Pseudononstationarity in the scaling exponents of finite-interval time series
Abstract
The accurate estimation of scaling exponents is central in the observational study of scale-invariant phenomena. Natural systems unavoidably provide observations over restricted intervals; consequently, a stationary stochastic process (time series) can yield anomalous time variation in the scaling exponents, suggestive of nonstationarity. The variance in the estimates of scaling exponents computed from an interval of N observations is known for finite variance processes to vary as ∼1/N as N→∞ for certain statistical estimators; however, the convergence to this behavior will depend on the details of the process, and may be slow. We study the variation in the scaling of second-order moments of the time-series increments with N for a variety of synthetic and “real world” time series, and we find that in particular for heavy tailed processes, for realizable N , one is far from this ∼1/N limiting behavior. We propose a semiempirical estimate for the minimum N needed to make a meaningful estimate of the scaling exponents for model stochastic processes and compare these with some “real world” time series.
- Publication:
-
Physical Review E
- Pub Date:
- March 2009
- DOI:
- 10.1103/PhysRevE.79.036109
- arXiv:
- arXiv:0808.2036
- Bibcode:
- 2009PhRvE..79c6109K
- Keywords:
-
- 89.75.Da;
- 05.45.Tp;
- 02.50.-r;
- Systems obeying scaling laws;
- Time series analysis;
- Probability theory stochastic processes and statistics;
- Physics - Data Analysis;
- Statistics and Probability
- E-Print:
- 12 pages, 5 figures, accepted Physical Review E