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The effect of long-term correlations on the return periods of rare events

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  • Bunde, Armin
  • F. Eichner, Jan
  • Havlin, Shlomo
  • Kantelhardt, Jan W.

Abstract

The basic assumption of common extreme value statistics is that different events in a time record are uncorrelated. In this case, the return intervals rq of events above a given threshold size q are uncorrelated and follow the Poisson distribution. In recent years there is growing evidence that several hydro-meteorological and physiological records of interest (e.g. river flows, temperatures, heartbeat intervals) exhibit long-term correlations where the autocorrelation function decays as Cx(s)∼s−γ, with γ between 0 and 1. Here we study how the presence of long-term correlations changes the statistics of the return intervals rq. We find that (a) the mean return intervals Rq=〈rq〉 are independent of γ, (b) the distribution of the rq follows a stretched exponential, lnPq(r)∼−(r/Rq)γ, and (c) the return intervals are long-term correlated with an exponent γ′ close to γ.

Suggested Citation

  • Bunde, Armin & F. Eichner, Jan & Havlin, Shlomo & Kantelhardt, Jan W., 2003. "The effect of long-term correlations on the return periods of rare events," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 330(1), pages 1-7.
    • Handle: RePEc:eee:phsmap:v:330:y:2003:i:1:p:1-7
    DOI: 10.1016/j.physa.2003.08.004
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    Citations

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    Cited by:

    1. Pushpa Dissanayake & Teresa Flock & Johanna Meier & Philipp Sibbertsen, 2021. "Modelling Short- and Long-Term Dependencies of Clustered High-Threshold Exceedances in Significant Wave Heights," Mathematics, MDPI, vol. 9(21), pages 1-33, November.
    2. M. Ghil & Pascal Yiou & Stéphane Hallegatte & B. D. Malamud & P. Naveau & A. Soloviev & P. Friederichs & V. Keilis-Borok & D. Kondrashov & V. Kossobokov & O. Mestre & C. Nicolis & H. W. Rust & P. Sheb, 2011. "Extreme events: dynamics, statistics and prediction," Post-Print hal-00716514, HAL.
    3. Ren, Fei & Gu, Gao-Feng & Zhou, Wei-Xing, 2009. "Scaling and memory in the return intervals of realized volatility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(22), pages 4787-4796.
    4. Czechowski, Zbigniew & Telesca, Luciano, 2024. "Effect of nonlinearity of discrete Langevin model on behavior of extremes in generated time series," Chaos, Solitons & Fractals, Elsevier, vol. 183(C).
    5. Ren, Fei & Guo, Liang & Zhou, Wei-Xing, 2009. "Statistical properties of volatility return intervals of Chinese stocks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(6), pages 881-890.
    6. Picoli, Sergio & Bombo, Giorgio & Santos, Edenize S.D. & Deprá, Pedro P. & Mendes, Renio S., 2022. "Characterizing postural sway signals by the analysis of zero-crossing patterns," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 596(C).
    7. Xie, Wen-Jie & Jiang, Zhi-Qiang & Zhou, Wei-Xing, 2014. "Extreme value statistics and recurrence intervals of NYMEX energy futures volatility," Economic Modelling, Elsevier, vol. 36(C), pages 8-17.
    8. Livina, V. & Tuzov, S. & Havlin, S. & Bunde, A., 2005. "Recurrence intervals between earthquakes strongly depend on history," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 348(C), pages 591-595.
    9. Karain, Wael I., 2019. "Investigating large-amplitude protein loop motions as extreme events using recurrence interval analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 520(C), pages 1-10.

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