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Native sparse multilevel FPCA qualification

This page documents deterministic known-truth qualification for the 1.1 development capability fit_sparse_multilevel_fpca().

Qualified target

The estimator targets the exchangeable two-level sparse functional model

\[ Y_{ij}(t)=\mu(t)+U_i(t)+V_{ij}(t)+\epsilon_{ij}(t), \]

with participant-level covariance \(K_B\), trial-within-participant covariance \(K_W\), and independent measurement error.

The implementation estimates:

  1. the grand mean from pooled retained native observations;
  2. latent total covariance from within-trial off-diagonal residual products;
  3. between-participant covariance from distinct-trial residual products within the same participant;
  4. within-participant covariance as smoothed total minus smoothed between covariance before level-specific PSD handling;
  5. participant and trial scores jointly by Gaussian BLUP using the full repaired covariance surfaces.

Raw sparse trajectories are never interpolated onto a common grid.

Methodological reference

The hierarchical covariance structure follows the sparse multilevel functional PCA framework of Di, Crainiceanu, and Jank (2014), Multilevel sparse functional principal component analysis, Statistics 3(1), 126–143, DOI 10.1002/sta4.50.

The current package tranche is narrower: it is univariate, two-level, exchangeable across repeated trials after any required fixed condition effects are handled upstream, and uses observation/raw-pair weighting only.

Known truth

The qualification simulator uses one known participant mode and one known trial mode:

\[ \phi_B(t)=\sqrt{2}\sin(\pi t), \qquad \lambda_B=1, \]

and

\[ \phi_W(t)=\sqrt{2}\sin(2\pi t), \qquad \lambda_W=0.45. \]

Each participant receives an independent participant score. Each retained trial receives an independent within-participant score. Native observation times are irregular and differ across trials.

Qualified scenarios

Balanced denser / low noise

  • 48 participants;
  • 3 trials per participant, 144 trials total;
  • 11–15 native observations per trial;
  • measurement-noise SD 0.08.

Observed qualification metrics:

  • between covariance relative RMSE: 0.2815;
  • within covariance relative RMSE: 0.3389;
  • between eigenfunction similarity: 0.9963;
  • within eigenfunction similarity: 0.9736;
  • participant-score absolute correlation: 0.9993;
  • trial-score absolute correlation: 0.9956;
  • failed participant BLUP systems: 0.

The between operator required a PSD projection with relative weighted-operator correction 0.0369; the within operator correction was 0.0287.

Unequal sparse / higher noise

  • 54 participants;
  • 122 trials total;
  • 7–10 native observations per trial;
  • measurement-noise SD 0.16;
  • 9 participants had a single retained trial;
  • 45 participants contributed repeated trials to between-covariance estimation.

Observed qualification metrics:

  • between covariance relative RMSE: 0.5553;
  • within covariance relative RMSE: 0.4886;
  • between eigenfunction similarity: 0.9844;
  • within eigenfunction similarity: 0.9709;
  • participant-score absolute correlation: 0.9712;
  • trial-score absolute correlation: 0.9602;
  • failed participant BLUP systems: 0.

The between operator required a PSD projection with relative weighted-operator correction 0.0683; the within operator correction was 0.0106.

The nine single-trial participants remained in the analysis and could be scored after the population hierarchy had been identified by repeated participants. Their between-covariance pair counts remain zero by construction.

Explicit non-identifiability check

A third deterministic dataset retains six participants but only two with repeated trials. The fitter is required to stop before smoothing between covariance. The qualification observed the declared failure:

insufficient_repeated_participants

This check prevents an underidentified hierarchy from being silently converted to a single-level model or from manufacturing between-participant covariance from cross-participant products.

Score-system contract

Qualification records confirm:

  • rank_k_covariance_used_for_scoring=False;
  • the participant observation covariance uses the full repaired between + within covariance surfaces plus declared measurement error;
  • participant scores borrow information across that participant's repeated trials;
  • each trial score's cross-covariance is confined to that trial's observation block;
  • deterministic replay reproduces the same scenario summary exactly;
  • raw_sparse_trajectory_interpolation_performed=False;
  • weighting="observation";
  • trial_fixed_effects_estimated=False.

The BLUP calculations are conditional on fitted population quantities. The qualification does not imply that population-estimation or bandwidth uncertainty has been propagated into score uncertainty.

Evidence identity

First completed B1 qualification on implementation head b654dcb184205097182248b0343b94767ba9dc44:

  • workflow run: 37295044677;
  • artifact: 11338301279;
  • artifact digest: sha256:59a0cbe5119ea9239dfc8071a081aca7f3cea35feba2d747ab4256b0f0d4f774.

A later documentation-only or corrective head must rerun this workflow before merge; the final PR evidence should point to the final qualified head.

Interpretation boundary

This evidence supports the implementation claim for the declared deterministic sparse two-level designs. It does not establish:

  • universal recovery across all sparse sampling mechanisms;
  • validity under unmodeled task/visit fixed effects;
  • robustness to informative observation processes beyond separately diagnosed missingness assumptions;
  • equal-participant weighting;
  • full inferential coverage after estimating covariance surfaces/eigenfunctions;
  • multivariate or asynchronous sparse multilevel FPCA.

Those require separate estimands and qualification programmes.