Volatility of linear and nonlinear time series.
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.
Kalisky
Tomer
T
Minerva Center and Department of Physics, Bar-Ilan University, Ramat-Gan, Israel.
Ashkenazy
Yosef
Y
Havlin
Shlomo
S
D001703:Biophysics / Q000379:methods*; D003627:Data Interpretation, Statistical; D017709:Fractals; D015233:Models, Statistical; D008962:Models, Theoretical; D017711:Nonlinear Dynamics; D016011:Normal Distribution; D009792:Oceans and Seas; D013269:Stochastic Processes; D013696:Temperature; D013995:Time; D013997:Time Factors