# Dresden 2009 – wissenschaftliches Programm

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# DY: Fachverband Dynamik und Statistische Physik

## DY 10: Brownian motion and transport I

### DY 10.7: Vortrag

### Dienstag, 24. März 2009, 15:45–16:00, HÜL 386

**Performance Tests for Techniques that measure long-range Persistence in Time Series** — •Annette Witt^{1,2} and Bruce D. Malamud^{2} — ^{1}Max-Planck-Institute for Dynamics and Self-organization, Bernstein Center for Computational Neuroscience Göttingen, Göttingen, Germany — ^{2}Department of Geography, King's College London, United Kingdom

Many time series of complex systems exhibit long-range persistence, where the power spectral density scales with a power law. The corresponding scaling exponent beta characterizes the "strength" of persistence. We compare four common techniques for quantifying long-range persistence in time series: (a) Power-spectral analysis, (b) Detrended fluctuation analysis, (c) Semivariogram analysis, and (d) Rescaled-Range (R/S) analysis. To evaluate these methods, we construct synthetic fractional noises with lengths between 512 and 4096, different persistence strengths, and different distributions (Gaussian, log-normal, Levy). We empirically find: (i) Power-spectral analysis and detrended fluctuation analysis are unbiased across all beta, although anti-persistence is over-estimated for asymmetric distributed time series; (ii) Detrended Fluctuation Analysis has larger random errors than power-spectral analysis, in particular for non-Gaussian signals. (iii) Semivariograms are appropriate for signals with long-range persistence strength between 1.0 and 2.8; it has large confidence intervals and systematically underestimates beta for asymmetric distributed time series in this range; (iv) Rescaled-Range Analysis is only accurate for beta of about 0.7, and systematically under- or over-estimates for other values.