Abstract
We review recent studies of the statistics of return intervals (i) in long-term correlated monofractal records and (ii) in multifractal records in the absence (or presence) of linear long-term correlations. We show that for the monofractal records which are long-term power-law correlated with exponent γ, the distribution density of the return intervals follows a stretched exponential with the same exponent γ and the return intervals are long-term correlated, again with the same exponent γ. For the multifractal record, significant differences in scaling behavior both in the distribuiton and correlation behavior of return intervals between large events of different magnitudes are demonstrated. In the absence of linear long-term correlations, the nonlinear correlations contribute strongly to the statistics of the return intervals such that the return intervals become long-term correlated even though the original data are linearly uncorrelated (i.e., the autocorrelation function vanishes). The distribution density of the return intervals is mainly described by a power law.
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References
Altmann, E. G. and Kantz, H. (2005), Recurrence time analysis, long-term correlations, and extreme events, Phys. Rev. E 71, doi:10.1103/PhysRevE.71.056106.
Arneodo, A., Auzy, J. F., and Sornette, D. (1998), “Direct” causal cascade in the stock market, Eur. Phys. J. B 2, 277–282.
Bacry, E., Delour, J., and Muzy, J. F. (2001), Multifractal random walk, Phys. Rev. E 64, doi:10.1103/PhysRevE.64.026103.
Bogachev, M. I., Eichner, J. F. and Bunde, A. (2007), The effect of nonlinear correlations on the statistics of return intervals in multifractal data sets, Phys. Rev. Lett. 99, doi:10.1103/PhysRevLett.99.240601.
Bogachev, M. I., Eichner, J. F. and Bunde, A. (2008), The effect of multifractality on the statistics of return intervals, Eur. Phys. J. Special Topics 161, 181–193.
Bogachev, M. I. and Bunde, A. (2008a), Interoccurrence times between large returns in financial markets, preprint.
Bogachev, M. I. and Aunde, A. (2008b), Risk Estimations in Multifractal Records: Applications to physiology and Financing, preprint.
Bouchaud, J.-P., Potters, M., and Meyer, M. (2000), Apparent multifractality in financial time series, Eur. Phys. J. B 13, 595–599.
Bunde, A., Eichner, J. F., Kantelhardt, J. W., and Havlin, S. (2005), Long-term memory: A natural mechanism for the clustering of extreme events and anomalous residual times in climate records, Phys. Rev. Lett. 94, doi:10.1103/PhysRevLett.94.048701.
Bunde, A., Eichner, J. F., Kantelhardt, J. W., and Havlin, S. (2002), The Science of Disasters — Climate Disruptions, Heart Attacks, and Market Crashes (eds. Bunde, A., Kropp, J., and Schellnhuber, H.-J., Springer-Verlag, Berlin, 2002).
Bunde, A. and Havlin, S. (1981), Fractals and Disordered Systems, (eds.: Bunde A., and Havlin, S., Springer-Verlag, Berlin, 1991).
Bunde, A., Eichner, J. F., Kantelhardt, J. W., and Havlin, S. (2003), The effect of long-term correlations on the return periods of rare events, Physica A 330, 1–7.
Bunde, A., Eichner, J. F., Kantelhardt J. W., and Havlin, S. (2004), Return intervals of rare events in records with long-term persistence, Physica A 342, 308–314.
Eichner, J. F., Koscielny-Bunde, E., Bunde, A., Havlin, S., and Schellnhuber, H.-J. (2003), Power-law persistence and trends in the atmosphere: A detailed study of long temperature records, Phys. Rev. E 68, doi:10.1103/PhysRevE.68.046133.
Eichner, J. F., Kantelhardt, J. W., Bunde, A., and Havlin, S. (2006a), Extreme value statistics in records with long-term persistence, Phys. Rev. E 73, doi:10.1103/PhysRevE.73.016130.
Eichner, J. F., Kantelhardt, J. W., Bunde, A., and Havlin, S. (2006b), Statistics of return intervals in long-term correlated records, Phys. Rev. E 75, doi:10.1103/PhysRevE.75.011128.
Feder, J., Fractals (New York, Plenum, 1989).
Fillol, J. (2003), Multifractality: Theory and evidence an application to the French stock market, Economics Bulletin 2, No. 31, 1–12.
Glaser, R., Klimageschichte Mitteleuropas (Wissenschaftliche Buchgesellschaft, Darmstadt, 2001).
Hurst, H. E., Black, R. P., and Simaika, Y. M., Long-term storage: An experimental study (Constable & Co. Ltd., London, 1965).
