Elena Farahbakhsh Touli: Graphical Models: Mathematical Foundation and Statistical Analysis

Abstract.

This thesis primarily focuses on addressing various problems in graph theory, particularly when combined with statistical methods.

In Paper I, we explored the concept of distance between trees and proposed a new definition for the interleaving distance, originally used to measure the distance between merge trees. Our new definition relies on mapping from one tree to another, enabling us to develop fixed-parameter tractable algorithms for determining the interleaving distance between merge trees under certain conditions.

In Paper II, we investigated the clustering coefficient in networks, which measures the tendency of network vertices to form triangles. We introduced a new measure, termed the Relative Clustering Coefficient, and highlighted its significance.

In Paper III, we analyzed the financial relationships between companies in Sweden using two methods: the Pearson Correlation Coefficient (PCC) and Generalized Variance Decomposition (GVD). These methods were applied to financial data comprising the daily returns of 28 stocks included in the OMX index (the index of the Swedish capital market).

Paper IV centers on the Gaussian Graphical Model. We examined three types of precision matrices and developed exact test theories corresponding to each type. Finally, we compared the new approaches to a benchmark method through an extensive simulation study.