How well a short-memory linear autoregressive model predicts the signal — and how much structure escapes it.
Fits an AR(10) model by Yule-Walker (each sample as a linear combination of the previous ten samples), then summarizes it. This is the classical linear-time-series view: it captures everything a linear predictor can, and whatever it cannot capture surfaces in the residual. Mirrors the meaningful part of tsfresh::ar_coefficient.
Lag-1 Yule-Walker coefficient (≈ lag-1 autocorrelation, the canonical short-memory parameter). High for smooth, slowly-varying signals (LIGO H1 Whitened 3.13, Bearing Ball 2.90, Clipped Sine 2.83); strongly negative for alternating, anti-correlated ones (Logistic Period-3 Window −5.0, Quantum Walk −1.68). Correlates with Spectral:spectral_slope (|r|≈0.89 — the AR(1)↔spectral-tilt identity); kept as the named, interpretable parameter. Clipped to |coef| ≤ 5 to neutralize a pseudo-inverse artifact on rank-deficient periodic signals.
Lag-2 coefficient. Positive where two-step memory adds information (Heisenberg Walk 0.69, Anderson 1D Localized 0.64); strongly negative for two-step anti-correlation (Bearing Ball −3.53, Ambient Microseism −2.79).
Whitened-residual variance fraction (σ²_AR / σ²_x). Near 1 means the signal looks white to a linear predictor (LIGO 1.00, Catalan digits 1.00, Rule 30 1.00 — Rule 30's structure is nonlinear, so AR sees noise); near 0 means the AR model captures it fully (Logistic Period-2 0.00, OTOC Growth 0.00). The strongest AR discriminator.
BIC-selected model order in 0–10. Saturates at 10 for richly-colored signals (Pink Noise, Blue Noise, Riemann Zeta Zeros, Devil's Staircase all 10); drops to 0 for signals an AR captures trivially or degenerately (LIGO 0).
| Source | Origin | Value |
|---|---|---|
| Pink Noise | stochastic-process | 10.0000 |
| Devil's Staircase | special-functions | 10.0000 |
| Thue-Morse | symbolic-dynamics | 10.0000 |
| ··· | ||
| Logistic Chaos | iterated-maps | 0.0000 |
| Tent Map | iterated-maps | 0.0000 |
| AES Encrypted | algorithmic-bytes | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| LIGO H1 Whitened (GW150914) | astrophysical | 3.1256 |
| Newton-Leipnik Attractor | continuous-flows | 2.8633 |
| Bearing Ball | machine-vibration | 2.8355 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | iterated-maps | -5.0000 |
| Quantum Walk | random-matrix-quantum | -1.6779 |
| Hofstadter Q | number-theory | -1.5848 |
| Source | Origin | Value |
|---|---|---|
| Fibonacci Tight-Binding | random-matrix-quantum | 1.5382 |
| Heisenberg Walk | symbolic-dynamics | 0.6900 |
| Anderson 1D Localized | random-matrix-quantum | 0.6349 |
| ··· | ||
| Bearing Ball | machine-vibration | -3.4367 |
| Ambient Microseism | geophysical | -2.7887 |
| LIGO H1 Whitened (GW150914) | astrophysical | -2.6788 |
| Source | Origin | Value |
|---|---|---|
| LIGO Livingston | astrophysical | 1.0000 |
| LIGO Hanford | astrophysical | 1.0000 |
| Catalan G Digits | number-theory | 0.9997 |
| ··· | ||
| OTOC Growth | random-matrix-quantum | 0.0000 |
| Periodic Word | symbolic-dynamics | 0.0000 |
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
Separates linearly-predictable short-memory processes from those whose structure escapes a linear model. ar_residual_frac is the workhorse: a near-1 value flags signals white to a linear predictor — genuine noise, but also algorithmically complex streams such as Rule 30 whose order is nonlinear and therefore invisible here.