Volatility of linear and nonlinear time series

Tomer Kalisky, Yosef Ashkenazy, Shlomo Havlin

Research output: Contribution to journalArticlepeer-review

51 Scopus citations


Previous studies indicated that nonlinear properties of Gaussian distributed time series with long-range correlations, ui, can be detected and quantified by studying the correlations in the magnitude series ui, the "volatility." However, the origin for this empirical observation still remains unclear and the exact relation between the correlations in ui and the correlations in ui is still unknown. Here we develop analytical relations between the scaling exponent of linear series ui and its magnitude series ui. Moreover, we find that nonlinear time series exhibit stronger (or the same) correlations in the magnitude time series compared with linear time series with the same two-point correlations. Based on these results we propose a simple model that generates multifractal time series by explicitly inserting long range correlations in the magnitude series; the nonlinear multifractal time series is generated by multiplying a long-range correlated time series (that represents the magnitude series) with uncorrelated time series [that represents the sign series sgn(ui)]. We apply our techniques on daily deep ocean temperature records from the equatorial Pacific, the region of the El-Ninõ phenomenon, and find: (i) long-range correlations from several days to several years with 1f power spectrum, (ii) significant nonlinear behavior as expressed by long-range correlations of the volatility series, and (iii) broad multifractal spectrum.

Original languageEnglish
Article number011913
JournalPhysical Review E
Issue number1
StatePublished - Jul 2005


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