(f) ~ f−β, with β the persistence strength. For modelling purposes, it is important to determine the strength of self-affine long-range persistence β as precisely as possible and to quantify the uncertainty of this estimate. After an extensive review and discussion of asymptotic and the more specific case of self-affine long-range persistence, we compare four common analysis techniques for quantifying self-affine long-range persistence: (a) rescaled range (R/S) analysis, (b) semivariogram analysis, (c) detrended fluctuation analysis, and (d) power spectral analysis. To evaluate these methods, we construct ensembles of synthetic self-affine noises and motions with different (1) time series lengths N = 64, 128, 256, …, 131,072, (2) modelled persistence strengths βmodel = −1.0, −0.8, −0.6, …, 4.0, and (3) one-point probability distributions (Gaussian, log-normal: coefficient of variation cv = 0.0 to 2.0, Levy: tail parameter a = 1.0 to 2.0) and evaluate the four techniques by statistically comparing their performance. Over 17,000 sets of parameters are produced, each characterizing a given process; for each process type, 100 realizations are created. The four techniques give the following results in terms of systematic error (bias = average performance test results for β over 100 realizations minus modelled β) and random error (standard deviation of measured β over 100 realizations): (1) Hurst rescaled range (R/S) analysis is not recommended to use due to large systematic errors. (2) Semivariogram analysis shows no systematic errors but large random errors for self-affine noises with 1.2 ≤ β ≤ 2.8. (3) Detrended fluctuation analysis is well suited for time series with thin-tailed probability distributions and for persistence strengths of β ≥ 0.0. (4) Spectral techniques perform the best of all four techniques: for self-affine noises with positive persistence (β ≥ 0.0) and symmetric one-point distributions, they have no systematic errors and, compared to the other three techniques, small random errors; for anti-persistent self-affine noises (β < 0.0) and asymmetric one-point probability distributions, spectral techniques have small systematic and random errors. For quantifying the strength of long-range persistence of a time series, benchmark-based improvements to the estimator predicated on the performance for self-affine noises with the same time series length and one-point probability distribution are proposed. This scheme adjusts for the systematic errors of the considered technique and results in realistic 95 % confidence intervals for the estimated strength of persistence. We finish this paper by quantifying long-range persistence (and corresponding uncertainties) of three geophysical time series—palaeotemperature, river discharge, and Auroral electrojet index—with the three representing three different types of probability distribution—Gaussian, log-normal, and Levy, respectively.
Highlights
Time series can be found in many areas of the Earth Sciences and other disciplines
In the two sections, we describe two techniques commonly found in the time series analysis literature for finding a best-fit power law to the power spectral density and will present the result of the power spectral analysis applied to the windowed and unwindowed time series examples discussed above
We have applied these techniques to many realizations of fractional noises and motions with well-defined properties, and after discussing systematic and random errors in the context of a specific example (Sect. 7.1) and confidence intervals (Sect. 7.2), we will present the overall results of our performance tests and the results of other studies (Sect. 7.3), along with reference to the supplementary material which contains all of our results
Summary
Time series can be found in many areas of the Earth Sciences and other disciplines. After obvious periodicities and trends have been removed from a time series, the stochastic component remains. In this introduction section we introduce long-range persistence and its importance in the Earth Sciences. These three time series—palaeotemperature, river discharge, and Auroral electrojet index—represent three different types of one-point probability distribution—Gaussian, log-normal, and Levy, respectively.
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