Flat Embeddings of Genetic and Distance Data

Abstract

The idea of displaying data in the plane is very attractive in many different fields of research. This thesis will focus on distance-based phylogenetics and multidimensional scaling (MDS). Both types of method can be viewed as a high-dimensional data reduction to pairwise distances and visualization of the data based on these distances. The difference between phylogenetics and multidimensional scaling is that the first one aims at finding a network or a tree structure that fits the distances, whereas MDS does not fix any structure and objects are simply placed in a low-dimensional space so that distances in the solution fit distances in the input as good as possible. Chapter 1 provides an introduction to the phylogenetics and multidimensional scaling. Chapter 2 focuses on the theoretical background of flat split systems (planar split networks). We prove equivalences between flat split systems, planar split networks and loop-free acyclic oriented matroids of rank three. The latter is a convenient mathematical structure that we used to design the algorithm for computing planar split networks that is described in Chapter 3. We base our approach on the well established agglomerative algorithms Neighbor-Joining and Neighbor-Net. In Chapter 4 we introduce multidimensional scaling and propose a new method for computing MDS plots that is based on the agglomerative approach and spring embeddings. Chapter 5 presents several case studies that we use to compare both of our methods and some classical agglomerative approaches in the distance-based phylogenetics.