ELS EVI ER Physica D Ill (1998)42-50 PHYSICA Analysis of positive Lyapunov exponents from random time series Toshiyuki Tanaka a.,, Kazuyuki Aihara b, Masao Taki a a Department of Electronics and Information Engineering, Faculty of Engineering, Toky'o Metropolitan University, 1-1 Minami Oosawa, Hachioji, Tok3.'o 192-03. Japan b Department of Mathematical Engineering and Information Physics, FaculO, of Engineering, Universit3" of Tok3'o, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan Received 24 June 1996; accepted 17 June 1997 Communicated by Y. Kuramoto Abstract The conventional method for estimating Lyapunov spectra can give spurious positive Lyapunov exponents when applied to random time series. We analyze this phenomenon by considering a situation in which the method is applied to completely random time series produced by a simple stochastic model. We show that the possible estimation of spurious positive Lyapunov exponents is due to the statistical nature and the finiteness of data. We also derive an upper bound of the largest Lyapunov exponent for the model, which is useful in testing positive Lyapunov exponents with random-shuffled surrogate data. The results suggest that the method should be applied very carefully to experimentally obtained chaotic time series with possible random contamination, so as to avoid spurious estimation of positive Lyapunov exponents as evidences of deterministic chaos. PACS: 05.45.+b; 05.40.÷j; 02.50.-r Keywords: Lyapunov spectra; Embedding; Time series 1. Introduction In studying the mechanism that generates a fluctu- ating time series, it is important to determine whether or not the series is produced from a deterministic sys- tem with chaotic dynamics and, if it is, to charac- terize the dynamics. The Lyapunov spectrum gives information useful for characterizing orbital instabil- ity of chaotic dynamics, and in recent years it has been increasingly applied to time series analysis, along with various methods concerning it [1]. When we want to estimate the Lyapunov spectrum from a given time series we can use the customary method consisting of the phase space reconstruction via embedding and * Corresponding author. E-mail: tanaka@eei.metro-u.ac.jp. the local function approximation. Knowing how this method behaves when it is applied to chaotic time se- ries contaminated by noise or applied improperly to random time series that are not chaotic would extend our understanding of the method and help to ensure that it is used properly. Ikeguchi and Aihara [2] have reported that naive application of the method to random time series (time interval data of gamma-ray emission by cobalt) can give spurious positive Lyapunov exponents. It should be noted that this result is itself of little practical im- portance because of the following two reasons: First, this result is, in a sense, commonly anticipated by most researchers in this field. It can be seen as an example of the well-known fact concerning any computational method, that garbage-in yields garbage-out. Second, 0167-2789/98/$19.00 Copyright © 1998 Elsevier Science B.V. All rights reserved Pll S0167-2789(97)001 48-6