Some Remarks on the Statistical Modeling of Chaotic Systems

  • Dominique Guegan


Many of the issues in modeling nonlinear systems are statistical ones. In this chapter we give a statistician’s viewpoint of the problems of studying and understanding data from real-world dynamical systems. We present some new insights into the possibilities of obtaining consistent estimates for the invariants of a dynamical system, and some new results concerning noise-removal.


Radial Basis Function Lyapunov Exponent Invariant Measure Chaotic System Master Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abarbanel H.D.I., Brown R., Sidorowitch, L.S. Tsimring (1993), The analysis of observed chaotic data in physical systems, Reviews of modern physics, 65, 1331–1392.MathSciNetCrossRefGoogle Scholar
  2. Andel J. (1986), Long Memory Time Series, Kybernetica, 22, 105–123.MathSciNetMATHGoogle Scholar
  3. Ashley, R.A., D.M. Patterson (1998), A direct comparison of the BDS, Hinich, and other tests for the presence of nonlinear dependence in time series, Preprint.Google Scholar
  4. Badel A.E., O. Michel, A. Hero (1997), Arbres de régression: modélisation non para- métrique et analyse de séries temporelles, Traitement du Signal, to appear.Google Scholar
  5. Baitlie R.T., Bolterslev T. and Mikkelsen H.O. (1996), Fractionally Integrated Generalized Autoregressive Conditional Heteroscedasticity, Journal of Econometrics, 31, 307–327.Google Scholar
  6. Blanke D., Bosq D. and Guégan D. (1999), Modelization and non-parametric estimation for a dynamical system with noise, Preprint CREST, Paris.Google Scholar
  7. Boilerslev T. (1986) Generalized Autoregressive Conditional Heteroscedasticity, Journal of Econometrics, 31, 307–259.MathSciNetCrossRefGoogle Scholar
  8. Bosq D. (1989) Non-parametric estimation of a nonlinear filter using a density estimator with a zero-one explosive behaviour in Rd, Statistics and Decisions 7 Google Scholar
  9. Bosq D., D. Guégan (1995) Non-parametric estimation of the chaotic function and the invariant measure of a dynamical system, Stat. Probe. Letters, 25, 201–212.MATHCrossRefGoogle Scholar
  10. Bosq D., D. Guégan, G. Léorat (1999), Statistical estimation of the embedding dimension of a dynamical system, To appear in International Journal of Bifurcations and Chaos. Google Scholar
  11. Bowen R. (1978), Israel J. Math., 28, 298–314.MathSciNetGoogle Scholar
  12. Brandstater A., J Swinney (1987), Phys. Rev. A, 35, 2207.CrossRefGoogle Scholar
  13. Brock W.A., W.D. Dechert, J.A. Scheinkman, B. LeBaron (1996), A test for independence based on the correlation dimension, Econometrics Review, 15, 197–235.MathSciNetMATHCrossRefGoogle Scholar
  14. Broomhead D.S., G.P. King (1986), Extracting qualitative dynamics from experimental data, Physica D, 20, 217–236.MathSciNetMATHCrossRefGoogle Scholar
  15. Casdagli M. (1989), Nonlinear prediction of chaotic time series, Physica D, 35, 335–356, 1989.Google Scholar
  16. Chan K.S. (1989), A note on the ergodicity of a Markov chain, Adv. in Appl. Prob., 21, 702–704.MATHCrossRefGoogle Scholar
  17. Chan K.S., H. Tong (1985), On the use of deterministic Lyapunov function for the ergodicity of stochastic differences equations, Adv. in Appl. Prob., 17, 666–678.MathSciNetMATHCrossRefGoogle Scholar
  18. Chauveau T., J. Damon, D. Guégan (1999) Testing for nonlinearity in intra-day finan- cial series: the case of two French stocks, Preprint Caisse des dépots, Paris.MATHGoogle Scholar
  19. Cutler C. (1997), A general approach to predictive and fractal scaling dimensions in discrete index time series, Fields Institute Communications, 11, 29–48.MathSciNetGoogle Scholar
  20. Cutler C. (1998), Determinism, (bad) Embeddings and scaling structures in time series, Presentation at the Workshop in Cambridge, U.K.Google Scholar
  21. Deissler M., D. Farmer (1992), Deterministic noise amplifiers, Physica D, 55, 155–165.MATHCrossRefGoogle Scholar
  22. Denker M. and Keller G. (1986), Rigorous Statistical Procedures for Data from Dynamical Systems, Journal of Statistical Physics, 44, 67–93.MathSciNetMATHCrossRefGoogle Scholar
