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Mathematical Problems in Engineering
Volume 2015, Article ID 485623, 7 pages
http://dx.doi.org/10.1155/2015/485623
Research Article

Correlation Properties of (Discrete) Fractional Gaussian Noise and Fractional Brownian Motion

EA 2991 Movement to Health, Euromov, University of Montpellier, 34090 Montpellier, France

Received 13 April 2015; Revised 9 July 2015; Accepted 2 August 2015

Academic Editor: Paul Bogdan

Copyright © 2015 Didier Delignières. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. L. Goldberger, L. A. N. Amaral, J. M. Hausdorff, P. C. Ivanov, C.-K. Peng, and H. E. Stanley, “Fractal dynamics in physiology: alterations with disease and aging,” Proceedings of the National Academy of Sciences of the United States of America, vol. 99, no. 1, pp. 2466–2472, 2002. View at Publisher · View at Google Scholar · View at Scopus
  2. S. Wallot, R. Fusaroli, K. Tylén, and E.-M. Jegindø, “Using complexity metrics with R-R intervals and BPM heart rate measures,” Frontiers in Physiology, vol. 4, article 211, 2013. View at Publisher · View at Google Scholar · View at Scopus
  3. P. Herman, B. G. Sanganahalli, F. Hyder, and A. Eke, “Fractal analysis of spontaneous fluctuations of the BOLD signal in rat brain,” NeuroImage, vol. 58, no. 4, pp. 1060–1069, 2011. View at Publisher · View at Google Scholar · View at Scopus
  4. T. Montez, S.-S. Poil, B. F. Jones et al., “Altered temporal correlations in parietal alpha and prefrontal theta oscillations in early-stage Alzheimer disease,” Proceedings of the National Academy of Sciences of the United States of America, vol. 106, no. 5, pp. 1614–1619, 2009. View at Publisher · View at Google Scholar · View at Scopus
  5. P. J. Fadel, S. M. Barman, S. W. Phillips, and G. L. Gebber, “Fractal fluctuations in human respiration,” Journal of Applied Physiology, vol. 97, no. 6, pp. 2056–2064, 12004. View at Publisher · View at Google Scholar
  6. M. Latka, M. Glaubic-Latka, D. Latka, and B. J. West, “Fractal rigidity in migraine,” Chaos, Solitons and Fractals, vol. 20, no. 1, pp. 165–170, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  7. J. J. Sosnoff, A. D. Valantine, and K. M. Newell, “The adaptive range of 1/f isometric force production,” Journal of Experimental Psychology: Human Perception and Performance, vol. 35, no. 2, pp. 439–446, 2009. View at Publisher · View at Google Scholar · View at Scopus
  8. D. G. Stephen and J. Anastas, “Fractal fluctuations in gaze speed visual search,” Attention, Perception, and Psychophysics, vol. 73, no. 3, pp. 666–677, 2011. View at Publisher · View at Google Scholar · View at Scopus
  9. D. L. Gilden, T. Thornton, and M. W. Mallon, “1/F Noise in human cognition,” Science, vol. 267, no. 5205, pp. 1837–1839, 1995. View at Publisher · View at Google Scholar · View at Scopus
  10. L. Lemoine, K. Torre, and D. Delignières, “Testing for the presence of 1/f noise in continuation tapping data,” Canadian Journal of Experimental Psychology, vol. 60, no. 4, pp. 247–257, 2006. View at Publisher · View at Google Scholar · View at Scopus
  11. K. Torre, D. Delignières, and L. Lemoine, “1/f β fluctuations in bimanual coordination: an additional challenge for modeling,” Experimental Brain Research, vol. 183, no. 2, pp. 225–234, 2007. View at Publisher · View at Google Scholar · View at Scopus
  12. J. U. Mafahim, D. Lambert, M. Zare, and P. Grigolini, “Complexity matching in neural networks,” New Journal of Physics, vol. 17, Article ID 015003, 2015. View at Publisher · View at Google Scholar · View at Scopus
  13. B. B. Mandelbrot and J. W. Van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Review, vol. 10, no. 4, pp. 422–437, 1968. View at Publisher · View at Google Scholar
  14. B. E. Hurst, “Long-term storage capacity of reservoirs,” Transactions of the American Society of Civil Engineers, vol. 116, pp. 770–799, 1951. View at Google Scholar
  15. M. Matsuzaki, “Fractals in earthquakes,” Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences, vol. 348, pp. 449–457, 1994. View at Publisher · View at Google Scholar
  16. W. E. Leland, M. S. Taqqu, W. Willinger, and D. V. Wilson, “On the self-similar nature of Ethernet traffic (extended version),” IEEE/ACM Transactions on Networking, vol. 2, no. 1, pp. 1–15, 1994. View at Publisher · View at Google Scholar · View at Scopus
  17. L. A. Lipsitz and A. L. Goldberger, “Loss of `complexity' and aging: potential applications of fractals and chaos theory to senescence,” The Journal of the American Medical Association, vol. 267, no. 13, pp. 1806–1809, 1992. View at Publisher · View at Google Scholar · View at Scopus
  18. J. Beran, Statistics for Long-Memory Processes, CRC Press, 1994.
  19. H. Qian, “Fractional Brownian motion and fractional Gaussian noise,” in Processes with Long-Range Correlations: Theory and Applications, G. Rangarajan and M. Ding, Eds., vol. 621 of Lecture Notes in Physics, pp. 22–33, Springer, Berlin, Germany, 2003. View at Publisher · View at Google Scholar
