Module pace_fpca 

PACE sparse FPCA for irregularly sampled functional data.

Implements the Yao–Müller–Wang (2005) PACE estimator via a six-step pipeline:

1. Mean: kernel-smoothed mean µ̂(t) on the work grid via mean_irreg.
2. Covariance surface: kernel-smoothed bivariate covariance Ĝ(s,t) via cov_irreg.
3. Eigendecomposition: symmetric eigendecomposition of W^{½} Ĝ W^{½} (Simpson-weighted) to obtain functional eigenvalues λ_k and orthonormal eigenfunctions φ_k.
4. BLUP scores: per-curve conditional-expectation (BLUP/PACE) scores ξ_ik = λ_k · φ_ik^T · Σ_yi^{-1} · (Y_i − µ_i), where Σ_yi = Φ_i diag(λ) Φ_i^T + σ²I_{n_i}.
5. Fitted trajectories: x̂_i(t) = µ̂(t) + Σ_k ξ_ik φ_k(t) on the work grid.
6. Confidence bands: pointwise bands from the BLUP prediction variance Ω (Yao et al. 2005, eq. 3.2).

Reference: Yao, Müller & Wang (2005), “Functional Data Analysis for Sparse Longitudinal Data”, JASA 100(470), 577–590.

Reuse-first: mean and covariance smoothing reuse crate::irreg_fdata; eigendecomposition uses nalgebra DMatrix::symmetric_eigen(); linear interpolation of eigenfunctions reuses crate::helpers::linear_interp; Cholesky solve reuses the crate-internal linalg::cholesky_solve. No new crate dependency is added.

Design note on σ²: The cov_irreg surface includes same-point pairs (j1 == j2), so its diagonal absorbs the measurement-error variance σ². Do NOT subtract σ² from the surface before eigendecomposition — σ² enters only as the ridge term σ²I in Σ_yi (step 4). This follows the standard PACE formulation (Yao et al. 2005, §2.2).

Size limits: For curves with large n_i, the n_i×n_i Σ_yi system can be expensive. Requiring σ² > 0 (strictly positive) ensures Σ_yi is positive-definite even for dense curves. Document n_i ≤ a few hundred as the expected regime for sparse functional data.

Structs

PaceFpcaConfig

Configuration for PACE sparse FPCA.

PaceFpcaResult

Result of PACE sparse FPCA.

Functions

pace_fpca

Fit PACE sparse FPCA for irregularly sampled functional data.