Abstract
The number and variety of methods used in dynamical analysis has increased dramatically during the last fifteen years, and the limitations of these methods, especially when applied to noisy biological data, are now becoming apparent. Their misapplication can easily produce fallacious results. The purpose of this introduction is to identify promising new methods and to describe safeguards that can be used to protect against false conclusions.
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References
Abarbanel, H.D.I., Brown, R. and Kennel, M.B. (1991). Variation of Lyapunov exponents on a strange attractor.Journal Nonlinear Science, 1: 175–199.
Abarbanel, H.D.I., Brown, R., Sidorowich, J.J. and Tsimiring, L.S. (1993). The analysis of observed chaotic data in physical systems.Reviews of Modern Physics, 65, 1331–1392.
Abarbanel, H.D.I., and Kennel, M.B. (1993). Local false nearest neighbors and dynamical dimensions from observed chaotic data.Physical Review, 47E, 3057–3068.
Albano, A.M., Muench, J., Schwartz, C., Mees, A.I. and Rapp, P.E. (1988). Singular-value decomposition and the Grassberger-Procaccia algorithm.Physical Review, 38A, 3017–3026.
Albano, A.M., Passamante, A. and Farrell, M.E. (1991). Using higher-order correlations to define an embedding window.Physica, 54D, 85–97.
Babloyantz, A. (1989). Some remarks on the nonlinear analysis of physiological time series. In:Measures of Complexity and Chaos. N.B. Abraham, A.M. Albano, A. Passamante and P.E. Rapp (Eds.), pp. 51–62. New York: Plenum Press.
Badii, R., Broggi, G., Derighetti, B., Ravani, M., Cilibertti, S., Politi, A. and Rubio, M.A. (1988). Dimension increase in filtered chaotic signals.Physical Review Letters, 60, 979–982.
Badii, R. and Politi, A. (1985). Statistical description of chaotic attractors: The dimension function.Journal of Statistical Physics, 40, 725–750.
Badii, R. and Politi, A. (1987). Renyi dimensions from local expansion rates.Physical Review, 35A, 1288–1293.
Barlow, J.S. (1985). Methods of analysis of nonstationary EEGs with emphasis on segmentation techniques: A comparative review.Journal of Clinical Neurophysiology, 2, 267–304.
Barlow, J.S., Creutzfeldt, O.D., Michael, D., Houchin, J. and Epelbaum, H, (1981). Automatic adaptive segmentation of clinical EEGs.Electroencephalography and Clinical Neurophysiology, 51, 512–525.
Barnard, G.A. (1963). Discussion on the spectral analysis of point processes by M.S. Bartlett.Journal of the Royal Statistical Society, 25B, 294.
Bendat, J.B. and Piersol, A.G. (1966).Measurement and Analysis of Random Data. New York: John Wiley.
Bennett, C.H. (1986). On the nature and origin of complexity in discrete, homogeneous, locally-interacting systems.Foundations Physics., 16, 585.
Berliner, L.M. (1992). Statistics, probability and chaos.Statistical Sciences, 7, 69–122.
Bodenstein, G. and Praetorious, H.M. (1977). Feature extraction from the electroencephalogram by adaptive segmentation.Proceedings IEEE, 65, 642–652.
Brock, W.A. and Dechert, W.E. (1989). Statistical inference theory for measures of complexity in chaos theory and nonlinear science. In:Measures of Complexity and Chaos. N.B. Abraham, A.M. Albano, A. Passamante and P.E. Rapp, (Eds.), pp. 79–98. New York: Plenum Press.
Brock, W.A. and Potter, S.M. (1992). Diagnostic testing for nonlinearity, chaos and general dependence in time series data. In:Nonlinear Modeling and Forecasting. M. Casdagli and S. Eubank, (Eds.), pp. 137–162. Reading, MA: Addison-Wesley.
Brock, W.A. and Sayers, C.L. (1988). Is the business cycle characterized by deterministic chaos.Journal of Monetary Economics, 22, 71–90.
Broomhead, D.S., Huke, J.P. and Muldoon, M.R. (1991). Linear filters and nonlinear systems.Journal of the Royal Statistical Society, 54B, 373–382.
