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Emergence of patterns in random processes
William I Newman1, Donald L Turcotte, Bruce D Malamud
1Department of Earth and Space Sciences, University of California, Los Angeles, California 90095-1567, USA. win@ucla.edu
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 26, 2012
Summary
A new method analyzes peak-to-peak sequences in time series to test for independent and identically distributed (i.i.d.) randomness. This statistical tool reveals patterns in diverse data, from earthquakes to stock market trends.
Area of Science:
- Statistical physics
- Time series analysis
- Complex systems
Background:
- Sixty years ago, a property of random variables (average of three events per peak-to-peak sequence) was linked to animal population cycles.
- This observation suggested randomness as a null hypothesis for cyclical patterns in natural phenomena.
- The need for robust statistical tests for randomness in time series data remains crucial.
Purpose of the Study:
- To derive a universal distribution for peak-to-peak sequence lengths in time series.
- To establish peak-to-peak sequence analysis as a rigorous test for the independent and identically distributed (i.i.d.) character of data.
- To investigate the influence of correlations on time series using this methodology.
Main Methods:
- Derivation of the universal distribution of peak-to-peak sequence lengths.
- Analysis of Gaussian white noise to validate the i.i.d. test.
- Examination of peak-to-peak and nearest-neighbor cluster statistics for random point processes.
- Application of the methodology to time series generated by the Langevin equation (Brownian motion).
Main Results:
- Demonstrated a universal distribution for peak-to-peak sequence lengths, applicable as an i.i.d. test for long datasets.
- Found good agreement with i.i.d. theory for earthquake magnitudes and interoccurrence times.
- Identified significant deviations from i.i.d. behavior in Old Faithful geyser intervals (antipersistence), geomagnetic substorms (mild persistence), and S&P 500 daily returns (persistence).
Conclusions:
- The peak-to-peak sequence analysis provides a powerful tool for assessing the i.i.d. nature of time series data.
- The methodology successfully distinguishes between random and correlated processes across various scientific domains.
- This approach has broad applicability for analyzing interoccurrence statistics and time series in numerous fields.
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