# Volatility of linear and nonlinear time series Academic Article

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### abstract

• 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 $\ensuremath{\mid}{u}_{i}\ensuremath{\mid}$, 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 $\ensuremath{\mid}{u}_{i}\ensuremath{\mid}$ is still unknown. Here we develop analytical relations between the scaling exponent of linear series ${u}_{i}$ and its magnitude series $\ensuremath{\mid}{u}_{i}\ensuremath{\mid}$. 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 $\mathrm{sgn}({u}_{i})$]. We apply our techniques on daily deep ocean temperature records from the equatorial Pacific, the region of the El-Nin\~o 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.

### publication date

• January 1, 2005