The modeling and uncertainty quantification of closed curves is an important problem in the field of shape analysis, and can have significant ramifications for subsequent statistical tasks. Many of these tasks involve collections of closed curves, which often exhibit structural similarities at multiple levels. Modeling multiple closed curves in a way that efficiently incorporates such between-curve dependence remains a challenging problem. In this work, we propose and investigate a multiple-output (a.k.a. multi-output), multi-dimensional Gaussian process modeling framework. We illustrate the proposed methodological advances, and demonstrate the utility of meaningful uncertainty quantification, on several curve and shape-related tasks. This model-based approach not only addresses the problem of inference on closed curves (and their shapes) with kernel constructions, but also opens doors to nonparametric modeling of multi-level dependence for functional objects in general.
翻译:封闭曲线的建模和不确定性量化是形状分析领域的一个重要问题,可能对随后的统计任务产生重大影响,其中许多任务涉及收集封闭曲线,这些曲线往往在多个层次上显示出结构上的相似性。建模多个封闭曲线,有效地纳入这种曲线之间的依赖性,这仍然是一个具有挑战性的问题。在这项工作中,我们提议并调查一个多产出(a.k.a.多产出),多维高斯进程建模框架。我们介绍了拟议的方法进展,并展示了在若干曲线和与形状有关的任务中有意义的不确定性量化的效用。这种建模方法不仅解决了以内核构造的封闭曲线(及其形状)的推论问题,而且还打开了一般功能物体多层次依赖性非参数建模的大门。