The statistical fingerprint beyond mean and variance — skewness, kurtosis, and nonlinear frequency coupling.
Projects pairs of consecutive values onto 20 directions in 2D and computes the skewness and kurtosis along each projection. Also computes permutation entropy (how many ordinal patterns appear) and bicoherence (whether triplets of frequencies are phase-locked). Together these detect non-Gaussianity from four independent angles: directional asymmetry, tail heaviness, ordinal structure, and nonlinear coupling.
Are any frequency triplets phase-locked? Measures whether the phases of frequencies f1, f2, and f1+f2 are correlated across segments, normalized by sqrt(P1*P2*P3) — a non-standard normalization that is unbounded above 1.0 (standard bicoherence uses a different denominator). Devil's Staircase scores 7.92: its self-similar staircase structure creates strong quadratic phase coupling between harmonics. Quantum Walk (5.54) is next — interference creates phase-locked frequency relationships. Linear systems and noise score near zero because their phases are independent.
Maximum projection kurtosis across 20 directions. Heavy tails in any direction push this up. Rainfall dominates at 1373: most hours are dry (near zero), but rare downpours create extreme outliers. Forest Fire (886) has similar bursty dynamics. Gaussian noise scores near 0 (the excess kurtosis baseline — Fisher's definition, not Pearson's). Constants and periodic orbits score 0.0 (degenerate single-point projections).
What fraction of ordinal patterns are observed, weighted by their frequencies? Pi Digits and XorShift32 score 0.999 (all 120 possible 5-element rank orderings appear, roughly equally). Forest Fire scores 0.008: its avalanche dynamics visit almost no ordinal patterns. This is encoding-invariant — it ignores absolute values entirely.
What fraction of ordinal patterns never appear? Logistic Period-2 scores 0.983 (only 2 of 120 patterns observed — alternating between two values constrains the rank structure almost completely). Noise, prime gaps, and pink noise score 0.0 (all patterns appear). This is a zero-training determinism detector: forbidden patterns are the fingerprint of low-dimensional dynamics.
Average absolute skewness across projections. Rainfall (20.8) and Forest Fire (8.32) have the most asymmetric distributions: rare large events create long right tails in nearly every direction. OpenBSD ELF (6.28) is skewed because binary executables have non-uniform byte distributions. Symmetric distributions (Gaussian noise, sine waves) score near zero.
Energy of the third-order cumulant (bispectrum). Measures nonlinear coupling strength beyond what bicoherence_max captures — the total energy rather than just the peak. Not yet in the atlas — recently added.
| Source | Domain | Value |
|---|---|---|
| De Bruijn Sequence | number_theory | 1.0000 |
| Logistic Edge-of-Chaos | chaos | 1.0000 |
| L-System (Dragon Curve) | exotic | 0.9999 |
| ··· | ||
| Logistic r=3.5 (Period-4) | chaos | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Thue-Morse | exotic | 0.0163 |
| Source | Domain | Value |
|---|---|---|
| Lotka-Volterra | bio | 3.7755 |
| Forest Fire | exotic | 3.3996 |
| Hodgkin-Huxley | bio | 2.7348 |
| ··· | ||
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Thue-Morse | exotic | 0.0001 |
| Noisy Period-2 | chaos | 0.0016 |
| Source | Domain | Value |
|---|---|---|
| Rainfall (ORD Hourly) | climate | 1400.3451 |
| Forest Fire | exotic | 885.8015 |
| Accel Sit | motion | 647.7791 |
| ··· | ||
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Neural Net (Dense) | binary | 0.0787 |
| ARMA(2,1) | noise | 0.0827 |
| Source | Domain | Value |
|---|---|---|
| Pi Digits | number_theory | 0.9994 |
| glibc LCG | binary | 0.9992 |
| XorShift32 | binary | 0.9992 |
| ··· | ||
| Forest Fire | exotic | 0.0084 |
| Square Wave | waveform | 0.0120 |
| Sawtooth Wave | waveform | 0.0128 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | chaos | 0.9833 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.9750 |
| Logistic r=3.5 (Period-4) | chaos | 0.9667 |
| ··· | ||
| Sunspot Number | astro | 0.0000 |
| Sandpile | exotic | 0.0000 |
| Coupled Map Lattice | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Rainfall (ORD Hourly) | climate | 20.6042 |
| Forest Fire | exotic | 8.3203 |
| OpenBSD ELF x86-64 | binary | 6.2805 |
| ··· | ||
| Logistic r=3.5 (Period-4) | chaos | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Thue-Morse | exotic | 0.0001 |
Higher-Order Statistics separates signals that look identical under mean-and-variance analysis. Two sources with the same entropy and autocorrelation can have completely different kurtosis profiles if one has heavy tails and the other doesn't. Bicoherence is the only metric in the framework that detects nonlinear frequency coupling — linear filters destroy it, so high bicoherence is a direct signature of nonlinear dynamics. In the atlas, the combination of kurt_max and perm_forbidden separates bursty processes (high kurtosis, low forbidden) from low-dimensional chaos (moderate kurtosis, high forbidden).