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# Practical implementation of nonlinear time series methods:
The TISEAN package

Rainer Hegger, Holger Kantz
Max Planck Institute for Physics of Complex Systems,
Nöthnitzer Str. 38, D-01187 Dresden
Thomas Schreiber
Physics Department, University of Wuppertal, D-42097 Wuppertal

### Abstract:

We describe the implementation of methods of nonlinear time series analysis which are based on the paradigm of deterministic chaos. A variety of algorithms for data representation, prediction, noise reduction, dimension and Lyapunov estimation, and nonlinearity testing are discussed with particular emphasis on issues of implementation and choice of parameters. Computer programs that implement the resulting strategies are publicly available as the TISEAN software package. The use of each algorithm will be illustrated with a typical application. As to the theoretical background, we will essentially give pointers to the literature.

### Note:

A version of this document has appeared as a journal article

> [CHAOS 9, 413-435 (1999)]

which now constitutes the correct reference to this work. Preprint paper versions are also available.

### Note:

Extensions to the package made after version 1.0 are not documented in this manual in order to keep it as close as possible to the paper version. Only minor updates have been made to the text and the link structure since then.

- Introduction
  - Philosophy of the TISEAN implementation
  - General computational issues
- Phase space representation
  - Delay coordinates
  - Embedding parameters
    - Mutual information
    - False nearest neighbors
  - Principal components
  - Poincaré sections
  - SVD filters
- Visualization, non-stationarity
  - Recurrence plots
  - Space-time separation plot
- Nonlinear prediction
  - Model validation
  - Simple nonlinear prediction
  - Finding unstable periodic orbits
  - Locally linear prediction
  - Global function fits
- Nonlinear noise reduction
  - Simple nonlinear noise reduction
  - Locally projective nonlinear noise reduction
  - Nonlinear noise reduction in a data stream
- Lyapunov exponents
  - The maximal exponent
  - The Lyapunov spectrum
- Dimensions and entropies
  - Correlation dimension
    - Takens-Theiler estimator
    - Gaussian kernel correlation integral
  - Information dimension
    - Entropy estimates
- Testing for nonlinearity
  - The concept of surrogate data
  - Iterative Fourier transform method
  - General constrained randomization
  - Measuring weak nonlinearity
- Conclusion and perspectives
  - Important methods which are (still) missing
- References
- About this document ...

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