Nonlinear Time-Series Analysis

This tutorial review presents an overview of the achievements and some present research activities in the field of state space based methods for nonlinear time-series analysis. In particular, questions of state space reconstruction, of modelling and prediction, of filtering and noise reduction, of detecting non-linearities in time series, and applications using chaotic synchronization are addressed. Furthermore, a new approach for modeling data from spatio-temporal systems is presented.

[1]  Farmer,et al.  Predicting chaotic time series. , 1987, Physical review letters.

[2]  L. Tsimring,et al.  Generalized synchronization of chaos in directionally coupled chaotic systems. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[3]  Mikhail I. Rabinovich,et al.  Nonlinearities in Action , 1992 .

[4]  D. Broomhead,et al.  Takens embedding theorems for forced and stochastic systems , 1997 .

[5]  R. Gencay,et al.  An algorithm for the n Lyapunov exponents of an n -dimensional unknown dynamical system , 1992 .

[6]  Theiler,et al.  Spurious dimension from correlation algorithms applied to limited time-series data. , 1986, Physical review. A, General physics.

[7]  J. Kurths,et al.  An attractor in a solar time series , 1987 .

[8]  Henry D. I. Abarbanel,et al.  Analysis of Observed Chaotic Data , 1995 .

[9]  H. Kantz A robust method to estimate the maximal Lyapunov exponent of a time series , 1994 .

[10]  K. Briggs An improved method for estimating Liapunov exponents of chaotic time series , 1990 .

[11]  Brown,et al.  Computing the Lyapunov spectrum of a dynamical system from an observed time series. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[12]  Rice,et al.  Method of false nearest neighbors: A cautionary note. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[13]  M. Paluš,et al.  Information theoretic test for nonlinearity in time series , 1993 .

[14]  Ljupco Kocarev,et al.  General approach for chaotic synchronization with applications to communication. , 1995, Physical review letters.

[15]  P. Grassberger,et al.  On noise reduction methods for chaotic data. , 1993, Chaos.

[16]  R. Savit,et al.  Time series and dependent variables , 1991 .

[17]  Ulrich Parlitz,et al.  Methods of chaos physics and their application to acoustics , 1988 .

[18]  André Longtin,et al.  Interspike interval attractors from chaotically driven neuron models , 1997 .

[19]  L. Tsimring,et al.  The analysis of observed chaotic data in physical systems , 1993 .

[20]  Nikolai F. Rulkov,et al.  Modeling and synchronizing chaotic systems from experimental data , 1994 .

[21]  Alfonso M Albano,et al.  Phase-randomized surrogates can produce spurious identifications of non-random structure , 1994 .

[22]  G. P. King,et al.  Extracting qualitative dynamics from experimental data , 1986 .

[23]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[24]  H. Kantz,et al.  Nonlinear time series analysis , 1997 .

[25]  Michael Peter Kennedy Chaos in the Colpitts oscillator , 1994 .

[26]  H. Abarbanel,et al.  Lyapunov exponents from observed time series. , 1990, Physical review letters.

[27]  Yasuji Sawada,et al.  Practical Methods of Measuring the Generalized Dimension and the Largest Lyapunov Exponent in High Dimensional Chaotic Systems , 1987 .

[28]  M. Rosenstein,et al.  Reconstruction expansion as a geometry-based framework for choosing proper delay times , 1994 .

[29]  D. T. Kaplan,et al.  Exceptional events as evidence for determinism , 1994 .

[30]  Grebogi,et al.  Synchronization of spatiotemporal chaotic systems by feedback control. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[31]  Kevin Judd,et al.  Reconstructing noisy dynamical systems by triangulations , 1997 .

[32]  Zoran Aleksic,et al.  Estimating the embedding dimension , 1991 .

[33]  A. Politi,et al.  Statistical description of chaotic attractors: The dimension function , 1985 .

[34]  A. Gallant,et al.  Convergence rates and data requirements for Jacobian-based estimates of Lyapunov exponents from data , 1991 .

[35]  Parlitz,et al.  Synchronization-based parameter estimation from time series. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[36]  Pierre Bergé,et al.  Order within chaos : towards a deterministic approach to turbulence , 1984 .

[37]  Mark R. Muldoon,et al.  Linear Filters and Non‐Linear Systems , 1992 .

[38]  V. I. Oseledec A multiplicative ergodic theorem: Lyapunov characteristic num-bers for dynamical systems , 1968 .

[39]  Ulrich Parlitz,et al.  Comparison of Different Methods for Computing Lyapunov Exponents , 1990 .

[40]  Thomas Schreiber,et al.  EFFICIENT NEIGHBOR SEARCHING IN NONLINEAR TIME SERIES ANALYSIS , 1995 .

[41]  T. Sauer,et al.  Correlation dimension of attractors through interspike intervals , 1997 .

[42]  Antonio Politi,et al.  Hausdorff Dimension and Uniformity Factor of Strange Attractors , 1984 .

[43]  I. Shimada,et al.  A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems , 1979 .

