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Volatility of linear and nonlinear time series Tomer Kalisky,1

Summary: Volatility of linear and nonlinear time series
Tomer Kalisky,1
Yosef Ashkenazy,2
and Shlomo Havlin1
Minerva Center and Department of Physics, Bar-Ilan University, Ramat-Gan, Israel
Solar Energy and Environmental Physics, BIDR, Ben-Gurion University, Midreshet Ben-Gurion, Israel
Received 14 June 2004; revised manuscript received 23 March 2005; published 21 July 2005
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 tech-
niques on daily deep ocean temperature records from the equatorial Pacific, the region of the El-Ninő phe-


Source: Ashkenazy, Yossi "Yosef" - Department of Solar Energy and Environmental Physics, Jacob Blaustein Institutes for Desert Research,Ben-Gurion University of the Negev


Collections: Physics; Environmental Management and Restoration Technologies