Boundary Detection Using a Bayesian Hierarchical Model for Multiscale Spatial Data

Spatial boundary analysis has attained considerable attention in several disciplines including engineering, shape analysis, spatial statistics, and computer science. The inferential question of interest is often to identify rapid surface change of an unobserved latent process. Curvilinear wombling and crisp wombling (or fuzzy) are two major approaches that have emerged in Bayesian spatial statistics literature. These methods are limited to a single spatial scale even though data with multiple spatial scales are often accessible. Thus, we propose a multiscale representation of the directional derivative Karhunen–Loéve expansion to perform directionally based boundary detection. Taking a multiscale spatial perspective allows us, for the first time, to consider the concept of curvilinear boundary fallacy (CBF) error, which is a boundary detection analog to the ecological fallacy that is often studied in spatial change of support literature. Furthermore, we propose a directionally based multiscale curvilinear boundary error criterion to quantify CBF. We refer to this metric as the criterion for boundary aggregation error (BAGE), and use it to perform boundary detection. Several theoretical results are derived to motivate BAGE. In particular, we show that no BAGE exists when the directional derivatives of eigenfunctions of a KL expansion are constant across spatial scales. We illustrate the use of our model through a simulated example and an analysis of Mediterranean wind measurements data. Supplementary materials for this article are available online.

空间边界分析在工程学、形状分析、空间统计学与计算机科学等多个学科中受到广泛关注。研究人员所关注的推断问题通常为识别未观测潜过程的快速表面变化。曲线型Wombling与清晰型Wombling（或称模糊型Wombling）是贝叶斯空间统计学研究领域中涌现出的两类主流方法。尽管实际中常可获取多空间尺度的数据，但现有此类方法仅局限于单一空间尺度。据此，本文提出基于方向导数的卡尔曼-洛埃（Karhunen–Loéve）展开多尺度表示方法，用于开展基于方向的边界检测。借助多尺度空间视角，我们首次得以探讨曲线边界谬误（CBF）的概念：该谬误是边界检测领域对应于生态谬误的问题，而生态谬误常被纳入空间支撑变化的相关研究中。此外，本文提出基于方向的多尺度曲线边界误差准则，用于量化CBF。我们将该指标命名为边界聚合误差准则（BAGE），并将其用于边界检测任务。本文推导了若干理论结果以支撑BAGE方法的合理性，具体而言，我们证明了当卡尔曼-洛埃展开的特征函数的方向导数在各空间尺度下保持恒定时，不存在边界聚合误差。我们通过模拟示例与地中海风速实测数据分析，演示了所提模型的应用。本文的补充材料可在线获取。