Fractal time series-a tutorial review

Mathematical Problems in Engineering

Volumn 2010, Issue , 2010, Pages

  Li, Ming ^a  

^a EAST CHINA NORMAL UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

AUTOCORRELATION FUNCTIONS; DIFFERENTIAL EQUATION OF FRACTIONAL ORDER; DIFFERENTIAL SYSTEMS; FRACTAL TIME SERIES; FRACTIONAL SYSTEMS; HEAVY-TAILED; INTEGER ORDER; LOCAL SELFSIMILARITY; LONG RANGE DEPENDENCE; POWER SPECTRUM FUNCTIONS; RANGE DEPENDENCE; STATISTICAL DEPENDENCE; STOCHASTIC PROCESS;

DIFFERENTIAL EQUATIONS; DISTRIBUTION FUNCTIONS; ELECTRONICS ENGINEERING; FRACTALS; PROBABILITY DENSITY FUNCTION; RANDOM PROCESSES; REGRESSION ANALYSIS; TIME SERIES; VIDEO STREAMING;

WHITE NOISE;

EID: 77951489276     PISSN: 1024123X     EISSN: 15635147     Source Type: Journal    
DOI: 10.1155/2010/157264     Document Type: Review

References (112)

1. D. H. Griffel, Applied Functional Analysis, Ellis Horwood Series in Mathematics and Its Applications, John Wiley & Sons, New York, NY, USA, 1981.
2. C. K. Liu, Applied Functional Analysis, Defence Industry Press, Beijing, China, 1986.
3. B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, San Francisco, Calif, USA, 1982.
4. G. Korvin, Fractal Models in the Earth Science, Elsevier, New York, NY, USA, 1992.
5. E. E. Peters, Fractal Market Analysis - Applying Chaos Theory to Investment and Economics, John Wiley & Sons, New York, NY, USA, 1994.
6. J. B. Bassingthwaighte, L. S. Liebovitch, and B. J. West, Fractal Physiology, Oxford University Press, New York, NY, USA, 1994.
7. W. A. Fuller, Introduction to Statistical Time Series, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, New York, NY, USA, 2nd edition, 1996.
8. G. E. P. Box, G. M. Jenkins, and G. C. Reinsel, Time Series Analysis: Forecasting and Control,Prentice Hall, Englewood Clif s, NJ, USA, 3rd edition, 1994.
9. S. K. Mitra and J. F. Kaiser, Handbook for Digital Signal Processing, John Wiley & Sons, New York, NY, USA, 1993.
10. J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedure, John Wiley & Sons, New York, NY, USA, 3rd edition, 2000.
11. B. B. Mandelbrot, Gaussian Self-Affinity and Fractals, Selected Works of Benoit B. Mandelbrot, Springer, New York, NY, USA, 2002.
12. J. Beran, Statistics for Long-Memory Processes, vol. 61 of Monographs on Statistics and Applied Probability, Chapman & Hall, New York, NY, USA, 1994.
13. M. Li and P. Borgnat, "Forward for the special issue on traffic modeling, its computations and applications," to appear in Telecommunication Systems.
14. J. Levy-Vehel, E. Lutton, and C. Tricot, Eds., Fractals in Engineering, Springer, New York, NY, USA, 1997.
15. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++. The Art of Scientific Computing, Cambridge University Press, Cambridge, UK, 2nd edition, 2002.
16. H. Y.-F. Lam, Analog and Digital Filters: Design and Realization, Prentice Hall, Englewood Clif s, NJ, USA, 1979.
17. A. Papoulis, The Fourier Integral and Its Applications, McGraw-Hill, New York, NY, USA, 1962.
18. R. O. Harger, An Introduction to Digital Signal Processing with MATHCAD, PWS Publishing, Boston, Mass, USA, 1999.
19. J. Van de Vegte, Fundamentals of Digital Signal Processing, Prentice Hall, Englewood Clif s, NJ, USA, 2003.
20. M. Li, "Comparative study of IIR notch filters for suppressing 60-Hz interference in electrocardiogram signals," International Journal of Electronics and Computers, vol. 1, no. 1, pp. 7-18, 2009.
21. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.
22. M. D. Ortigueira, "Introduction to fractional linear systems - part 1: continuous-time case," IEE Proceedings: Vision, Image and Signal Processing, vol. 147, no. 1, pp. 62-70, 2000.
23. M. D. Ortigueira, "Introduction to fractional linear systems - part 2: discrete-time case," IEE Proceedings: Vision, Image and Signal Processing, vol. 147, no. 1, pp. 71-78, 2000.
