How rough is the signal's path, as characterized by the critical exponent where cumulative variation transitions from finite to infinite.
Computes the sum of |increments|^p for multiple values of p. For smooth signals, all p-variations are finite. For Brownian motion, the 2-variation is finite but the 1-variation diverges. For rougher paths (Levy flights), even the 2-variation diverges. The variation index — the critical p at the transition — is a fundamental path invariant that characterizes roughness independently of amplitude.
The 2-variation (sum of squared increments, normalized). Fibonacci Word (0.764) scores highest: its binary substitution sequence produces increments of size 1 at almost every step, maximizing the normalized quadratic variation. Thue-Morse (0.667) and Collatz Parity (0.663) are similar — all are binary-valued with frequent transitions. Persistent fBm scores 0.000005 (nearly zero — its smooth, correlated increments have tiny squared variation). This metric separates "jumpy" binary signals from "smooth" continuous ones.
The estimated critical p. Accelerometer Jog, Accelerometer Walk, and Wind Speed all hit 4.0 (maximum — their paths are extremely rough, with variation diverging even at high p). Mu-law Sine scores 1.0 (smooth path with finite 1-variation — the slope transition happens at p=1). The variation index is invariant under affine transformations (shift and scale) of the signal but not under general monotone transforms.
Lag-1 autocorrelation of increments. Clipped Sine (0.98) and Mackey-Glass (0.97) have strongly persistent increments (smooth, trending dynamics). Logistic period-2 (-1.0) has perfectly anti-persistent increments (each step reverses the previous one). This is the path-roughness analog of the Hurst exponent: positive = smooth trending, negative = rapidly alternating.
Autocorrelation of absolute increments (|Δx_t|). Lotka-Volterra (0.95) and Clipped Sine (0.94) score highest — their increment magnitudes are strongly correlated over time (large changes cluster together). Phyllotaxis (-0.62) and Circle Map QP (-0.62) score most negative — their increment sizes anti-alternate. This is the classic GARCH-like volatility clustering signature, measured directly from the path's p-variation structure.
| Source | Origin | Value |
|---|---|---|
| Logistic Chaos | iterated-maps | 6.0000 |
| Henon Map | iterated-maps | 6.0000 |
| White Noise | stochastic-process | 6.0000 |
| ··· | ||
| μ-law Sine | signal-synthesis | 1.0001 |
| OTOC Growth | random-matrix-quantum | 1.0003 |
| Van der Pol Oscillator | continuous-flows | 1.0005 |
| Source | Origin | Value |
|---|---|---|
| Magnetic Pendulum (3-Magnet) | continuous-flows | 0.9998 |
| OTOC Growth | random-matrix-quantum | 0.9997 |
| Sine Wave | signal-synthesis | 0.9996 |
| ··· | ||
| Phyllotaxis | number-theory | -0.6180 |
| Circle Map Quasiperiodic | iterated-maps | -0.6180 |
| Hilbert Walk | symbolic-dynamics | -0.6000 |
p-Variation captures path roughness in a way that's complementary to Holder Regularity. Holder measures local smoothness point by point; p-Variation measures global path roughness as a summability property. In the atlas, var_p2 strongly separates binary/symbolic signals (Fibonacci, Thue-Morse, Collatz Parity: high var_p2) from continuous oscillations (fBm, sine waves: near-zero var_p2). The variation_index provides the roughness invariant that connects the framework to stochastic analysis: Brownian motion has index 2, and deviations from 2 indicate either smoother-than-Brownian (index < 2) or rougher-than-Brownian (index > 2) dynamics.