Characterizing the structure of nonlinear systems using gradual wavelet reconstruction

Nonlinear Processes in Geophysics

Volumn 17, Issue 6, 2010, Pages 615-632

  Keylock, C J ^a  

^a UNIVERSITY OF SHEFFIELD   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

BROWNIAN MOTION; FOURIER TRANSFORM; GEOPHYSICS; HYPOTHESIS TESTING; NONLINEARITY; SIGNAL PROCESSING; TIME SERIES ANALYSIS; WAVELET ANALYSIS;

EID: 78649382060     PISSN: 10235809     EISSN: 16077946     Source Type: Journal    
DOI: 10.5194/npg-17-615-2010     Document Type: Article

References (62)

1. Bacry, E., Muzy, J. F., and Arnéodo, A.: Singularity spectrum of fractal signals: exact results, J. Stat. Phys. 70, 635-674, 1993.
2. Barrio, R. and Serrano, S.: A three-parametric study of the Lorenz model, Physica D, 229, 43-51, 2007.
3. Barrio, R. and Serrano, S.: Bounds for the chaotic region in the Lorenz model, Physica D, 238, 1615-1624, 2009.
4. Basu, S., Foufoula-Georgiou, E., Lasheremes, B., and Arnéodo, A.: Estimating intermittency exponent in neutrally stratified atmospheric surface layer flows: A robust framework based on magnitude cumulant analysis and surrogate analyses, Phys. Fluids, 19, 115102, doi:10.1063/1.2786001, 2007. (Pubitemid 350226423)
5. Benzi, R., Biferale, L., Crisanti, A., Paladin, G., Vergassola, M., and Vulpiani, A.: A random process for the construction of multiaffine fields, Physica D, 65, 352-358, 1993.
6. Borgnat, P. and Flandrin, P.: Stationarization via surrogates, J. Stat. Mech., P01001, doi:10.1088/1742-5468/2009/01/P01001, 2009.
7. Borgnat, P., Flandrin, P., Honeine, P., Richard, C., and Xiao, J.: Testing Stationarity With Surrogates: A Time-Frequency Approach, IEEE Trans. Sig. Proc., 58, 3459-3470, 2010.
8. Breakspear, M., Brammer, M., and Robinson, P. A.: Construction of multivariate surrogate sets from nonlinear data using the wavelet transform, Physica D, 182, 1-22, 2003.
9. Daubechies, I.: Orthonormal bases of compactly supported wavelets: II. variations on a theme, SIAM J. Math. Anal. 24, 499-519, 1993.
10. FRACLAB: A fractal analysis toolbox for signal and image processing, http://fraclab.saclay.inria.fr/, 2006.
11. Frascati, A. and Lanzoni, S.: Long-term river meandering as a part of chaotic dynamics' A contribution from mathematical modelling, Earth Surf. Proc. Land., 35, 791-802, 2010.
12. Frisch, U., Sulem, P. L., and Nelkin, M.: Simple dynamical model of intermittent fully developed turbulence, J. Fluid Mech., 87, 719-736, 1978.
13. Fritsch, F. N. and Carlson, R. E.: Monotone Piecewise Cubic Interpolation, SIAM J. Numerical Analysis, 17, 238-246, 1980.
14. Gagnon, J. S., Lovejoy, S., and Schertzer, D.: Multifractal surfaces and terrestrial topography, Europhys. Lett., 62, 801-807, 2003.
15. Gottwald, G. A. and Melbourne, I.: Testing for chaos in deterministic systems with noise, Physica D, 212, 100-110, 2005.
16. Grassberger, P. and Procaccia, I.: Characterization of strange attractors, Phys. Rev. Lett., 50, 346-349, 1983.
17. Huebner, U., Abraham, N. B., and Weiss, C. O.: Dimensions and entropies of chaotic intensity pulsations in a single-mode farinfrared NH3 laser, Phys. Rev. A, 40, 6354-6365, 1989.
18. Jaffard, S.: Multifractal formalism for functions: Parts I and II, SIAM J. Numerical Anal. 28, 944-998, 1997.
19. Johnson, J. R. and Wing, S. A.: Solar cycle dependence of nonlinearity in magnetospheric activity, J. Geophys. Res., 110, A04211, doi:10.1029/ 2004JA010638, 2005. (Pubitemid 44228240)