24. M. D. Ortigueira, "An introduction to the fractional continuous-time linear systems: the 21st century systems," IEEE Circuits and Systems Magazine, vol. 8, no. 3, pp. 19-26, 2008.
25. Y. Q. Chen and K. L. Moore, "Discretization schemes for fractional-order differentiators and integrators," IEEE Transactions on Circuits and Systems I, vol. 49, no. 3, pp. 363-367, 2002.
26. B. M. Vinagre, Y. Q. Chen, and I. Petras, "Two direct Tustin discretization methods for fractional- order differentiator/integrator," Journal of the Franklin Institute, vol. 340, no. 5, pp. 349-362, 2003.
27. B. B. Mandelbrot and J. W. van Ness, "Fractional Brownian motions, fractional noises and applications," SIAM Review, vol. 10, pp. 422-437, 1968.
28. T. Hida, Brownian Motion, vol. 11 of Applications of Mathematics, Springer, New York, NY, USA, 1980.
29. I. M. Gelfand and K. Vilenkin, Generalized Functions, vol. 1, Academic Press, New York, NY, USA, 1964.
30. J. Mikusinski, Operational Calculus, International Series of Monographs on Pure and Applied Mathematics, Vol. 8, Pergamon Press, New York, NY, USA, 1959.
31. M. Li and C.-H. Chi, "A correlation-based computational model for synthesizing long-range dependent data," Journal of the Franklin Institute, vol. 340, no. 6-7, pp. 503-514, 2003.
32. M. D. Ortigueira, "Comments on "Modeling fractional stochastic systems as non-random fractional dynamics driven Brownian motions"," Applied Mathematical Modelling, vol. 33, no. 5, pp. 2534-2537, 2009.
33. M. D. Ortigueira and A. G. Batista, "On the relation between the fractional Brownian motion and the fractional derivatives," Physics Letters A, vol. 372, no. 7, pp. 958-968, 2008.
34. M. D. Ortigueira and A. G. Batista, "A fractional linear system view of the fractional brownian motion," Nonlinear Dynamics, vol. 38, no. 1-4, pp. 295-303, 2004.
35. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
36. S. C. Lim, M. Li, and L. P. Teo, "Locally self-similar fractional oscillator processes," Fluctuation and Noise Letters, vol. 7, no. 2, pp. L169-L179, 2007.
37. S. C. Lim, M. Li, and L. P. Teo, "Langevin equation with two fractional orders," Physics Letters A, vol. 372, no. 42, pp. 6309-6320, 2008.
38. M. Li and S. C. Lim, "A rigorous derivation of power spectrum of fractional Gaussian noise," Fluctuation and Noise Letters, vol. 6, no. 4, pp. C33-C36, 2006.
39. M. Li and S. C. Lim, "Power spectrum of generalized Cauchy process," to appear in Telecommunication Systems.
40. B. B. Mandelbrot, Multifractals and 1/f Noise, Selected Works of B. B. Mandelbrot, Springer, New York, NY, USA, 1999.
41. M. Li, "Fractional Gaussian noise and network traffic modeling," in Proceedings of the 8th International Conference on Applied Computer and Applied Computational Science (WSEAS '09), pp. 34-39, Hangzhou, China, May 2009.
42. M. Li, "Self-similarity and long-range dependence in teletraffic," in Proceedings of the 9th International Conference on Applied Computer and Applied Computational Science (WSEAS '09), pp. 19-24, Hangzhou, China, May 2009.
43. P. Abry, P. Borgnat, F. Ricciato, A. Scherrer, and D. Veitch, "Revisiting an old friend: on the observability of the relation between long range dependence and heavy tail," to appear in Telecommunication Systems.
44. M. Li and S. C. Lim, "Modeling network traffic using generalized Cauchy process," Physica A, vol. 387, no. 11, pp. 2584-2594, 2008.
45. J. T. Kent and A. T. Wood, "Estimating the fractal dimension of a locally self-similar Gaussian process by using increments," Journal of the Royal Statistical Society B, vol. 59, no. 3, pp. 679-699, 1997.
46. P. Hall and R. Roy, "On the relationship between fractal dimension and fractal index for stationary stochastic processes," The Annals of Applied Probability, vol. 4, no. 1, pp. 241-253, 1994.
