Uncertainty quantification

Every estimator in sid returns confidence information alongside its point estimates. The non-parametric estimators use asymptotic variance formulas (Ljung, 1999) per frequency; the COSMIC state-space identifier propagates a Bayesian posterior through to A(k), B(k) standard deviations and, optionally, to frozen transfer functions G(ω, k).

Frequency-domain

Per-frequency variance for the Blackman-Tukey estimator follows from the Hann window's effective number of independent averages and the input-output coherence. See Specification §3.

State-space (COSMIC)

The COSMIC posterior is derived in Uncertainty derivation. Activate it by passing Uncertainty=True (Python) / 'Uncertainty', true (MATLAB) to ltv_disc / sidLTVdisc, which adds AStd, BStd, and a per-step row covariance P to the result.