How the signal's local geometric character changes over time.
Computes a 5D descriptor (entropy, lag-1 autocorrelation, variance, kurtosis, permutation entropy) on sliding windows and tracks its trajectory through descriptor space. The other geometries compute static, full-sequence summaries. This one measures the derivative: how fast the local character is changing, how bursty that change is, and how much of the descriptor space the trajectory explores.
Mean speed of movement through descriptor space (z-scored). Triangle Wave (3.31) and Clipped Sine (3.27) score highest because their periodic structure creates rapid, repeated transitions between distinct geometric regimes. Logistic Period-5 (2.95) is similar. Devil's Staircase scores 0.0: its local geometry is constant within each plateau, and the jumps between plateaus are too rare to raise the mean speed.
How long do geometric regimes last? Measured by the autocorrelation decay time of the descriptor trajectory. Rossler Hyperchaos, Quantum Walk, and Lotka-Volterra all score 1.0 (maximum persistence — once they enter a geometric regime, they stay). Rossler Attractor scores 0.033 (regimes change rapidly as the trajectory spirals between lobes). High persistence signals piecewise-stationary dynamics.
PCA participation ratio of the descriptor cloud, normalized by 5. How many independent descriptor axes does the trajectory use? Zipf Distribution (0.904) and Noisy Sine (0.896) explore nearly the full 5D space. Devil's Staircase scores 0.0 (the trajectory is confined to a single point in descriptor space). High trajectory_dim means the signal's local geometry changes in multiple independent ways simultaneously.
Coefficient of variation of the descriptor speed. Is the rate of geometric change itself stable or bursty? Gaussian Collatz (2.08) and Thue-Morse (2.01) score highest: their geometric changes come in bursts separated by calmer intervals. This is the actual "volatility of volatility" — a second-order nonstationarity measure. Van der Pol (1.74) scores high because its relaxation oscillations create alternating fast and slow geometric evolution.
| Source | Domain | Value |
|---|---|---|
| Triangle Wave | waveform | 3.3135 |
| Clipped Sine | waveform | 3.2654 |
| Logistic r=3.74 (Period-5 Window) | chaos | 2.9489 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Devil's Staircase | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Rössler Hyperchaos | chaos | 1.0000 |
| Quantum Walk | quantum | 1.0000 |
| Lotka-Volterra | bio | 1.0000 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Rossler Attractor | chaos | 0.0333 |
| Source | Domain | Value |
|---|---|---|
| Zipf Distribution | exotic | 0.9039 |
| Noisy Sine (SNR 3 dB) | waveform | 0.8957 |
| Gaussian Noise | noise | 0.8907 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Devil's Staircase | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Gaussian Collatz Orbit | number_theory | 2.0839 |
| Thue-Morse | exotic | 2.0135 |
| Van der Pol Oscillator | exotic | 1.7395 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Devil's Staircase | exotic | 0.0000 |
Nonstationarity detects regime switching and concatenation that static metrics miss entirely. A signal made by splicing together two different sources will score high on vol_of_vol (bursty regime changes) and trajectory_dim (multiple descriptors change) while possibly looking unremarkable to any single static geometry. In the atlas, regime_persistence separates the dynamical view's "coherent chaos" cluster (Rossler Hyperchaos, Lotka-Volterra: chaotic but geometrically stable) from "incoherent chaos" (Rossler Attractor: chaotic and geometrically unstable).
# Scale Geometries