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 $\widetilde{y}_n(i)$ is defined as $1/n \sum_{k=0}^{n-1} y(i-k)$, $n$ is the dimension of moving average box and $N_{max}$ 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 $C_n(i)= y(i)-\widetilde{y}_n(i)$ 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.