Whether the signal's roughness is uniform or varies from place to place.
A monofractal signal (like fractional Brownian motion) has the same scaling behavior everywhere — zoom in anywhere and you see the same texture. A multifractal signal (like turbulence or financial returns) has smooth regions and rough regions interspersed. This geometry computes the singularity spectrum f(α) via structure functions and measures how wide it is.
The range of local scaling exponents (α_max − α_min). Zero means monofractal: every point has the same roughness. Seismic noise (ANMO) dominates the atlas at 0.78 — earthquake background has wildly varying local regularity from microseisms to cultural noise. GOES X-ray flux (0.49) is next: solar flares inject bursts of irregular scaling into a smooth background. Sawtooth wave (0.45) is surprisingly wide because the discontinuous resets create a sharp transition between smooth ramps and singular jumps. Constants score 0.0 (no scaling structure at all).
The self-affinity exponent H = τ(2)/2. Sine wave scores 0.90 (strongly persistent, smooth). Chua's circuit (0.86) is also highly persistent — its double-scroll attractor stays in each lobe for extended stretches. Prime gaps and divisor count score 0.0: number-theoretic sequences have no self-affine scaling.
| Source | Domain | Value |
|---|---|---|
| Sine Wave | waveform | 0.9006 |
| Chua's Circuit | exotic | 0.8571 |
| Projectile with Drag | motion | 0.8243 |
| ··· | ||
| Divisor Count | number_theory | 0.0000 |
| Prime Gaps | number_theory | 0.0000 |
| Constant 0xFF | noise | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Seismic Noise (ANMO) | geophysics | 0.7795 |
| GOES X-Ray Flux | astro | 0.4935 |
| Sawtooth Wave | waveform | 0.4529 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Collatz Parity | number_theory | 0.0000 |
Multifractal width is the primary separator between "simple" dynamics (periodic, noise, low-dimensional chaos) and "complex" dynamics (turbulence, geophysical processes, financial markets). In the seismic P-wave investigation, multifractal metrics contributed to the top discriminators — earthquake arrivals temporarily broaden the local spectrum width by injecting a new scaling regime into the ambient noise.