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Optimizing Time Delays for Reconstructing Attractors of Dynamical Systems from Time Series Signals

Research Project

Project/Area Number 07832019
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section時限
Research Field 非線形科学
Research InstitutionScience University of Tokyo

Principal Investigator

IKEGUCHI Tohru  Science University of Tokyo, Research Associate, 基礎工学部, 助手 (30222863)

Project Period (FY) 1995 – 1996
Project Status Completed (Fiscal Year 1997)
Budget Amount *help
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 1996: ¥400,000 (Direct Cost: ¥400,000)
Fiscal Year 1995: ¥1,900,000 (Direct Cost: ¥1,900,000)
Keywordschaos / time series analysis / attractor / delay coordinate / reconstruction / higher order correlation / nonlinear dynamical systems / cumulants / 埋め込み / フラクタル / 軌道不安定性 / 長期予測不能性 / 帰無仮説 / 検定 / 時間遅れ / 非線形
Research Abstract

Identifying deterministic chaos and its quantitative characterization are very important from the viewpoint of time series analysis based on nonlinear dynamical systems theory. For quantitative characterization of deterministic chaos, there are many statistics, for example, the fractal dimensions, the Lyapunov exponents, the metric entropies and so on.
In order to estimate these nonlinear statistics, the first step is the reconstructior of attractors from observed time series'. In order to accomplish the above issue, the embedding theorems by Takens (1981) and Embedology theory by Sauer et al.(1991) are lmportant from theoretical justification.
For practical reconstruction of dynamical systems from an observed single variable time series, the theory of embedding proved that the transformation by the time delay coordinates is an embedding if the dimension of reconstructing attractors is at least larger than twice of the box counting dimension of an underlying dynamical system. Although the time delay value in this reconstrution is an arbitrary value under the theoretical situation, in the case of analyzing real time series, which is corrupted by noise and whose length and measurement precision are finite, it is important to set an appropriate value, particularly for continuous data.
We propose a novel criterion of deciding time delays for reconstructing attractors from a single variable time series. Our criterion considers higher order correlation functions, and finds the convergence values of the extrema. In order to see how our method works well, we analyze two numerical examples by observing the shapes of reconstructed attractors and calculating phase space continuities.Our method can be a good criterion to set appropriate time delays on reconstructing attractors as a result.

Report

(3 results)
  • 1997 Final Research Report Summary
  • 1996 Annual Research Report
  • 1995 Annual Research Report
  • Research Products

    (38 results)

All Other

All Publications (38 results)

