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# Surrogate time series

Thomas Schreiber and Andreas Schmitz
Physics Department, University of Wuppertal, D-42097 Wuppertal, Germany

Abstract Before we apply nonlinear techniques, for example those inspired by chaos theory, to dynamical phenomena occurring in nature, it is necessary to first ask if the use of such advanced techniques is justified by the data. While many processes in nature seem very unlikely a priori to be linear, the possible nonlinear nature might not be evident in specific aspects of their dynamics. The method of surrogate data has become a very popular tool to address such a question. However, while it was meant to provide a statistically rigorous, foolproof framework, some limitations and caveats have shown up in its practical use. In this paper, recent efforts to understand the caveats, avoid the pitfalls, and to overcome some of the limitations, are reviewed and augmented by new material. In particular, we will discuss specific as well as more general approaches to constrained randomisation, providing a full range of examples. New algorithms will be introduced for unevenly sampled and multivariate data and for surrogate spike trains. The main limitation, which lies in the interpretability of the test results, will be illustrated through instructive case studies. We will also discuss some implementational aspects of the realisation of these methods in the TISEAN software package.

### Note:

A version of this document has been submitted as a journal article to

> [Physica D (1999)]

which will eventually - if it gets accepted - constitute the correct reference to this work. Preprint paper versions are also available.

- Introduction
- Detecting weak nonlinearity
  - Higher order statistics
  - Phase space observables
- Surrogate data testing
  - Typical vs. constrained realisations
  - The null hypothesis: model class vs. properties
  - Test design
- Fourier based surrogates
  - Rescaled Gaussian linear process
  - Flatness bias of AAFT surrogates
  - Iteratively refined surrogates
  - Example: Southern oscillation index
  - Periodicity artefacts
  - Iterative multivariate surrogates
- General constrained randomisation
  - Null hypotheses, constraints, and cost functions
  - Computational issues of simulated annealing
  - Example: avoiding periodicity artefacts
  - Combinatorial minimisation and accuracy
  - The curse of accuracy
- Various Examples
  - Including non-stationarity
  - Multivariate data
  - Uneven sampling
  - Spike trains
- Questions of interpretation
  - Non-dynamic nonlinearity
  - Non-stationarity
- Conclusions: Testing a Hypothesis
  vs. Testing Against Surrogates
- The TISEAN implementation
  - Measures of nonlinearity
  - Iterative FFT surrogates
  - Annealed surrogates
- References
- About this document ...

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