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.
Mutual information between the 5 descriptor components along the trajectory. L-System Dragon (7.75) and Clipped Sine (6.89) score highest — their descriptor dimensions co-vary strongly (when entropy changes, so does autocorrelation, kurtosis, etc.). Constants score 0.0 (no trajectory, no coupling). High coupling means the signal's nonstationarity is coordinated across all descriptor dimensions; low coupling means each descriptor changes independently.
Signed slope of the lag-1 autocorrelation series across sliding windows. Positive = AC1 is rising through time (the signal is becoming smoother / more memory-bearing). Critical Transition Fold (+0.49) and Aliquot Orbit Lengths (+0.43) lead — both are textbook examples of critical slowing-down (AC1 → 1 as the system approaches a tipping point). Spectral Form Factor (-0.47) and NASDAQ Returns (-0.45) sit at the antipersistent end: their memory structure decays through time. A signed counterpart to regime_persistence (which sees magnitude only).
Augmented Dickey-Fuller test p-value: how confidently can the unit-root null be rejected? Critical Transition Fold (0.99), Logistic r=3.5 Period-4 (0.96), and fBm Persistent (0.80) saturate near 1.0 — ADF cannot reject unit-root behavior (the signal is integrated or near-integrated). Logistic Chaos and Henon Map collapse to 0.0 — clearly stationary by ADF. The classic econometric stationarity test, embedded as a window-trajectory feature rather than a global pass/fail.
Signed slope of the windowed-variance series. Positive = variance is growing through time (heteroskedasticity, regime broadening). Goldbach r(2n) (+0.94), Exponential Chirp (+0.73), and Critical Transition Fold (+0.45) lead — all have systematic amplitude growth across the window. Sine Map Feigenbaum (-0.55) and Langton's Ant (-0.52) sit at the contracting end. Distinct from change_quantiles_*: variance_trend reads the second moment globally; change_quantiles reads localized quantile excursions.
Mean absolute first-difference of the signal restricted to the bottom value-quantile band. Henon Map (0.175) and ECG Beat Conformity (0.167) lead — their attractors revisit the low band frequently with non-trivial increments each time. The bottom-band leg of a catch22-family quantile-band triplet.
Same statistic restricted to the middle value-quantile band. Logistic Period-2 saturates at 1.0 because its 2-cycle straddles the median; Thue-Morse and Kolakoski Sequence (~0.67) follow. Logistic Chaos and Tent Map collapse to 0.0 — their orbits traverse the middle band continuously without discrete jumps.
Same statistic restricted to the top value-quantile band. Earthquake Depths (0.282) and Continued Fractions (0.244) lead — both have heavy upper-tail activity with large jumps between top-band visits. Together with the low and mid variants, the triplet fingerprints whether the signal's nonstationarity is concentrated in tails, in the body, or distributed.
| Source | Origin | Value |
|---|---|---|
| Critical Transition (Fold) | iterated-maps | 0.4892 |
| Aliquot Orbit Lengths | number-theory | 0.4348 |
| Exponential Chirp | signal-synthesis | 0.3768 |
| ··· | ||
| Spectral Form Factor | random-matrix-quantum | -0.4697 |
| NASDAQ Returns | financial | -0.4470 |
| Nikkei Returns | financial | -0.2371 |
| Source | Origin | Value |
|---|---|---|
| Critical Transition (Fold) | iterated-maps | 0.9904 |
| Logistic r=3.5 (Period-4) | iterated-maps | 0.9585 |
| fBm (Persistent) | stochastic-process | 0.7993 |
| ··· | ||
| Logistic Chaos | iterated-maps | 0.0000 |
| Henon Map | iterated-maps | 0.0000 |
| Tent Map | iterated-maps | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| Earthquake Depths | geophysical | 0.2817 |
| Look-and-Say | symbolic-dynamics | 0.2448 |
| Continued Fractions | number-theory | 0.2440 |
| ··· | ||
| Logistic Chaos | iterated-maps | 0.0000 |
| Henon Map | iterated-maps | 0.0000 |
| Tent Map | iterated-maps | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| Henon Map | iterated-maps | 0.1748 |
| ECG Beat Conformity | physiological | 0.1666 |
| Stern-Brocot Walk | number-theory | 0.1412 |
| ··· | ||
| DNA SARS-CoV-2 | genomic | 0.0000 |
| DNA Centromere | genomic | 0.0000 |
| DNA Phage Lambda | genomic | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | iterated-maps | 1.0000 |
| Periodic Word | symbolic-dynamics | 0.8572 |
| Thue-Morse | symbolic-dynamics | 0.6667 |
| ··· | ||
| Logistic Chaos | iterated-maps | 0.0000 |
| Uniform Chaos (Logistic Scramble) | iterated-maps | 0.0000 |
| Penrose Substitution | symbolic-dynamics | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| Prime Indicator | number-theory | 9.4469 |
| Gap-Word β-Shift | symbolic-dynamics | 9.3779 |
| Golden-Mean β-Shift | symbolic-dynamics | 9.3620 |
| ··· | ||
| Logistic r=3.5 (Period-4) | iterated-maps | 0.0000 |
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
| Logistic Edge-of-Chaos | iterated-maps | 0.9768 |
| Source | Origin | Value |
|---|---|---|
| Clipped Sine | signal-synthesis | 3.4598 |
| Triangle Wave | signal-synthesis | 3.4401 |
| Pell Word | symbolic-dynamics | 3.2873 |
| ··· | ||
| Logistic r=3.5 (Period-4) | iterated-maps | 0.0000 |
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
| Sine Map (Feigenbaum) | iterated-maps | 0.0373 |
| Source | Origin | Value |
|---|---|---|
| Spectral Form Factor | random-matrix-quantum | 1.0000 |
| Copeland-Erdős | number-theory | 0.9967 |
| Gaussian Collatz Orbit | number-theory | 0.9700 |
| ··· | ||
| Tribonacci Word | symbolic-dynamics | 0.0333 |
| Pell Word | symbolic-dynamics | 0.0333 |
| Collatz Cycle Word | number-theory | 0.0333 |
| Source | Origin | Value |
|---|---|---|
| Zipf Distribution | stochastic-process | 0.8527 |
| Geometric Waiting Times | stochastic-process | 0.8220 |
| Poisson Spacings | random-matrix-quantum | 0.8197 |
| ··· | ||
| Logistic r=3.5 (Period-4) | iterated-maps | 0.0000 |
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
| Sine Map (Feigenbaum) | iterated-maps | 0.2000 |
| Source | Origin | Value |
|---|---|---|
| Goldbach r(2n) | number-theory | 0.9387 |
| Exponential Chirp | signal-synthesis | 0.7280 |
| Critical Transition (Fold) | iterated-maps | 0.4497 |
| ··· | ||
| Sine Map (Feigenbaum) | iterated-maps | -0.5545 |
| Langton's Ant | symbolic-dynamics | -0.5150 |
| Fibonacci Tight-Binding | random-matrix-quantum | -0.3267 |
| Source | Origin | Value |
|---|---|---|
| Sine Map (Feigenbaum) | iterated-maps | 5.7662 |
| Gaussian Collatz Orbit | number-theory | 3.4781 |
| Devil's Staircase | special-functions | 2.9184 |
| ··· | ||
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
| Logistic r=3.5 (Period-4) | iterated-maps | 0.0000 |
| Logistic r=3.74 (Period-5 Window) | iterated-maps | 0.2402 |
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).