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Extracting Knowledge From Time Series pp 275–308Cite as

Model Equations: “Black Box” Reconstruction

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Abstract

Black box reconstruction is both the most difficult and the most tempting modelling problem when any prior information about an appropriate model structure is lacking. An intriguing thing is that a model capable of reproducing an observed behaviour or predicting further evolution should be obtained only from an observed time series, i.e. “from nothing” at first sight. Chances for a success are not large. Even more so, a “good” model would become a valuable tool to characterise an object and understand its dynamics. Lack of prior information causes one to utilise universal model structures, e.g. artificial neural networks, radial basis functions and algebraic polynomials are included in the right-hand sides of dynamical model equations. Such models are often multi-dimensional and involve quite many free parameters.

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Notes

  1. 1.

    The set M is a compact smooth manifold and the quantity d is its topological dimension (Sect. 10.1.1). There are generalisations of the theorem to the case of non-smooth sets M and fractal dimension d, which are beyond the scope of our discussion (Sauer et al., 1991). We note that the set M mentioned in the theorems does not inevitably correspond to the motion on an attractor. For instance, let an attractor be a limit cycle C “reeled” on a torus M. If one is interested only in the description of an established periodic motion on the cycle C, then it is sufficient to use \(D>2\) model variables for reconstruction according to Takens’ theorems. If one needs to describe motions on the entire torus M, including transient processes, then it is necessary to use \(D>4\) variables. In practice, one often has a single realisation corresponding to established dynamics. Therefore, one usually speaks of the reconstruction of an attractor.

  2. 2.

    An equivalent description of a motion on a limit cycle is assured for \(D=3\) even if the cycle “lives” in an infinite-dimensional phase space.

  3. 3.

    If an object is a map \({\textbf{y}}(t_{n+1})=\textbf{F}(\textbf{y}(t_{n}))\), then an evolution operator \(\Phi_{\Delta t}(\textbf{y}(t_{0}))\) is just the function F. If an object is a set of ODEs \({\mathrm{d}}\textbf{y}/{\mathrm{d}}t=\textbf{F}(\textbf{y}(t))\), then the function \(\Phi_{t}(\textbf{y}(t_{0}))\) is the result of the integration of the ODEs over a time interval of length t. If an original system is given by a partial differential equation \(\partial\textbf{y}/\partial t=\textbf{F}(\textbf{y},\partial\textbf{y}/\partial\textbf{r},\partial^{2}\textbf{y}/\partial\textbf{r}^{2},\ldots)\), where r is a spatial coordinate, then y is a vector belonging to an infinite-dimensional space of functions y(r) and Φ t is an operator acting in that space.

  4. 4.

    It means that for the same segments \([\eta(t),\eta(t+\tau),\ldots,\eta(t+(D-1)\tau)]\) encountered at different time instants t, one observes the same continuation (i.e. the same future). It gives a justification to the predictive method of analogues applied already by E. Lorenz. The method is based on the search of the time series segments, which “resemble” a current segment, in the past and subsequent usage of a combination of their “futures” as a forecast. In a modern formulation, it is realised with local models (Sect. 10.2.1).

  5. 5.

    For \(l=1\) and a small sampling interval Δ t, a reconstructed phase orbit stretches along the main diagonal, since it appears that \(x_{1}\approx x_{2}\approx \ldots\approx x_{D}\).

References

  • Abarbanel, H.D.I., Brown, R., Kadtke, J.B.: Prediction and system identification in chaotic nonlinear systems: time series with broadband spectra. Phys. Lett. A. 138, 401–408 (1989)

    Article  ADS  Google Scholar 

  • Anosov, O.L., Butkovsky, O.Ya., Kravtsov, Yu.A.: Limits of predictability for linear autoregression models. J. Communications Technol. Electron. 40(12), 1866–1873, (in Russian) (1995)

    Google Scholar 

  • Bär, M., Hegger, R., Kantz, H.: Fitting partial differential equations to space-time dynamics. Phys. Rev. E. 59, 337–343 (1999)

    Article  ADS  Google Scholar 

  • Bezruchko, B.P., Smirnov, D.A.: Constructing nonautonomous differential equations from a time series. Phys. Rev. E. 63, 016207, (2001)

    Article  MathSciNet  ADS  Google Scholar 

  • Broomhead, D.S., King, G.P.: Extracting qualitative dynamics from experimental data. Physica D. 20, 217–236 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Broomhead, D.S., Lowe, D.: Multivariable functional interpolation and adaptive networks. Complex Syst. 2, 321–355 (1988)

    MathSciNet  MATH  Google Scholar 

  • Brown, R., Rulkov, N.F., Tracy, E.R.: Modeling and synchronizing chaotic systems from experimental data. Phys. Lett. A. 194, 71–76 (1994)