Ivanov, P. Ch., Amaral, L. A. N., Goldberger, A. L., Havlin, S., Rosenblum, M. G., Stanley, H. E., and Struzik, Z. R. (2001), From 1/f noise to multifracial cascades in heartbeat dynamics, Chaos 11, 641–652.
Kantelhardt, J. W., Penzel, T., Rostig, S., Becker, H. F., Havlin, S., and Bunde, A. (2003), Breathing during REM and non-REM sleep: Correlated versus uncorrelated behaviour, Physica A 319, 447–457.
Kantelhardt, J. W., Koscielny-Bunde, E., Rybski, D., Braun, P., Bunde, A., and Havlin, S. (2006), Long-term persistence and multifractality of precipitation and river runoff records, J. Geophys. Res. (Atmos.) 111, D01106, doi:10.1029/2005JD005881.
Kantelhardt, J. W., Zschiegner, S. A., Koscielny-Bunde, E., Havlin, S., Bunde, A., and Stanley, H. E. (2002), Multifractal detrended fluctuation analysis of nonstationary time series, Physica A 316, 87–114.
Koscielny-Bunde, E., Bunde, A., Havlin, S., Roman, H. E., Goldreich, Y., and Schellnhuber, H.-J.(1998), Indication of a universal persistence law governing atmospheric variability, Phys. Rev. Lett. 81, doi:10.1103/ PhysRevLett.81.729.
Koscielny-Bunde, E., Kantelhardt, J. W., Braun, P., Bunde, A., and Havlin, S. (2006), Long-term persistence and multifractality of river runoff records: Detrended fluctuation studies, J. Hydrol. Vol. 322, 120–137.
Fractals in Biology and Medicine (ed. by Losa, G. A., Merlini, D., Nannenmacher, T. F., and Weibel, E. R.) (Birkhäuser, Basel, 2005).
Lovejoy, S. and Schertzer, D., Scaling nonlinear variability in geodynamics: Multiple singularities, observais, universality classes. In Nonlinear Variability in Geophysics: Scaling and Fractals (eds: Lovejoy, S., and Schertzer, D.) (Kluwer Academic Publ., Dordecht, Netherlands, 1991).
Makse, H. A., Havlin, S., Schwartz, M., and Stanley, H. E. (1996), Method for generating long-range correlations for large systems, Phys. Rev. E 53, doi:10.1103/PhysRevE.53.5445.
Mandelbrot, B. B. (1974), Intermittent turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier, J. Fluid Mech. 62, 331–358.
Mandelbrot, B. B., Fisher, A., and Calvet, L. (1997), A multifractal model for asset returns, Cowles Foundation Discussion Paper No. 1164, Working Paper, Yale University.
Mudelsee, M., Börngen, M., Tetzlaff, G., and Grünwald, U. (2003), No upward trends in the occurrence of extreme floods in central Europe, Nature, 425, 166–169.
Muzy, J. F., Bacry, E., and Kozhemyak, A. (2006), Extreme values and fat tails of multifractal fluctuations, Phys. Rev. E 73, doi:10.1103/PhysRevE.73.066114.
Pfister, C., Wetternachhersage, 500 Jahre Klimavariationen und Naturkatastrophen 1496–1995, (Verlag Paul Haupt, Bern, 1998).
Schreiber, T., and Schmitz, A. (1996), Improved surrogate data for nonlinearity tests, Phys. Rev. Lett. 77, doi:10.1103/PhysRevLett.77.635.
Stanley, H. E., Amaral, L. A. N., Goldberger, A. L., Havlin, S., Ivanov, P. Ch. and Peng, C.-K. (1999), Statistical physics and physiology: Monofractal and multifractal approaches, Physica A 270, 309–324.
Turcotte, D. L., Fractals and Chaos in Geology and Geophysics (Cambridge Univ. Press, 1992).
v. Storch, H. and Zwiers, F. W., Statistical Analysis in Climate Research (Cambridge Univ. Press, 2001).
Yamasaki, K., Muchnik, L., Havlin, S., Bunde, A., and Stanley, H. E. (2005), Scaling and memory in volatility return intervals in financial markets, PNAS 102, 9424–9428.
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Bogachev, M.I., Eichner, J.F., Bunde, A. (2008). On the Occurence of Extreme Events in Long-term Correlated and Multifractal Data Sets. In: Camacho, A.G., Díaz, J.I., Fernández, J. (eds) Earth Sciences and Mathematics. Pageoph Topical Volumes. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8907-9_11
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