Annals of Statistics

On the optimal reconstruction of partially observed functional data

Alois Kneip and Dominik Liebl

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We propose a new reconstruction operator that aims to recover the missing parts of a function given the observed parts. This new operator belongs to a new, very large class of functional operators which includes the classical regression operators as a special case. We show the optimality of our reconstruction operator and demonstrate that the usually considered regression operators generally cannot be optimal reconstruction operators. Our estimation theory allows for autocorrelated functional data and considers the practically relevant situation in which each of the $n$ functions is observed at $m_{i}$, $i=1,\dots ,n$, discretization points. We derive rates of consistency for our nonparametric estimation procedures using a double asymptotic. For data situations, as in our real data application where $m_{i}$ is considerably smaller than $n$, we show that our functional principal components based estimator can provide better rates of convergence than conventional nonparametric smoothing methods.

Article information

Ann. Statist., Volume 48, Number 3 (2020), 1692-1717.

Received: September 2018
Revised: May 2019
First available in Project Euclid: 17 July 2020

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11] 62H25: Factor analysis and principal components; correspondence analysis 62G05: Estimation 62G08: Nonparametric regression

Functional data analysis functional principal components incomplete functions


Kneip, Alois; Liebl, Dominik. On the optimal reconstruction of partially observed functional data. Ann. Statist. 48 (2020), no. 3, 1692--1717. doi:10.1214/19-AOS1864.

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Supplemental materials

  • Supplemental paper. The supplemental paper contains the proofs of our theoretical results.