Long-Range Persistence Techniques Evaluated

Many time series in the Earth Sciences exhibit persistence (memory) where large values (small values) `cluster' together. Here we examine long-range persistence, where one value is correlated with all others in the time series. A time series is long-range persistent (a self-affine fractal) if the power spectral density scales with a power law. The scaling exponent beta characterizes the `strength' of persistence. We compare five common analysis techniques for quantifying long-range persistence: (a) Power-spectral analysis, (b) Wavelet variance analysis, (c) Detrended Fluctuation analysis, (d) Semivariogram analysis, and (e) Rescaled-Range (R/S) analysis. To evaluate these methods, we construct 26,000 synthetic fractional noises with lengths between 512 and 4096, different persistence strengths, different distributions (normal, log-normal, levy), and using different construction methods: Fourier filtering, discrete wavelets, random additions, and Mandelbrot `cartoon' Brownian motions. We find: (a) Power-spectral and wavelet analyses are the most robust for measuring long-range persistence across all beta, although `antipersistence' is over-estimated for non- Gaussian time series. (b) Detrended Fluctuation Analysis is appropriate for signals with long-range persistence strength beta between -0.2 and 2.8 and has very large 95% confidence intervals for non-Gaussian signals. (c) Semivariograms are appropriate for signals with long-range persistence strength between 1.0 and 2.8; it has large confidence intervals and systematically underestimates log-normal noises in this range. (d) Rescaled- Range Analysis is only accurate for beta of about 0.7. We conclude some techniques are much better suited than others for quantifying long-range persistence, and the resultant beta (and associated error bars on them) are sensitive to the one point probability distribution, the length of the time series, and the techniques used.