Ridge reconstruction of partially observed functional data is asymptotically optimal

Abstract

When functional data are observed on parts of the domain, it is of interest to recover the missing parts of curves. Kraus (2015) proposed a linear reconstruction method based on ridge regularization. Kneip and Liebl (2020) argue that an assumption under which Kraus (2015) established the consistency of the ridge method is too restrictive and propose a principal component reconstruction method that they prove to be asymptotically optimal. In this note we relax the restrictive assumption that the true best linear reconstruction operator is Hilbert-Schmidt and prove that the ridge method achieves asymptotic optimality under essentially no assumptions. The result is illustrated in a simulation study.

Multiple imputation in the functional linear model with partially observed covariate and missing values in the response

“…Several recent works have begun addressing the estimation of covariance functions for short functional segments observed at sparse and irregular grid points, called functional snippets (Lin and Wang, 2020;Lin et al, 2021) or for fragmented functional data observed on small subintervals (Delaigle et al, 2020). For densely observed partial data, existing studies have focused on estimating the unobserved part of curves (Kneip and Liebl, 2020;Kraus and Stefanucci, 2020), prediction (Goldberg et al, 2014), classification (Kraus and Stefanucci, 2018;Park and Simpson, 2019), functional regression (Gellar et al, 2014), and inferences (Kraus, 2019;Park et al, 2022).…”

Functional linear model with partially observed covariate and missing values in the response

“…For the missing data mechanism of the functional covariate, we adopt the paradigm of partially observed functions as in Kneip and Liebl (2020) or Kraus (2015). We also refer the reader to Delaigle et al (2020) or Kraus and Stefanucci (2020) for recent contributions on this topic. More precisely, for each curve X i , i = 1, .…”