Abstract
The fact that even though when we will not know the equations defining an underlying dynamical system and we are not able to measure all state space variables, we may be able to find a one-to one correspondence between the original state space and a reconstructed space using few variables means that it is possible to identify unambiguously the original state space from measurements. This has open a new field of research: non-linear time series analysis. The objective of this Chapter is to provide the reader with an overall picture of the embedding theory and time-delay state space reconstruction techniques. We hope that this introductory chapter will guide the reader in understanding the subsequent chapters where different relevant aspects of dynamical systems theory are going to be discussed in more depth and detail.
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References
Abarbanel, H.D.I., Analysis of Observed Chaotic Data, 1996, Springer-Verlag, New-York.
Abarbanel, H. D. I., R. Brown, J. J. Sidorowich and L. Sh. Tsimring, 1993, The analysis of observed chaotic data in physical systems, Rev. Mod. Phys. 65, 1331–1392.
Abarbanel, H. D. I. and Kennel, M. B., 1993, Local false neighbours and dynamical dimension from observed chaotic data. Phys. Rev. E 47, 3057–3068.
Badii, R., Broggi, G., Derighetti, B., Ravani, M., Ciliberto, S., Politi, A. and Rubio, M. A., 1988, Dimension increase in filtered signals. Phys. Rev. Lett. 60, 979–982.
Breeden, J. L. and N. H. Packard, 1994, A learning algorithm for optimal representation of experimental data, Int. J. of Bifurcations and Chaos 4, 311–326.
Burden, R. L. and Faires, J. D., Numerical Analysis, 3rd Ed., PWS, Boston.
Broomhead, D. S. and G. P. King, 1986, Extracting qualitative dynamics from experimental data, Physica D 20, 217–236.
Cannon, M. J., Percival, D. B., Caccia, D. C, Raymond, G. M. and Bassingthwaighte, J. B., 1997, Evaluating scaled windowed variance methods for estimating the Hurst coefficient of time series. Physica A 241, 606–626.
Casdagli, M., 1989, Nonlinear prediction of chaotic time series. Physica D 35, 335–356.
Casdagli, M., Eubank, S., Farmer, J. D., Gibson, J., 1991, State space reconstruction in the presence of noise. Physica D 51, 52–98.
Davies, M., 1994, Noise reduction schemes for chaotic time series, Physica D 79, 174–192.
Diks, C., 1999, Nonlinear Time Series Analysis: Methods and applications. World Scientific, Singapore.
Eckmann, J. P., 1981, Roads to turbulence in dissipative dynamical systems, Rev. Mod. Phys. 53, 643–654.
Eckmann, J. P., Kamphorst, S. O. and Ruelle, D., 1987, Recurrence plots of dynamical systems, Europhys. Lett. 4, 973–977.
Eckmann, J. P. and Ruelle, D., 1985, Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656.
Farmer, J.D., Ott, E., Yorke, J.A., 1983, The dimension of chaotic attractors, Physica D 7, 153–180
Farmer, J. D. and Sidorowich, J. J., 1987, Predicting chaotic time series. Phys. Rev. Lett. 59, 845–848.
Fraser, A. and Swinney, H., 1986, Independent coordinates for strange attractors from mutual information. Phys. Rev. A 33, 1134–1140.
Gilmore, R., 1998, Topological analysis of chaotic dynamical systems. Rev. Mod. Phys. 70, 1455–1529,
Grassberger, P., 1983, Generalized dimension of strange attractors. Phys. Lett A 97, 227–230.
Grassberger, P. and Procaccia, I., 1983, Characterization of strange attractors, Phys. Rev. Lett. 50, 346–349
Haken, H., 1983, Synergetics: An Introduction, Springer-Verlag, Berlin.
Hentschel, H.G.E. and Procaccia, I., 1983, The infinite number of generalized dimensions of fractals and strange attractors, Physica D 8, 435–444.
Hurst, H. E., 1951, Long-term storage capacity of reservoirs, Trans. Am. Soc. Civ. Eng. 116, 770–779.
Jensen, H. J., 1998, Self-Organized criticality, Cambridge University Press.
Kantz H., Shreiber T., Nonlinear Time Series Analysis, 1997, Cambridge University Press
Kaplan, J. L. and Yorke, J. A., 1979, Chaotic behaviour in multidimensional difference equations. In Peitgen H. O. and Walther, H. O. (eds.) Functional Differential Equations and Approximation of Fixed Points, Springer, Berlin, 204–227.
Kennel, M. B., R. Brown and H. D. I. Abarbanel, 1992, Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phys. Rev. A 45, 3403–3411.
Kolmogorov, A.N., 1958, A new invariant for transitive dynamical systems, Dokl. Akad. Nauk SSSR 119, 861–864.
Kondratieff, N. D., 1935, The long wave in economic life, Review of Economic Statistics 17, 105–115.