  23. Devaney R.L. (1989), An introduction to chaotic dynamical systems, Addison Wesley Pul. Comp., N.Y.Google Scholar
  24. Dieholt J. and Guégan D. (1991), Le modèle fl-ARCH, CRAS, Série I, 312, 625–630.Google Scholar
  25. Diebolt J. and Guégan D. (1993), Tail Behavior of the Stationary Density of General Nonlinear Autoregressive Process of Order One, J. Appl. Prob., 30, 315–329.CrossRefGoogle Scholar
  26. Dingh Z. and Granger C.W.J. (1996), Modelling Volatility Persistence of Speculative Returns: A New Approach, Journal of Econometrics, 73, 185–215.MathSciNetCrossRefGoogle Scholar
  27. Dingh Z., Granger C.W.J. and Engle R.F. (1993), A Long Memory Property of Stock Market Return and a New Model, Journal of Empirical Finance, 1, 83–106.CrossRefGoogle Scholar
  28. Eckmann J.P., Ruelle D. (1985), Ergodic theory of chaos and strange attractors, Reviews of Modern Physics, 57, 617–656.MathSciNetCrossRefGoogle Scholar
  29. Engle R.F. (1982), Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation, Econometrica, 50, 987–1007.MathSciNetMATHCrossRefGoogle Scholar
  30. Farmer J.D., V. Sidorovich (1987), Predicting chaotic time series, Physical Review Letters, 59, 845–848.MathSciNetCrossRefGoogle Scholar
  31. Farmer J.D., V. Sidorovich (1988), Exploiting chaos to predict the future and reduce noise, in Evolution, Learning and cognition , eds. Lee, World Scientific press.Google Scholar
  32. Ferrara L. and Guégan D. (1999), Forecasting with k-factor Gegenbauer Processes, Preprint CREST, Paris.Google Scholar
  33. Ferrara L. and Guégan D. (2000a), Gegenbauer Processes: Estimation and Applications. To appear in Journal of econometrics. Google Scholar
  34. Ferrara L. and Guégan D. (2000b), An approach to financial time series by generalized long memory processes. To appear in Advances in Asset Managment, ed. C. Dunis, Kluver Academic Publishers.Google Scholar
  35. Finkenstadt B., P. Kuhbier (1995) Forecasting nonlinear economic time series: a simple test to accompany the nearest neighbor method, Empirical economics, 20, 243–263.Google Scholar
  36. Flandrin P., Michel O., P. Ruiz (1993), Chaos et analyse non linéaire du signal, Rapport Interne Laboratoire de Physique, E.N.S. Lyon. Dance.Google Scholar
  37. Gayraud G. and Guégan D. (1999), Estimation of the chaotic map by regressogram, Preprint CREST, Paris.Google Scholar
  38. Giraitis L. and Leipus R. (1995), A Generalized Fractionally Differencing Approach in Long Memory Modelling, Lithuanian Math. Journ., 35, 65–81.MathSciNetGoogle Scholar
  39. Girosi F., G., Anzelotti (1993), Rates of convergence for radial basis functions and neural networks, in Artificial Neural Network for speech and Vision, R.J. Mammone ed., Chapman and Hall, 97–114.Google Scholar
  40. Granger C.W.J. and Joyeux R. (1980), An Introduction to Long Memory Time Series Models and Fractional Differencing, J. T.S.A., 1, 15–29.MathSciNetMATHGoogle Scholar
  41. Grassberger P. and I. Procaccia (1983), Measuring the strangeness of strange attractors, Physica D, 9, 189–208.MathSciNetMATHCrossRefGoogle Scholar
  42. Gray H.L., Zhang N. and Woodward A. (1989), On Generalized Fractional Processes, J. T. S. A., 10, 233–257.MathSciNetMATHGoogle Scholar
  43. Guégan D. (1983), Une condition d’ergodicité pour des modèles bilinéaires à temps discret„ CRAS Série I, 297–300.Google Scholar
  44. Guégan D. (1994), Séries chronologiques non linéaires it temps discret, Ecanomica, Paris.Google Scholar
  45. Guégan D. (1997) Non-parametric methods for time series and dynamical systems, in Statistical Challenges in Modern Astronomy II, G.J. Babu and E.D. Feigelson eds., Springer Verlag.Google Scholar
  46. Guégan D. (1999), Note on Long Memory Processes with Cyclical Behavior and Heteroscedasticity, Prepublication 99 - -08, 1–21, University of Reims, France.Google Scholar
  47. Guégan D. (2000), A new model: the k-factor GIGARCH process. To appear in Journal of signal processing. Google Scholar
  48. Guégan D., J. Diebolt (1994), Probabilistic properties of the β-ARCH model, Statistica Sinica, 2, 157–174.Google Scholar
