The point correlation dimension: Performance with nonstationary surrogate data and noise

  • James E. Skinner
  • Mark Molnar
  • Claude Tomberg
Papers Principles And Algorithms


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.

Key Words

Chaos theory nonlinear dynamics deterministic models 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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.Google Scholar
  2. 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.Google Scholar
  3. 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.PubMedGoogle Scholar
  4. Farmer, J.D., Ott, E. and Yorke, J.A. (1983). Dimension of chaotic attractors.Physica 7D, 153–180.Google Scholar
  5. Grassberger, P. and Procaccia, I. (1983). Characterization of strange attractors. Physical Review Letters, 50(5), 346–349.CrossRefGoogle Scholar
  6. 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.CrossRefGoogle Scholar
  7. 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.CrossRefGoogle Scholar
  8. Molnar, M., and Skinner, J.E. (1992). Low-dimensional Chaos in Event-Related Brain Potentials.Intern. J. Neuroscience 66, 263–276.Google Scholar
  9. 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.PubMedCrossRefGoogle Scholar
  10. 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.CrossRefGoogle Scholar
  11. Pais, A. (1982). The Science and the Life of Albert Einstein. New York, Oxford University Press, 440–469.Google Scholar
  12. Powell, C.S. (1992). The golden age of cosmology.Sci. Amer. 267, 17–22.CrossRefGoogle Scholar
  13. 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.PubMedCrossRefGoogle Scholar
  14. 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.CrossRefGoogle Scholar
  15. 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.Google Scholar
  16. 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.PubMedGoogle Scholar
  17. 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.Google Scholar
  18. 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.PubMedCrossRefGoogle Scholar
  19. Skinner, J.E. (1993b). Neurocardiology: Brain Mechanism Underlying Fatal Cardiac Arrhythmias.Neurol. Clin., 11, 325–351.PubMedGoogle Scholar
  20. 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).Google Scholar
  21. 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).Google Scholar
  22. 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).Google Scholar
  23. Theiler, J. (1986). Spurious dimension from correlation algorithms applied to limited time-series data.Phys. Rev. A, 34, 2427–2432.PubMedCrossRefGoogle Scholar
  24. Theiler, J. (1988). Quantifying chaos: Practical estimation of the correlation dimension. Thesis. California Institute of Technology, Pasadena, California.Google Scholar
  25. Theiler, J. (1990). Estimating the fractal dimension of chaotic time series.The Lincoln Lab. J., 3, 63–86.Google Scholar
  26. Wolf, S. (1971). The artery and the process of arteriosclerosis: Measurement and modification, 235–237.Google Scholar

Copyright information

© Springer 1994

Authors and Affiliations

  • James E. Skinner
    • 1
    • 2
  • Mark Molnar
    • 3
  • Claude Tomberg
    • 4
  1. 1.Totts Gap Medical Research LabsBangor
  2. 2.Baylor College of MedicineHouston
  3. 3.Hungarian Academy of SciencesBudapestHungary
  4. 4.University of Brussels Medical SchoolBrusselsBelgium

Personalised recommendations