Sparse joint-PACE score-uncertainty qualification¶

This page documents the deterministic known-population qualification for the 1.1 development capability sparse_mfpca_score_uncertainty().

What is being qualified?¶

For a retained joint score vector \(\boldsymbol\xi_i\) and paired native planar observations, the capability reports

\[ \operatorname{Var}(\boldsymbol\xi_i\mid\mathbf Y_i,\text{ fitted population}) =\mathbf\Lambda -\mathbf\Lambda\mathbf\Phi_i^\top \mathbf\Sigma_i^{-1} \mathbf\Phi_i\mathbf\Lambda. \]

The claim is deliberately narrow: this is conditional-on-fitted-population uncertainty for native sparse planar joint-PACE scores. It is not full uncertainty for sparse-MFPCA estimation.

The qualification therefore supplies the true Gaussian population mean, joint covariance operator, eigensystem, and measurement-error covariance to the score system. This isolates the conditional covariance calculation from sparse mean/covariance estimation, smoothing-bandwidth choice, PSD repair, and measurement-error estimation.

Exact score system¶

The validation uses the same primitives as native joint_pace_scores():

\[ \mathbf\Sigma_i =\mathbf C_i^{TM} +I_{m_i}\otimes\mathbf R_\epsilon +\gamma I_{2m_i}, \]

where the fitted joint covariance is evaluated at the curve's retained native times in time_major_interleaved_xy order. The operator itself is retained in channel-major form. The posterior calculation uses a linear solve; it never constructs an explicit matrix inverse.

The evidence contract requires:

the full fitted joint covariance rather than a rank-truncated reconstruction;
paired synchronous x/y native observations;
the declared 2x2 measurement-error covariance, including correlated-error cases;
the same score ridge and condition policy as the fitted joint-PACE system;
no raw sparse-trajectory interpolation;
exact curve/order/support alignment with the fit.

Oracle scenarios¶

The dedicated sparse-mfpca-score-uncertainty-validation workflow runs four deterministic scenarios with 500 curves each:

Scenario	Native samples per curve	Measurement error
`diagonal_sparse`	5-7	diagonal 2x2 covariance
`diagonal_dense`	14-18	same diagonal covariance
`correlated_sparse`	5-7	positive correlated 2x2 covariance
`correlated_dense`	14-18	same correlated covariance

The latent process has two known orthonormal planar modes and fixed eigenvalues. The simulator generates the latent scores and irregular native observations; the oracle scorer then receives the same known population functions and covariance used to generate those data.

Qualification guards¶

The workflow fails if any declared guard is violated. It checks that:

every joint conditional score system is solved successfully;
each posterior covariance is symmetric and positive semidefinite within numerical tolerance;
marginal score-error MSE is broadly consistent with mean reported conditional variance;
standardized score errors have approximately zero mean and unit scale;
nominal 68% and 95% marginal conditional intervals have broadly compatible empirical coverage;
denser native observation reduces mean total conditional variance within each fixed measurement-error regime;
both diagonal and correlated measurement-error contracts are exercised;
deterministic replay under the same seed is exact;
the uncertainty provenance declares that a rank-K covariance reconstruction was not used; and
population-estimation, bandwidth-selection, and measurement-error-estimation uncertainty are explicitly excluded.

The numerical acceptance windows are intentionally broad Monte Carlo qualification guards, not inferential confidence intervals or universal performance guarantees.

First exact-head qualification evidence¶

The first complete qualification run on PR #170 head 06c0f61dd94a3b008fc223c94ca0bf6fe4a45ce0 passed all guards. GitHub Actions run 37288035380 produced artifact 11334658246, digest sha256:2799a46491c43ec702df55b247b39187fb59d066727a53a60d7c32d697c0ae1d.

Across the eight component-by-scenario calibration cells:

MSE / mean reported conditional variance ranged from 0.966 to 1.089;
standardized-error means ranged from -0.093 to 0.056;
standardized-error standard deviations ranged from 0.975 to 1.045;
empirical 68% marginal conditional coverage ranged from 0.660 to 0.704; and
empirical 95% marginal conditional coverage ranged from 0.932 to 0.956.

No joint score system failed in any scenario. Posterior covariance symmetry error was exactly 0.0 throughout, and the smallest retained posterior eigenvalue across all scenarios was approximately 9.58e-05, so every retained posterior covariance was positive definite in this qualification run.

The information comparison behaved in the expected direction. Mean total conditional variance fell from 0.002157 to 0.000488 when the diagonal-error design moved from 5-7 to 14-18 native observations per curve, and from 0.002352 to 0.000474 under the correlated-error design. Deterministic replay passed for all four scenarios. The artifact also records rank_k_covariance_used_for_uncertainty=false, population_estimation_uncertainty_included=false, and bandwidth_uncertainty_included=false for every scenario.

These are qualification observations for the declared oracle designs, not distribution-free guarantees for arbitrary fitted sparse-MFPCA analyses.

What this evidence supports¶

A passing run supports the implementation claim that, when the fitted joint Gaussian population objects are treated as fixed and correctly specified, sparse_mfpca_score_uncertainty() reproduces the expected conditional score-covariance behavior for the qualified planar designs.

It also supports the software contracts for joint observation ordering, correlated measurement error, native-time evaluation, deterministic replay, failure retention, and posterior covariance diagnostics.

What it does not support¶

This qualification does not establish:

full frequentist coverage after estimating sparse mean/covariance surfaces;
uncertainty from smoothing-bandwidth selection;
uncertainty from joint PSD repair;
uncertainty from estimating or misspecifying the 2x2 measurement-error covariance;
asynchronous x/y observation grids;
validity after silently interpolating raw sparse trajectories;
automatic propagation of score uncertainty into downstream regression or prediction; or
equivalence to dense-grid projection scores.

Those are separate statistical targets and require separate estimands and qualification programmes.

Reproducible evidence¶

The workflow runs:

python scripts/run_sparse_mfpca_score_uncertainty_validation.py \
  --n-curves 500 \
  --output sparse-mfpca-score-uncertainty-validation.json

and uploads sparse-mfpca-score-uncertainty-validation.json as the sparse-mfpca-score-uncertainty-validation artifact. The JSON retains scenario-level calibration summaries, posterior PSD/symmetry diagnostics, information comparisons, measurement-error mode, deterministic-replay status, and the explicit uncertainty-scope flags.