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Volatility of linear and nonlinear time series
Tomer Kalisky1, Yosef Ashkenazy, Shlomo Havlin
1Minerva Center and Department of Physics, Bar-Ilan University, Ramat-Gan, Israel.
Summary
This study reveals the origin of nonlinear properties in time series by analyzing correlations in magnitude (volatility) and sign series. A new model generates multifractal time series, validated on El Niño data.
Area of Science:
- Nonlinear time series analysis
- Statistical physics
- Geophysics
Background:
- Nonlinear properties of Gaussian time series with long-range correlations are detectable via volatility.
- The origin of this empirical observation and the precise relationship between correlations in u(i) and |u(i)| remain unclear.
Purpose of the Study:
- Develop analytical relations between scaling exponents of linear series u(i) and its magnitude series |u(i)|.
- Propose a model for generating nonlinear multifractal time series.
- Apply the techniques to deep ocean temperature records.
Main Methods:
- Derivation of analytical relations between scaling exponents.
- Development of a multiplicative model for multifractal time series generation.
- Analysis of daily deep ocean temperature data from the equatorial Pacific.
Main Results:
- Established analytical relations between scaling exponents of linear and magnitude series.
- Demonstrated that nonlinear time series exhibit stronger or equal correlations in magnitude series compared to linear series.
- Generated multifractal time series by combining long-range correlated magnitude series with uncorrelated sign series.
- Observed long-range correlations (1/f power spectrum) and nonlinear behavior in El Niño temperature records.
Conclusions:
- The study clarifies the origin of nonlinear time series properties and their relation to volatility.
- The proposed model successfully generates multifractal time series.
- Empirical data from the El Niño phenomenon exhibit long-range correlations, nonlinear behavior, and a broad multifractal spectrum.
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