Bayesian Framework for Simultaneous Registration and Estimation of Noisy, Sparse and Fragmented Functional Data

Abstract

In many applications, smooth processes generate data that is recorded under a variety of observation regimes, such as dense, sparse or fragmented observations that are often contaminated with error. The statistical goal of registering and estimating the individual underlying functions from discrete observations has thus far been mainly approached sequentially without formal uncertainty propagation, or in an applicationspecific manner. We propose a unified Bayesian framework for simultaneous registration and estimation, which is flexible enough to accommodate inference on individual functions under general observation regimes. Our ability to do this relies on the specification of strongly informative prior models over the amplitude component of function variability. We provide two strategies for this critical choice: a data-driven approach that defines an empirical basis for the amplitude subspace based on training data, and a shape-restricted approach when the relative location and number of local extrema is well-understood. The proposed methods build on elastic functional data analysis, which separately models amplitude and phase variability inherent in functional data. We emphasize the importance of uncertainty quantification and visualization of these two components as they provide complementary information about the estimated functions. We validate the framework using simulations and real applications to medical imaging and biometrics.

Registration for Incomplete Non-Gaussian Functional Data

“…Cheng et al (2016), Kurtek (2017) and Lu et al (2017) adapted the SRVF approach to perform registration in a Bayesian setting. Bayesian approaches were also introduced for data settings with stronger noise under informative priors (Matuk et al, 2019;Tucker et al, 2021). These Bayesian approaches can provide a full representation of the joint phase and amplitude uncertainty, but are computationally very demanding.…”

“…These Bayesian approaches can provide a full representation of the joint phase and amplitude uncertainty, but are computationally very demanding. The method of Matuk et al (2019) handles sparse and fragmented Gaussian functional data where measurements are only available over some parts of the observation domain, but no software implementation was publicly available at the time of writing.…”

“…The observed curve domains are constrained so that they dilate or extend the domain by a factor 0.9 − 1.1 to ensure reasonable warpings. As noted above, Matuk et al (2019) allows to analyze fragmented functional data, but their approach is feasible only for small to intermediate sets of Gaussian data. Bryner and Srivastava (2021) introduced an approach for elastic partial matching to tackle trailing incompleteness.…”

“…Especially the registration of incomplete curves, i.e., curves whose underlying process is not observed from its natural starting point all the way to its natural endpoint, has received only limited attention so far (see e.g. Matuk et al, 2019;Bryner and Srivastava, 2021). However, such data arise in many fields.…”