20. Kaplan, J. L. and Yorke, J. A.: Preturbulence: A regime observed in a fluid flow model of Lorenz, Commun. Math. Phys., 67, 93-108, 1979.
21. Keylock, C. J.: Constrained surrogate time series with preservation of the mean and variance structure, Phys. Rev. E, 73, 036707, doi:10.1103/PhysRevE. 73.036707, 2006.
22. Keylock, C. J.: A wavelet-based method for surrogate data generation, Physica D, 225, 219-228, 2007.
23. Keylock, C. J.: Improved preservation of autocorrelative structure in surrogate data using an initial wavelet step, Nonlin. Processes Geophys., 15, 435-444, doi:10.5194/npg-15-435-2008, 2008a.
24. Keylock, C. J.: A criterion for delimiting active periods within turbulent flows, Geophys. Res. Lett., 35, L11804, doi:10.1029/2008GL033858, 2008b.
25. Keylock, C. J.: Evaluating the dimensionality and significance of active periods in turbulent environmental flows defined using Lipshitz/Hölder regularity, Environ. Fluid Mech., 9, 509-523, 2009.
26. Khan, S., Ganguly, A. R., and Saigal, S.: Detection and predictive modeling of chaos in finite hydrological time series, Nonlinear Proc. Geophys., 12, 41-53, 2005.
27. Kolmogorov, A. N.: A refinement of previous hypotheses concerning the local structure of turbulence in a viscous, incompressible fluid at high Reynolds number, J. Fluid Mech., 13, 82-85, 1962.
28. Kolwankar, K. M. and Lévy Vehel, J.: A time domain characterization of the fine local regularity of functions, J. Fourier Anal. Appl., 8, 319-334, 2002.
29. Kreuz, T., Mormann, F., Andrzejak, R.G., Kraskov, A., Lehnertz, K., and Grassberger, P.: Measuring synchronization in coupled model systems: A comparison of different approaches, Physica D, 225, 29-42, 2007.
30. Kwasniok, F. and Lohmann, G.: Deriving dynamical models from paleoclimatic records: Application to glacial millennial-scale climate variability, Phys. Rev. E, 80, 066104, 2009.
31. Liang, J., and Parks, T. W.: A translation-invariant wavelet representation algorithm with applications, IEEE T. Signal Proces., 44, 225-232, 1996.
32. Lorenz, E. N.: Deterministic nonperiodic flow, J. Atmos. Sci., 20, 130-141, 1963.
33. Lux, T.: Detecting multifractal properties in asset returns: The failure of the "scaling estimator", Int. J. Mod. Phys. C, 15, 481-491, 2004.
34. Mallat, S.: A wavelet tour of signal processing, Academic Press, 637 pp., 1999.
35. Manneville, P. and Pomeau, Y.: Intermittency and the Lorentz model, Phys. Lett. A, 75, 1-2, 1979.
36. Meneveau, C. and Sreenivasan, K. R.: Simple multifractal cascademodel for fully developed turbulence, Phys. Rev. Lett., 59, 1424-1427, 1987.
37. Mormann, F., Kreuz, T., Rieke, C., Andrzejak, R. G., Kraskov, A., David, P., Elger, C. E., and Lehnertz, K.: On the predictability of epileptic seizures, Clin. Neurophys., 116, 569-587, 2005.
38. Muzy, J. F., Bacry, E., and Arnéodo, A.: Wavelets and multifractal formalism for singular signals: Application to turbulence data, Phys. Rev. Lett., 67, 3515-3518, 1991.
39. Nakaya, S. and Hashimoto, T.: Temporal variation of multifractal properties of seismicity in the region affected by the mainshock of the October 6, 2000 Western Tottori Prefecture, Japan, earthquake (M=7.3), Geophys. Res. Lett., 29, 1495, doi:10.1029/2001GL014216, 2002.
40. Palus, M.: Bootstrapping multifractals: Surrogate data from random cascades on wavelet dyadic trees, Phys. Rev. Lett., 101, 134101, doi:10.1103/PhysRevLett.101.134101, 2008.
