Rényi Entropy & Multifractal Singularity Spectrum

Multifractal measures have a spectrum of local scaling exponents α, described by the f(α) singularity spectrum. Rényi entropies H_q = log(Σpᵢq)/(1−q) probe different moments of the measure.

Measure type

p (cascade parameter)0.30

Resolution levels8

H_q = log(Σᵢ pᵢ^q)/(1−q)
τ(q) = lim_{ε→0} log(Σpᵢ^q)/log(ε)
α = −dτ/dq
f(α) = qα − τ(q) [Legendre]
D_q = τ(q)/(q−1) (generalized dims)