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

In many applications, smooth processes generate data that are 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.

在众多应用场景中，平滑的过程产生的数据通常在多种观测机制下进行记录，包括密集采样以及稀疏或破碎的观测，而这些观测往往受到误差的污染。因此，从离散观测中注册和估计个体潜在函数的统计目标，至今主要采取顺序进行，缺乏正式的不确定性传播，或者以特定应用的方式通过汇总不同观测对象的信息来实现。我们提出了一种统一的贝叶斯框架，用于同时进行注册和估计，该框架足够灵活，能够适应在不同观测机制下对单个函数的推理。我们实现这一能力依赖于对函数变异性振幅成分的强烈信息先验模型的指定，采用两种策略：一种是以训练数据为基础，定义振幅子空间的经验基础的基于数据的方法；另一种是在相对位置和极值数量被充分理解的情况下，采用形状限制的方法。所提出的方法建立在弹性功能数据分析框架之上，以分别对功能数据中的振幅和相位变异性进行建模。我们强调不确定性量化以及这两个组成部分的可视化的重要性，因为它们为估计函数提供了互补信息。我们通过多个模拟研究和实际应用验证了所提出的框架。