  49. Guégan D. and Léorat G. (1997), Consistent estimation to determine embedding dimension in financial data„ The European Journal of Finance, 231–242.Google Scholar
  50. Guégan D. and Lisi F. (1997), Predictive dimension: an alternative definition of the embedding dimension„ Preprint CREST 9749, Paris.Google Scholar
  51. Guégan D. and Merder L. (1997), Prediction in chaotic time series: Methods and comparisons using simulations, in Signal Analysis and Prediction I, A. Prochazka, J. Uhlir, P. Sovka eds., EURASIP, ICP Press, Prague 1997, 215–218.Google Scholar
  52. Guégan D. and Mercier L. (1998a), Stochastic or chaotic dynamics in hight frequency financial data, in Nonlinear modelling hight frequency financial time series, C. Dunis and B. Zhou eds., John Wiley and Sons, 87–107.Google Scholar
  53. Guégan D. and Mercier L. (1998b), Forecasting and non-parametric techniques for mixing chaotic time series, in Signal analysis and prediction, Chapter 25, Birkauser, Boston, 363–372.Google Scholar
  54. Guégan D. and Nguyen J.M. (1998), The Multiplicative Threshold Model: An Alternative to Detect Breaks and Hidden Cycles on Real Data, Preprint CREST n°9839, 1–13, Paris.Google Scholar
  55. Guégan D. and Nguyen J.M. (1999), COMTAR Models: Theory and Applications, Preprint CREST, Paris.Google Scholar
  56. Guégan D. and Tchernig R. (2000), Prediction of chaotic time series in presence of measurement error: the importance of initial conditions, to appear in Statistics and Competing.Google Scholar
  57. Guégan D., R. Wolff (1999), On non parametric recursive estimators of invariant densities of chaotic maps, Preprint, University of Brisbane, Australia.Google Scholar
  58. Hall P. and Wolff R.C.L. (1995), Properties of Invariant Distributions and Lyapunov Exponents for Chaotic Logistic Maps, J.R.S.S., B, 57.Google Scholar
  59. Hofbauer F. and Keller G. (1982), Math. Zeitschrift, 180,119–140.MathSciNetMATHCrossRefGoogle Scholar
  60. Hardie W., A.B. Tsybakov (1997) Local polynomial estimators of the volatility function in non-parametric autoregression, J. of Econometrics, 81, 223–242.Google Scholar
  61. Hosking J.R.M. (1981), Fractional Diferencing, Biometrika, 88, 165–176.MathSciNetCrossRefGoogle Scholar
  62. Katok A., Hasselblatt B. (1995) Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its applications, 54, Cambridge University Press.Google Scholar
  63. Kohda T. and Murao K. (1990), Approach of to Time Series Analysis of One Dimensional Chaos Based on Frobenuis-Perron Operator, Trans. Inst. Electron. Inf. and Comm. Eng., E73, 6, 793–800, Japan.Google Scholar
  64. Krzyzewski K., Szlenk W. (1969), On invariant measures for expanding differential mappings, Stud. math., 33, 83–92.MathSciNetMATHGoogle Scholar
  65. Ladoucette S. (1999), Etude d’un système dynamique chaotique: approche probabiliste, Mémoire de DEA, Université de Reims.Google Scholar
  66. Lasota, J. and Mac Key, M.C. (1994), Chaos, fractals and noise: stochastic aspects of dynamics, 2nd ed., Springer Verlag, N.Y.MATHGoogle Scholar
  67. Lasota A., Yorke J. A. (1973), Trans. Am. Math. Soc., 186, 481.MathSciNetCrossRefGoogle Scholar
  68. Lawrance A.J. and Spencer N.M. (1996), Statistical Aspects of Curved Chaotic Map Models and their Stochastic Reversibility, Preprint, University of Birmingham, United Kingdom.Google Scholar
  69. Lim K.S., H. Tong, (1980), Threshold Autoregression, limit cycles and cyclical data, J.R.S.S., B 42, 245–292.Google Scholar
  70. Lisi F., Nicolis O., Sandri M. (1995), Combining singular - spectrum analysis and neural networks for time series forecasting, Neural processing letters, 2, 6–10.CrossRefGoogle Scholar
  71. Lu Z.Q., Smith R.L. (1992), Estimating local Lyapunov exponents. To appear in Fields Institute Communication Google Scholar
  72. Mac Caffrey D.F., S. Ellner, A.R. Gallant, D.W. Nychka (1992), Estimating Lyapunov exponents of a chaotic system with non parametric regression, J. T.S.A., 87, 682–695.Google Scholar
  73. MacKernan D. N. (1997), Generalized Markov coarse graining and the observables of chaos, Thèse de doctorat, U.L.B., Brussels.Google Scholar
