Functional principal components (FPC) analysis is widely used to decompose and express functional observations. Curve estimates implicitly condition on basis functions and other quantities derived from FPC decompositions; however these objects are unknown in practice. In this paper, we propose a method for obtaining correct curve estimates by accounting for uncertainty in FPC decompositions. Additionally, pointwise and simultaneous confidence intervals that account for both model- based and decomposition-based variability are constructed. Standard mixed-model representations of functional expansions are used to construct curve estimates and variances conditional on a specific decomposition. A bootstrap procedure is implemented to understand the uncertainty in principal component decomposition quantities. Iterated expectation and variance formulas combine both sources of uncertainty by combining model-based conditional estimates across the distribution of decompositions. Our method compares favorably to competing approaches in simulation studies that include both densely- and sparsely-observed functions. We apply our method to sparse observations of CD4 cell counts and to dense white-matter tract profiles. Code for the analyses and simulations is publicly available, and our method is implemented as the IVfpca() function in the R package refund on CRAN