  • [Publications] 池口 徹, 合原 一幸: "力学系の埋め込み定理と時系列データからのアトラクタ再構成" 日本応用数理学会誌. 7,4. 260-270 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] T.Ikeguchi, K.Aihara: "Lyapunov Spectral Analysis on Random Data" International Journal of Bifurcation and Chaos. 7,6. 1267-1282 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] T.Ikeguchi, K.Aihara: "Difference correlation can distinguish deterministic chaos from 1/f^d type colored noise" Dhysical Review E. 55・3. 2530-2538 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] T.Ikeguchi, K.Aihara: "On Dimension Estimates with Surrogate Data Sets" IEICE Trans.Fundamentals. E80-A・5. 859-868 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] T.Ikeguchi, K.Aihara: "Estimating Correlation dimensions of biological time series using a reliable method" Journal of Intelligent and Fuzzy Systems. 5.1. 33-52 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] M.Hasegawa, T.Ikeguchi et al.: "An Analysis of Additire Effects of Nonlinear Dynamics for the Combinatorial Optimization" IEICE Trans.Fundamentals. E80-A・1. 206-213 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] T.Ikeguchi et al.: "Reconstructing Attractors with Higher Order Conelation" Proc.International Symposium on Nonlinear Theory and its Applications. 2・1. 1149-1152 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] T.Ikeguchi et al.: "An Analysis on Nonlinear Statistics of Biological Time Series" Proc.International Symposium on Nonlinear Theory and its Applications. 2・1. 1169-1172 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] S.Ogawa, T.Ikeguchi et al.: "Nonlinear modeling by radial basis function networks" IEICE Trans.Fundamentals. E79-A・10. 1608-1617 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] T.Ikeguchi, K.Aihara: "Nonlinear Prediction on Seismic Events in Japan" Proc.International Symposium on Nonlinear Theory and its Applications. 1,1. 285-288 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] A.Tomisaki, T.Ikeguchi et al.: "Deterministic Nonlinear Prediction of Time Series by Radial Basis Function Networks" Proc.International Symposium on Nonlinear Theory and its Applications. 1・1. 341-344 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] 池口 徹, 合原 一幸: "カオスと時系列解析" 電子情報通信学会誌. J79・8. 874-879 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] T.Ikeguchi and K.Aihara: "Embedding theory of dynamical systems and reconstructing attractors from time series data" Bulletin of the Japan Society for Industrial and Applied Mathematics. Vol.7, No.4. 260-270 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] T.Ikeguchi and K.Aihara: "Lyapunov Spectral Analysis on Random Data" International Journal of Bifurcation and Chaos. Vol.7, No.6. 1267-1282 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] T.Ikeguchi and K.Aihara: "Difference correlation can distinguish deterministic chaos from 1/ f^<alpha> type colored noise" Physical Review E. Vol.55, No.3. 2530-2538 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] T.Ikeguchi and Aihara: "On Dimension Estimates with Surrogate Data Sets" IEICE Trans.Vol.E80-A,No.5. 859-868 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] T.Ikeguchi and K.Aihara: "Estimating correlation dimensions of biological time series using a reliable method" Journal of Intellingent and Fuzzy Systems. 33-52 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] M.Hasegawa, T.Ikeguchi, T.Matozaki and K.Aihara: "An Analysis of Additive Effects of Nonliner Dynamics for Combinatorial Optimization" IEICE Trans.Fundamentals. Vol.E-80-A,No.1. 206-213 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] T.Ikeguchi, M.Takakura and K.Aihara: "Reconstructing Attractors with Higher Order Correlation" Proceeding of International Symposium on Nonlinear Theory and Its Application. Vol.2. 1149-1152 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] T.Ikeguchi, C.Kitawaki and K.Aihara: "An Analysis on Nonlinear Statistics of Biological Timeseries" Proceedings of International Symposium on Nonlinear Theory and Its Applications. Vol.2. 1169-1172 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] S.Ogawa, T.Ikeguchi, T.Matozaki and K.Aihara: "Nonlinear Modeling by Radial Basis Function Networks" Transactions IEICE. Vol.E79, No.10. (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] T.Ikeguchi and K.Aihara: "Nonlinear Prediction on Seismic Events in Japan" Proceedings of 1996 International Symposium on Nonlinear Theory and Its Applications. 285-288 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] A.Tomisaki, T.Ikeguchi and T.Matozaki: "Deterministic Nonlinear Prediction of Time Series by Radial Basis Function Networks" Proceedings of the International Symposium on Nonlinear Theory and Its Applications. 341-344 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] T.Ikeguchi and K.Aihara: "Deterministic Chaos and Time Series Analysis" The Journal of IEICE. Vol.J79, No.8. 814-819 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] T.Ikeguchi and K.Aihara: "Local versus Global Plots on Lyapunov Spectra" Technical Report of IEICE. Vol.96, No.477. 97--104 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] T.Ikeguchi, M Nakashima, M.hasegawa, M.Kimura, T.Matozaki and K.Aihara: "Quantifying Recurrence Plots by Texture Analysis" Technical Report of IEICE. Vol.95, No.505. 23--30 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] T.Ikeguchi & K.Aihara: "On dimension estimates with Surrogate data sets" IEICE Trans.Fundamentals. E80-A(印刷中). (1997)

    • Related Report
      1996 Annual Research Report
  • [Publications] T.Ikeguchi & K.Aihara: "Lyapunov spectral analysis on random data" International Journal of Bifurcation and Chaos. 7・6(印刷中). (1997)

    • Related Report
      1996 Annual Research Report
  • [Publications] T.Ikeguchi & K.Aihara: "Difference Correlation can distinguish deterministic chaos from 1/f^α type colored noise" Physical Review E. 55・3. 2530-2538 (1997)

    • Related Report
      1996 Annual Research Report
  • [Publications] T.Ikeguchi & K.Aihara: "Estimating conelation dimensions of biological timeseries usiyng a relialde method" Journal of Intelliguent and Fuzzy Systems. 5・1(印刷中). (1997)

    • Related Report
      1996 Annual Research Report
  • [Publications] S.Ogawa,T.Ikeguchi et al.: "Nonlinear modeling by Radial Basis Function Netwooks" Trans.LEICE Fundamentals. E79A・10. 1608-1617 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] M.Hasegawa,T.Ikeguchi et al.: "An Analysis of Additive Effects of Nonlinear Dynamics for ConbinatorialOptimization" Trans.LEICE Fundamentals. E80A・1. 206-213 (1997)

    • Related Report
      1996 Annual Research Report
  • [Publications] T.Ikeguchi: "Prediction of Chaotic Time Series with Noise" IEICE Trans.Fundamentals. E78. 1291-1298 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] 寺崎 健: "経済時系列データの非線形ダイナミカル特性に関する解析" 電子情報通信学会論文誌. J78-A. 1601-1617 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] T.Ikeguchi: "Estimating corvelation dimensions of biological time series using ardiable method" Journal of Intelligent and Fuzzy systems. 印刷中. (1996)

    • Related Report
      1995 Annual Research Report
  • [Publications] T.Ikeguchi: "Nonlinear Prediction on Squid Axon Response" Proc.IEEE,International Conferone on Neural Networks. 6. 3134-3139 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] T.Ikeguchi: "Nonlinear Prediction on Interspike Intervals Produced from Dynamical Systems" Proc.International Symposium on Nonlinear Theory andits Application. 1. 121-126 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] T.Ikeguchi: "On Dimension Estimates with Surrogate Data Sets" IEICE Technical Report. 95-482. 47-54 (1996)

    • Related Report
      1995 Annual Research Report

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Published: 1995-04-01   Modified: 2016-04-21  

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