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An investigation of average stable ranks: On plane geometric objects and financial transaction data
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2020 (English)Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesisAlternative title
En undersökning av den genomsnittliga stabila rangen hos plana geometriska figurer och finansiella transaktioner (Swedish)
Abstract [en]

This thesis concerns the topological features of plane 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 financial transaction data. 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.

Abstract [en]

I denna uppsats analyseras finansiell transaktionsdata samt plana geometriska objekt med hjälp av verktyg inom Topologisk Dataanalys. Topologiska egenskaper såsom homologi samt stabil rang analyseras och det undersöks hur en matematiskt kan beskriva skillnaden mellan geometriska objekt. Det visar sig att simplistiska komplex och dess motsvarande stabila rang kan användas för att beskriva dessa skillnader. Det undersöks även hur stokastiska metoder kan appliceras på geometrisk data och begreppet genomsnittlig stabil rang introduceras. Känsligheten för brus hos den genomsnittliga stabila rangen undersöks för plana objekt och det undersöks hur den genomsnittliga stabila rangen av en datamängd ändras om en datapunkt läggs till. En metod för att beskriva avstånd på kategorisk data introduceras eftersom analysen av stabil rang kräver ett definierat avstånd mellan datapunkter. Det undersöks huruvida det finns topologiska skillnader mellan bedrägliga och icke-bedrägliga transaktioner, samt om det finns skillnader mellan olika typer av bedrägliga transaktioner.

Place, publisher, year, edition, pages
2020.
Series
TRITA-SCI-GRU ; 2020:012
Keywords [en]
Topological data analysis, TDA, Mathematics, Stable rank, Topology, Data analysis, Financial transactions
Keywords [sv]
Topologisk dataanalys, Topologi, TDA, Matematik, Teoretisk Matematik, Dataanalys, Finans, Stabil rang
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-266894OAI: oai:DiVA.org:kth-266894DiVA, id: diva2:1391482
External cooperation
Svenska Handelsbanken AB
Subject / course
Mathematics
Educational program
Master of Science - Mathematics
Supervisors
Examiners
Available from: 2020-02-04 Created: 2020-02-04 Last updated: 2020-02-04Bibliographically approved

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CiteExportLink to record
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Citation style
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