| """Exact OU innovation variance calibration and bounded-drift tube budgets. |
| |
| Calibration assumes known kappa, independent exact-state observations, |
| Gaussian innovations, and fixed (not data-selected) pairs. It is not valid |
| for censored, noisy-camera, or locally stopped trajectories without a new |
| observation likelihood. Units: time seconds, distance arbitrary consistent. |
| """ |
| import numpy as np |
| from scipy.stats import chi2 |
|
|
|
|
| def _finite(*values): |
| if not all(np.isfinite(v) for v in values): |
| raise ValueError("all scalar parameters must be finite") |
|
|
|
|
| def diffusion_interval(innovations, kappa, dt, alpha=0.05): |
| """Two-sided (1-alpha) CI for d in dZ=-k Z dt+sqrt(2d)dB. |
| |
| Each flattened value must be an independent scalar innovation with |
| zero known mean. Degrees of freedom are not adjusted for fitted kappa. |
| """ |
| _finite(kappa, dt, alpha) |
| values = np.asarray(innovations, dtype=float).ravel() |
| if values.size < 1 or not np.isfinite(values).all(): |
| raise ValueError("finite nonempty innovations required") |
| if kappa <= 0 or dt <= 0 or not 0 < alpha < 1: |
| raise ValueError("kappa, dt > 0 and alpha in (0,1) required") |
| scale = -np.expm1(-2 * kappa * dt) / kappa |
| ss = float(values @ values) |
| df = values.size |
| return (ss / (scale * chi2.ppf(1-alpha/2, df)), |
| ss / (scale * chi2.ppf(alpha/2, df))) |
|
|
|
|
| def diffusion_upper(innovations, kappa, dt, alpha=0.05): |
| """Exact one-sided variance bound; Bonferroni handles dependent pairs.""" |
| _finite(kappa, dt, alpha) |
| values = np.asarray(innovations, dtype=float).ravel() |
| if values.size < 1 or not np.isfinite(values).all(): |
| raise ValueError("finite nonempty innovations required") |
| if kappa <= 0 or dt <= 0 or not 0 < alpha < 1: |
| raise ValueError("kappa, dt > 0 and alpha in (0,1) required") |
| scale = -np.expm1(-2*kappa*dt) / kappa |
| return float(values @ values) / (scale * chi2.ppf(alpha, values.size)) |
|
|
|
|
| def ou_innovations(positions, kappa, dt): |
| """Residuals from equally spaced exact observations of a global OU.""" |
| _finite(kappa, dt) |
| z = np.asarray(positions, dtype=float) |
| if z.ndim < 1 or z.shape[0] < 2 or kappa <= 0 or dt <= 0 or not np.isfinite(z).all(): |
| raise ValueError("at least two times and positive kappa,dt required") |
| return z[1:] - np.exp(-kappa*dt)*z[:-1] |
|
|
|
|
| def drift_tube(kappa, horizon, initial_error, drift_bound): |
| """Maximum scalar convolution bound on [0,T], with nonnegative inputs.""" |
| _finite(kappa, horizon, initial_error, drift_bound) |
| if kappa <= 0 or horizon < 0 or initial_error < 0 or drift_bound < 0: |
| raise ValueError("invalid tube parameters") |
| end = initial_error*np.exp(-kappa*horizon) |
| end += drift_bound/kappa * (-np.expm1(-kappa*horizon)) |
| return max(float(initial_error), float(end)) |
|
|
|
|
| def robust_margin(clearance, kappa, horizon, initial_error=0, drift_bound=0): |
| """Remaining OU margin; nonpositive output rejects the certificate.""" |
| _finite(clearance, kappa, horizon, initial_error, drift_bound) |
| if clearance <= 0: |
| raise ValueError("clearance must be positive") |
| return clearance-drift_tube(kappa, horizon, initial_error, drift_bound) |
|
|
|
|
| def girsanov_event_allowance(kappa, d, horizon, drift_bound): |
| """Pinsker event allowance under a common full-rank isotropic diffusion. |
| |
| kappa is unused in the formula but retained for common calling convention. |
| Bound: KL <= b^2 T/(4d), TV <= b sqrt(T/(8d)). This is a |
| same-diffusion result, not a bound for diffusion-coefficient mismatch. |
| """ |
| _finite(kappa, d, horizon, drift_bound) |
| if kappa <= 0 or d <= 0 or horizon < 0 or drift_bound < 0: |
| raise ValueError("invalid parameters") |
| return min(1.0, drift_bound*np.sqrt(horizon/(8*d))) |
|
|
