Scaling Properties of Long-Range Correlated Noisy Signals

The Hurst coefficient $H$ of a stochastic fractal signal is estimated using the function $\sigma_{MA}^2=\frac{1}{N_{max}-n}\sum_{i=n}^{N_{max}} [y(i)-\widetilde{y}n(i)]^2$,where$\widetilde{y}n(i)$isdefinedas$1/n \sum{k=0}^{n-1} y(i-k)$,$n$isthedimensionofmovingaverageboxand$N{max}$ is the dimension of the stochastic series. The ability to capture scaling properties by ${\sigma }_{MA}^2$ can be understood by observing that the function $C_n(i)=y(i)-{\tilde{y}}_n(i)$ generates a sequence of random clusters having power-law probability distribution of the amplitude and of the lifetime, with exponents equal to the fractal dimension $D$ of the stochastic series.