Abstract

In recent years there has been noticeable interest in the study of the “shape of data.” Among the many ways a “shape” could be defined, topology is the most general one, as it describes an object in terms of its connectivity structure: connected components (topological features of dimension 0), cycles (features of dimension 1) and so on. There is a growing number of techniques, generally denoted as Topological Data Analysis, or TDA for short, aimed at estimating topological invariants of a fixed object; when we allow this object to change, however, little has been done to investigate the evolution in its topology. In this work we define the Persistence Flamelet, a multiscale version of one of the most popular tool in TDA, the Persistence Landscape. We examine its theoretical properties and we show its performance as both an exploratory and inferential tool. In addition, we provide open source implementation of the objects and methods presented in the R-package pflamelet. Supplementary materials for this article are available online.

摘要

近年来，人们对“数据形状”的研究产生了明显的兴趣。在定义“形状”的多种方式中，拓扑是最通用的一种，因为它根据对象的连通性结构来描述对象：连通分量（0维拓扑特征）、环路（1维特征）等在。越来越多的技术通常被称为拓扑数据分析（Topological Data Analysis），简称 TDA，旨在估计固定对象的拓扑不变量；然而，当我们允许这个对象发生变化时，我们几乎没有采取任何措施来研究其拓扑结构的演变。在这项工作中，我们定义了Persistence Flamelet，TDA 中最流行的工具之一持久性景观的多尺度版本。我们研究了它的理论特性，并展示了它作为探索性和推理工具的性能。此外，我们还提供 R 包 pflamelet 中提供的对象和方法的开源实现。本文的补充材料可在线获取。