Volatility of Linear and Nonlinear Time Series

Tomer Kalisky; Yosef Ashkenazy; Shlomo Havlin

doi:10.1103/PhysRevE.72.011913

paper

Volatility of Linear and Nonlinear Time Series

arXiv:cond-mat/0406310 · doi:10.1103/PhysRevE.72.011913

Abstract

Previous studies indicate that nonlinear properties of Gaussian time series with long-range correlations, $u_{i}$ , can be detected and quantified by studying the correlations in the magnitude series $∣ u_{i} ∣$ , i.e., 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 find 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 to 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 $s g n (u_{i})$ ]. Our results of magnitude series correlations may help to identify linear and nonlinear processes in experimental records.

7 pages, 5 figures

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Multifractal detrended fluctuation analysis of nonstationary time series

Volatility of Linear and Nonlinear Time Series

Abstract

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