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Detrending Moving Average variance: a derivation of the scaling law

Sergio Arianos·Anna Carbone
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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 $ σ_{DMA}^2={1}/{(N-n)}\sum_i [y(i)-\widetilde{y}_n(i)]^2\:$, with $\widetilde{y}_n(i)= {1}/{n}\sum_{k}y(i-k)$ 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 $σ_{DMA}^2$ to scale as $n^{2H}$. Here, we analytically show that $σ_{DMA}^2$ is equivalent to $C_H n^{2H}$ for $n\gg 1$ and provide an explicit expression for $C_H$.

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Sergio Arianos, Anna Carbone. 2006-08-31. Detrending Moving Average variance: a derivation of the scaling law. https://doi.org/10.1016/j.physa.2007.02.074

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