Evaluating Cooling Center Coverage Using Persistent Homology of a Filtered Witness Complex

TL;DR

This study addresses the problem of identifying geographic areas at risk from heat-related mortality by examining holes in cooling-center coverage through persistent homology (PH). It introduces a filtered witness complex built from census-block landmarks and cooling-center witnesses, producing persistence diagrams that reveal coverage gaps across multiple spatial scales, and compares this topological view to a standard heat vulnerability index (HVI). The results across four cities show that PH and HVI identify different vulnerable regions, suggesting that combining both perspectives yields a more comprehensive vulnerability assessment and more informed planning of cooling-center distributions. The proposed approach is adaptable to other resources and spatial scales, highlighting the value of topological data analysis in urban heat risk assessment.

Abstract

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.

Figures (19)

• Figure 1: An example of 11 census tract centroids (landmarks) and 7 cooling centers (witnesses). Note: this image was created based on a small subset of Boston, MA census tracts and the witness locations were generated randomly. To see the full city of Boston with the true locations of cooling centers, see Figure \ref{['data']}.
• Figure 2: Example of select steps in the filtration. As shown in Figure \ref{['RandomExample']}, light green points represent the centroids (landmarks) of the geographic region and dark green diamonds (witnesses) represent the cooling center locations. All vertices (landmarks) appear in (a). (b)-(d) show the simplicial complex (in black) at various stages of the filtration. Edges are drawn between vertices when they are within a distance $\alpha$ of the same witness. Green discs show the radii at that filtration value. Note that the witnesses are drawn for easier visualization but they are not vertices in the simplicial complex.
• Figure 3: Heat vulnerability index (HVI) score maps for the four regions of interest.
• Figure 4: Persistence diagram for our four geographic locations. The birth and death times are in units of kilometers. Note that the axes are not on the same scale.
• Figure 5: Box plots for each of the four geographic locations of interest of (a) death times of the 0D, (b) death times of 1D homology classes, and (c) the lifetimes of the 1D homology classes. A tables of the corresponding means and standard deviations can be found in Table \ref{['tab:deathstats']}.

Theorems & Definitions (1)

• Definition 3.1