Rényi Entropy & Multifractals

Generalized entropy H_q interpolates between min-entropy (q→∞), Shannon entropy (q→1), and Hartley entropy (q→0), each weighting rare vs. common events differently.

H_q vs q for different distributions — drag q-slider to see highlighted value
Current distribution (edit by dragging bars)
Multifractal spectrum f(α)
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q=0: H_0 = log₂(N), q=1: Shannon, q=∞: -log₂(max p)
Rényi entropy: H_q = (1/(1-q))·log₂(Σᵢ pᵢ^q) for q≠1. Special cases: H_0 = log₂(support size) — Hartley entropy; H_1 = −Σ pᵢ log₂ pᵢ — Shannon entropy (L'Hôpital limit); H_2 = −log₂(Σ pᵢ²) — collision entropy; H_∞ = −log₂(max pᵢ) — min-entropy. All satisfy H_∞ ≤ H_2 ≤ H_1 ≤ H_0. The multifractal spectrum f(α) = q·α − (q-1)·D_q relates Rényi exponents D_q = H_q(scale→0)/log(scale) to local Hölder exponents α via Legendre transform — it characterizes the distribution of singularities in a fractal measure.