The Intersection of Statistics and Computational Topology

Data is ubiquitous in today’s society: we track our locations with GPS; we record the number of steps each day using pedometers; we digitally record daily life with pictures and videos; etc. To interpret and to understand the data requires both technical theory to find important features of the data and application expertise to interpret those features. My research is primarily in the technical theory used to find important features of data. In particular, my research is an interdisciplinary approach to topological data analysis (TDA). TDA refers to a collection of methods for extracting topological structure from data. My approach to TDA employs statistics and is application-oriented. In this statement, I describe two current research projects illustrating both the broad spectrum of possible applications for my research as well as a specific application. In both projects, I have made novel contributions and published in premiere peer-reviewed venues. As a faculty member, I plan to continue both developing the theoretical foundations and employing the theory in practical applications. In the first project, I describe a statistical approach to persistent homology. We can summarize data in various ways. One way is to use persistent homology, a description of the (in nontechnical words) components, tunnels, and voids of the topological space represented by the data. However, persistent homology is cumbersome to use directly in order to compare data sets or to recognize meaningful features. I turn to statistics to overcome this problem. In the second project, I work on evaluating road network reconstructions from GPS trajectory data based on both the geometry and the topology of the embedding. The datasets studied in this application are the maps created from trajectory data. I have helped develop the theory to measure the difference between two datasets, and have applied this to various real and hypothetical datasets.