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Sparse MFPCA comparator sensitivity

PR E characterizes the native direct sparse MFPCA estimator against two non-canonical comparator routes. This is sensitivity evidence, not an equivalence or architecture-selection exercise.

Internal two-stage marginal-basis benchmark

The benchmark fits native univariate sparse FPCA/PACE separately to x and y using declared marginal ranks. The marginal score matrices are concatenated,

\[ Z_i = [\xi^{(x)}_{i1},\ldots,\xi^{(x)}_{ir_x}, \xi^{(y)}_{i1},\ldots,\xi^{(y)}_{ir_y}], \]

their empirical covariance is diagonalized, and the resulting loading vectors combine the marginal eigenfunctions into joint vector-valued functions.

This route is useful because it asks how much of the direct joint result can be recovered after committing first to separate marginal bases. It is not the canonical estimator and is never exposed through the public API.

The deterministic sensitivity fixture fixes rho_xy=0.6 and evaluates declared marginal ranks 1 and 2. No automatic rank selection is performed. Evidence retains:

  • named covariance-block ISE against truth;
  • joint functional-subspace principal cosines against truth;
  • joint functional-subspace principal cosines against the direct estimator;
  • score-subspace principal cosines against truth;
  • score-subspace principal cosines against the direct estimator;
  • latent reconstruction ISE;
  • the fraction of curves with finite marginal-score systems.

Marginal-basis truncation can change the estimand represented by the two-stage route. A smaller covariance or reconstruction error in one finite sample is not interpreted as an architecture winner.

External mGSFPCA sensitivity

The external comparator uses CRAN mGSFPCA version 0.2.2 and mGSFPCA::spMultFPCA() on the same frozen irregular rho_xy=0.6 fixture used by the native direct fit.

The package is installed only in the validation workflow and is not a runtime scientific dependency. The frozen eyetrajectoriespy curve IDs remain unchanged; the existing comparator-boundary adapter maps them to contiguous integer IDs only for the R call and persists that mapping.

Rank and basis settings are explicit. Automatic model selection is not treated as equivalent to the package contract.

The evaluator compares three invariant relationships:

  1. native direct versus known truth;
  2. mGSFPCA versus known truth;
  3. mGSFPCA versus native direct.

For each relationship it reports functional-subspace and score-subspace principal cosines. Raw component signs and implementation-specific normalizations are not comparison targets.

Observed PR E evidence

For the internal two-stage benchmark, marginal rank 1 produced a two-dimensional joint basis with truth/direct minimum functional-subspace cosines 0.9992/0.9970 and truth/direct minimum score-subspace cosines 0.9999/0.9999, but latent reconstruction ISE was 1.3485. This is the intended truncation warning: excellent alignment of the retained subspace does not imply that the omitted temporal mode is negligible.

At marginal rank 2, the benchmark retained four joint components. Its truth/direct minimum functional-subspace cosines were 0.9983/0.9938 and truth/direct minimum score-subspace cosines were 0.9996/0.9998. Reconstruction ISE fell to 0.0226, compared with 0.0153 for the canonical direct fit on the same fixture. Total two-stage covariance ISE fell from 0.8385 at marginal rank 1 to 0.4742 at marginal rank 2. These values characterize truncation sensitivity; they are not an automatic architecture-selection rule.

For the exact mGSFPCA 0.2.2 external run, the native direct fit had minimum functional/score subspace cosines to truth of 0.9910/0.9997. mGSFPCA had 0.9386/0.9930 to truth, while mGSFPCA versus native direct was 0.9388/0.9925. The mGSFPCA and native retained eigenvalues differed materially, which is consistent with their different smoothing, basis, likelihood and normalization contracts. The evidence therefore supports broad invariant compatibility, not exact equivalence.

Interpretation boundary

The evidence explicitly records

equivalence_claim = false
architecture_winner_selected = false
automatic_rank_selection = false

because sparse smoothing, marginal truncation, likelihood, basis construction, normalization and score contracts are not identical across implementations.

PR E therefore answers:

Are the main joint functional and score subspaces broadly compatible with reasonable internal and external comparator routes under a controlled common fixture, and how sensitive is the two-stage route to marginal truncation?

It does not answer whether one implementation is universally superior.

Observation-process stress and dedicated 0.12 performance evidence remain separate downstream gates before a release-candidate decision.