Training-only error estimation and prediction uncertainty

All 26 deterministic controls pass. A research tool now compares a fixed list of error covariances using training observations only, and includes uncertainty from fitted linear parameters in later predictions. It has not been applied to NASA recordings or changed their model preferences.

An IMU (inertial measurement unit) measures turning and acceleration. In the archive analysis, instrument calibration and starting conditions are estimated from observations. Predictions must account for that estimation rather than treating the fitted values as perfectly known.

What the controls show

With independent unit-variance errors, fitting a three-axis mean from two epochs adds half a unit of variance to a later epoch's prediction error: total variance is 1.5 rather than 1. That is an analytic example, not a NASA uncertainty estimate.

With correlated errors, simply adding the fitted parameters' covariance gives the wrong answer. The correct adjustment accounts for shared information between the training and prediction errors. Independently generated finite Gaussian moment controls reproduce the predicted covariance in both independent and correlated examples. There are no random campaigns or fitted Earth models in these tests.

The covariance selector uses restricted likelihood (REML): it accounts for the parameters estimated from the training data and the determinant of the error covariance, rather than rewarding a covariance merely for making residuals smaller. Candidates and their order are supplied before selection. Equal scores are marked ambiguous; score differences are not statistical significance levels. No future observations are accepted by the estimator or prediction API.

The tests also cover actual irregular timestamps, the unchanged full-record drift anchor, rank-deficient designs, different parameter units, malformed inputs, and prediction sensitivity to every supplied covariance candidate. No covariance option is discarded silently, averaged into a pseudo-confidence interval, or estimated separately for each physical model.

Formula and limits

For a linear mean with training design \(X_t\), future design \(X_p\), training covariance \(C_{tt}\), and cross-boundary covariance \(C_{pt}\), let

The future prediction-error covariance is

This is checked for an unconstrained, identifiable linear mean and each supplied covariance. The covariance itself is not assumed to have been measured perfectly: predictions are returned separately for all choices, making sensitivity visible. Their spread is a development diagnostic, not integrated covariance- parameter uncertainty or calibrated coverage.

The NASA fits are nonlinear and bounded. These controls do not propagate those fits' calibration uncertainty, resolve the integration clock, or replace their likelihood. A local linear approximation requires its own adequacy and active-bound checks. Weak or rank-deficient linear designs are rejected rather than producing apparently precise parameter estimates.

Next step

Prepare a separately versioned, bounded comparison using designated development training intervals and one common error specification for all physical candidates. Before launch, specify how nonlinear calibration uncertainty and active bounds will be handled, preserve covariance-choice sensitivity, and obtain a finite allowance for observed fits. Do not reinterpret the original costs under the new likelihood. Independent statistical calibration and replication remain separate work.

These controls add zero archive reads, zero observed fit starts and zero native predictions. They leave all completed studies and unresolved labels intact. See the shared error-model plan and preceding covariance checks.

Reproduce

env PYTHONPATH=analysis OPENBLAS_NUM_THREADS=2 OMP_NUM_THREADS=2 MKL_NUM_THREADS=2 ~/venv/bin/python docs/ilvis0-error-training-controls-20261012/build_report.py

The report receipt hashes the implementation, tests, plan and outputs and records the numerical environment. The fixed controls use only small matrix calculations.