Broomhead, D.S., Jones, R. and King, G.P. (1987). Topological dimension and local coordinates from time series data.Journal d’Physique, 20A, L563-L569.
Broomhead, D.S. and King, G.P. (1986). Extracting qualitative dynamics from experimental data.Physica, 20D, 217–236.
Bryan, P., Brown, R. and Abarbanel, H.D.I. (1990). Lyapunov exponents from observed time series.Physical Review Letters, 65, 1523–1526.
Buzug, T. and Pfister, G. (1992). Comparison of algorithms calculating optimal embedding parameters for delay time coordinatesPhysica, 58D, 127–137.
Casdagli, M. (1989). Nonlinear prediction of chaotic time series.Physica, 35D, 335–356.
Casdagli, M. (1991). Chaos and deterministic versus stochastic nonlinear modelling.Journal of the Royal Statistical Society, 54B, 303–328.
Casdagli, M. and Eubank, S., (Eds.). (1992).Nonlinear Modeling and Forecasting. Reading, MA: Addison Wesley.
Chaitin, G.J. (1987).Algorithmic Information Theory. Cambridge: Cambridge University Press.
Childers, D.G. (1978).Modern Spectrum Analysis. New York: IEEE Press.
Chua, L.O., Komuro, M. and Matsumoto, T. (1986). The double scroll family.IEEE Transactions on Circuits and Systems, CAS-33, 1073–1117.
Crutchfield, J.P. (1992). Semantics and thermodynamics. In:Nonlinear Modeling and Forecasting. M. Casdagli and S. Eubank, (Eds.), pp. 317–360. Reading, MA: Addison Wesley.
Crutchfield, J.P. and Young, K. (1989). Inferring statistical complexity.Physical Review Letters, 63, 105–108.
Dressler, U. and Farmer, J.D. (1992). Generalized Lyapunov exponents corresponding to higher derivatives.Physica, 59D, 365–377.
Eckmann, J.-P., Kamphorst, S.O. and Ruelle, D. (1987). Recurrence plots of dynamical systems.Europhysics Letters, 1, 973–977.
Eckmann, J.-P., Kamphorst, S.O. and Ruelle, D. and Cilibertti, S. (1986). Liapunov exponents from time series.Physical Review, 34A, 4971–4979.
Eckmann, J.-P., and Ruelle, D. (1985). Ergodic theory of chaos and strange attractors.Reviews of Modern Physics, 57, 617–656.
Eckmann, J.-P., and Ruelle, D. (1992). Fundamental limitations for estimating dimensions and Liapunov exponents in dynamical systems.Physica, 56D, 185–187.
Ellner, S. (1988). Estimating attractor dimensions from limited data. A new method with error estimates.Physics Letters, 133A, 128–133.
Ellner, S., Gallant, A.R., McCaffrey, D. and Nychka, D. (1991). Convergence rates and data requirements for Jacobian-based estimates of Lyapunov exponents from data.Physics Letters, 153A, 357–363.
Essex, C. and Nerenberg, M.A.H. (1991). Comments on “Deterministic chaos: The science and the fiction” by David Ruelle.Proceedings of the Royal Society (London), 435A, 287–292.
Farmer, J.D., Ott, E. and Yorke, J.A. (1983). The dimension of chaotic attractors.,Physica, 7D, 153–180.
Farmer, J.D. and Sidorowich, J.J. (1987). Predicting chaotic time series.Physical Review Letters, 59, 845–848.
Farmer, J.D. and Sidorowich, J.J. (1988). Exploiting chaos to predict the future and reduce noise. In:Evolution, Learning and Cognition, Y.C. Lee, (Ed.), pp. 277–300., Singapore: World Scientific.
Ferber, G. (1987). Treatment of some nonstationarities in the EEG.Neuropsychobiology, 17, 100–104.
Fraser, A.M. (1989a). Information storage and entropy in strange attractors.IEEE Transactions on Information Theory, 35, 245–262.
Fraser, A.M. (1989b). Reconstructing attractors from scalar time series: A comparison of singular system and redundancy criteria.Physica, 34D, 391–404.
