简单英文维基百科条目的拓扑数据分析
Topological Data Analysis on Simple English Wikipedia Articles
摘要：
单参数持续同调是拓扑数据分析中的一个重要工具，它与量化结果意义的统计技术一起被广泛应用于数据问题。相比之下，统计技术的两个参数的持久性，而非常可取的现实世界中的应用，几乎没有被考虑。我们提出了三种统计方法比较几何数据，使用两个参数的持久同源性，这些方法依赖于希尔伯特函数，匹配距离，和条形码从两个参数的持久性模块计算的点云数据。我们的统计方法是广泛适用于分析的几何数据索引的实值参数。我们应用这些方法来分析从简单的英语维基百科文章中获得的高维点云数据。特别是，我们展示了如何利用我们的方法来区分维基百科数据的某些子集，并与随机数据进行比较。这些结果产生的见解零分布的建设和稳定性，我们的方法相对于嘈杂的数据。
英文摘要：
Single-parameter persistent homology, a key tool in topological data analysis, has been widely applied to data problems along with statistical techniques that quantify the significance of the results. In contrast, statistical techniques for two-parameter persistence, while highly desirable for real-world applications, have scarcely been considered. We present three statistical approaches for comparing geometric data using two-parameter persistent homology; these approaches rely on the Hilbert function, matching distance, and barcodes obtained from two-parameter persistence modules computed from the point-cloud data. Our statistical methods are broadly applicable for analysis of geometric data indexed by a real-valued parameter. We apply these approaches to analyze high-dimensional point-cloud data obtained from Simple English Wikipedia articles. In particular, we show how our methods can be utilized to distinguish certain subsets of the Wikipedia data and to compare with random data. These results yield insights into the construction of null distributions and stability of our methods with respect to noisy data.
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