2004/06/14 by Tomer Kalisky, Yosef Ashkenazy, Shlomo Havlin
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Chaos control and synchronization #Complex Systems and Time Series Analysis #Econometrics #Economics #Financial Risk and Volatility Modeling #Geology #Mathematics #Nonlinear system #Physics #Series (stratigraphy) #Volatility (finance) #cond-mat.other #cond-mat.stat-mech #q-fin.ST
paper · pdf · doi:10.1103/physreve.72.011913
7 pages, 5 figures
arxiv created 2004/06/14 · openalex publication_date 2005/07/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Previous studies indicated that nonlinear properties of Gaussian distributed time series with long-range correlations, u(i), can be detected and quantified by studying the correlations in the magnitude series |u(i)|, the "volatility." However, the origin for this empirical observation still remains unclear and the exact relation between the correlations in u(i) and the correlations in |u(i)| is still unknown. Here we develop analytical relations between the scaling exponent of linear series u(i) and its magnitude series |u(i)|. 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 (u(i))]. 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 1/f power spectrum, (ii) significant nonlinear behavior as expressed by long-range correlations of the volatility series, and (iii) broad multifractal spectrum.