Module partial 

Partial-domain monitoring for functional data.

Monitors processes where only a partial observation of the functional domain is available (e.g., real-time monitoring of an ongoing process). Supports three strategies for handling the unobserved domain:

• Conditional expectation (BLUP)
• Partial projection (scaled inner products)
• Zero padding

Mathematical framework

The BLUP (Best Linear Unbiased Predictor) conditional expectation formula is:

xi_hat = Lambda · Phi_obs^T · (Phi_obs · Lambda · Phi_obs^T + sigma^2 I)^{-1} · y_obs

where Lambda = diag(eigenvalues) is the ncomp x ncomp prior covariance of the FPC scores, Phi_obs is the eigenfunction matrix restricted to observed grid points (n_obs x ncomp), sigma^2 is estimated from the median SPE (robust to outliers), and y_obs is the centered partial observation. Under Gaussian assumptions, this is the BLUP (Yao et al., 2005, section 3, pp. 580–583, Eq. 6).

The domain fraction is computed as (argvals[n_obs-1] - argvals[0]) / (argvals[m-1] - argvals[0]), representing the proportion of the domain range covered by observed points. For a single observed point (range = 0), the point-count fraction n_obs/m is used as a fallback, consistent with the PACE framework where prediction from a single observation reduces to the marginal BLUP.

Numerical stability

The ncomp x ncomp system matrix Phi_obs · Lambda · Phi_obs^T + sigma^2 I (equivalently, Lambda^{-1} + sigma^{-2} Phi_obs^T Phi_obs via the Woodbury identity) is solved via Cholesky factorization. If the matrix is near-singular (condition number proxy diag_max/diag_min > 10^12), progressive Tikhonov regularization is applied: the diagonal loading is increased by factors of 10 until the factorization succeeds (up to 5 retries).

References

• Yao, F., Muller, H.-G. & Wang, J.-L. (2005). Functional data analysis for sparse longitudinal data. Journal of the American Statistical Association, 100(470), 577–590, section 3, pp. 580–583 (PACE framework and BLUP derivation).

Structs

PartialDomainConfig

Configuration for partial-domain monitoring.

PartialMonitorResult

Result of partial-domain monitoring for a single observation.

Enums

DomainCompletion

Strategy for handling unobserved domain.

Functions

spm_monitor_partial

Monitor a single partially-observed curve.

spm_monitor_partial_batch

Monitor a batch of partially-observed curves.