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.

在诸多应用场景中，平滑过程会生成数据，这类数据会通过多种观测模式（observational regimes）被记录，涵盖密集采样、稀疏或碎片化观测，且此类观测往往带有误差污染。目前，从离散观测中配准（registration）并估计个体潜在函数的统计目标，主要通过两种路径实现：一是采用顺序方法，但未开展严格的不确定性传播（uncertainty propagation）；二是采用特定应用场景下的方式，通过跨个体整合信息完成该目标。我们提出了一种用于同步配准与估计的统一贝叶斯框架（Bayesian framework），该框架具备足够灵活性，可适配一般观测模式下的个体函数推断任务。我们得以实现该目标，依赖于针对函数变异性的振幅分量（amplitude component）构建强信息先验模型，具体采用两种策略：一是数据驱动方法，基于训练数据为振幅子空间（amplitude subspace）定义经验基；二是形状约束方法，适用于极值点相对位置与数量已知的场景。所提方法以弹性函数数据分析（Elastic Functional Data Analysis）框架为基础，对函数数据中固有的振幅变异性与相位变异性（phase variability）分别进行建模。我们着重强调不确定性量化（uncertainty quantification）以及这两个分量的可视化的重要性，因为二者可提供关于估计函数的互补信息。我们通过多项模拟研究与实际应用，对所提框架进行了验证。