On optimal prediction of missing functional data with memory

TL;DR

This paper develops a theoretical framework for optimally reconstructing missing parts of functions based on observed segments, especially for Gaussian processes with transformations, outperforming traditional interpolation methods.

Contribution

It provides explicit integral equations for optimal reconstruction of missing functional data and analyzes convergence rates for approximations, including cases with memory effects.

Findings

Proposed methods outperform conventional interpolation in simulations.

Explicit solutions derived for Gaussian processes with transformations.

Convergence rates established for approximation methods.

Abstract

This paper considers the problem of reconstructing missing parts of functions based on their observed segments. It provides, for Gaussian processes and arbitrary bijective transformations thereof, theoretical expressions for the $L^2$-optimal reconstruction of the missing parts. These functions are obtained as solutions of explicit integral equations. In the discrete case, approximations of the solutions provide consistent expressions of all missing values of the processes. Rates of convergence of these approximations, under extra assumptions on the transformation function, are provided. In the case of Gaussian processes with a parametric covariance structure, the estimation can be conducted separately for each function, and yields nonlinear solutions in presence of memory. Simulated examples show that the proposed reconstruction indeed fares better than the conventional interpolation methods in various situations.