Module stats 

Hotelling T-squared and SPE (Squared Prediction Error) statistics.

These are the two fundamental monitoring statistics for functional statistical process control: T-squared captures systematic variation in the principal component subspace, while SPE captures residual variation outside it.

Finite-sample distribution

For finite calibration samples of size n, T² follows (n·ncomp/(n−ncomp))·F(ncomp, n−ncomp) exactly under Gaussian scores. The χ²(ncomp) limit is the large-sample limit as n → ∞.

Numerical stability

If the ratio max(eigenvalue)/min(eigenvalue) exceeds ~10^6, the T² statistic becomes numerically sensitive. Use hotelling_t2_regularized in such cases.

Quadrature

The Simpson’s rule quadrature used for SPE integration has error O(h^4 * max|f’‘’’|) where h is the grid spacing, giving excellent accuracy for smooth functional data.

References

• Hotelling, H. (1947). Multivariate quality control. Techniques of Statistical Analysis, pp. 111–113.
• Bersimis, S., Psarakis, S. & Panaretos, J. (2007). Multivariate statistical process control charts: an overview. Quality and Reliability Engineering International, 23(5), 517–543. §2.1, pp. 519–522.

Assumptions

The statistics in this module assume the FPCA scores are approximately uncorrelated (diagonal covariance). This holds by construction when eigenvalues come from PCA/SVD. For non-PCA score vectors, use the full covariance Mahalanobis distance instead.

Functions

hotelling_t2

Compute Hotelling T-squared statistic for each observation.

hotelling_t2_regularized

Compute Hotelling T-squared with eigenvalue regularization.

spe_multivariate

Compute SPE for multivariate functional data.

spe_univariate

Compute SPE (Squared Prediction Error) for univariate functional data.