    Article  ADS  Google Scholar 

  • Cao, L., Mees, A.I., Judd, K.: Dynamics from multivariate time series. Phys. D. 121, 75–88 (1998)

    Article  MATH  Google Scholar 

  • Casdagli, M.: Nonlinear prediction of chaotic time series. Phys. D. 35, 335–356 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  • Casdagli, M., Eubank, S., Farmer, J.D., Gibson, J.: State space reconstruction in presence of noise. Phys. D. 51, 52–98 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  • Cellucci, C.J., Albano, A.M., Rapp, P.E.: Comparative study of embedding methods. Phys. Rev. E. 67, 066210 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  • Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  • Farmer, J.D., Sidorowich, J.J.: Predicting chaotic time series. Phys. Rev. Lett. 59, 845–848 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  • Fraser, A.M., Swinney, H.L.: Independent coordinates for strange attractors from mutual information. Phys. Rev. A. 33, 1131–1140 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  • Gerschenfeld, N.A., Weigend, A.S. (eds.): Time Series Prediction: Forecasting the Future and Understanding the Past. SFI Studies in the Science of Complexity, Proc. V. XV. Addison-Wesley, New York (1993)

    Google Scholar 

  • Gibson, J.F., Farmer, J.D., Casdagli, M., Eubank, S.: An analytic approach to practical state space reconstruction. Phys. D. 57, 1–30 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Giona, M., Lentini, F., Cimagalli, V.: Functional reconstruction and local prediction of chaotic time series. Phys. Rev. E. 44, 3496–3502 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  • Gliklikh Yu.E.: What a smooth manifold is. Soros Educ. J. 11, 155–159, (in Russian) (1998)

    Google Scholar 

  • Gorban’ A.N.: Multivariable functions and neural networks. Soros Educational J. No. 12, 105–112, (in Russian) (1998)

    Google Scholar 

  • Gouesbet, G.: Reconstruction of the vector fields of continuous dynamical systems from scalar time series. Phys. Rev. A. 43, 5321–5331 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  • Gouesbet, G., Letellier, C.: Global vector-field approximation by using a multivariate polynomial \(L_{2}\) approximation on nets. Phys. Rev. E. 49, 4955–4972 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  • Gouesbet, G., Meunier-Guttin-Cluzel, S., Ménard, O.: Global reconstructions of equations of motion from data series, and validation techniques, a review. In: Gouesbet, G., Meunier-Guttin-Cluzel, S., Ménard, O. (eds.) Chaos and Its Reconstructions, pp. 1–160. Nova Science Publishers, New York, (2003)

    Google Scholar 

  • Grassberger, P., Procaccia, I.: Measuring the strangeness of strange attractors. Phys. D. 9, 189–208 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  • Gribkov, D.A., Gribkova, V.V., Kravtsov Yu.A., et al. Reconstructing structure of dynamical system from time series. J. Commun. Technol. Electron. 39(2), 269–277, (in Russian) (1994)

    Google Scholar 

  • Hubner, U., Weiss, C.-O., Abraham, N.B., Tang, D.: Lorenz-like chaos in \(\mathit{NH}_{3}\) – FIR lasers (data set A). In: Gerschenfeld, N.A., Weigend, A.S. (eds.) Time Series Prediction: Forecasting the Future and Understanding the Past. SFI Studies in the Science of Complexity, Proc. V. XV, pp. 73–104. Addison-Wesley, Reading, MA (1993)

    Google Scholar 

  • Janson, N.B., Pavlov, A.N., Anishchenko, V.S.: One method for restoring inhomogeneous attractors. Int. J. Bif. Chaos. 8, 825–833 (1998)

    Article  MATH  Google Scholar 

  • Johnson, L.W., Riess, R.D.: Numerical Analysis, 2nd edn. Addison-Wesley, Reading, MA (1982)

    MATH  Google Scholar 

  • Judd, K., Mees, A.I. On selecting models for nonlinear time series. Phys. D. 82, 426–444 (1995)

    Article  MATH  Google Scholar 

  • Judd, K., Mees, A.I.: Embedding as a modeling problem. Phys. D. 120, 273–286 (1998)

    Article  MATH  Google Scholar 

  • Judd, K., Small, M.: Towards long-term prediction. Phys. D. 136, 31–44 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Kalitkin, N.N.: Numerical Methods. Nauka, Moscow, (in Russian) (1978)

    Google Scholar 

  • Kantz, H., Schreiber, T.: Nonlinear Time Series Analysis. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  • Kartsev, V.P.: Adventures of Great Equations. Znaniye, Moscow, (in Russian) (1976)

