Robust Inference for Partially Observed Functional Response Data

Abstract

Irregular functional data in which densely sampled curves are observed over different ranges pose a challenge for modeling and inference, and sensitivity to outlier curves is a concern in applications. Motivated by applications in quantitative ultrasound signal analysis, this paper investigates a class of robust M-estimators for partially observed functional data including functional location and quantile estimators. Consistency of the estimators is established under general conditions on the partial observation process. Under smoothness conditions on the class of M-estimators, asymptotic Gaussian process approximations are established and used for large sample inference. The large sample approximations justify a bootstrap approximation for robust inferences about the functional response process.The performance is demonstrated in simulations and in the analysis of irregular functional data from quantitative ultrasound analysis.

A data-adaptive dimension reduction for functional data via penalized low-rank approximation

“…Instead, we can calculate the n × L proxy matrix, denoted as Φ Y , by computing basis coefficients for ψ l (t) on the expansion of demeaned Y i (t), i.e., I {Y i (t) − μ(t)}ψ l (t)dt, where μ(t) = n −1 n i=1 Y i (t). We note that nonparametric methods, such as local or spline smoothing regression models, or robust mean estimates under functional M-estimator (Park et al, 2022), can be adopted in estimating µ(t). When considering functional data collected over individual-specific domains, Y i (t), for t ∈ I i ⊂ I, the basis coefficients are calculated by Ii {Y i (t) − μ(t)}ψ l (t)dt, for t ∈ I i .…”

“…As the first step, we estimate the mean function by calculating the Hubertype location estimator Park et al (2022), a class of functional M-estimator applicable to the data containing atypically behaved trajectories with missing segments. We then implement the proposed algorithm to demeaned trajectories by employing 51 Fourier basis functions for orthonormal basis functions ψ l (t) with q = 10.…”

“…Our proposed approach has several advantages. First, it is applicable to data with random missing segments, often referred to as partially observed functional data (Kraus, 2015;Park et al, 2022), where each trajectory includes individual-specific missing segments while relatively rich trajectory information is available over observed segments. Our projection-based approach, paired with the matrix recovery algorithm, enables handling missing information and outlying behaviors simultaneously in the recovery process.…”

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).…”