Typed Topological Structures Of Datasets

A datatset X on R2 is a finite topological space. Current research of a dataset focuses on statistical methods and the algebraic topological method citecarlsson. In citehu, the concept of typed topological space was introduced and showed to have the potential for studying finite topological spaces, such as a dataset. It is a new method from the general topology perspective. A typed topological space is a topological space whose open sets are assigned types. Topological concepts and methods can be redefined using open sets of certain types. In this article, we develop a special set of types and its related typed topology on a dataset X. Using it, we can investigate the inner structure of X. In particular, R2 has a natural quotient space, in which X is organized into tracks, and each track is split into components. Those components are in a order. Further, they can be represented by an integer sequence. Components crossing tracks form branches, and the relationship can be well represented by a type of pseudotree (called typed-II pseudotree). Such structures provide a platform for new algorithms for problems such as calculating convex hull, holes, clustering and anomaly detection.