The Horizontal Tunnelability Graph is Dual to Level Set Trees

Abstract

Time series data, reflecting phenomena like climate patterns and stock prices, offer key insights for prediction and trend analysis. Contemporary research has independently developed disparate geometric approaches to time series analysis. These include tree methods, visibility algorithms, as well as persistence-based barcodes common to topological data analysis. This thesis enhances time series analysis by innovatively combining these perspectives through our concept of horizontal tunnelability. We prove that the level set tree gotten from its Harris Path (a time series), is dual to the time series' horizontal tunnelability graph, itself a subgraph of the more common horizontal visibility graph. This technique extends previous work by relating Merge, Chiral Merge, and Level Set Trees together along with visibility and persistence methodologies. Our method promises significant computational advantages and illuminates the tying threads between previously unconnected work. To facilitate its implementation, we provide accompanying empirical code and discuss its advantages