This paper presents a new way to compute simplicial and Tukey data depths using Open Multi‐Processing parallelization, which makes it practical to compute point depths for tens of thousands of points. The definition of point depth is the order statistic depth of a single point, here in two dimensions. Using the point depths, the regional depth characteristics of the dataset as a whole can be explored. Using this new methodology, fast parallel computation of both simplicial depth and Tukey depth for a dataset of n points has time complexity O(n²logn) with O(n) space, which is practical for n up to 100,000. Obtaining depths for a large number of points in a faster manner by parallel computation supports identifying the central region quickly, because the points of maximum depth are known. The point depth computation identifies the depths of selected spoke segments around each origin point. These spoke depths are used to create new visualizations of depth characteristics and contour depths without adding virtual points.

中文翻译：

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双变量点数据深度的并行计算和内在深度段的显示

本文提出了一种使用开放式多处理并行化计算简单和Tukey数据深度的新方法，这使得计算成千上万点的点深度变得实用。点深度的定义是单个点的顺序统计深度，此处为二维。使用点深度，可以探索整个数据集的区域深度特征。使用这种新方法，对于n个点的数据集，简单深度和Tukey深度的快速并行计算具有时间复杂度O（n ² log n）和O（n）空间，这对于n高达100,000。由于已知最大深度的点，因此通过并行计算以更快的方式获得大量点的深度支持快速识别中心区域。点深度计算可识别每个原点周围选定辐条段的深度。这些轮辐深度用于创建深度特征和轮廓深度的新可视化效果，而无需添加虚拟点。