Fraser, A.M. and Swinney, H.L. (1986). Independent coordinates for strange attractors from mutual information.Physical Review, 33A, 1134–1140.
Froehling, H., Crutchfield, J.P., Farmer, D., Packard, N.H. and Shaw, R. (1981). On determining the dimension of chaotic flows.Physica, 3D, 605–617.
Gao, J., and Zheng, Z. (1993). Local exponential divergence plot and optimal embedding of a chaotic time series.Physics Letters, 181A, 153–158.
Gerschenfeld, N.A. (1992). Dimension measurement on high dimensional systems.Physica, 55D, 135–154.
Gibson, J.F., Farmer, J.D., Casdagli, M. and Eubank, S. (1992). An analytic approach to practical state space construction.Physica, 57D, 1–30.
Gollub, J.P. and Benson, S.V. (1980). Many routes to turbulent convection,Journal of Fluid Mechanics, 100, 449–470.
Gollub, J.P., Romer, E.G. and Socolar, J.E. (1980). Trajectory divergence for coupled relaxation oscillators: Measurements and models.Journal of Statistical Physics, 23, 321–333.
Gollub, J.P. and Swinney, H.L. (1975). Onset of turbulence in a rotating fluid.Physical Review Letters, 35, 927–930.
Golub, G.H. and Reinsch, C. (1970). Singular value decomposition and least squares solutions.Numerical Mathematics, 14, 403–420.
Golub, G.H. and Reinsch, C. (1971). Singular value decomposition and least squares solutions. Handbook for Automatic Computation. Vol. II. Linear Algebra. Heidelberg: Springer.
Grassberger, P. and Procaccia, I. (1983). Measuring the strangeness of strange attractors.Physica, 9D, 189–208.
Grassberger, P., Schreiber, T. and Schaffrath, C. (1991). Nonlinear time sequence analysis.International Journal of Bifurcation and Chaos, 1, 521–548.
Greenside, H.S., Wolf, A., Swift, J. and Pignataro, T. (1982). Impracticality of a box counting algorithm for calculating the dimensionality of strange attractors.Physical Review, 25A, 3453–3456.
Hasman, A., Jansen, B.H., Landeweerd, G.H. and van Blokland-Vogelsang, A.W. (1978). Demonstration of segmentation techniques for EEG records.International Journal of BioMedical Computing, 9, 311–321.
Havstad, J.W. and Ehlers, C.L. (1989). Attractor dimension of nonstationary dynamical systems from small data sets.Physical Review, 39A, 845–853.
Hediger, T., Passamante, A. and Farrell, M.E. (1990). Characterizing attractors using local intrinsic dimension calculated by singular value decomposition and information-theoretic criteria.Physical Review, 41A, 5325–5332.
Holzfuss, J. and Mayer-Kress, G. (1986). An approach to error-estimation in the application of dimension algorithms. In:Dimensions and Entropies in Chaotic Systems. G. Mayer-Kress, (Ed.), pp. 114–122. Berlin: Springer-Verlag.
Hope, A.C.A. (1968). A simplified Monte Carlo significance test procedure.Journal of the Royal Statistical Society, 30B, 582–598.
Huberman, B.A. and Hogg, T. (1986). Complexity and adaptation.Physica, 22D, 376–384.
Jensen, B.H. and Cheng, W.K. (1988). Structural EEG analysis: An explorative study.International Journal of Biomedical Computing, 23, 221–237.
Jiménez-Montaño, M.A. (1984). On the syntactic structure of protein sequences and the concept of complexity.Bulletin of Mathematical Biology, 46, 641–659.
Judd, K. (1992). An improved estimator of dimension and some comments on providing confidence intervals.Physica, 56D, 216–228.
Judd, K. and Mees, A.I. (1991). Estimating dimensions with confidence.International Journal of Bifurcation and Chaos, 1, 467–470.
Kaplan, D.T. and Glass, L. (1992). Direct test for determinism in a time series.Physical Review Letters, 68, 427–430.
Kaplan, D.T. and Glass, L. (1993). Coarse-grained embeddings of time series: Random walks, Gaussian random processes, and deterministic chaos.Physica, 64D, 431–454.