  74. MacKernan D., Nicolis G. (1993), Generalized Markov Coarse graining and spectral decomposition of chaotic piecewise linear maps, Preprint ULB,Brussels. Google Scholar
  75. Masry E. (1991) Multivariate probability density deconvolution for stationary processes, IEEE Trans. Inform. Theory, 37, 1105–1115.MathSciNetMATHCrossRefGoogle Scholar
  76. Mayer D.H. (1984), Approach to Equilibrium for Locally Expanding Maps in Ek, Comm. Math. Phys., 95, 1–15.MathSciNetMATHCrossRefGoogle Scholar
  77. Meyn S.P., R. Tweedie (1994), Markov chains and stochastic stability, Springer Verlag, N.Y.Google Scholar
  78. Meyer D.H. and Roepstorff G. (1983), Strange Attractors and Asymptotic Measures of Discrete-Time Dissipative Systems, Journal of Statistical Physics, 31, 309–326.MathSciNetCrossRefGoogle Scholar
  79. Mercier L. (1998) Séries temporelles chaotiques appliquées à la finance: problèmes statistiques et algorithmiques., These de Doctorat, Université Paris IX.Google Scholar
  80. Michel O., A. Hero (1995), The structured nonlinear signal modelling and prediction, Preprint.Google Scholar
  81. Nicolis G., Nicolis C. (1988), Master equation approach to deterministic chaos, Physical Review A, 38, 427–433.MathSciNetMATHCrossRefGoogle Scholar
  82. Nicolis G., Nicolis C. and MacKernan D. (1993), Stochastic Resonance in Chaotic Dynamics, Journal of Statistical Physics, 70, 127–139.Google Scholar
  83. Pesaran M.H., Potter S.M. (1993), Nonlinear dynamics, chaos and econometrics, J. Wiley, New York.MATHGoogle Scholar
  84. Petrucelli J.D., S.W. Woolford (1984), A threshold AR(1) model, J. Appt. Prob., 21, 270–286.CrossRefGoogle Scholar
  85. Pham D.T. (1986) The mixing property of bilinear and generalized random coefficient autoregressive models, Stoch. Proc. and their Appl.,23, 291–300.MathSciNetMATHCrossRefGoogle Scholar
  86. Robinson P.M. (1983), Non-parametric estimation for time series models, J. T.S. A. 4, 185–208.MATHGoogle Scholar
  87. Robinson P.M. (1991), Testing for Strong Serial Correlation and Dynamics Conditional Heteroscedasticity in Multiple Regression, Journal of Econometrics, 47, 67–84.MathSciNetMATHCrossRefGoogle Scholar
  88. Ruelle D. (1977), Applications conservant une mesure absolument continue par rapport à dx sur [0,1], Commun. Math. Phys., 55, 477–493.MathSciNetCrossRefGoogle Scholar
  89. Samorodnisky V. and Taqqu M. (1994), a-Stable Processes, Chapman and Hall, New York.Google Scholar
  90. Smale S. (1967), Bull. Am. Math. Soc., 73,817.MathSciNetCrossRefGoogle Scholar
  91. Sauer T. (1998), Embedology, Eckmann-Ruelle matrices and Lyapunov exponents, Presentation to the Workshop in Cambridge, U.K..Google Scholar
  92. Silverman B.W. (1993), Density estimation, Chapman and Hall, N.Y.Google Scholar
  93. Shub M. (1969), Am. J. Math., 91, 175–199.MathSciNetCrossRefGoogle Scholar
  94. Smith L.A. (1992), Identification and prediction of law dimensional dynamics, Physics D, 58, 50–76.MATHGoogle Scholar
  95. Stark J., D.S. Broomhead, M.E. Davies, J. Huke (1998)? Tokens embedding theorems for forced and stochastic systems, Presentation to the Workshop in Cambridge,U.K..Google Scholar
  96. Stockbro K., D.K. Umberger (1992), Forecasting with weighted maps, in Nonlinear modelling, eds. Casdagli and Eubank, Addison-Wesley.Google Scholar
  97. Subba Rao T., M.M. Gabr (1984), Bilinear time series, Lecture notes in Statistics,10, Springer.Google Scholar
  98. Taylor S. (1986), Modelling Financial Time Series, J. Wiley, Chichester.Google Scholar
  99. Ulam S.M., J. Von Neuman (1947), On the combination of stochastic and deterministic properties, Bull. Am. Math. Soc., 53, 1120.Google Scholar
  100. Wayne A., Woodward A., Cheng A.C. and Gray H.L. (1998), A k-Factor GARMA Long Memory Model, J.T.S.A., 19, 485–504.MATHGoogle Scholar
  101. R. Wolff (1992), Local Lyapunov exponents: looking closely at chaos, J.R.S.S B, 54, 353–372.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Dominique Guegan

There are no affiliations available

Personalised recommendations