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 observational regimes, including dense sampling and 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 application-specific manner by pooling information across subjects. We propose a unified Bayesian framework for simultaneous registration and estimation, which is flexible enough to accommodate inference on individual functions under general observational regimes. Our ability to do this relies on the specification of strongly informative prior models over the amplitude component of function variability using two strategies: 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 extrema is well-understood. The proposed methods build on the elastic functional data analysis framework to separately model 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 proposed framework using multiple simulation studies and real applications.

Shape-Based Classification of Partially Observed Curves, With Applications to Anthropology

“…Knowledge about information content in an observed partial curve for classification can be obtained either from a training dataset consisting of fully observed curves with class labels or from a subject matter expert. In such cases, a Bayesian classification model with a judicious choice of prior on class-specific templates can be developed; such an approach will extend the one recently proposed for univariate functional data [23] to the curve setting, and constitutes ongoing work.…”

“…An attractive model-based approach is to not just estimate the missing piece of the partially observed curve, but instead estimate an entire template that has a portion that is very similar in shape to the partially observed curve. Such an approach has recently been used for traditional univariate functional data under a Bayesian formulation [23] and appears promising.…”

“…They formulate a Bayesian classification model that also heavily relies on prior shape information. Finally, there is some recent work on missing data techniques for functional data analysis [21,22,23].…”

Sequential Bayesian Registration for Functional Data

“…Finally, Cheng et al, and Bharath and Kurtek used the Dirichlet distribution as a prior model on consecutive increments of discretized phase functions [28,29]. An extension of Bayesian registration to sparse or fragmented functional data was recently developed by Matuk et al [30]. Importantly, all of the aforementioned model-based approaches rely on batch learning, and in particular MCMC, for inference.…”

“…For comparison, we implement MCMC-based batch learning given the full dataset f 1:100 , which includes both Gibbs steps and adaptive Metropolis-Hastings updates [26,30]. We use a total of 50, 000 MCMC iterations, with a burn-in period of 40, 000.…”