In light of the increase in frequency of extreme heat events, there is a critical need to develop tools to identify geographic locations that are at risk of heat-related mortality. This paper aims to identify locations by assessing holes in cooling-center coverage using persistent homology (PH), a method from topological data analysis (TDA). Persistent homology has shown promising results in identifying holes in coverage of specific resources. We adapt these methods using a witness complex construction to study the coverage of cooling centers. We test our approach on four locations (central Boston, MA; central Austin, TX; Portland, OR; and Miami, FL) and use death times, a measurement of the size and scale of the gap in coverage, to identify most at risk regions. For comparison, we implement a standard technique for studying the risk of heat-related mortality called a heat vulnerability index (HVI). The HVI is a numerical score calculated for a geographic area based on demographic information. PH and the HVI identify different locations as vulnerable, thus indicating a potential value of assessing vulnerability from multiple perspectives. By using the regions identified by both persistent homology and the HVI, we provide a more holistic understanding of coverage.

翻译：鉴于极端高温事件频发，亟需开发工具以识别存在热相关死亡风险的地理位置。本文旨在通过使用持续同调——一种拓扑数据分析方法——评估降温中心覆盖范围的空洞来定位风险区域。持续同调在识别特定资源覆盖空洞方面已展现出良好效果。我们采用见证复形构造方法调整这些技术，以研究降温中心的覆盖情况。我们在四个地区（马萨诸塞州波士顿市中心、德克萨斯州奥斯汀市中心、俄勒冈州波特兰市及佛罗里达州迈阿密市）测试了该方法，并利用死亡时间（衡量覆盖缺口规模与尺度的指标）来识别最高风险区域。作为对比，我们实施了研究热相关死亡风险的标准方法——热脆弱性指数。该指数是基于人口统计信息为地理区域计算的数值评分。持续同调与热脆弱性指数识别出的脆弱区域存在差异，这表明从多视角评估脆弱性具有潜在价值。通过综合持续同调与热脆弱性指数共同识别的区域，我们对覆盖范围提供了更全面的理解。