[44]  Bulsara,et al.  Array enhanced stochastic resonance and spatiotemporal synchronization. , 1995, Physical review letters.

[45]  G. Benettin,et al.  Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application , 1980 .

[46]  P. F. Meier,et al.  Evaluation of Lyapunov exponents and scaling functions from time series , 1988 .

[47]  James P. Crutchfield,et al.  Geometry from a Time Series , 1980 .

[48]  A. Fraser Reconstructing attractors from scalar time series: A comparison of singular system and redundancy criteria , 1989 .

[49]  Thomas Schreiber,et al.  Constrained Randomization of Time Series Data , 1998, chao-dyn/9909042.

[50]  Joachim Holzfuss,et al.  Approach to error-estimation in the application of dimension algorithms , 1986 .

[51]  Eckmann,et al.  Liapunov exponents from time series. , 1986, Physical review. A, General physics.

[52]  L. A. Aguirre,et al.  GLOBAL NONLINEAR POLYNOMIAL MODELS: STRUCTURE, TERM CLUSTERS AND FIXED POINTS , 1996 .

[53]  Parlitz,et al.  Predicting low-dimensional spatiotemporal dynamics using discrete wavelet transforms. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[54]  Parlitz,et al.  Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems. , 1996, Physical review letters.

[55]  Passamante,et al.  Recognizing determinism in a time series. , 1993, Physical review letters.

[56]  Ruedi Stoop,et al.  Calculation of Lyapunov exponents avoiding spurious elements , 1991 .

[57]  Kurt Wiesenfeld,et al.  Disorder-enhanced synchronization , 1995 .

[58]  Martin Casdagli,et al.  Nonlinear prediction of chaotic time series , 1989 .

[59]  José R. Dorronsoro,et al.  Local State-Space Reconstruction Using Averaged Scalar Products of the Dynamical-System Flow Vectors , 1995 .

[60]  Mees,et al.  Mutual information, strange attractors, and the optimal estimation of dimension. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[61]  Holger Kantz,et al.  Effective deterministic models for chaotic dynamics perturbed by noise , 1997 .

[62]  Ulrich Parlitz,et al.  Identification of True and Spurious Lyapunov Exponents from Time Series , 1992 .

[63]  J. Fell,et al.  Resonance-like phenomena in Lyapunov calculations from data reconstructed by the time-delay method , 1994 .

[64]  Michael Rosenblum,et al.  Time series analysis for system identification and diagnostics , 1991 .

[65]  F. Takens Detecting strange attractors in turbulence , 1981 .

[66]  H. Abarbanel,et al.  Generalized synchronization of chaos: The auxiliary system approach. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[67]  Carroll,et al.  Synchronization in chaotic systems. , 1990, Physical review letters.

[68]  James Theiler,et al.  Estimating fractal dimension , 1990 .

[69]  P. Grassberger,et al.  NONLINEAR TIME SEQUENCE ANALYSIS , 1991 .

[70]  W. Singer Synchronization of cortical activity and its putative role in information processing and learning. , 1993, Annual review of physiology.

[71]  D. Ruelle,et al.  Ergodic theory of chaos and strange attractors , 1985 .

[72]  Markus Eiswirth,et al.  Computation of Lyapunov spectra: effect of interactive noise and application to a chemical oscillator , 1993 .

[73]  Stephen A. Billings,et al.  Identification of models for chaotic systems from noisy data: implications for performance and nonlinear filtering , 1995 .

[74]  Ivan Dvořák,et al.  Singular-value decomposition in attractor reconstruction: pitfalls and precautions , 1992 .

[75]  James B. Kadtke,et al.  GLOBAL DYNAMICAL EQUATIONS AND LYAPUNOV EXPONENTS FROM NOISY CHAOTIC TIME SERIES , 1993 .

[76]  Roger A. Pielke,et al.  EXTRACTING LYAPUNOV EXPONENTS FROM SHORT TIME SERIES OF LOW PRECISION , 1992 .

[77]  Peter Grassberger,et al.  Generalizations of the Hausdorff dimension of fractal measures , 1985 .

[78]  Ulrich Parlitz Lyapunov exponents from Chua's Circuit , 1993, J. Circuits Syst. Comput..

[79]  Parlitz,et al.  Encoding messages using chaotic synchronization. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[80]  James Theiler,et al.  On the evidence for low-dimensional chaos in an epileptic electroencephalogram , 1995 .

[81]  Ioannis G. Kevrekidis,et al.  DISCRETE- vs. CONTINUOUS-TIME NONLINEAR SIGNAL PROCESSING OF Cu ELECTRODISSOLUTION DATA , 1992 .

[82]  Schreiber,et al.  Noise reduction in chaotic time-series data: A survey of common methods. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[83]  M. Rosenstein,et al.  A practical method for calculating largest Lyapunov exponents from small data sets , 1993 .

[84]  L. Cao Practical method for determining the minimum embedding dimension of a scalar time series , 1997 .