41. Pavlos, G. P., Kugiumtzis, D., Athanasiu, M. A., Hatzigeorgiu, N., Diamantidis, D., and Sarris, E. T.: Nonlinear analysis of magnetospheric data Part II. Dynamical characteristics of the AE index time series and comparison with nonlinear surrogate data, Nonlin. Processes Geophys., 6, 79-98, doi:10.5194/npg-6-79-1999, 1999. (Pubitemid 29465217)
42. Percival, D. B. and Walden, A. T.: Wavelet methods for time series analysis, Cambridge University Press, Cambridge, 594 pp., 2000.
43. Pikovsky, A. S., Zaks, M. A., Feudel, U., and Kurths, J.: Singular continuous spectra in dissipative dynamics, Phys. Rev. E, 52, 285-296, 1995.
44. Pomeau, Y. and Manneville, P.: Intermittent transition to turbulence in dissipative dynamical systems, Commun. Math. Phys., 74, 189-197, 1980.
45. Roux, S. G., Venugopal, V., Fienberg, K., Arneodo, A., and Foufoula-Georgious, E.: Evidence for inherent nonlinearity in temporal rainfall, Adv. Water Resour., 32, 41-48, 2009.
46. Schertzer, D., Tchiguirinskaia, I., Lovejoy, S., Hubert, P., Bendjoudi, H., and Larcheveque, M.: Discussion of "Evidence of chaos in the rainfall-runoff process" - Which chaos in the rainfall-runoff process', Hydrol. Sci. J., 47, 139-149, 2002.
47. Schertzer, D. and Lovejoy, S.: Hard and soft multifractal processes, Physica A, 185, 187-194, 1992.
48. Schmitt, F., Lavallee, D., Schertzer, D., and Lovejoy, S.: Empirical determination of unviersal multifractal exponents in turbulent velocity fields, Phys. Rev. Lett., 68, 305-308, 1992.
49. She, Z. S. and Leveque, E.: Universal scaling laws in fully developed turbulence, Phys. Rev. Lett., 72, 336-339, 1994.
50. Schreiber, T. and Schmitz, A.: Improved surrogate data for nonlinearity tests, Phys. Rev. Lett., 77, 635-638, 1996.
51. Schreiber, T. and Schmitz, A.: Discrimination power of measures for nonlinearity in a time series, Phys. Rev. E, 55, 5443-5447, 1997. (Pubitemid 127630247)
52. Seuret, S.: Detecting and creating oscillations using multifractal methods, Math. Nachr. 279, 1195-1211, 2006.
53. Takens, F.: Detecting strange attractors in turbulence, Lect. Notes Math., 898, 366-381, 1981.
54. Tessier, Y., Lovejoy, S., and Schertzer, D.: Universal multifractals -Theory and observations for rain and clouds, J. Appl. Meteorol., 32, 223-250, 1993.
55. Theiler, J., Eubank, S., Longtin, A., Galdrikian, B., and Farmer, J. D.: Testing for nonlinearity in time-series the method of surrogate data, Physica D, 58, 77-94, 1992.
56. Vela-Arevalo, L. V. and Marsden, J. E.: Time-frequency analysis of the restricted three-body problem: transport and resonance transitions, Classical Quant. Grav., 21, S351-S375, 2004.
57. Venema, V., Ament, F., and Simmer, C.: A Stochastic Iterative Amplitude Adjusted Fourier Transform algorithm with improved accuracy, Nonlin. Processes Geophys., 13, 321-328, doi:10.5194/npg-13-321-2006, 2006.
58. Venugopal, V. and Foufoula-Georgiou, E.: Energy decomposition of rainfall in the time-frequency-scale domain using wavelet packets, J. Hydrol., 187, 3-27, 1996.
59. Venugopal, V., Roux, S. G., Foufoula-Georgiou, E., and Arneodo, A.: Revisiting multifractality of high-resolution temporal rainfall using a wavelet-based formalism, Water Resour. Res., 42, doi:10.1029/2005WR004489, 2006. (Pubitemid 44156595)
60. WMTSA: Wavelet Methods for Time Series Analysis, http://www. atmos.washington.edu/-wmtsa/, 2006.
61. Wolf, A., Swift, J. B., Swinney, H. L., and Vastano, J. L.: Determining Lyapunov exponents from a time-series, Physica D, 16, 285-317, 1985.
62. Zeglache, H. and Mandel, P.: Influence of detuning on the properties of laser equations, J. Opt. Soc. Am. B, 2, 18-22, 1985.