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What is a linear process?

P J Bickel1, P Bühlmann

  • 1Department of Statistics, University of California, Berkeley 94720-3860, USA.

Proceedings of the National Academy of Sciences of the United States of America
|October 29, 1996
PubMed
Summary
This summary is machine-generated.

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It is impossible to perfectly distinguish between linear and nonlinear processes, even with infinite data. This study characterizes the closure of linear processes, revealing it includes complex nonergodic processes.

Area of Science:

  • Statistics
  • Time Series Analysis
  • Dynamical Systems

Background:

  • Distinguishing between linear and nonlinear time series is crucial in many scientific fields.
  • Chaotic processes, despite being nonlinear, can exhibit behavior difficult to differentiate from linear models.

Purpose of the Study:

  • To demonstrate the inherent difficulty in perfectly distinguishing linear from nonlinear processes using any test statistic, regardless of data length.
  • To precisely characterize the closure of the set of moving-average (linear) processes under a suitable metric.

Main Methods:

  • Consideration of the set of moving-average (linear) processes.
  • Analysis of the closure of this set using a specific metric.
  • Characterization of the resulting set, including its non-stationary and non-ergodic components.

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Main Results:

  • It is impossible to perfectly distinguish between linear and nonlinear processes, even with an infinitely long data sequence.
  • The closure of the set of moving-average processes is unexpectedly large.
  • This closure includes nonergodic processes, specifically Poisson sums of independent and identically distributed stationary processes.

Conclusions:

  • The theoretical limits of distinguishing linear and nonlinear time series are fundamental.
  • The mathematical framework reveals a broader-than-expected set of processes that cannot be definitively classified as purely linear.
  • Further theoretical development is needed to fully understand the implications of these findings for time series analysis.