点集分层和Delaunay深度
Point set stratification and Delaunay depth
摘要：
在深度函数的研究中，重要的是决定我们是否希望这样的函数对多峰敏感。在本文中，我们分析了Delaunay深度函数，这是敏感的多模态和比较这种深度与其他人，作为凸深度和位置深度。我们研究了Delaunay深度在点集（层）和整个平面（水平）中引起的分层，并且我们开发了一种用于计算Delaunay深度轮廓的算法，该算法与平面中的点集相关联，运行时间为O（n log^2 n）。查询点P的深度相对于一个数据集S在平面上是深度P的联盟S和P。当S和P是给定的输入的Delaunay深度可以计算在O（n log n），我们证明了这个值是最佳的。
英文摘要：
In the study of depth functions it is important to decide whether we want such a function to be sensitive to multimodality or not. In this paper we analyze the Delaunay depth function, which is sensitive to multimodality and compare this depth with others, as convex depth and location depth. We study the stratification that Delaunay depth induces in the point set (layers) and in the whole plane (levels), and we develop an algorithm for computing the Delaunay depth contours, associated to a point set in the plane, with running time O(n log^2 n). The depth of a query point p with respect to a data set S in the plane is the depth of p in the union of S and p. When S and p are given in the input the Delaunay depth can be computed in O(n log n), and we prove that this value is optimal.
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