  20. O. Magre and M. Guglielmi, “Approximation de l'autocorrélation des incréments du (fbm) de Mandelbrot par modélisation de Barnes et Allan,” Colloques sur le Traitement du Signal et des Images, vol. 15, pp. 5–8, 1995. View at Google Scholar
  21. A. M. Wing and A. B. Kristofferson, “Response delays and the timing of discrete motor responses,” Perception & Psychophysics, vol. 14, no. 1, pp. 5–12, 1973. View at Publisher · View at Google Scholar · View at Scopus
  22. D. Delignières and V. Marmelat, “Theoretical and methodological issues in serial correlation analysis,” in Progress in Motor Control: Neural, Computational and Dynamic Approaches, vol. 782 of Advances in Experimental Medicine and Biology, pp. 127–148, Springer, Berlin, Germany, 2013. View at Publisher · View at Google Scholar
  23. A. Eke, P. Hermán, J. B. Bassingthwaighte et al., “Physiological time series: distinguishing fractal noises from motions,” Pflügers Archiv—European Journal of Physiology, vol. 439, no. 4, pp. 403–415, 2000. View at Publisher · View at Google Scholar · View at Scopus
  24. E. J. Wagenmakers, S. Farrell, and R. Ratcliff, “Estimation and interpretation of 1/f(alpha) noise in human cognition,” Psychonomic Bulletin & Review, vol. 11, pp. 579–615, 2004. View at Publisher · View at Google Scholar
  25. B. J. West, E. L. Geneston, and P. Grigolini, “Maximizing information exchange between complex networks,” Physics Reports, vol. 468, no. 1–3, pp. 1–99, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  26. C.-K. Peng, J. Mietus, J. M. Hausdorff, S. Havlin, H. E. Stanley, and A. L. Goldberger, “Long-range anticorrelations and non-Gaussian behavior of the heartbeat,” Physical Review Letters, vol. 70, no. 9, pp. 1343–1346, 1993. View at Publisher · View at Google Scholar · View at Scopus
  27. V. Marmelat, K. Torre, and D. Delignières, “Relative roughness: an index for testing the suitability of the monofractal model,” Frontiers in Physiology, vol. 3, article 208, 2012. View at Publisher · View at Google Scholar · View at Scopus
  28. S. Das, Functional Fractional Calculus, Springer, Berlin, Germany, 2011. View at Publisher · View at Google Scholar
  29. D. Delignieres, S. Ramdani, L. Lemoine, K. Torre, M. Fortes, and G. Ninot, “Fractal analyses for ‘short’ time series: a re-assessment of classical methods,” Journal of Mathematical Psychology, vol. 50, no. 6, pp. 525–544, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. J. W. Kantelhardt, “Fractal and multifractal time series,” in Mathematics of Complexity and Dynamical Systems, R. A. Meyers, Ed., pp. 463–487, Springer, Berlin, Germany, 2011. View at Google Scholar
  31. R. B. Davies and D. S. Harte, “Tests for hurst effect,” Biometrika, vol. 74, no. 1, pp. 95–101, 1987. View at Publisher · View at Google Scholar · View at Scopus
  32. D. Saupe, “Algorithms for random fractals,” in The Science of Fractal Images, H. O. Peitgen and D. Saupe, Eds., pp. 71–136, Springer, Berlin, Germany, 1988. View at Publisher · View at Google Scholar
  33. A. F. Farag, S. M. El-Metwally, and A. A. A. Morsy, “Automated sleep staging using detrended fluctuation analysis of sleep EEG,” in Soft Computing Applications, V. E. Balas, J. Fodor, A. R. Várkonyi-Kóczy, J. Dombi, and L. C. Jain, Eds., Advances in Intelligent Systems and Computing, pp. 501–510, Springer, Berlin, Germany, 2013. View at Publisher · View at Google Scholar
  34. R.-G. Yeh, J.-S. Shieh, Y.-Y. Han, Y.-J. Wang, and S.-C. Tseng, “Detrended fluctuation analyses of short-term heart rate variability in surgical intensive care units,” Biomedical Engineering—Applications, Basis and Communications, vol. 18, no. 2, pp. 67–72, 2006. View at Publisher · View at Google Scholar · View at Scopus
  35. A. Hartmann, P. Mukli, Z. Nagy, L. Kocsis, P. Hermán, and A. Eke, “Real-time fractal signal processing in the time domain,” Physica A: Statistical Mechanics and its Applications, vol. 392, no. 1, pp. 89–102, 2013. View at Publisher · View at Google Scholar · View at Scopus
  36. C. Kamath, “A new approach to detect congestive heart failure using sequential spectrum of electrocardiogram signals,” Medical Engineering and Physics, vol. 34, no. 10, pp. 1503–1509, 2012. View at Publisher · View at Google Scholar · View at Scopus
  37. A. K. Kiefer, C. K. Rhéa, and W. H. Warren, “VR-based assessment and rehabilitation of functional mobility,” in Human Walking in Virtual Environments, F. Steinicke, Y. Visell, J. Campos, and A. Lecuyer, Eds., pp. 333–350, Springer, Berlin, Germany, 2013. View at Google Scholar
  38. D. Katsavelis, M. Mukherjee, L. Decker, and N. Stergiou, “The effect of virtual reality on gait variability,” Nonlinear Dynamics, Psychology, and Life Sciences, vol. 14, no. 3, pp. 239–256, 2010. View at Google Scholar · View at Scopus
  39. C. K. Rhea, A. W. Kiefer, M. W. Wittstein et al., “Fractal gait patterns are retained after entrainment to a fractal stimulus,” PLoS ONE, vol. 9, no. 9, Article ID e106755, 2014. View at Publisher · View at Google Scholar · View at Scopus