标题： 带有周期性边界条件的领域中的DBSCAN 显示英文标题

标题： DBSCAN in domains with periodic boundary conditions

摘要： 许多科学问题涉及嵌入在具有周期性边界条件的空间中的数据。这可能与数据中的固有循环或旋转对称性或空间上扩展的周期性有关。在分析此类数据时，需要定制的方法来获得符合问题周期性边界条件的高效方法。在本工作中，我们提出了一种方法，基于DBSCAN算法（一种广泛使用的无监督机器学习方法，用于识别数据中的聚类），将聚类算法应用于嵌入在周期性域中的数据。所提出的方法内部利用了适用于开放边界域的传统DBSCAN算法，从而保持与开放域中所有优化的邻域搜索实现的兼容性。这样，它保留了相同的优化运行时间复杂度$O(N\log N)$。我们使用一维、二维和三维的合成数据展示了所提出方法的工作原理，并将其应用于一个现实世界中的例子，即湍流中气泡的聚类。所提出的方法已实现为一个可以直接使用的Python包，我们将其公开提供。 显示英文摘要

摘要： Many scientific problems involve data that is embedded in a space with periodic boundary conditions. This can for instance be related to an inherent cyclic or rotational symmetry in the data or a spatially extended periodicity. When analyzing such data, well-tailored methods are needed to obtain efficient approaches that obey the periodic boundary conditions of the problem. In this work, we present a method for applying a clustering algorithm to data embedded in a periodic domain based on the DBSCAN algorithm, a widely used unsupervised machine learning method that identifies clusters in data. The proposed method internally leverages the conventional DBSCAN algorithm for domains with open boundaries, such that it remains compatible with all optimized implementations for neighborhood searches in open domains. In this way, it retains the same optimized runtime complexity of $O(N\log N)$. We demonstrate the workings of the proposed method using synthetic data in one, two and three dimensions and also apply it to a real-world example involving the clustering of bubbles in a turbulent flow. The proposed approach is implemented in a ready-to-use Python package that we make publicly available.