Kember, G. and Fowler, A.C. (1993). A correlation function for choosing time delays in phase portrait reconstructions.Physics Letters, 179A, 72–80.
Kennel, M.D., Brown, R. and Abarbanel, H.D.I. (1992). Determining minimum embedding dimension using a geometrical construction.Physical Review, 45A, 3404–3411.
Klemm, W.R. and Sherry, C.J. (1982). Do neurons process information by relative intervals in spike trains?Neuroscience and Biobehavioral Research, 6, 429–437.
Koebbe, M. and Mayer-Kress, G. (1992). Use of recurrence plots in the analysis of time series data. In:Nonlinear Modeling and Forecasting, M. Casdagli and S. Eubank (Eds.), pp. 361–378. Reading, MA: Addison Wesley.
Kolmogorov, A.N. (1958). A metric invariant of transient dynamical systems and automorphisms in Lebsegue spaces. Dokl. Acad. Nauk USSR. 119, 861–864. (English summary:Mathematical Reviews, 21, 386.)
Kolmogorov, A.N. (1965). Three approaches to the definition of the concept of quantity of information.IEEE Transactions on Information Theory, IT14, 662–669.
Kostelich, E.J. and Yorke, J.A. (1988). Noise reduction in dynamical systems.Physical Reviews, 38A, 1649–1652.
Kuskowski, M.A., Mortimer, J.A., Morley, G.K., Malone, S.M. and Okaya, A.J. (1993). Rate of cognitive decline in Alzheimer’s disease is associated with EEG alpha power.Biological Psychiatry, 33, 659–662.
Lachenbruch, P.A. (1975).Discriminant Analysis. New York: Hafner Press.
Libchaber, A. (1983). Experimental aspects of the period doubling scenario.Lecture Notes in Physics, 179, 157–164.
Libchaber, A. and Mauer, J. (1982). A Rayleigh-Benard experiment: Helium in a small box. In:Nonlinear Phenomena at Phase Transitions and Instabilities, T. Riske (Ed.), pp. 259–286. New York: Plenum.
Liebert, W. and Schuster, H.G. (1988). Proper choice for time delay for the analysis of chaotic time series.Physics Letters A., 142, 107–111.
Linsay, P.S. (1991). An efficient method of forecasting chaotic time series using linear interpolation.Physics Letters, 153A, 353–356.
Lopes da Silva, F.H. (1978). Analysis of EEG non-stationarities. In:Contemporary Clinical Neurophysiology, (EEG Suppl. No. 34). W.A. Cobb and H. van Duijn (Eds.), pp. 165–179. Amsterdam: Elsevier.
Mañé, R. (1980). On the dimension of the compact invariant sets of certain nonlinear maps. In:Dynamical Systems and Turbulence. Lecture Notes in Mathematics. Volume 898. D.A. Rand and L.S. Young (Eds.), pp. 230–242. New York: Springer-Verlag.
Martinerié, J., Albano, A.M., Mees, A.I. and Rapp, P.E. (1992). Mutual information, strange attractors and optimal estimation of dimension.Physical Review, 45A, 7058–7064.
Mees, A.I. (1991). Dynamical systems and tesselations: Detecting determinism in data.International Journal of Bifurcation and Chaos, 1, 777–794.
Mees, A.I. and Judd, K. (1993). Dangers of geometric filtering.Physica, 68D, 427–436.
Mees, A.I., Rapp, P.E. and Jennings, L.S. (1987). Singular value decomposition and embedding dimension.Physical Review, 36A, 340–346.
Michael, D. and Houchin, J. (1979). Automatic EEG analysis: A segmentation procedure based on the autocorrelation function.Electroencephalography and Clinical Neurophysiology, 46, 232–239.
Mitra, M., and Skinner, J.E. (1992). Low-dimensional chaos maps learning in a model neuropil (olfactory bulb).Integrative Physiological and Behavioral Science, 27, 304–322.
Mitschke, F. (1990). Acausal filters for chaotic signals.Physical Review, 41A, 1169–1171.
Mitschke, F., Müller, M. and Lange, W. (1988b). Measuring filtered chaotic signals.Physical Review, 37A, 4518–4521.