Kostelich, E. J. and Schreiber, T., 1993, Noise reduction in chaotic time-series data: A survey of common methods. Phys. Rev. E 48, 1752–1763.
Kuznets, S., 1973, Moders economic growth: Findings and reflections, American Economic Review 63, 247–258.
Mandelbrot, B. B., 1983, The Fractal Geometry of Nature, W. H. Freeman. New York.
Mitchel, W. C., 1927, Bussines Cycles: The Problem and its Setting. National Bureau of Economic Research, New York.
Nicolis, G. and Prigogine, I., 1977, Self-organization in Complex Systems, Wiley, New York.
Parker, Y.S. and Chua, L.O., 1989, Practical Numerical Algorithms for Chaotic Systems, Springer-Verlag, New York.
Packard, N., Crutchfield, J., Farmer, D. and Shaw, R., 1980, Geometry from a time series. Phys. Rev. Lett. 45, 712–715.
Pesin, Y.B., 1977, Lyapunov characteristic exponents and smooth ergodic theory, Russian Math Survey 32, 55–104.
Peters, E. E., 1996, Chaos and Order in the Capital Markets: a New View of Cycles, Prices and Volatility, 2nd Edition, Wiley, New York.
Provenzale, A., Smith, L. A., Vio, R. and Murante, G., 1992, Distinguishing between low-dimensional dynamics and randomness in measured time series. Physica D 58, 31–49.
Rasmussen, S., Mosekilde, E. and Sterman, J. D., 1985, Bifurcations and chaotic behaviour in a simple model of the economic wave, Systems Dynamics Review 1, 92–110.
Ruelle, D. and Takens F., 1971, On the nature of turbulence. Comm. Math. Phys. 20, 167–192.
Ruelle, D., 1990, Deterministic chaos: the science and the fiction. Proc. R. Soc. Lond. A 427, 241–248.
Sauer, T., Yorke, Y and Casdagli, M., 1991, Embedology, J. Stat. Phys. 65, 579–616.
Schreiber, T., 1998, Interdisciplinary application of nonlinear time series methods. Physics Reports.
Schuster H.G., Deterministic Chaos: an introduction, 1995, VCH
Stark, J., Broomhead, D.S., Davies, M.E. and Huke, J. 1997, Takens embedding theorems for forced and stochastic systems. Nonlinear Analysis 30, 5303–5314.
Sterman, J. D., 1985, A behavioral model of the economic long wave. J. of Economic Behavior and Organization 6, 17–53.
Sterman, J. D., 1986, The economic long wave: theory and evidence. Systems Dynamics Review 2, 87–125.
Takens, F., 1981, in Dynamical Systems and Turbulence, Warwick 1980, vol. 898 of Lecture Notes in Mathematics, edited by A. Rand and L.S Young, Springer, Berlin, pp. 366–381.
Takens, F., 1996, The effect of small noise on systems with chaotic dynamics. In Stochastic and Spatial Structures of Dynamical Systems, S. J. van Strien and S. M. Verduyn Lunel, Verhandelingen KNAW, Afd. Natuurkunde, vol. 45, pp. 3–15. North_Holland, Amsterdam.
Theiler, J., Galdrikian, B., Longtin, A, Eubank, S. and Farmer, J. D., 1992, Using surrogate data to detect nonlinearity in time series, in Nonlinear Modeling and Forecasting, Casdagli, M. and Eubank, S. (eds.), Addison-Wesley, Redwood City, California, pp. 163–188.
Tong, H., 1990, Nonlinear Time Series: a Dynamical System Approach. Oxford University Press. Oxford.
Trulla, L.L, A. Giuliani, J.P. Zbilut, and C.L. Webber, Jr. (1996). Recurrence quantification analysis of the logistic equation with transients. Phys. Lett. A 223, 255–26.
Waldrop, M., 1993, Complexity: The Emerging Science at the Edge of Chaos, Touchstone Books.
Webber Jr. C. L. and Zbilut, J. P., 1994, Dynamical assessment of physiological systems and states using recurrence plot strategies. J. Appl. Physiol. 76, 965–973.
Whitney, H., 1936, Differentiate manifolds. Ann. Math. 37, 645–680.
Wolf A., J. B. Swift, H. R. Swinne, J. A. Vastan, 1985, Determining Lyapunov exponents from a time series, Physica 16D, 285–317.
Zbilut, J. P. and Webber Jr. C. L., 1992, Embeddings and delays as derived from quantification of recurrence plots. Phys. Lett. A 171, 199–203
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Strozzi, F., Zaldivar, J.M. (2002). Embedding Theory: Introduction and Applications to Time Series Analysis. In: Soofi, A.S., Cao, L. (eds) Modelling and Forecasting Financial Data. Studies in Computational Finance, vol 2. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-0931-8_2
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