Persistent Homology for Analyzing Environmental Lake Monitoring Data

Abstract

Topological data analysis (TDA) is a new method for analyzing large, high-dimensional, heterogeneous, and noisy data that are characteristic of modern scientific and engineering applications. One major tool in TDA is persistent homology, wherein a filtration of a simplicial complex is generated from point clouds and subsequently analyzed for topological features. Betti numbers are computed across varying spatial resolutions, based on a proximity parameter R, where the n-th Betti number equals the rank of the n-th homology group. In this paper, persistent homology is applied to lake environmental monitoring data collected from a sonde sensor attached to a commercial cruise vessel, and to weather station observations. A modified form of the witness complex described by de Silva is used in an attempt to eliminate the need for persistence and thus to reduce computation time. From preliminary results, witness complexes are very promising in capturing the shape of the data and for detecting patterns. It is therefore proposed that TDA, combined with standard statistical techniques and interactive visualizations, enable insights into observations collected from environmental monitoring sensors.