Publication:
Detection of chaos in time series. Application to spanish sea swell

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1995
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Facultad de Ciencias Económicas y Empresariales. Decanato
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En esta memoria se revisan tres herramientas complementarias para el Análisis No-Lineal de Series Temporales: dimensión fractal, entropía, y exponentes de Liapunov. Se hace una revisión crítica del estado de la cuestión en el área, señalando el fundamento teórico de las técnicas aplicadas cuando éste existe, el campo de aplicación admitido por los especialistas cuando no existe base teórica, las limitaciones, etc. En ocasiones se aportan pruebas rigurosas, o razonamientos heurísticos, nuevos en la literatura.
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Abarbanel, H.D., R. Brown & M.B. Kennel (1991) "Liapunov Exponents in Chaotic Systems: their Importance and their Evaluation using Observed Data", Int. J. of Mod. Physics B, 5,9,1347-75. Arnold, V.I. (1980) Chapitres Suplémentaires de la Théorie des Equations Différentielles Ordinaires, Mir, 1980. Atmanspacher, H., H. Scheingraber & W. Voges (1988) "Global Scaling Properties of a Chaotic Attractor Reconstructed from Experimental Data", Phys. Rev. A 37, 1314. Besicovitch, A.S. (1934) "On the Sum of Digits of Real Numbers Represented in the Dyadic System", Math. Annalen 110, 321-30. Bajo, O., F. Fernández & S. Sosvilla (1992) "Volatilidad y Predecibilidad en las Series de Tipo de Cambio Peseta-Dólar: Un Enfoque basado en el Caos Determinista", Revista Española de Economía, 1992, Monográfico Mercados Financieros, 91-109. Barnsley, M. (1988) Fractals Everywhere, Academic Press, 1988. Bartlett, M.S. (1990) "Chance or Chaos? (with discussion)", J.R. Statist. Soc. A, 153, 3, 321-347. Ben-Mizrachi, A., P. Grassberger & I. Proccacia (1984) "Characterization of experimental (noisy) strange attractors", Phys. Rev. A (3) 29, 975-77. Billingsley, P. (1978) Ergodic Theory and Information, Krieger, New York, 1978. Billingsley, P. (1979) Probability and Measure, Wiley, New York, 1979. Box, G.E.P. & G.M. Jenkins, (1976) Time Series Analysis, Holden-Day, San Francisco, 1976. Brandstäter, A., J. Swift, R.J. Swinney, A. Wolf, J.D. Farmer, E. Jen & J.P. Crutchfield (1983) "Low-Dimensional Chaos in a Hydrodynamical System", Phys. Rev. Lett. 51, 1492-95. Brock, W.A. (1986) "Distinguishing Random and Deterministic Systems", Journal of Economic Theory 40, 168-195. Brock, W.A. & W.D. Dechert, (1991) "Non-Linear Dynamical Systems: Instability and Chaos in Economics", in Handbook of Mathematical Economics, vol. IV, W. Hildebrand and H. Sonnenschein eds., Elsevier Science Publishers BV, 1991. Brock, W.A., W.D. Dechert & J.A. Scheinkman (1987) "A Test for Independence based upon the Correlation Dimension", Department of Economics, University of Wisconsin, Madison, University of Houston and University of Chicago. Brock, W.A. & C. Sayers (1988) "Is the Business Cycle Characterized by Deterministic Chaos?", J. Monetary Econ. 22, 71-90. Brock, W.A., D.A. Hsieh & B. LeBaron, (1992) Nonlinear Dynamics, Chaos, and Instability: Statistical Theory and Economic Evidence, The MIT Pres, 1992. Broomhead, D.S. & P. King (1986) "Extracting Qualitative Dynamics from Experimental Data", Physica 20D, 217-36. Buzug, T., T. Reimers & G. Pfister (1990) "Optimal Reconstructions of Strange Attractors from Purely Geometrical Arguments", Europhys. Lett., 13 (7), 605-10. Canizzo, J.K., D.A. Goodings & J.A. Mattei, (1990) "A Search for Chaotic Behavior in the Light Curves of three Long-Period Variables", The Astrophysical Journal, 357, 235-242. Caswell, W.E. & J.A. Yorke (1986) "Invisible Errors in Dimension Calculation: Geometrical and Sistematic Effects", in Dimension and