We develop a method for analyzing spatiotemporal anomalies in geospatial data using topological data analysis (TDA). To do this, we use persistent homology (PH), a tool from TDA that allows one to algorithmically detect geometric voids in a data set and quantify the persistence of these voids. We construct an efficient filtered simplicial complex (FSC) such that the voids in our FSC are in one-to-one correspondence with the anomalies. Our approach goes beyond simply identifying anomalies; it also encodes information about the relationships between anomalies. We use vineyards, which one can interpret as time-varying persistence diagrams (an approach for visualizing PH), to track how the locations of the anomalies change over time. We conduct two case studies using spatially heterogeneous COVID-19 data. First, we examine vaccination rates in New York City by zip code. Second, we study a year-long data set of COVID-19 case rates in neighborhoods in the city of Los Angeles.

中文翻译：

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使用持久同源性分析时空异常：使用 COVID-19 数据进行案例研究

我们开发了一种使用拓扑数据分析 (TDA) 分析地理空间数据中时空异常的方法。为此，我们使用了持久同源性 (PH)，这是一种来自 TDA 的工具，它允许通过算法检测数据集中的几何空隙并量化这些空隙的持久性。我们构建了一个有效的过滤单纯复形（FSC），使得我们 FSC 中的空隙与异常一一对应。我们的方法不仅仅是识别异常；它还对有关异常之间关系的信息进行编码。我们使用葡萄园，可以将其解释为随时间变化的持久性图（一种可视化 PH 的方法），来跟踪异常位置如何随时间变化。我们使用空间异构的 COVID-19 数据进行了两个案例研究。第一的，我们按邮政编码检查纽约市的疫苗接种率。其次，我们研究了洛杉矶市社区中为期一年的 COVID-19 病例率数据集。