47. A. J. Adler, The Geometry of Random Fields, John Wiley & Sons, New York, NY, USA, 1981.
48. T. Gneiting and M. Schlather, "Stochastic models that separate fractal dimension and the Hurst effect," SIAM Review, vol. 46, no. 2, pp. 269-282, 2004.
49. S. C. Lim and M. Li, "A generalized Cauchy process and its application to relaxation phenomena," Journal of Physics A, vol. 39, no. 12, pp. 2935-2951, 2006.
50. M. S. Taqqu, V. Teverovsky, and W. Willinger, "Estimators for long-range dependence: an empirical study," Fractals, vol. 3, no. 4, pp. 785-798, 1995.
51. M. Li and W. Zhao, "Detection of variations of local irregularity of traffic under DDOS flood attack," Mathematical Problems in Engineering, vol. 2008, Article ID 475878, 11 pages, 2008.
52. M. Li, "An approach to reliably identifying signs of DDOS flood attacks based on LRD traffic pattern recognition," Computers & Security, vol. 23, no. 7, pp. 549-558, 2004.
53. G. M. Raymond, D. B. Percival, and J. B. Bassingthwaighte, "The spectra and periodograms of anticorrelated discrete fractional Gaussian noise," Physica A, vol. 322, no. 1-4, pp. 169-179, 2003.
54. J. Beran, "Fitting long-memory models by generalized linear regression," Biometrika, vol. 80, no.4, pp. 817-822, 1993.
55. J. Beran, "On parameter estimation for locally stationary long-memory processes," Journal of Statistical Planning and Inference, vol. 139, no. 3, pp. 900-915, 2009.
56. M. J. Cannon, D. B. Percival, D. C. Caccia, G. M. Raymond, and J. B. Bassingthwaighte, "Evaluating scaled windowed variance methods for estimating the Hurst coefficient of time series," Physica A, vol. 241, no. 3-4, pp. 606-626, 1997.
57. M. J. Cannon, D. B. Percival, D. C. Caccia, G. M. Raymond, and J. B. Bassingthwaighte, "Evaluating scaled windowed variance methods for estimating the Hurst coefficient of time series," Physica A, vol. 241, no. 3-4, pp. 606-626, 1997.
58. J. B. Bassingthwaighte and G. M. Raymond, "Evaluating rescaled range analysis for time series," Annals of Biomedical Engineering, vol. 22, no. 4, pp. 432-444, 1994.
59. S. E. Schepers, J. H. G. M. van Beek, and J. B. Bassingthwaighte, "Four methods to estimate the fractal dimension from self-affine signals," IEEE Engineering in Medicine and Biology Magazine, vol. 11, no. 2, pp. 57-64, 1992.
60. J. Mielniczuk and P. Wojdyłło, "Estimation of Hurst exponent revisited," Computational Statistics & Data Analysis, vol. 51, no. 9, pp. 4510-4525, 2007.
61. D. O. Cajueiro and B. M. Tabak, "The rescaled variance statistic and the determination of the Hurst exponent," Mathematics and Computers in Simulation, vol. 70, no. 3, pp. 172-179, 2005.
62. G. M. Raymond and J. B. Bassingthwaighte, "Deriving dispersional and scaled windowed variance analyses using the correlation function of discrete fractional Gaussian noise," Physica A, vol. 265, no. 1-2, pp. 85-96, 1999.
63. J. B. Bassingthwaighte and G. M. Raymond, "Evaluation of the dispersional analysis method for fractal time series," Annals of Biomedical Engineering, vol. 23, no. 4, pp. 491-505, 1995.
64. C. M. Kendziorski, J. B. Bassingthwaighte, and P. J. Tonellato, "Evaluating maximum likelihood estimation methods to determine the Hurst coefficient," Physica A, vol. 273, no. 3-4, pp. 439-451, 1999.
65. A. Guerrero and L. A. Smith, "A maximum likelihood estimator for long-range persistence," Physica A, vol. 355, no. 2-4, pp. 619-632, 2005.
66. D. Veitch and P. Abry, "A wavelet-based joint estimator of the parameters of long-range dependence," IEEE Transactions on Information Theory, vol. 45, no. 3, pp. 878-897, 1999.
67. C. Cattani, "Shannon wavelets theory," Mathematical Problems in Engineering, vol. 2008, Article ID 164808, 24 pages, 2008.
