STATISTICAL MECHANICS OF BIOLOGICAL AND OTHER COMPLEX EXPERIMENTAL TIME SERIES: ASSESSING GEOMETRICAL AND DYNAMICAL PROPERTIES

Biological and other experimental time series often exhibit complex and possibly chaotic behavior that may not be completely deterministic or completely random. Particularly problematic is the fact that measures of chaos such as the dynamical or geometrical invariants, e.g. the correlation dimension, Lyapunov exponents, or Kolmogorov entropy, often cannot be calculated from short, noisy, and possibly highly discretized experimental time series. Here, it is argued that nonrandom structure in the data may be uncovered by using a conceptual framework based on statistical mechanics and the standard correlation integral as a computational tool. A new use of the generalized correlation integral is proposed to assess statistically the occurrence of nonrandom spatiotemporal patterns in experimental data. We argue that nonrandomness of a time series can be assessed by the statistics of the topology of the reconstructed state space distribution, which we quantify via the generalized correlation integral. This approach provides a simple, graphical tool which can yield immediate information about the length scales and sequence lengths where the data may appear to be different from random, and also may provide a data classification tool based on spatiotemporal patterns. We demonstrate the usefulness of this approach using several numerical examples, including data from experimental biological systems. Finally, we propose that particular characteristics of such patterns imply considerable macroscopic information about the behavior of the generating system, and qualitative changes in the time series.