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Abstract In many modern applications, discretely-observed data may be naturally understood as a set of functions. Functional data often exhibit two confounded sources of variability: amplitude (y-axis) and phase (x-axis). The extraction of amplitude and phase, a process known as registration, is essential in exploring the underlying structure of functional data in a variety of areas, from environmental monitoring to medical imaging. Critically, such data are often gathered sequentially with new functional observations arriving over time. Despite this, existing registration procedures do not sequentially update inference based on the new data, requiring model refitting. To address these challenges, we introduce a Bayesian framework for sequential registration of functional data, which updates statistical inference as new sets of functions are assimilated. This Bayesian model-based sequential learning approach utilizes sequential Monte Carlo sampling to recursively update the alignment of observed functions while accounting for associated uncertainty. Distributed computing significantly reduces computational cost relative to refitting the model using an iterative method such as Markov chain Monte Carlo on the full data. Simulation studies and comparisons reveal that the proposed approach performs well even when the target posterior distribution has a challenging structure. We apply the proposed method to three real datasets: (1) functions of annual drought intensity near Kaweah River in California, (2) annual sea surface salinity functions near Null Island, and (3) a sequence of repeated patterns in electrocardiogram signals. 

Abstract Proliferation of high‐resolution imaging data in recent years has led to substantial improvements in the two popular approaches for analyzing shapes of data objects based on landmarks and/or continuous curves. We provide an expository account of elastic shape analysis of parametric planar curves representing shapes of two‐dimensional (2D) objects by discussing its differences, and its commonalities, to the landmark‐based approach. Particular attention is accorded to the role of reparameterization of a curve, which in addition to rotation, scaling and translation, represents an important shape‐preserving transformation of a curve. The transition to the curve‐based approach moves the mathematical setting of shape analysis from finite‐dimensional non‐Euclidean spaces to infinite‐dimensional ones. We discuss some of the challenges associated with the infinite‐dimensionality of the shape space, and illustrate the use of geometry‐based methods in the computation of intrinsic statistical summaries and in the definition of statistical models on a 2D imaging dataset consisting of mouse vertebrae. We conclude with an overview of the current state‐of‐the‐art in the field. This article is categorized under: Image and Spatial Data < Data: Types and StructureComputational Mathematics < Applications of Computational Statistics 

In this work, we develop a new set of Bayesian models to perform registration of real-valued functions. A Gaussian process prior is assigned to the parameter space of time warping functions, and a Markov chain Monte Carlo (MCMC) algorithm is utilized to explore the posterior distribution. While the proposed model can be defined on the infinite-dimensional function space in theory, dimension reduction is needed in practice because one cannot store an infinite-dimensional function on the computer. Existing Bayesian models often rely on some pre-specified, fixed truncation rule to achieve dimension reduction, either by fixing the grid size or the number of basis functions used to represent a functional object. In comparison, the new models in this paper randomize the truncation rule. Benefits of the new models include the ability to make inference on the smoothness of the functional parameters, a data-informative feature of the truncation rule, and the flexibility to control the amount of shape-alteration in the registration process. For instance, using both simulated and real data, we show that when the observed functions exhibit more local features, the posterior distribution on the warping functions automatically concentrates on a larger number of basis functions. Supporting materials including code and data to perform registration and reproduce some of the results presented herein are available online.