    Google Scholar 

  • Kennel, M.B., Brown, R., Abarbanel, H.D.I.: Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phys. Rev. A. 45, 3403–3411 (1992)

    Article  ADS  Google Scholar 

  • Kravtsov Yu.A.: Randomness, determinacy, predictability. Phys. Uspekhi. 158(1), 93–115, (in Russian) (1989)

    Google Scholar 

  • Kugiumtzis, D., Lingjaerde, O.C., Christophersen, N.: Regularized local linear prediction of chaotic time series. Phys. D. 112, 344–360 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Landa, P.S., Rosenblum, M.G.: Comparison of methods for construction of phase space and determination of attractor dimension from experimental data. Tech. Phys. 59(11), 1–8, (in Russian) (1989)

    Google Scholar 

  • Letellier, C., Aguirre, L.A.: Investigating nonlinear dynamics from time series: The influence of symmetries and the choice of observables. Chaos. 12, 549–558 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Letellier, C., Le Sceller, L., Gouesbet, G., et al.: Recovering deterministic behavior from experimental time series in mixing reactor. AIChE J. 43(9), 2194–2202 (1997)

    Article  Google Scholar 

  • Letellier, C., Le Sceller, L., Maréchal, E., et al.: Global vector field reconstruction from a chaotic experimental signal in copper electrodissolution. Phys. Rev. E. 51, 4262–4266 (1995)

    Article  ADS  Google Scholar 

  • Letellier, C., Maquet, J., Labro, H., et al. Analyzing chaotic behavior in a Belousov-Zhabotinskyi reaction by using a global vector field reconstruction. J. Phys. Chem. 102, 10265–10273 (1998a)

    Article  Google Scholar 

  • Letellier, C., Macquet, J., Le Sceller, L., et al. On the non-equivalence of observables in phase space reconstructions from recorded time series. J. Phys. A: Math. Gen. 31, 7913–7927 (1998b)

    Article  ADS  MATH  Google Scholar 

  • Liebert, W., Schuster, H.G.: Proper choice the of time delay for the analysis of chaotic time series. Phys. Lett. A. 142, 107–111 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  • Makarenko, N.G.: Fractals, attractors, neural networks and so forth. Procs. IV All-Russian Conf. “Neuroinformatics-2002”. Part 2, pp. 121–169. Moscow, (in Russian) (2002)

    Google Scholar 

  • Makarenko, N.G.: Embedology and Neuro-Prediction. Procs. V All-Russian Conf. “Neuroinformatics-2003”. Part 1, pp. 86–148. Moscow, (in Russian) (2003)

    Google Scholar 

  • Malinetsky, G.G., Potapov, A.B.: Contemporary Problems of Nonlinear Dynamics. Editorial URSS, Moscow, (in Russian) (2000)

    Google Scholar 

  • Parlitz, U., Mayer-Kress, G.: Predicting low-dimensional spatiotemporal dynamics using discrete wavelet transforms. Phys. Rev. E. 51, R2709–R271 (1995)

    Article  ADS  Google Scholar 

  • Parlitz, U., Merkwirth, C.: Prediction of spatiotemporal time series based on reconstructed local states. Phys. Rev. Lett. 84, 1890–1893 (2000)

    Article  ADS  Google Scholar 

  • Pavlov, A.N., Janson, N.B., Anishchenko, V.S.: Application of statistical methods to solve the problem of global reconstruction. Tech. Phys. Lett. 23(4), 297–299 (1997)

    Article  ADS  Google Scholar 

  • Pavlov, A.N., Janson, N.B., Anishchenko, V.S.: Reconstruction of dynamical systems. J. Commun. Technol. Electron. 44(9), 999–1014 (1999)

    Google Scholar 

  • Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes in C. Cambridge University Press, Cambridge (1988)

    Google Scholar 

  • Rulkov, N.F., Sushchik, M.M., Tsimring, L.S., Abarbanel, H.D.I.: Generalized synchronization of chaos in directionally coupled chaotic systems. Phys. Rev. E. 51, 980–994 (1995)

    Article  ADS  Google Scholar 

  • Samarsky, A.A.: Introduction to Numerical Methods. Nauka, Moscow, (in Russian) (1982)

    Google Scholar 

  • Sauer, T., Yorke, J.A., Casdagli, M.: Embedology. J. Stat. Phys. 65(3–4), 579–616 (1991)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Sauer, T.: Time series prediction by using delay coordinate embedding. in Time Series Prediction: Forecasting the Future and Understanding the Past. Eds. N.A. Gerschenfeld, A.S. Weigend. SFI Studies in the Science of Complexity, Proc. V. XV, pp. 175–193. Addison-Wesley (1993)