68. C. Cattani, "Harmonic wavelet approximation of random, fractal and high frequency signals," to appear in Telecommunication Systems.
69. C. Cattani, "Harmonic wavelet analysis of a localized fractal," International Journal of Engineering and Interdisciplinary Mathematics, vol. 1, no. 1, pp. 35-44, 2009.
70. E. G. Bakhoum and C. Toma, "Relativistic short range phenomena and space-time aspects of pulse measurements," Mathematical Problems in Engineering, vol. 2008, Article ID 410156, 20 pages, 2008.
71. G. W. Wornell, "Wavelet-based representations for the 1/f family of fractal processes," Proceedings of the IEEE, vol. 81, no. 10, pp. 1428-1450, 1993.
72. P. Abry, D. Veitch, and P. Flandrin, "Long-range dependence: revisiting aggregation with wavelets," Journal of Time Series Analysis, vol. 19, no. 3, pp. 253-266, 1998.
73. Y.-Q. Chen, R. Sun, and A. Zhou, "An improved Hurst parameter estimator based on fractional Fourier transform," to appear in Telecommunication Systems.
74. R. B. Govindan, J. D. Wilson, H. Preifißl, H. Eswaran, J. Q. Campbell, and C. L. Lowery, "Detrended fluctuation analysis of short datasets: an application to fetal cardiac data," Physica D, vol. 226, no. 1, pp. 23-31, 2007.
75. G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes, Stochastic Modeling, Chapman & Hall, New York, NY, USA, 1994.
76. T. G. Sinai, "Distribution of the maximum of fractional Brownian motion," Russian Mathematical Surveys, vol. 52, no. 2, pp. 119-138, 1997.
77. T. G. Sinai, "Self-similar probability distributions," Theory of Probability & Its Applications, vol. 21, no. 1, pp. 63-80, 1976.
78. V. M. Sithi and S. C. Lim, "On the spectra of Riemann-Liouville fractional Brownian motion," Journal of Physics A, vol. 28, no. 11, pp. 2995-3003, 1995.
79. S. V. Muniandy and S. C. Lim, "Modeling of locally self-similar processes using multifractional Brownian motion of Riemann-Liouville type," Physical Review E, vol. 63, no. 4, Article ID 046104, 7 pages, 2001.
80. D. Feyel and A. de La Pradelle, "On fractional Brownian processes," Potential Analysis, vol. 10, no. 3, pp. 273-288, 1999.
81. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill, New York, NY, USA, 1961.
82. S. C. Lim and S. V. Muniandy, "On some possible generalizations of fractional Brownian motion," Physics Letters A, vol. 266, no. 2-3, pp. 140-145, 2000.
83. P. Flandrin, "On the spectrum of fractional Brownian motions," IEEE Transactions on Information Theory, vol. 35, no. 1, pp. 197-199, 1989.
84. R. F. Peltier and J. Levy-Vehel, "Multifractional Brownian motion: definition and preliminaries results," INRIA TR 2645, 1995.
85. R. F. Peltier and J. Levy-Vehel, "A new method for estimating the parameter of fractional Brownian motion," INRIA TR 2696, 1994.
86. M. Li, S. C. Lim, and W. Zhao, "Investigating multi-fractality of network traffic using local Hurst function," Advanced Studies in Theoretical Physics, vol. 2, no. 10, pp. 479-490, 2008.
87. S. V. Muniandy, S. C. Lim, and R. Murugan, "Inhomogeneous scaling behaviors in Malaysian foreign currency exchange rates," Physica A, vol. 301, no. 1-4, pp. 407-428, 2001.
88. M. Li, "Modeling autocorrelation functions of long-range dependent teletraffic series based on optimal approximation in Hilbert space - a further study," Applied Mathematical Modelling, vol. 31, no. 3, pp. 625-631, 2007.
89. J.-P. Chiles and P. Delfiner, Geostatistics, Modeling Spatial Uncertainty, Wiley Series in Probability and Statistics: Applied Probability and Statistics, John Wiley & Sons, New York, NY, USA, 1999.
90. P. Todorovic, An Introduction to Stochastic Processes and Their Applications, Springer Series in Statistics: Probability and Its Applications, Springer, New York, NY, USA, 1992.
