On the Optimal Reconstruction of Partially Observed Functional Data

TL;DR

This paper introduces an optimal reconstruction operator for partially observed functional data, outperforming traditional regression methods, especially when data points per function are limited, with proven consistency and improved convergence rates.

Contribution

The paper presents a novel, optimal reconstruction operator within a broad class of functional operators, extending estimation theory to autocorrelated data and demonstrating superior convergence in practical scenarios.

Findings

The new operator is proven to be optimal among a large class of functional operators.

The estimation method achieves consistency with derived convergence rates.

In real data applications, the proposed estimator outperforms conventional smoothing methods.

Abstract

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.