Detrending Moving Average variance: a derivation of the scaling law

Abstract

The Hurst exponent $H$ of long range correlated series can be estimated by means of the Detrending Moving Average (DMA) method. A computational tool defined within the algorithm is the generalized variance ${\sigma }_{DMA}^2=1/(N-n){\sum }_i[y(i)-{\tilde{y}}_n(i){]}^2$, with ${\tilde{y}}_n(i)=1/n{\sum }_{k}y(i-k)$ the moving average, $n$ the moving average window and $N$ the dimension of the stochastic series $y(i)$. This ability relies on the property of ${\sigma }_{DMA}^2$ to scale as $n^{2H}$. Here, we analytically show that ${\sigma }_{DMA}^2$ is equivalent to $C_H{n}^{2H}$ for $n\gg 1$ and provide an explicit expression for $C_H$.

11pages, 3 figures. Presented at Int. Conf. on Application of Physics in Financial Analisys (APFA5), June 29 - July 1, 2006 Torino, Italy

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