91. S. C. Lim and S. V. Muniandy, "Generalized Ornstein-Uhlenbeck processes and associated self- similar processes," Journal of Physics A, vol. 36, no. 14, pp. 3961-3982, 2003.
92. R. L. Wolpert and M. S. Taqqu, "Fractional Ornstein-Uhlenbeck Levy processes and the telecom process: upstairs and downstairs," Signal Processing, vol. 85, no. 8, pp. 1523-1545, 2005.
93. M. Li and S. C. Lim, "Modeling network traffic using cauchy correlation model with long-range dependence," Modern Physics Letters B, vol. 19, no. 17, pp. 829-840, 2005.
94. M. Li and W. Zhao, "Representation of a stochastic traffic bound," to appear in IEEE Transactions on Parallel and Distributed Systems.
95. S. C. Lim and L. P. Teo, "Gaussian fields and Gaussian sheets with generalized Cauchy covariance structure," Stochastic Processes and Their Applications, vol. 119, no. 4, pp. 1325-1356, 2009.
96. P. Vengadesh, S. V. Muniandy, and W. H. Abd.Majid, "Fractal morphological analysis of Bacteriorhodopsin (bR) layers deposited onto Indium Tin Oxide (ITO) electrodes," Materials Science and Engineering C, vol. 29, no. 5, pp. 1621-1626, 2009.
97. A. Scherrer, N. Larrieu, P. Owezarski, P. Borgnat, and P. Abry, "Non-Gaussian and long memory statistical characterizations for internet traffic with anomalies," IEEE Transactions on Dependable and Secure Computing, vol. 4, no. 1, pp. 56-70, 2007. (Pubitemid 46384621)
98. A. Karasaridis and D. Hatzinakos, "Network heavy traffic modeling using α-stable self-similar processes," IEEE Transactions on Communications, vol. 49, no. 7, pp. 1203-1214, 2001.
99. G. E. Uhlenbeck and L. S. Ornstein, "On the theory of the Brownian motion," Physical Review, vol. 36, no. 5, pp. 823-841, 1930.
100. D.-X. Lu, Stochastic Processes and Their Applications, Tsinghua University Press, Beijing, China, 2006.
101. L. Valdivieso, W. Schoutens, and F. Tuerlinckx, "Maximum likelihood estimation in processes of Ornstein-Uhlenbeck type," Statistical Inference for Stochastic Processes, vol. 12, no. 1, pp. 1-19, 2009.
102. M. Shlesinger, G. M. Zaslavsky, and U. Frisch, Eds., Lévy Flights and Related Topics in Physics,Springer, New York, NY, USA, 1995.
103. R. N. Mantegna and H. E. Stanley, "Stochastic process with ultraslow convergence to a Gaussian: the truncated Lévy flight," Physical Review Letters, vol. 73, no. 22, pp. 2946-2949, 1994.
104. J.-P. Bouchaud and A. Georges, "Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications," Physics Reports, vol. 195, no. 4-5, pp. 127-293, 1990.
105. D. Applebaum, "Levy processes - from probability to finance and quantum groups," Notices of the American Mathematical Society, vol. 51, no. 11, pp. 1336-1347, 2004.
106. R. J. Martin and A. M. Walker, "A power-law model and other models for long-range dependence," Journal of Applied Probability, vol. 34, no. 3, pp. 657-670, 1997.
107. R. J. Martin and J. A. Eccleston, "A new model for slowly-decaying correlations," Statistics & Probability Letters, vol. 13, no. 2, pp. 139-145, 1992.
108. M. Li, W. Jia, and W. Zhao, "Correlation form of timestamp increment sequences of self-similar traffic on Ethernet," Electronics Letters, vol. 36, no. 19, pp. 1668-1669, 2000.
109. F. Chapeau-Blondeau, "(max, +) dynamic systems for modeling traffic with long-range dependence," Fractals, vol. 6, no. 4, pp. 305-311, 1998. (Pubitemid 128535290)
110. B. Minasny and A. B. McBratney, "The Matérn function as a general model for soil variograms," Geoderma, vol. 128, no. 3-4, pp. 192-207, 2005.
111. W. Z. Daoud, J. D. W. Kahl, and J. K. Ghorai, "On the synoptic-scale Lagrangian autocorrelation function," Journal of Applied Meteorology, vol. 42, no. 2, pp. 318-324, 2003.
112. T. von Karman, "Progress in the statistical theory of turbulence," Proceedings of the National Academy of Sciences of the United States of America, vol. 34, pp. 530-539, 1948.