Linn Odelius: An investigation of average stable ranks on plane geometric objects and financial transaction data

Abstract

This thesis concerns the topological features of two dimensional geometric shapes and financial transaction data. Topological properties of the data such as homology groups and their stable ranks are analysed. It is investigated how to mathematically describe differences between data sets and it is found that stable ranks can be used to capture these differences. Sub sampling is introduced as a way to apply stochastic methods to geometric structures. It is found that the average stable rank can be used to differentiate data sets. Furthermore, the sensitivity of average stable ranks to random noise is explored and it is studied how a single point changes the average stable ranks of geometric shapes and sets of financial transactions. A method to incorporate categorical data within the analysis is introduced. The theory is applied to financial transaction data with the objective to understand if there are topological differences between fraudulent and legit transactions which can be used to classify them.