Eliminating finite-size effects and detecting the amount of white noise in short records with long-term memory

Details Eliminating finite-size effects and detecting the amount of white noise in short records with long-term memory Eliminating finite-size effects and detecting the amount of white noise in short records with long-term memory

Jun 2009

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2009phrve..79f6101l&link_type=abstract

Physical Review E, vol. 79, Issue 6, id. 066101

Physics

7

Systems Obeying Scaling Laws, Finite-Size Systems, Time Series Analysis, Time Variability

Scientific paper

Long-term memory is ubiquitous in nature and has important consequences for the occurrence of natural hazards, but its detection often is complicated by the short length of the considered records and additive white noise in the data. Here we study synthetic Gaussian distributed records xi of length N that consist of a long-term correlated component (1-a)yi characterized by a correlation exponent γ , 0<γ<1 , and a white-noise component aηi , 0≤a≤1 . We show that the autocorrelation function CN(s) has the general form CN(s)=[C∞(s)-Ea]/(1-Ea) , where C∞(0)=1 , C∞(s>0)=Bas-γ , and Ea={2Ba/[(2-γ)(1-γ)]}N-γ+O(N-1) . The finite-size parameter Ea also occurs in related quantities, for example, in the variance ΔN2(s) of the local mean in time windows of length s : ΔN2(s)=[Δ∞2(s)-Ea]/(1-Ea) . For purely long-term correlated data B0≅(2-γ)(1-γ)/2 yielding E0≅N-γ , and thus CN(s)=[((2-γ)(1-γ))/(2)s-γ-N-γ]/[1-N-γ] and ΔN2(s)=[s-γ-N-γ]/[1-N-γ] . We show how to estimate Ea and C∞(s) from a given data set and thus how to obtain accurately the exponent γ and the amount of white noise a .

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