zbMATH — the first resource for mathematics

Self-affine time series: Measures of weak and strong persistence. (English) Zbl 1045.62529
Summary: In this paper, we examine self-affine time series and their persistence. Time series are defined to be self-affine if their power-spectral density scales as a power of their frequency. Persistence can be classified in terms of range, short or long range, and in terms of strength, weak or strong. Self-affine time series are scale-invariant, thus they always exhibit long-range persistence. Synthetic self-affine time series are generated using the Fourier power-spectral method. We generate fractional Gaussian noises (fGns), \(1\leq\beta\leq 1\), where \(\beta\) is the power-spectral exponent. These are summed to give fractional Brownian motions (fBms), \(1\leq\beta\leq 3\), where the series are self-affine fractals with fractal dimension \(1\leq D\leq2\); \(\beta=2\) is a Brownian motion. With \(\beta>1\), the time series are non-stationary and moments of the time series depend upon its length; with \(\beta<1\) the time series are stationary. We define self-affine time series with \(\beta>1\) to have strong persistence and with \(\beta<1\) to have weak persistence. We use a variety of techniques to quantify the strength of persistence of synthetic self-affine time series with \(3\leq\beta\leq 5\). These techniques are effective in the following ranges: (1) semivariograms, \(1\leq\beta\leq 3\), (2) rescaled-range (R/S) analyses, \(1\leq\beta\leq 1\), (3) Fourier spectral techniques, all values of \(\beta\), and (4) wavelet variance analyses, all values of \(\beta\). Wavelet variance analyses lack many of the inherent problems that are found in Fourier power-spectral analysis.

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
28A80 Fractals
longmemo; sapa
Full Text: DOI
[1] Ahnert, F., Local relief and the height limits of mountain ranges, Am. J. sci., 284, 1035-1055, (1984)
[2] Bassingthwaighte, J.B., Liebovitch, L.S., West, B.J., 1994. Fractal Physiology. Oxford Univ. Press, New York.
[3] Bassingthwaighte, J.B.; Raymond, G.M., Evaluating rescaled range analysis for time series, Ann. biomed. eng., 22, 432-444, (1994)
[4] Bassingthwaighte, J.B.; Raymond, G.M., Evaluation of the dispersional analysis method for fractal time series, Ann. biomed. eng., 23, 491-505, (1995)
[5] Beran, J., 1994. Statistics for Long-Memory Processes. Monographs on Statistics and Probability 61. Chapman & Hall, New York.
[6] Box, G.E.P., Jenkins, G.M., Reinsel, G.C., 1994. Time Series Analysis: Forecasting and Control, 3rd ed. Prentice-Hall, Englewood Cliffs, NJ. · Zbl 0858.62072
[7] Chatfield, C., 1996. The Analysis of Time Series, 5th ed. Chapman & Hall, London. · Zbl 0870.62068
[8] Daubechies, I., Orthonormal bases of compactly supported wavelets, Commun. pure appl. math., 41, 909-996, (1988) · Zbl 0644.42026
[9] Family, F., Scaling of rough surfaces: effects of surface diffusion, J. phys. A math. gen., 19, L441-L446, (1986)
[10] Fasham, M.J.R., The statistical and mathematical analysis of plankton patchiness, Oceanogr. mar. biol. ann. rev., 16, 43-79, (1978)
[11] Feder, J., 1988. Fractals. Plenum Press, New York. · Zbl 0648.28006
[12] Flandrin, P., 1993. Fractional Brownian motion and wavelets. In: Farge, M., Hunt, J.C.R., Vassilicos, J.C. (Eds.), Wavelets, Fractals, and Fourier Transforms, Oxford Univ. Press, Oxford, pp. 109-123.
[13] Gallant, J.C.; Moore, I.D.; Hutchinson, M.F.; Gessler, P., Estimating fractal dimension of profiles: a comparison of methods, Math. geol., 26, 455-481, (1994)
[14] Grossmann, A.; Morlet, J., Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. math. anal., 15, 723-736, (1984) · Zbl 0578.42007
[15] Hastings, H.M., Sugihara, G., 1993. Fractals: A User’s Guide for the Natural Sciences. Oxford Univ. Press, Oxford. · Zbl 0820.28003
[16] Houghton, J.T., Callendar, B.A. (Eds.), 1995. Climate Change, The IPCC Scientific Assessment. Cambridge Univ. Press, New York.
