Corrected confidence bands for functional data using principal components.

Area of Science:

• Statistics
• Functional Data Analysis

Background:

• Functional principal components (FPC) analysis is a common technique for decomposing functional observations.
• Estimating functional curves often relies on basis functions and FPC decompositions, but these components introduce uncertainty as they are unknown in practice.

Purpose of the Study:

• To propose a novel method for obtaining correct functional curve estimates by accounting for uncertainty in FPC decompositions.
• To construct pointwise and simultaneous confidence intervals that incorporate both model-based and decomposition-based variability.

Main Methods:

• Utilizing standard mixed model representations for functional expansions to derive curve estimates and variances conditional on a specific decomposition.
• Employing iterated expectation and variance formulas to integrate conditional estimates across the distribution of decompositions.
• Implementing a bootstrap procedure to quantify uncertainty in principal component decomposition quantities.

Main Results:

• The proposed method demonstrates favorable comparisons against existing approaches in simulation studies involving both dense and sparse functional data.
• The method was successfully applied to real-world datasets, including sparse CD4 cell counts and dense white-matter tract profiles.

Conclusions:

• The developed method provides a robust way to handle uncertainty in FPC decompositions for improved functional curve estimation.
• The availability of public code and an R package facilitates the application of this method in statistical research and practice.