Abstract
The dynamics of many biological systems have recently been attributed to low-dimensional chaos instead of high-dimensional noise, as previously thought. Because biological data are invariably nonstationary, especially when recorded over a long interval, the conventional measures of low-dimensional chaos (e.g., the correlation dimension algorithms) cannot be applied. A new algorithm, the point correction dimension (PD2i) was developed to deal with this fundamental problem. In this article we describe the details of the algorithm and show that the local mean PD2i will accurately track dimension in nonstationary surrogate data.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Albano, A.M., Abraham, N.B., Guzman de, G.C., Tarropja, M.F.H., Bandy, D.K., Gioggia, R.S., Rapp., P.E., Zimmerman, I.D., Greenbaun, N.N. and Bashore, T.R. (1986). Lasers and brains: Complex systems with low-dimensional attractors. In G. Mayer-Kress (Ed.), Dimensions and entropies in chaotic systems, 231–240. Berlin: Springer.
Babloyantz, A.: Strange attractors in the dynamics of brain activity. (1985). In: H. Haken (Ed.), Complex Systems—Operational approaches in neurobiology, physics, and computers, 116–122. Berlin: Springer.
Elbert, T., Ray, W.J., Kowalik, Z.J., Skinner, J.E., Graf, K.E., and Birbaumer, N. (1994). Chaos and physiology. Physiol. Rev. 74, 1–47.
Farmer, J.D., Ott, E. and Yorke, J.A. (1983). Dimension of chaotic attractors.Physica 7D, 153–180.
Grassberger, P. and Procaccia, I. (1983). Characterization of strange attractors. Physical Review Letters, 50(5), 346–349.
Kleiger, R.E., Miller, J.P., Bigger, J.T., Moss, A.J., and the Multicenter Post-Infarction Research Group. (1988). Decreased heart rate variability and its association with increased mortality after acute myocardial infarction.Am J Cardiol, 59, 256–262.
Mayer-Kress, G., Yates, F.E., Benton, L., Keidel, M., Tirsch, W., Poppl, S.J. and Geist, K. (1988). Dimensional analysis of non-linear oscillations in brain, heart and muscle.Mathematical Biosciences, 90, 155–182.
Molnar, M., and Skinner, J.E. (1992). Low-dimensional Chaos in Event-Related Brain Potentials.Intern. J. Neuroscience 66, 263–276.
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.
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.
Pais, A. (1982). The Science and the Life of Albert Einstein. New York, Oxford University Press, 440–469.
Powell, C.S. (1992). The golden age of cosmology.Sci. Amer. 267, 17–22.
Rapp, P.E., Bashore, T.R., Martineire, J.M., Albano, A.M., Zimmerman, I.D. and Mees, A.I. (1989). Dynamics of brain electrical activity.Brain Topography, 2, 99–118.
Skinner, J.E.; Goldberger, A.L., Mayer-Kress, G. and Ideker, R.E. (1990a). Chaos in the heart: implications for clinical cardiology.Biotechnology, 8, 1018–1024.
Skinner, J.E., Martin, J.L., Landisman, C.E., Mommer, M.M., Fulton, K., Mitra, M., Burton, W.D. and Saltzberg, B. (1990b). Chaotic attractors in a model of neocortex: Dimensionalities of olfactory bulb surface potentials are spatially uniform and event related. In: E. Basar (Ed.),Chaos in brain function, 119–134. Berlin: Springer.
Skinner, J.E., Carpeggiani, C., Landisman, C.E. and Fulton, K.W. (1991). The correlation-dimension of the heartbeat is reduced by myocardial ischemia in conscious pigs.Circulation Research, 68, 966–976.
Skinner, J.E., Molnar, M., Vybiral, T., and Mitra M. (1992). Application of chaos theory to biology and medicine.Integrative Physiological Behavioral Science, 27, 43–57.
Skinner, J.E., Pratt, C.M., Vybiral, T. (1993a). A reduction in the correlation dimension of heart beat intervals proceeds imminent ventricular fibrillation in human subjects.Am. Heart J., 125, 731–743.
Skinner, J.E. (1993b). Neurocardiology: Brain Mechanism Underlying Fatal Cardiac Arrhythmias.Neurol. Clin., 11, 325–351.
Skinner, J.E. (1993c). Neurocardiology: How stress produces fatal cardiac arrhythmias. In: P.J. Podrid and P.R. Kowey (Eds.),Arrhythmia: A Clinical Approach, Williams & Wilkins, Baltimore (in press).
Skinner, J.E., Braun, C., Miltner, W. and Birbaumer, N. (1993d). Calculation of the point correlation dimension of event-related auditory potentials in humans (in press).
Takens, F. Detecting strange attractors in turbulance. (1981). Lecture Notes in Mathematics, 898, 366–381. See also, Takens, F.: On the numerical determination of the dimension of an attractor. (1985).Lecture Notes in Mathematics, 1125, 99–106 (same publication).
Theiler, J. (1986). Spurious dimension from correlation algorithms applied to limited time-series data.Phys. Rev. A, 34, 2427–2432.
Theiler, J. (1988). Quantifying chaos: Practical estimation of the correlation dimension. Thesis. California Institute of Technology, Pasadena, California.
Theiler, J. (1990). Estimating the fractal dimension of chaotic time series.The Lincoln Lab. J., 3, 63–86.
Wolf, S. (1971). The artery and the process of arteriosclerosis: Measurement and modification, 235–237.
Author information
Authors and Affiliations
Additional information
Research support by NIH, NS 27745.
Rights and permissions
About this article
Cite this article
Skinner, J.E., Molnar, M. & Tomberg, C. The point correlation dimension: Performance with nonstationary surrogate data and noise. Integrative Physiological and Behavioral Science 29, 217–234 (1994). https://doi.org/10.1007/BF02691327
Issue Date:
DOI: https://doi.org/10.1007/BF02691327