The shape and locality of phase coupling between frequency triplets (f₁, f₂, f₁ + f₂) — orthogonal extensions of Higher-Order Statistics' peak-bicoherence channel.
The bispectrum B(f₁, f₂) = E[X(f₁) X(f₂) X*(f₁ + f₂)] is the Fourier transform of the third-order cumulant. Linear Gaussian processes have identically zero bispectrum (Liouville: the third cumulant vanishes for Gaussian distributions, and linear filtering preserves Gaussianity); quadratic nonlinearity and phase-locked triplets produce non-zero bispectral mass at the coupled frequencies. Normalized via Kim-Powers 1979 bicoherence² ∈ [0, 1] = |B|² / (E[|X(f₁)X(f₂)|²] · E[|X(f₁ + f₂)|²]), which removes amplitude dependence and isolates phase-locking. Computed via segment-averaged Hann-windowed direct estimation; requires ≥16 segments for reliable phase averaging. Not encoding-invariant — uses raw amplitudes.
Gini coefficient of the bicoherence² distribution over the principal triangle. 0 = uniform spread (noise-like, mass everywhere); 1 = all mass concentrated at one triplet (pure tone, sawtooth). Replaces the earlier Shannon-entropy formulation which saturated near 1.0 for ~93% of sources because normalized-entropy of an O(K²) noisy array sits very close to max log(K²/2) for almost any input. Gini doesn't saturate. White noise sits at ~0.5, designed (a,b,a+b) triplets sit at ~0.85, constants at 0.
mean(bicoherence² off the f₁=f₂ diagonal) / mean(bicoherence² on diagonal + off). Distinguishes self-harmonic coupling (single fundamental with harmonics → mass concentrated on the diagonal) from cross-mode coupling (multiple oscillators interacting → off-diagonal mass). Rössler Hyperchaos (0.88), Weierstrass (0.85), and the 3/4/5-Torus Quasiperiodic sources (0.70–0.73) score highest — genuine multi-mode interactions. Period-3 (0.11) and Euler Totient Ratio (0.19) sit at the diagonal-dominated end (single harmonic series). Constants are 0 (no bicoherence at all).
Mass-weighted mean of (f₁ + f₂) over the principal triangle, normalized to the Nyquist bin. Captures where in frequency the phase coupling lives — orthogonal to amount-of-coupling metrics (bicoherence_concentration, off_diagonal_ratio, HOS:bicoherence_max). Logistic period-3 and period-5 windows top out at 0.71–0.72 (windowed-periodic dynamics couple at high harmonics, beating with the chaotic backbone); broadband chaos (Ikeda, Lorenz-96) sits at 0.36–0.42; Blood Pressure Waveform at 0.27 and Hodgkin-Huxley at 0.32 (low-frequency cardiac / bursty neural coupling). The uniform-on-triangle baseline is ~0.50, where Sine, White Noise, AES, and Pi Digits cluster — they have no preferred coupling-frequency location. F-by-domain is low (1.77) because the axis cuts across domains rather than along them: chaos has the widest within-domain std (0.077) in the atlas.
| Source | Domain | Value |
|---|---|---|
| Riemann-Hardy-Littlewood | exotic | 0.9426 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.9420 |
| Weierstrass | exotic | 0.9294 |
| ··· | ||
| Constant 0x00 | noise | 0.0000 |
| Logistic Edge-of-Chaos | chaos | 0.0002 |
| Sawtooth Wave | waveform | 0.0050 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.83 (Period-3 Window) | chaos | 0.7152 |
| Logistic r=3.74 (Period-5 Window) | chaos | 0.7146 |
| Logistic r=3.9 (Near-Full Chaos) | chaos | 0.6148 |
| ··· | ||
| Blood Pressure Waveform | medical | 0.2853 |
| Hodgkin-Huxley | bio | 0.3123 |
| Weierstrass | exotic | 0.3319 |
| Source | Domain | Value |
|---|---|---|
| Rössler Hyperchaos | chaos | 0.8806 |
| Weierstrass | exotic | 0.8486 |
| 4-Torus Quasiperiodic | chaos | 0.7282 |
| ··· | ||
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.1062 |
| Euler Totient Ratio | number_theory | 0.1862 |
Bispectrum is the textbook complement to the power spectrum and a direct supplement to Higher-Order Statistics' peak-bicoherence channel. HOS:bicoherence_max captures the intensity of the strongest phase-coupled triplet; Bispectrum's three metrics capture the shape (bicoherence_concentration), locality (off_diagonal_ratio), and frequency location (coupling_frequency_centroid) of the full coupling distribution. The geometry is atlas-orthogonal: max |r| with any other metric is 0.378 (centroid with Hodge-Laplacian:spectral_anisotropy); entropy and off_diagonal_ratio stay at max |r| ≤ 0.30. Two earlier candidate metrics — peak_bicoherence (r = +0.752 with HOS:bicoherence_max) and phase_coupled_fraction (r = +0.898 intra, +0.757 cross) — were dropped at the 2026-05-17 redundancy bar; a coupling_anisotropy candidate (λ-ratio of the b² covariance) was dropped at the 2026-05-18 pilot for r = -0.77 with off_diagonal_ratio and r = -0.73 with HOS:bicoherence_max.