Scaling Properties of Long-Range Correlated Noisy Signals

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The Hurst coefficient H of a stochastic fractal signal is estimated using the function σMA²=1Nmax-n∑ᵢ₌ₙNmax [y(i)-yₙ(i)]², where yₙ(i) is defined as 1/n ∑ₖ₌₀ⁿ⁻¹ y(i-k), n is the dimension of moving average box and Nmax is the dimension of the stochastic series. The ability to capture scaling properties by σMA² can be understood by observing that the function Cₙ(i)= y(i)-yₙ(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.