Quantification of LongRange Persistence in Geophysical Time Series: Conventional and BenchmarkBased Improvement Techniques
 Annette Witt,
 Bruce D. Malamud
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Abstract
Time series in the Earth Sciences are often characterized as selfaffine longrange persistent, where the power spectral density, S, exhibits a powerlaw dependence on frequency, f, S(f) ~ f ^{−β }, with β the persistence strength. For modelling purposes, it is important to determine the strength of selfaffine longrange 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 selfaffine longrange persistence, we compare four common analysis techniques for quantifying selfaffine longrange 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 selfaffine noises and motions with different (1) time series lengths N = 64, 128, 256, …, 131,072, (2) modelled persistence strengths β _{model} = −1.0, −0.8, −0.6, …, 4.0, and (3) onepoint probability distributions (Gaussian, lognormal: coefficient of variation c _{v} = 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 selfaffine noises with 1.2 ≤ β ≤ 2.8. (3) Detrended fluctuation analysis is well suited for time series with thintailed probability distributions and for persistence strengths of β ≥ 0.0. (4) Spectral techniques perform the best of all four techniques: for selfaffine noises with positive persistence (β ≥ 0.0) and symmetric onepoint distributions, they have no systematic errors and, compared to the other three techniques, small random errors; for antipersistent selfaffine noises (β < 0.0) and asymmetric onepoint probability distributions, spectral techniques have small systematic and random errors. For quantifying the strength of longrange persistence of a time series, benchmarkbased improvements to the estimator predicated on the performance for selfaffine noises with the same time series length and onepoint 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 longrange 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, lognormal, and Levy, respectively.
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 Title
 Quantification of LongRange Persistence in Geophysical Time Series: Conventional and BenchmarkBased Improvement Techniques
 Open Access
 Available under Open Access This content is freely available online to anyone, anywhere at any time.
 Journal

Surveys in Geophysics
Volume 34, Issue 5 , pp 541651
 Cover Date
 20130901
 DOI
 10.1007/s1071201292178
 Print ISSN
 01693298
 Online ISSN
 15730956
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Fractional noises and motions
 Selfaffine time series
 Longrange persistence
 Hurst rescaled range (R/S) analysis
 Semivariogram analysis
 Detrended fluctuation analysis
 Power spectral analysis
 Random and systematic errors
 Rootmeansquared error
 Confidence intervals
 Benchmarkbased improvements
 Geophysical time series
 Industry Sectors
 Authors

 Annette Witt ^{(1)} ^{(2)}
 Bruce D. Malamud ^{(2)}
 Author Affiliations

 1. Department of Nonlinear Dynamics, Max–Planck Institute for Dynamics and SelfOrganization, Postfach 2853, 37018, Göttingen, Germany
 2. Department of Geography, Earth and Environmental Dynamics Research Group, King’s College London, Strand, London, WC2R 2LS, UK