[17] Hsui, A.T.; Rust, K.A.; Klein, G.D., A fractal analysis of quaternary, cenozoic-mesozoic, and late Pennsylvanian sea-level changes, J. geophys. res., 98, 21, 963-967, (1993)
[18] Hubbard, B.B., 1996. The World According to Wavelets: The Story of a Mathematical Technique in the Making. A.K. Peters, Wellesley. · Zbl 0855.42020
[19] Hurst, H.E., Long-term storage capacity of reservoirs, Trans. amer. soc. civil engrs., 116, 770-808, (1951)
[20] Hurst, H.E., Black, R.P., Simaika, Y.M., 1965. Long-term Storage. Constable, London.
[21] Lambeck, K., Lateral density anomalies in the upper mantle, J. geophys. res., 81, 6333-6340, (1976)
[22] Li, W.K.; McLeod, A.I., Fractional time series modeling, Biometrika, 73, 217-221, (1986)
[23] Malamud, B.D., Turcotte, D.L., 1999. Self-affine time series: I. Generation and analyses. Adv. Geophys., in press. · Zbl 1045.62529
[24] Mandelbrot, B.B., How long is the coast of britain? statistical self-similarity and fractional dimension, Science, 156, 636-638, (1967)
[25] Mandelbrot, B.B., 1982. The Fractal Geometry of Nature. Freeman, New York. · Zbl 0504.28001
[26] Mandelbrot, B.B., Self-affine fractals and fractal dimension, Phys. scripta, 32, 257-260, (1985) · Zbl 1063.28500
[27] Mandelbrot, B.B.; Van Ness, J.W., Fractional Brownian motions, fractional noises and applications, SIAM rev., 10, 422-437, (1968) · Zbl 0179.47801
[28] Mandelbrot, B.B.; Wallis, J.R., Noah, Joseph and operational hydrology, Water resour. res., 4, 909-918, (1968)
[29] Mandelbrot, B.B.; Wallis, J.R., Computer experiments with fractional Gaussian noises. parts I, II, and III, Water resour. res., 5, 228-267, (1969)
[30] Mandelbrot, B.B.; Wallis, J.R., Some long-run properties of geophysical records, Water resour. res., 5, 321-340, (1969)
[31] Mandelbrot, B.B.; Wallis, J.R., Robustness of the rescaled range R/S in the measurement of noncyclic long run statistical dependence, Water resour. res., 5, 967-988, (1969)
[32] Matheron, G., Principles of geostatistics, Econom. geol., 58, 1246-1266, (1963)
[33] McKinney, M.L.; Frederick, D., Extinction and population dynamics: new methods and evidence from paleogene foraminifera, Geology, 20, 343-346, (1992)
[34] McLeod, M.G., Spatial and temporal power spectra of the geomagnetic field, J. geophys. res., 101, 2745-2763, (1996)
[35] McLeod, A.I., Hipel, K.W., 1995. The McLeod-Hipel Time-Series Datasets Collection. Electronic data available from the StatLib electronic database, Carnegie Mellon University Statistics Department. The datasets collection accompanies Hipel, K.W., McLeod, A.I., 1994. Time Series Modeling of Water Resources and Environmental Systems, Elsevier, Amsterdam.