    Google Scholar 

  • Schroer, C., Sauer, T., Ott, E., Yorke, J.: Predicting chaos most of the time from embeddings with self-intersections. Phys. Rev. Lett. 80, 1410–1413 (1998)

    Article  ADS  Google Scholar 

  • Sitz, A., Kurths, J., Voss, H.U.: Identification of nonlinear spatiotemporal systems via partitioned filtering. Phys. Rev. E. 68, 016202 (2003)

    Article  ADS  Google Scholar 

  • Sitz, A., Schwartz, U., Kurths, J., Voss, H.U.: Estimation of parameters and unobserved components for nonlinear systems from noisy time series. Phys. Rev. E. 66, 016210 (2002)

    Article  ADS  Google Scholar 

  • Small, M., Judd, K., Mees, A.: Modelling continuous processes from data. Phys. Rev. E. 65, 046704 (2002)

    Article  ADS  Google Scholar 

  • Small, M., Judd, K.: Comparisons of new nonlinear modeling techniques with applications to infant respiration. Physica, D. 117, 283–298 (1998)

    Article  ADS  MATH  Google Scholar 

  • Small, M., Tse, C.K.: Optimal embedding: A modelling paradigm. Phys. D. 194, 283–296 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Smirnov, D.A., Bezruchko, B.P., Seleznev Ye.P.: Choice of dynamical variables for global reconstruction of model equations from time series. Phys. Rev. E. 65, 026205 (2002)

    Article  ADS  Google Scholar 

  • Smirnov, D.A., Bezruchko, B.P.: Nonlinear dynamical models from chaotic time series: methods and applications. In: Winterhalder, M., Schelter, B., Timmer, J. (eds.) Handbook of Time Series Analysis, pp. 181–212. Wiley-VCH, Berlin (2006)

    Chapter  Google Scholar 

  • Smith, L.A.: Identification and prediction of low-dimensional dynamics. Phys. D. 58, 50–76 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Smith, L.A.: Maintenance of uncertainty. Proc. Int. School of Physics “Enrico Fermi”, Course CXXXIII, pp. 177–246. Italian Physical Society, Bologna, (1997). Available at http://www.maths.ox.ac.uk/∼lenny

    Google Scholar 

  • Stark, J., Broomhead, D.S., Davies, M., Huke, J.: Takens embedding theorem for forced and stochastic systems. Nonlinear Analysis. Theory, Methods, and Applications. Proc. 2nd Congress on Nonlinear Analysis. Elsevier Science Ltd., 30(8), 5303–5314 (1997)

    Google Scholar 

  • Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, New York (1980)

    Google Scholar 

  • Takens, F.: Detecting strange attractors in turbulence. Lect. Notes Math. 898, 366–381 (1981)

    Article  MathSciNet  Google Scholar 

  • Timmer, J.: Parameter estimation in nonlinear stochastic differential equations. Chaos, Solitons Fractals 11, 2571–2578 (2000)

    Article  ADS  MATH  Google Scholar 

  • Tong, H.: Nonlinear Time Series Analysis: a Dynamical System Approach. Oxford University Press, Oxford (1990)

    Google Scholar 

  • Voss, H.U., Bünner, M., Abel, M.: Identification of continuous, spatiotemporal systems. Phys. Rev. E. 57, 2820–2823 (1998)

    Article  ADS  Google Scholar 

  • Wan, E.A.: Time series prediction by using a connectionist network with internal delay lines. In: Gerschenfeld, N.A., Weigend, A.S. (eds.) Time Series Prediction: Forecasting the Future and Understanding the Past. SFI Studies in the Science of Complexity, Proc. V. XV, pp. 195–217. Addison-Wesley (1993)

    Google Scholar 

  • Yule, G.U.: On a method of investigating periodicities in disturbed series, with special reference to Wolfer’s sunspot numbers. Phil. Trans. R. Soc. Lond. A. 226, 267–298 (1927)

    Article  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Saratov State University, Astrakhanskaya Street 83, 410012, Saratov, Russia

    Prof. Boris P. Bezruchko

  2. Russian Academy of Science, V.A. Kotel’nikov Institute of RadioEngineering and Electronics, Saratov Branch, Zelyonaya Str. 38, 410019, Saratov, Russia

    Dr. Dmitry A. Smirnov

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  1. Prof. Boris P. Bezruchko
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  2. Dr. Dmitry A. Smirnov
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Correspondence to Boris P. Bezruchko .

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Bezruchko, B.P., Smirnov, D.A. (2010). Model Equations: “Black Box” Reconstruction. In: Extracting Knowledge From Time Series. Springer Series in Synergetics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12601-7_10

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