[85]  Brown,et al.  Modeling and synchronizing chaotic systems from time-series data. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[86]  Gao,et al.  Direct dynamical test for deterministic chaos and optimal embedding of a chaotic time series. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[87]  Schreiber,et al.  Signal separation by nonlinear projections: The fetal electrocardiogram. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[88]  Lauterborn,et al.  Liapunov exponents from a time series of acoustic chaos. , 1989, Physical review. A, General physics.

[89]  George G. Szpiro Forecasting chaotic time series with genetic algorithms , 1997 .

[90]  Schreiber,et al.  Improved Surrogate Data for Nonlinearity Tests. , 1996, Physical review letters.

[91]  Jianbo Gao,et al.  Local exponential divergence plot and optimal embedding of a chaotic time-series , 1993 .

[92]  U. Parlitz,et al.  Lyapunov exponents from time series , 1991 .

[93]  Gerd Pfister,et al.  Optimal Reconstruction of Strange Attractors from Purely Geometrical Arguments , 1990 .

[94]  D. Ruelle,et al.  Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems , 1992 .

[95]  Ulrich Parlitz,et al.  Experimental Nonlinear Physics , 1997 .

[96]  Carroll,et al.  Synchronous chaos in coupled oscillator systems. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[97]  Antanas Cenys,et al.  Estimation of the number of degrees of freedom from chaotic time series , 1988 .

[98]  I. Stewart,et al.  Bubbling of attractors and synchronisation of chaotic oscillators , 1994 .

[99]  J. Yorke,et al.  HOW MANY DELAY COORDINATES DO YOU NEED , 1993 .

[100]  L. Sirovich Chaotic dynamics of coherent structures , 1989 .

[101]  Leonard A. Smith Identification and prediction of low dimensional dynamics , 1992 .

[102]  H. Abarbanel,et al.  LYAPUNOV EXPONENTS IN CHAOTIC SYSTEMS: THEIR IMPORTANCE AND THEIR EVALUATION USING OBSERVED DATA , 1991 .

[103]  Schreiber,et al.  Nonlinear noise reduction: A case study on experimental data. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[104]  Sawada,et al.  Measurement of the Lyapunov spectrum from a chaotic time series. , 1985, Physical review letters.

[105]  Ljupco Kocarev,et al.  Subharmonic Entrainment of Unstable Period Orbits and Generalized Synchronization , 1997 .

[106]  Leonard A. Smith,et al.  Distinguishing between low-dimensional dynamics and randomness in measured time series , 1992 .

[107]  H. Abarbanel,et al.  Determining embedding dimension for phase-space reconstruction using a geometrical construction. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[108]  Cawley,et al.  Smoothness implies determinism: A method to detect it in time series. , 1994, Physical review letters.

[109]  P. Grassberger,et al.  Characterization of Strange Attractors , 1983 .

[110]  Mike E. Davies,et al.  Linear Recursive Filters and Nonlinear Dynamics , 1996 .

[111]  G. Broggi,et al.  Evaluation of dimensions and entropies of chaotic systems , 1988 .

[112]  H. Schuster,et al.  Proper choice of the time delay for the analysis of chaotic time series , 1989 .

[113]  Floris Takens,et al.  DETECTING NONLINEARITIES IN STATIONARY TIME SERIES , 1993 .

[114]  A. Fowler,et al.  A correlation function for choosing time delays in phase portrait reconstructions , 1993 .

[115]  A. Mees,et al.  On selecting models for nonlinear time series , 1995 .

[116]  Andrew M. Fraser,et al.  Information and entropy in strange attractors , 1989, IEEE Trans. Inf. Theory.

[117]  Fedor Mitschke,et al.  Estimation of Lyapunov exponents from time series: the stochastic case , 1993 .

[118]  K. Pawelzik,et al.  Optimal Embeddings of Chaotic Attractors from Topological Considerations , 1991 .

[119]  R. Eykholt,et al.  Estimating the Lyapunov-exponent spectrum from short time series of low precision. , 1991, Physical review letters.

[120]  Sauer,et al.  Reconstruction of dynamical systems from interspike intervals. , 1994, Physical review letters.

[121]  Francis C. Moon,et al.  Chaotic and fractal dynamics , 1992 .

[122]  E. Ott Chaos in Dynamical Systems: Contents , 1993 .

[123]  H. Fujisaka,et al.  Stability Theory of Synchronized Motion in Coupled-Oscillator Systems , 1983 .

[124]  James Theiler,et al.  Testing for nonlinearity in time series: the method of surrogate data , 1992 .

[125]  Pfister,et al.  Optimal delay time and embedding dimension for delay-time coordinates by analysis of the global static and local dynamical behavior of strange attractors. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[126]  J. D. Farmer,et al.  State space reconstruction in the presence of noise" Physica D , 1991 .

[127]  Fraser,et al.  Independent coordinates for strange attractors from mutual information. , 1986, Physical review. A, General physics.

[128]  Martin Casdagli,et al.  An analytic approach to practical state space reconstruction , 1992 .