[36] Meynadier, L.; Valet, J.-P.; Bassonot, F.C.; Shackleton, N.J.; Guyodo, Y., Asymmetrical saw-tooth pattern of the geomagnetic field intensity from equatorial sediments in the Pacific and Indian oceans, Earth plan. sci. lett., 126, 109-127, (1994)
[37] Muller, P., 1996. Stochastic forcing of quasi-geostrophic eddies. In: Adler, R.J., Muller, P., Rozovskii, B. (Eds.), Stochastic Modeling in Physical Oceanography, Birkhauser, Boston, pp. 381-395. · Zbl 0865.76077
[38] Musha, T.; Higuchi, H., The 1/f fluctuation of a traffic current of an expressway, Jpn. J. appl. phys., 15, 1271-1275, (1976)
[39] Osborne, A.R.; Provenzale, A., Finite correlation dimension for stochastic systems with power-law spectra, Physica, D35, 357-381, (1989) · Zbl 0671.60030
[40] Palmer, M.W., Fractal geometry: a tool for describing spatial patterns of plant communities, Vegetatio, 75, 91-102, (1988)
[41] Passier, M.L.; Snieder, R.K., On the presence of intermediate-scale heterogeneity in the upper mantle, Geophys. J. int., 123, 817-837, (1995)
[42] Pelletier, J.D., Variations in solar luminosity from time scales of minutes to months, Astrophys. J., 463, L41-L45, (1996)
[43] Pelletier, J.D., Analysis and modeling of the natural variability of climate, J. climate, 10, 1331-1342, (1997)
[44] Percival, D.B., Walden, A.T., 1993. Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques. Cambridge Univ. Press, Cambridge. · Zbl 0796.62077
[45] Platt, T.; Denman, K.L., Spectral analysis in ecology, Ann. rev. ecol. systems, 6, 189-210, (1975)
[46] Plotnick, R.E., Prestegaard, K., 1993. Fractal analysis of geologic time series. In: De Cola, L., Lam, S. (Eds.), Fractals in Geography, Prentice-Hall, Englewood Cliffs, NJ, pp. 193-210.
[47] Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 1994. Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. Cambridge Univ. Press, Cambridge. · Zbl 0661.65001
[48] Priestley, M.B., 1981. Spectral Analysis and Time Series. Academic Press, London. · Zbl 0537.62075
[49] Schepers, H.E., van Beek, J.H.G.M., Bassingthwaighte, J.B., 1992. Four methods to estimate the fractal dimension from self-affine signals. IEEE Eng. Med. Biol. 11, 57-64, 71.
[50] Schmittbuhl, J.; Vilotte, J-P.; Roux, S., Reliability of self-affine measurements, Phys. rev., E51, 131-147, (1995)
[51] Sugihara, G.; May, R.M., Applications of fractals in ecology, Trends in ecol. evol., 5, 79-86, (1990)
[52] Takayasu, M.; Takayasu, H., 1/f noise in a traffic model, Fractals, 1, 860-866, (1993) · Zbl 0906.90072
[53] Tapiero, C.S.; Vallois, P., Run length statistics and the Hurst exponent in random and birth-death random motions, Chaos, solitons fractals, 7, 1333-1341, (1996) · Zbl 1080.60506
[54] Tjemkes, S.A.; Visser, M., Horizontal variability of temperature, specific humidity, and cloud liquid water as derived from spaceborne observations, J. geophys. res., 99, 23089-23105, (1994)
[55] Turcotte, D.L., A fractal interpretation of topography and geoid spectra on the Earth, Moon, venus, and Mars, J. geophys. res., 92, 597-601, (1987)
[56] Turcotte, D.L., 1997. Fractals and Chaos in Geology and Geophysics, 2nd ed., Cambridge Univ. Press, Cambridge. · Zbl 0785.58005
[57] Van Kampen, N.G., 1981. Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam. · Zbl 0511.60038
[58] Voss, R.F., 1985a. Random fractals: characterization and measurement. In: Pynn, R., Skjeltorp, A. (Eds.), Scaling Phenomena in Disordered Systems. Plenum Press, New York, pp. 1-11.
[59] Voss, R.F., 1985b. Random fractal forgeries: from mountains to music. In: Nash, S. (Ed.), Science and Uncertainty. Science Reviews, London, pp. 69-85.
[60] Voss, R.F., 1985c. Random fractal forgeries. In: Earnshaw, R.A. (Ed.), Fundamental Algorithms for Computer Graphics, NATO ASI Series. Springer, Berlin, F17, pp. 805-835.
[61] Voss, R.F., 1988. Fractals in nature: from characterization to simulation. In: Peitgen, H.-O., Saupe, D. (Eds.), The Science of Fractal Images. Springer, New York, pp. 22-70.
[62] Wornell, G.W., 1996. Signal Processing with Fractals: A Wavelet-based Approach. Prentice-Hall, Englewood Cliffs, NJ.
[63] Wunsch, C.; Stammer, D., The global frequency-wavenumber spectrum of oceanic variability estimated from TOPEX/POSEIDON altimetric measurements, J. geophys. res., 100, 24895-24910, (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.