Nonlinear time-series analysis

Project Description:

Nonlinear time-series analysis (NLTSA) is a powerful methodology for studying dynamical systems. The foundation for this field is delay embedding, which allows one to reconstruct the full dynamics of a system, up to diffeomorphism, from a scalar time series. Check out the 2015 review paper in CHAOS in the list below for some details on the procedure.

There are a number of free parameters in this methodology that you have to get right in order for things to work, notably the delay and the dimension. We've done some work on methods for choosing those values, which are described in various papers below (Physica D, 2023; CHAOS, 2020; Phys Rev E, 2016).

We've also worked on formal methods for identifying and characterizing scaling regions (CHAOS, 2021), looked into what happens when you use fewer than the theoretically required number of embedding dimensions (CHAOS, 1998 & 2015; Physica D, 2016)

On the applications side, we've worked on a number of different kinds of systems, ranging from digital computers (CHAOS 2009; IDA, 2013) to human gait (CHAOS, 2013).

Papers and code:

• V. Deshmukh, R. Meikle, E. Bradley, J. D. Meiss, and J. Garland, "Using scaling-region distributions to select embedding parameters," Physica D 446:133674 (2023). Preprint available on arXiv.
• V. Deshmukh, E. Bradley, J. Garland, and J. D. Meiss, "Towards automated extraction and characterization of scaling regions in dynamical systems," CHAOS 31:123102 (2021). DOI: 10.1063/5.0069365. Preprint available on arXiv.
• A Python notebook containing all of the code for that 2021 CHAOS paper. To cite, use doi.org/10.5281/zenodo.7962698.
• V. Deshmukh, E. Bradley, J. Garland, and J. D. Meiss, "Using curvature to select the time lag for delay reconstruction," CHAOS 30:053108 (2020). Preprint available on arXiv.
• J. Garland, E. Bradley, and J. Meiss, "Exploring the topology of dynamical reconstructions," Physica D 334:49-59 (2016). Preprint available on arXiv.
• J. Garland, R. James, and E. Bradley, "Leveraging information storage to select forecast-optimal parameters for delay-coordinate reconstructions," Physical Review E 93:022221 (2016). Preprint available on arXiv (with a different title: "A new method for choosing parameters in delay reconstruction-based forecast strategies")
• J. Garland and E. Bradley, "Prediction in projection," Chaos 25:123108 (2015); http://dx.doi.org/10.1063/1.4936242. Preprint available on arXiv.
• E. Bradley and H. Kantz, "Nonlinear time-series analysis revisited" Chaos 25: 097610 (2015). DOI: 10.1063/1.4917289. Preprint available on arXiv.
• N. Look, C. Arellano, A. Grabowski, W. McDermott, R. Kram, and E. Bradley, "Dynamic stability of running: The effects of speed and leg amputations on the maximal Lyapunov exponent," Chaos 23:043131 (2013)
• J. Garland and E. Bradley, "On the importance of nonlinear modeling in computer performance prediction," IDA-13 (Proceedings of the 12th International Symposium on Intelligent Data Analysis), London, October 2013. Preprint available on arXiv.
• T. Mytkowicz, A. Diwan, and E. Bradley, "Computers are dynamical systems," Chaos 19:033124 (2009); doi:10.1063/1.3187791
• E. Bradley, "Time-series analysis," in M. Berthold and D. Hand, editors, Intelligent Data Analysis: An Introduction, Springer Verlag, 1999.
• J. Iwanski and E. Bradley, "Recurrence plot analysis: To embed or not to embed?," Chaos, 8:861-871 (1998).

Links:

• Kantz & Schreiber, the bible for this field.
• The TISEAN software package, which instantiates pretty much everything in Holger's book.
• I teach a MOOC on nonlinear dynamics through the Santa Fe Institute's Complexity Explorer platform. Units 7-9 in that MOOC go through the basics of NLTSA.

Support:

• This material is based upon work supported by the NSF. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the NSF.