Molnar, M. and Skinner, J.E. (1992). Low-dimensional chaos in event-related brain potentials.International Journal of Neuroscience, 66, 263–276.
Mosteller, F. and Tukey, J.W. (1977),Data Analysis and Regression. Reading, MA: Addison-Wesley.
Nagase, Y., Okubo, Y., Matsuura, M. and Kojima, T. (1992). Topographical changes in alpha power in medicated and unmedicated schizophrenics during digits span reverse matching test.Biological Psychiatry, 32, 870–879.
Nerenberg, M.A.H. and Essex, C. (1990). Correlation dimension and systematic geometric effects.Physical Review, 42A, 7065–7074.
Noakes, L. (1991). The Takens embedding theorem.International Journal of Bifurcation and Chaos, 1, 867–872.
Nychka, D., Ellner, S., Gallant, A.R. and McCaffrey, D. (1992). Finding chaos in noisy systems.Journal of the Royal Statistical Society, 52B, 399–426.
Packard, N.H., Crutchfield, J.P., Farmer, J.D. and Shaw, R.S. (1980). Geometry from a time series.Physical Review Letters, 45, 712–716.
Parlitz, U. (1992). Identification of true and spurious Lyapunov exponents from time series.International Journal of Bifurcation and Chaos, 2, 155–166.
Passamante, A., Hediger, T. and Gollub, M. (1989). Fractual dimension and local intrinsic dimension.Physical Review, 39A, 3640–3645.
Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vetterling, W.T. (1986).Numerical Recipes. The Art of Scientific Computing. Cambridge: Cambridge University Press.
Ramsey, J.B. and Yuan, H.-J. (1989). Bias and error bars in dimension calculations and their evaluation in some simple models.Physics Letters, 134A, 287–297.
Rapp, P.E. (1993). Chaos in the neurosciences: Cautionary tales from the frontier.The Biologist (London: Institute of Biology), 40, 89–94.
Rapp, P.E., Bashore, T.R., Martinerié, J.M., Albano, A.M. and Mees, A.I. (1989). Dynamics of brain electrical activity.Brain Topography, 2, 99–118.
Rapp, P.E., Jiménez-Montaño, M.A., Langs, R.J. and Thomson, L. (1990). Quantitative characterization of patient-therapist communication.Mathematical Biosciences, 105, 207–227.
Rapp, P.E., Albano, A.M., Schmah, T.I. and Farwell, L.A. (1993a). Filtered noise can mimic low dimensional chaotic attractors.Physical Review, 47E, 2289–2297.
Rapp, P.E., Goldberg, G., Albano, A.M., Janicki, M.B., Murphy, D., Niemeyer, E. and Jiménez-Montaño, M.A. (1993b). Using coarse-grained measures to characterize electromyographic signals.International Journal of Bifurcation and Chaos, 3, 525–542.
Rapp, P.E., Zimmerman, I.D., Vining, E.P., Cohen, N., Albano, A.M. and Jiménez-Montaño, M.A. (1994a). The algorithmic complexity of neural spike trains increases during focal seizures.Journal of Neuroscience (in press).
Rapp, P.E., Albano, A.M., Zimmerman, I.D. and Jiménez-Montaño, M.A. (1994b). Phase-randomized surrogates can produce spurious identifications of non-random structure.Physics Letters (in press).
Rissanen, Y. (1992).Stochastic Complexity in Statistical Inquiry. Singapore: World Scientific.
Rosenstein, M.T., Collins, J.J. and DeLuca, C.J. (1993a). A practical method for calculating largest Lyapunov exponents from small data sets.Physica, 65D, 117–134.
Rosenstein, M.T., Collins, J.J. and DeLuca, C.J. (1993b). Reconstruction expansion as a geometry-based framework for choosing proper delay times.Physica, 73D, 82–98.
Sano, M. and Sawada, Y. (1985). Measurement of the Lyapunov spectrum from chaotic time series.Physical Review Letters, 55, 1082.
Sauer, T. (1992). A noise reduction method for signals from nonlinear systems.Physica, 58D, 193–201.