Entropies in Chaotic Dynamical Systems-Quantification of Complex Behavior", G. Mayer-Kress ed.,(Springer Series in Synergetics, vol. 32) Springer-Verlag, 1986. Conde, J.J. & J. Calderón (1986) "Análisis Convencional de un Registro de Oleaje", Programa de Clima Marítimo. Publicación Nº 10. Crutchfield, J.S. & B.S. McNamara (1987) "Equations of Motion from Data Series", Compl. Syst. 1, 417-52. Cutler, C.D. (1990) "Connecting Ergodicity and Dimension in Dynamical Systems", Ergodic Th. Dynam. Syst. 10, 451-62. Cutler, C.D. (1990) "A Dynamical System with Integer Information Dimension and Fractal Correlation Exponent", Comm. Math. Phys. 129, 621-29. Cutler, C.D. (1991) "Some Results on the Behaviour and Estimation of the Fractal Dimension of Distributions on Attractors", J. Stat. Phys., 62, 651-708. Deliu, A., J.S. Geronimo, R. Shonkwiler & D. Hardin (1992) "Dimensions Associated with Recurrent Self-Similar Sets", Math. Proc. Camb. Phil. Soc. 110, 327-36. Denker, M. & G. Keller (1986) "Rigorous Statistical Procedures for Data from Dynamical Systems", J. Stal. Physics, 44, 1/2, 67-93. Devaney, R.L. (1987) An Introduction to Chaotic Dynamical Systems, New York, Addison-Wesley, 1987. Eckmann, J.P., S.O. Kamphorst, D. Ruelle & S. Ciliberto (1986) "Liapunov Exponents from Time Series", Physical Review A, 34, 6, 4971. Eckmann, J.P. & D. Ruelle (1985) "Ergodic Theory of Chaos and Strange Attractors", Rev. Mod. Phys., 57, 617. Essex, C. & M. Nerenberg, (1991) "Comments on 'Deterministic Chaos: The Science and The Fiction' by D. Ruelle", Proc. R. Soc. Lond. A 435, 287-92. Falconer, K.J., (1990) Fractal Geometry. Mathematical Foundations and Applications, Wiley, New York, 1990. Farmer, J.D., E. Ott & J.A. Yorke (1983) "The Dimension of Chaotic Attractors", Physica 7D, 153-80. Fraser, A.M. (1989) "Reconstructing Attractors from Scalar Time Series: a Comparison of Singular System and Redundancy Criteria", Physica 34D, 391-404. Fraser, A.M. & H.L. Swinney (1986) "Independent Coordinates for Strange Attractors from Mutual Information", Phys. Rev. A 33, 2, 1134-40. Frederickson, P., J.L. Kaplan, E.D. Yorke & J.A. Yorke (1983) "The Liapunov Dimension of Strange attractors", J. Diff. Equ., 49, 185. Froehling, H., J. Crutchfield, J.D. Farmer, N. Packard & R. Shaw (1981) "On Determining the Dimension of Chaotic Flows", Physica D 3, 605. Gantmacher, F.R. (1977) Matrix Theory, Chelsea, 1977. Guillemin, V. & A. Pollac, (1974) Differential Topology, Prentice Hall, 1974. Gonin, R. & A.H. Money (1989) Non-Linear Lp-Norm Estimation, Marcel Dekker, Inc. Statistics Texbooks and Monographs, vol. 100. Grassberger, P. (1983) "Generalized Dimensions of Strange Attractors", Phys. Lett. A, 97, 227. Grassberger, P. (1988) "Finite Sample Corrections to Entropy and Dimension Estimates", Physics Letters A 128, 369-73. Grassberger, P. (1990) "An Optimized Box-Assisted Algorithm for Fractal Dimensions", Phys. Lett. A, 148, 63. Grassberger, P., T. Schreiber & C. Schaffrath (1991) "Non-linear Time Sequence Analysis", Int. J. Bif. Chaos, 1, 3, 521-547. Grassberger, P. & I. Procaccia (1983a) "Characterization of Strange Attractors", Phys. Rev. Lett., 50, 346-49. Grassberger, P. & I. Procaccia (1983) "Estimation of the Kolmogorov Entropy from a Chaotic Signal", Phys. Rev. A, 28, 2591-93. Grassberger, P. & I. Procaccia (1983c) "Measuring the Strangeness of Strange Attractors", Physica D, 9, 189-208. Greenside, H., A. Wolf, J. Swift & T. Pignataro (1982) "Impracticality of a Box Counting Algorithm for Calculating the Dimensionality of Strange Attractors", Phys. Rev. A25, 3453. Haken, H. (1983) "At Least One Liapunov Exponent Vanishes if the Trayectory of an Attractor does not Contain a Fixed Point", Physics Letters A, 94, 2, 71-72. Halsey, T.C., M.H. Jensen, L.P. Kadanoff, I. Procaccia & B.I. Shraiman (1986) "Fractal Measures and