Physical Sciences and Mathematics Commons^™

Data Analysis Methods Using Persistence Diagrams, Andrew Marchese

Doctoral Dissertations

In recent years, persistent homology techniques have been used to study data and dynamical systems. Using these techniques, information about the shape and geometry of the data and systems leads to important information regarding the periodicity, bistability, and chaos of the underlying systems. In this thesis, we study all aspects of the application of persistent homology to data analysis. In particular, we introduce a new distance on the space of persistence diagrams, and show that it is useful in detecting changes in geometry and topology, which is essential for the supervised learning problem. Moreover, we introduce a clustering framework directly …

Dependence Structures In Lévy-Type Markov Processes, Eddie Brendan Tu

Doctoral Dissertations

In this dissertation, we examine the positive and negative dependence of infinitely divisible distributions and Lévy-type Markov processes. Examples of infinitely divisible distributions include Poissonian distributions like compound Poisson and α-stable distributions. Examples of Lévy-type Markov processes include Lévy processes and Feller processes, which include a class of jump-diffusions, certain stochastic differential equations with Lévy noise, and subordinated Markov processes. Other examples of Lévy-type Markov processes are time-inhomogeneous Feller evolution systems (FES), which include additive processes. We will provide a tour of various forms of positive dependence, which include association, positive supermodular association (PSA), positive supermodular dependence (PSD), and positive …

Statistical Computational Topology And Geometry For Understanding Data, Joshua Lee Mike

Doctoral Dissertations

Here we describe three projects involving data analysis which focus on engaging statistics with the geometry and/or topology of the data.

The first project involves the development and implementation of kernel density estimation for persistence diagrams. These kernel densities consider neighborhoods for every feature in the center diagram and gives to each feature an independent, orthogonal direction. The creation of kernel densities in this realm yields a (previously unavailable) full characterization of the (random) geometry of a dataspace or data distribution.

In the second project, cohomology is used to guide a search for kidney exchange cycles within a kidney paired …