Sauer, T., Yorke, J.A. and Casdagli, M. (1991). Embedology.Journal of Statistical Physics, 65, 579–616.
Schellenberg, R., Schwarz, A., Knorr, W. and Haufe, C. (1992). EEG-brain mapping. A method to optimize therapy in schizophrenics using absolute power and center frequency values.Schizophrenia Research, 8, 21–29.
Schmitt, A.O., Herzel, H. and Ebeling, W. (1993). A new method to calculate higher-order entropies from finite samples.Europhysics Letters, 23, 303–309.
Schreiber, T. and Grassberger, P. (1991). A simple noise reduction method for real data.Physics Letters, 160A, 411–418.
Sherry, C.J. and Klemm, W.R. (1984). What is the meaningful measure of neuronal spike train activity?Journal of Neuroscience Methods, 10, 205–213.
Skinner, J.E., Pratt, C.M. and Vybiral, T. (1993). A reduction in the correlation dimension of heartbeat intervals precedes imminent ventricular-fibrillation in human-subjects.American Heart Journal, 125, 731–743.
Sloan, E.P. and Fenton, G.W. (1993). EEG power spectra and cognitive change in geriatric psychiatry: A longitudinal study.Electroencephalography and Clinical Neurophysiology, 86, 361–367.
Smith, L.A. (1988). Intrinsic limits on dimension calculations.Physics Letters, 133A, 283–288.
Smith, R.L. (1992). Estimating dimension in noisy chaotic time series.Journal of the Royal Statistical Society, 54B, 329–351.
Somorjai, R.L. and Ali, M.K. (1988). An efficient algorithm for estimating dimensionalities.Canadian Journal of Chemistry, 66, 979–982.
Sugihara, G., Grenfell, B. and May, R.M. (1990). Distinguishing error from chaos in ecological time series.Philosophical Transactions Royal Society, 330B, 235–251.
Sugihara, G. and May, R.M. (1990). Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series.Nature, (London) 344, 734–741.
Takens, F. (1980). Detecting strange attractors in turbulence. Lecture Notes in Mathematics. Volume 898. D.A. Rand and L.S. Young (Eds.), pp. 365–381. New York: Springer-Verlag.
Termonia, Y. and Alexandrowicz, Z. (1983). Fractal dimension of strange attractors from radius versus size of arbitrary clusters.Physical Review Letters, 51, 1265–1268.
Theiler, J. (1986). Spurious dimensions from correlation algorithms applied to limited time series data.Physical Review, 34A, 2427–2433.
Theiler, J., Eubank, S., Longtin, A., Galdrikian, B. and Farmer, J.D. (1992). Testing for nonlinearity in time series: The method of surrogate data.Physica, 58D, 77–94.
Wayland, R., Bromley, D., Pickett, D. and Passamante, A. (1993). Recognizing determinism in a time series.Physical Review Letters, 70, 580–582.
Wiesel, W.E. (1992). Extended Lyapunov exponents.Physical Review, 46A, 7480–7491.
Wolf, A., Swift, J.B., Swinney, H.L. and Vastano, J.A. (1985). Determining Lyapunov exponents from a time series.Physica, 16D, 285–317.
Yip, K.-P., Marsh, D.J. and Holstein-Rathlou, N.-H. (1994). Low dimensional chaos in renal blood flow control in genetic and experimental hypertension.Physica D, in press.
Zbilut, J.P., Koebbe, M., Loeb, H. and Mayer-Kress, G. (1991). Use of recurrence plots in the analysis of heart beat intervals.Proceedings Computers in Cardiology, Washington: IEEE.
Zbilut, J.P. and Webber, C.L. (1992). Embeddings and delays derived from quantification of recurrence plots.Physics Letters, 171A, 199–203.
Zeng, X., Eykholt, R. and Pielke, R.A. (1991). Estimating the Lyapunov exponent spectrum from short time series of low precision.Physical Review Letters, 66, 3229–3232.
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Rapp, P.E. A guide to dynamical analysis. Integrative Physiological and Behavioral Science 29, 311–327 (1994). https://doi.org/10.1007/BF02691335
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DOI: https://doi.org/10.1007/BF02691335