their Singularities: the Characterization of Strange Sets", Phys. Rev. A, 33, 1141-51. Henon, M (1976) "A Two-Dimensional Mapping with a Strange Attractor", Comm. Math. Physics 50, 69. Holden, A.V. & M.A. Muhamad (1986) "A Graphical Zoo of Strange and Peculiar Attractors", in Chaos, A.V. Holden ed., Princeton Univ. Press. Holzfuss, J. & G. Mayer-Kress (1986) "An Approach to Error-Estimation in the Application of Dimension Algorithms", in Dimension and Entropies in Chaotic Dynamical Systems-Quantijication of Complex Behavior", G. Mayer-Kress ed.,(Springer Series in Synergetics, vol. 32) Springer-Verlag, 1986. Horn, R.A. & C.R. Johnson (1985) Matrix Analysis, Cambridge, 1985. Hunt, B., T. Sauer & J.A. Yorke "Prevalence: a Traslation-Invariant 'Almost Every' on Infinite-Dimensional Spaces", preprint. Ilyachenko, Y.S. (1983) "On the Dimension of Attractors of k-Contracting Systems in an Infinite-Dimensional space", Vestn. Mosk. Univ. Ser. 1 Mekh, 3, 52-58. Irwin, M.C., (1980) Smooth Dynamical Systems, Academic Press, 1980. Isham, V. (1993) "Statistical Aspects of Chaos: A Review" in Network and Chaos-Statistical and Probabilistic Aspects, O.E. Barndorff-Nielsen, J.L. Jensen and W.S. Kendall eds., Chapman & Hall, 1993. Jensen, J.L. (1993) "Chaotic Dynamical Systems with a View towards Statistics: A Review", in Networks and Chaos-Statistical and Probabilistic Aspects, O.E. Barndorff-Nielsen, J.L. Jensen and W.S. Kendall eds., Chapman & Hall, 1993. Johnson, R.A., K.J. Palmer & R. Sell (1987) "Ergodic Properties of Linear Dynamical Systems", SIAM J. Math. An., 18, 1-33. Kaplan, J. & J.A. Yorke (1978) "Functional Differential Equations and the Approximation of Fixed Points", Lecture Notes in Math. 730, 228 (Peitgen & Walters eds.), Springer. Koopmans, L.H. (1974) The Spectral Analysis of Time Series, Academic Press, New York, 1974. Krengel, U. (1985) Ergodie Theorems, De Gruyter Studies in Mathematics, W. de Gruyter, 1985. Ledrappier, F. & L.S. Young (1984) "The Metric Entropy of Diffeormorphisms", Bull. Am. Math. Soc. (New Ser.) 11, 343. Liebert, W., K. Pawelzik & H.G. Schuster (1991) "Optimal Embeddings of Chaotic Attractors from Topological Considerations", Europhys. Lett. 14 (6), 521-26. Liebert, W. & H.G. Schuster (1939) "Proper Choice of the Time Delay for the Analysis of Chaotic Time Series", Phis. Lett. A, 142(2,3), 107-11. Lorenz, E.N. (1963) "Deterministic Nonperiodic Flow", J. Atmos. Sci. 20, 130. Lorenz, E.N. (1990) "Dimension of Weather and Climate Attractors", Nature 353, 241-44. Mattila, P. (1975) "Hausdorff Dimension, Orthogonal Projections and Intersection with Planes", Ann. Acad. Sci. Fenn. Ser. A1, Math., 1, 227. Mañé, R. (1987) Ergodie Theory and Differentiable Dynamics, Springer-Verlag, 1987. Mayer-Kress, G. (1987) "Application of Dimension Algorithms to Experimental Chaos" in Chaos, Hao Bai-Lin ed., World Scientfic, 1987. Mera, E., M. Morán & J.M. Rey, (1993) "Cálculo de la Dimension Box-Counting en Bajas Dimensiones", unpublished. Mera, E. & M. Morán "Convergence of the Eckmann-Ruelle algorithm for computing Liapunov Exponents", manuscript in preparation. Morán, M & J.M. Rey (1994) "Singularity of Self-Similar Measures with respect to Hausdorff Measures", preprint. Osborne, A.R., A.D. Kirwan, A. Provenzale & L. Bergamasco (1986) "A Search for Chaotic Behavior in Large and Mesoscale Motions in the Pacific Ocean", Physica D 35, 357. Osborne, A.R. & A. Provenzale (1989) "Finite Correlation Dimension for Stochastic Systems with Power Law Spectra", Physica D, 35, 357-381. Oseledec, V.I. (1968) "A Multiplicative Ergodic Theorem. Liapunov Characteristic Exponents", Trans. Moscow Math. Soc., 19, 179-231. Packard, N.H., J.P. Crutchfield, J.D. Farmer & R.S. Shaw (1980) "Geometry from a Time Series", Phys. Rev. Lett. 45, 712-716. Priestley, M.B. (1987) Speciral Analysis and