The Accumulated Persistence Function, a New Useful Functional Summary Statistic for Topological Data Analysis, With a View to Brain Artery Trees and Spatial Point Process Applications
The Accumulated Persistence Function, a New Useful Functional Summary Statistic for Topological Data Analysis, With a View to Brain Artery Trees and Spatial Point Process Applications
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DOI：
10.1080/10618600.2019.1573686
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发表时间：
2019-04-29
影响因子：
2.4
通讯作者：
Moller, Jesper
中科院分区：
文献类型：
--
作者：
Biscio, Christophe A. N.;Moller, Jesper
我们首先简单介绍拓扑数据分析，其中最流行的工具称为持久性图。简单地说，持久性图是平面上的多点集，描述了当尺度参数变化时，紧集的拓扑特征的持久性。由于统计方法很难直接应用于持久性图，因此提出了各种替代的功能摘要统计，但它们要么不包含持久性图的全部信息，要么是二维函数。我们建议一个新的功能汇总统计，是一维的，因此更容易处理，并在温和的条件下包含的持久性图的全部信息。它的有用性说明在统计设置与点云和脑动脉树。其中包括额外的方法和示例，技术细节以及用于所有示例的R代码。
We start with a simple introduction to topological data analysis where the most popular tool is called a persistence diagram. Briefly, a persistence diagram is a multiset of points in the plane describing the persistence of topological features of a compact set when a scale parameter varies. Since statistical methods are difficult to apply directly on persistence diagrams, various alternative functional summary statistics have been suggested, but either they do not contain the full information of the persistence diagram or they are two-dimensional functions. We suggest a new functional summary statistic that is one-dimensional and hence easier to handle, and which under mild conditions contains the full information of the persistence diagram. Its usefulness is illustrated in statistical settings concerned with point clouds and brain artery trees. The include additional methods and examples, technical details, and the R code used for all examples.