Long memory time series



Hurst exponents, non-stationarity etc. I used to do a lot of work in this area, but have not now for so long that I no longer claim any authority.

TBD.

Many interesting things here, but for now, note that many natural generic models of long-memory in time series turn out ot be fractal models, so note power laws, \(1/f\) noise, fractional brownian motion etc. Link to branching processes.

References

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———. 1994. Statistics for Long-Memory Processes. CRC Press.
———. 2010. “Long-Range Dependence.” Wiley Interdisciplinary Reviews: Computational Statistics 2 (1): 26–35. https://doi.org/10.1002/wics.52.
Beran, Jan, and Norma Terrin. 1996. “Testing for a Change of the Long-Memory Parameter.” Biometrika 83 (3): 627–38. https://doi.org/10.1093/biomet/83.3.627.
Berkes, István, Lajos Horváth, Piotr Kokoszka, and Qi-Man Shao. 2006. “On Discriminating Between Long-Range Dependence and Changes in Mean.” The Annals of Statistics 34 (3): 1140–65. https://doi.org/10.1214/009053606000000254.
Brouste, Alexandre, Jacques Istas, and Sophie Lambert-Lacroix. 2016. “Conditional Fractional Gaussian Fields with the Package FieldSim.” R JOURNAL 8 (1): 38–47. http://perso.univ-lemans.fr/~abrouste/work/BIL15.pdf.
Chen, Yanguang. 2011. “Zipf’s Law, 1/f Noise, and Fractal Hierarchy.” Chaos, Solitons & Fractals, November, S0960077911001901. https://doi.org/10.1016/j.chaos.2011.10.001.
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Dieker, Ton. 2004. “Simulation of Fractional Brownian Motion.” MSc Theses, University of Twente, Amsterdam, The Netherlands. http://www.columbia.edu/~ad3217/fbm/thesisold.pdf.
Doukhan, P, G Oppenheim, and M S Taqqu. 2003. Theory and Applications of Long-Range Dependence. Birkhauser.
Giraitis, L, and D Surgailis. 1999. “Central Limit Theorem for the Empirical Process of a Linear Sequence with Long Memory.” Journal of Statistical Planning and Inference 80 (1-2): 81–93.
Granger, C. W. J., and Roselyne Joyeux. 1980. “An Introduction to Long-Memory Time Series Models and Fractional Differencing.” Journal of Time Series Analysis 1 (1): 15–29. https://doi.org/10.1111/j.1467-9892.1980.tb00297.x.
Greaves-Tunnell, Alexander, and Zaid Harchaoui. 2019. “A Statistical Investigation of Long Memory in Language and Music.” arXiv:1904.03834 [cs, Eess, Stat], June. http://arxiv.org/abs/1904.03834.
Hurvich, Clifford M. 2002. “Multistep Forecasting of Long Memory Series Using Fractional Exponential Models.” International Journal of Forecasting, Forecasting Long Memory Processes, 18 (2): 167–79. https://doi.org/10.1016/S0169-2070(01)00151-0.
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Lahiri, S N. 1993. “On the Moving Block Bootstrap Under Long Range Dependence.” Statistics & Probability Letters 18 (5): 405–13. https://doi.org/10.1016/0167-7152(93)90035-H.
Mandelbrot, Benoit B. 1999. Multifractals and 1/ƒ Noise: Wild Self-Affinity in Physics (1963–1976). New York: Springer-Verlag. https://doi.org/10.1007/978-1-4612-2150-0.
McCauley, Joseph L. 2008. “Time Vs. Ensemble Averages for Nonstationary Time Series.” Physica A: Statistical Mechanics and Its Applications 387 (22): 5518–22. https://doi.org/10.1016/j.physa.2008.05.057.
McLeod, A. Ian. 1998. “Hyperbolic Decay Time Series.” Journal of Time Series Analysis 19 (4): 473–83. https://doi.org/10.1111/1467-9892.00104.
Papavasiliou, Anastasia, and Kasia B. Taylor. 2016. “Approximate Likelihood Construction for Rough Differential Equations.” arXiv:1612.02536 [math, Stat], December. http://arxiv.org/abs/1612.02536.
Pipiras, Vladas, and Murad S. Taqqu. 2017. Long-Range Dependence and Self-Similarity. Cambridge Series in Statistical and Probabilistic Mathematics 45. Cambridge, United Kingdom ; New York, NY, USA: Cambridge University Press.
Saichev, A. I., and D. Sornette. 2010. “Generation-by-Generation Dissection of the Response Function in Long Memory Epidemic Processes.” The European Physical Journal B 75 (3): 343–55. https://doi.org/10.1140/epjb/e2010-00121-7.
Schmitt, Francois G., and Yongxiang Huang. 2016. Stochastic Analysis of Scaling Time Series: From Turbulence Theory to Applications. Cambridge: Cambridge University Press.

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