Time Series, vols. I and II, Academic, London, 1987. Provenzale, A. & A.R. Osborne (1989) "Finite Correlation Dimension for Stochastic Systems with Power-Law spectra", Physica D, 35, 357-381. Provenzale, A., A.R. Osborne & R. Soj (1991) "Convergence of the K2-Entropy for Random Noises with Power Law Spectra", Physica D, 47, 361-72. Provenzale, A., L.A. Smith, R. Vio & G. Murante (1992) "Distinguishing between Low-dimensional Dynamics and Randomness in Measured Time Series", preprint. Royal Statistical Society Meeting on Chaos (Discussion),(1992) J. Roy. Stat. Soc. B 54, 451-474. Raghunathan, M.S. (1978) "A Proof of Oseledec's Multiplicative Ergodic Theorem", Israel J. Math., 32, 4, 356-62. Ramsey, J.B. & H.J. Yuan (1990) "The statistical Properties of Dimension Calculation using Small Data Sets", Nonlinearity 3, 155-176. Renyi, A. (1970) Probability Theory, North-Holland, Amsterdam, 1970. Rossler, O.E. (1976) "An Equation for Continuous Chaos", Phys. Lett. A 54, 397. Ruelle, D. (1979) "Ergodic Theory of Differentiable Dynamical Systems", Publ. Math. IHES, 50, 275-307. Ruelle, D. (1993) Chance and Chaos, Penguin Books, 1993. Ruelle, D. (1987) Chaotic Evolution and Strange Attractors., Cambridge University Press, 1987. Ruelle, D. (1990) "Deterministic Chaos: the Science and the Fiction", Prac. R. Soc. London A, 427, 241. Ruelle, D. & F. Takens (1971) "On the Nature of Turbulence", Comm. Math. Physics., 21, 12. Russell, D.A., J.D. Hansen & E. Ott (1980) Dimensionality and Liapunov Numbers of Strange Attractors", Phys. Rev. Lett 45, 1175. Sakai, H. & H. Tokumaru (1980) "Autocorrelations of a Certain Chaos", IEEE Transactions on Acoustic, Speech and Signal Processing V.I. ASSP-28, 5, 588-90. Sano, M. & Y. Sawada (1985) "Measurement of the Liapunov Spectrum from a Chaotic Time Series", Phys. Rev. Lett., 55, 10, 1082-85. Sato, S., M. Sano & Y. Sawada (1987) "Practical Methods of Measuring the Generalized Dimension and the Largest Liapunov Exponent in High Dimensional Chaotic Systems", Prog. Theor. Phys., 77, 1. Sauer, T., J.A. Yorke, & M. Casdagli (1991) "Embedology" J. Stat. Physics, 65, 579-616. Scheinkman, J. & B. Lebaron (1989) "Nonlinear Dynamics and Stock Returns", J. Business, 62, 311. Takens, F. (1981) "Detecting Strange Attractors in Turbulence", Dynamical Systems and Turbulence (Rand & Young eds.) Lecture Notes in Mathematics 898, 366 (Spririger). Takens, F. (1983) "Invariants related to Dimension and Entropy", in Atas do 13. Col. Brasiliero de Matematicas, Rio de Janeiro. Theiler, J. (1986) "Spurious Dimension from Correlation Algorithms applied to Limited Time-Series Data", Pys. Rev. A 34, 3, 2427-32. Theiler, J. (1990) "Estimating Fractal Dimension", J. Opt. Soc. Am. A, 7, 1055-73. Theiler, J. (1990) "Statistical Precision of Dimension Estimators", Phys. Rev. A 41, 6, 3038-51. Theiler, J. (1991) "Somme Comments on the Correlation Dimension of 1/f noise", Phys. Lett. A, 155, 480. Bernard, P. & T. Ratiu (eds.), Turbulence Seminar, Proceedings 1976/77, Lecture Notes in Mathematics 615, Springer-Verlag. Tong, H. (1990) Non-Linear Time Series Analysis, Oxford Univ. Press, 1990. Tricot, C. (1982) "Two definitions of Fractional Dimension" Proc. Camb. Phil. Soc., 91, 57-74. Vastano, J.A. & E.J. Kostelich (1985) "Comparison of Algorithms for Determining the Liapunov Exponents from Experimental Data", in Dimensions and Entropies in Chaotic Systems, G. Mayer-Kress, ed. (Springer). Walters, P. (1982) An Introduction to Ergodic Theory, Springer-Verlag, 1982. Wolf, A., J.B. Swift, H.L. Swinney & A. Vastano (1984) "Determining Liapunov Exponents from Time Series", Physica D, 16, 285-317. Wolff, R. (1992) "Local Liapunov Exponents: Looking Closely at Chaos", J.R. Stat. Soc. B 54, 353-71. Young, L.S. (1982) "Dimension, Entropy and Liapunov Exponents", Ergod. Theor. & Dyn. Sys., 2, 109.