pqc.detect.ou

Compute OU innovations and auxiliary noise-scale estimates.

This module provides the core OU-statistics used by bad-measurement detection. For irregularly sampled residuals, it computes normalized innovations under an Ornstein–Uhlenbeck (OU) model and estimates a non-negative extra variance parameter q using robust scale matching.

Notes

Definition (OU process)

Continuous-time mean-reverting Gaussian process: \(dX_t = -\theta X_t dt + \sigma dW_t\), with \(\tau = 1/\theta\) as correlation timescale.

Discretization used here

For intervals \(\Delta t_i\), use \(\phi_i = \exp(-\Delta t_i/\tau)\) and innovation \(e_i = y_i - \phi_i y_{i-1}\).

Why used here

Residuals are often serially correlated on short scales; innovation normalization approximates whitened residual surprises.

Assumptions

• Locally stationary correlation structure represented by OU.

• Approximate Gaussianity of innovations for downstream p-values.

• Correct time ordering of observations.

References

[1]

Uhlenbeck, G. E., & Ornstein, L. S. (1930), Physical Review, 36, 823.

[2]

Gardiner, C. (2009), Stochastic Methods, 4th ed.

[3]

Rousseeuw, P. J., & Croux, C. (1993), JASA, 88(424), 1273-1283.

Functions

estimate_q(t_days, y, sigma, tau_days[, ...])	Estimate q by matching robust innovation scale to unity.
ou_innovations_z(t_days, y, sigma, tau_days, q)	Compute normalized OU innovations for irregularly sampled data.