Detrending moving average algorithm: A closed-form approximation of the scaling law

The Hurst exponent H of long range correlated series can be estimated by means of the detrending moving average (DMA) method. The computational tool, on which the algorithm is based, is the generalized variance sigma(2)(DMA) = 1/(N - n)Sigma(N)(i=n)[v(i) - y(n)(i)](2), with y(n)(i) = 1/n Sigma(n)(k=0)y(i - k) being the average over the moving window n and N the dimension of the stochastic series y(i). The ability to yield H relies on the property of sigma(2)(DMA) to vary as n(2H) over a wide range of scales [E. Alessio, A. Carbone, G. Castelli, V. Frappietro, Eur. J. Phys. B 27 (2002) 197]. Here, we give a closed form proof that sigma(2)(DMA) is equivalent to C(H)n(2H) and provide an explicit expression for C(H). We furthermore compare the values of C(H) with those obtained by applying the DMA algorithm to artificial self-similar signals. (C) 2007 Published by Elsevier B.V.