OU Innovations

What it is

PQC models short-timescale correlated residual noise with an Ornstein-Uhlenbeck (OU) process. The OU process is a mean-reverting Gaussian stochastic process:

\[dX_t = -\frac{1}{\tau}X_t\,dt + \sigma\,dW_t\]

Why PQC uses it

TOA residuals are often not independent at short lags. OU innovations provide a local whitening transform so pointwise surprises are measured against a correlated-noise baseline rather than white-noise assumptions.

Discrete innovation form used in PQC

For irregular samples at times \(t_i\):

\[\phi_i = \exp\left(-\frac{t_i-t_{i-1}}{\tau}\right)\]
\[e_i = y_i - \phi_i y_{i-1}\]
\[v_i = \sigma_i^2 + \phi_i^2 \sigma_{i-1}^2 + q(1-\phi_i^2)\]
\[z_i = \frac{e_i}{\sqrt{v_i}}\]

where q is an additional variance term estimated robustly.

Interpretation

  • \(|z_i| \approx 0\): consistent with the OU-noise prediction.

  • large \(|z_i|\): locally inconsistent, candidate bad measurement.

Assumptions and caveats

  • OU is only an approximation of residual correlation.

  • Innovation tails are treated approximately Gaussian.

  • Non-monotonic timestamps or severe model mismatch can distort scores.

Small worked example

If \(\tau=10\) d and \(\Delta t=1\) d, then \(\phi=\exp(-0.1)=0.905\). With \(y_{i-1}=1.0\), \(y_i=0.2\): \(e_i = 0.2 - 0.905 = -0.705\). If \(v_i=0.25\), then \(z_i=-1.41\).

References

[UO1930]

Uhlenbeck, G. E., & Ornstein, L. S. (1930). “On the theory of the Brownian motion.” Physical Review, 36, 823-841.

[Gardiner2009]

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