Rényi Entropy
Rényi entropy Hq generalizes Shannon (q→1), max entropy (q=0), and min-entropy (q→∞). For multifractal measures, the singularity spectrum f(α) — Hausdorff dimension of sets with local Hölder exponent α — encodes the full scaling hierarchy.
H_q = log(Σpᵢq)/(1-q) [q≠1]
H_1 = -Σpᵢlog(pᵢ) (Shannon)
τ(q) = (q-1)D_q
α = dτ/dq, f(α) = qα - τ(q)
H_1 = -Σpᵢlog(pᵢ) (Shannon)
τ(q) = (q-1)D_q
α = dτ/dq, f(α) = qα - τ(q)
Measure Type
Binomial
Cantor
Logistic
Uniform
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H₁ (Shannon)
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D₀ (Box dim)
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D₂ (Corr. dim)
—
f(α) width Δα