Analysis of Clusters Formed by the Moving Average of a Long-Range Correlated Time Series
A. Carbone, Harry Eugene Stanley · PORTO Publications Open Repository TOrino (Politecnico di Torino) · 2004
We analyze the stochastic function C-n(i)equivalent toy(i)-y(n)(i), where y(i) is a long-range correlated time series of length N-max and y(n)(i)equivalent to(1/n)Sigma(k=0)(n-1)y(i-k) is the moving average with window n. We argue that C-n(i) generates a stationary sequence of self-affine clusters C with length l, lifetime tau, and area s. The length and the area are related to the lifetime by the relationships lsimilar totau(psi)l and ssimilar totau(s)(psi), where psil=1 and psi(s)=1+H. We also find that l, tau, and s are power law distributed with exponents depending on H: P(l)similar tol(-alpha), P(tau)similar totau(-beta), and P(s)similar tos(-gamma), with alpha=beta=2-H and gamma=2/(1+H). These predictions are tested by extensive simulations on series generated by the midpoint displacement algorithm of assigned Hurst exponent H (ranging from 0.05 to 0.95) of length up to N-max=2(21) and n up to 2(13).