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An exploration of topological properties of high-frequency one-dimensional financial time series data using TDA
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
2017 (English)Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesisAlternative title
An exploration of topological properties of high-frequency one-dimensional financial time series data using TDA (Swedish)
Abstract [en]

Topological data analysis has been shown to provide novel insight in many natural sciences. To our knowledge, the area is however relatively unstudied on financial data. This thesis explores the use of topological data analysis on one dimensional financial time series. Takens embedding theorem is used to transform a one dimensional time series to an $m$-dimensional point cloud, where $m$ is the embedding dimension. The point cloud of the time series represents the states of the dynamical system of the one dimensional time series. To see how the topology of the states differs in different partitions of the time series, sliding window technique is used. The point cloud of the partitions is then reduced to three dimensions by PCA to allow for computationally feasible persistent homology calculation. Synthetic examples are shown to illustrate the process. Lastly, persistence landscapes are used to allow for statistical analysis of the topological features. The topological properties of financial data are compared with quantum noise data to see if the properties differ from noise. Complexity calculations are performed on both datasets to further investigate the differences between high-frequency FX data and noise. The results suggest that high-frequency FX data differs from the quantum noise data and that there might be some property other than mutual information of financial data which topological data analysis uncovers.

Abstract [sv]

    Topologisk dataanalys har visat sig kunna ge ny insikt i många naturvetenskapliga discipliner. Till vår kännedom är tillämpningar av metoden på finansiell data relativt ostuderad. Uppsatsen utforskar topologisk dataanalys på en endimensionell finanstidsserie. Takens inbäddningsteorem används för att transformera en endimensionell tidsserie till ett $m$-dimensionellt punktmoln, där $m$ är inbäddningsdimensionen. Tidsseriens punktmoln representerar tillstånd hos det dynamiska systemet som associeras med den endimensionella tidsserien. För att undersöka hur topologiska tillstånd varierar inom tidsserien används fönsterbaserad teknik för att segmentera den endimensionella tidsserien. Segmentens punktmoln reduceras till 3D med PCA för att göra ihållande homologi beräkningsmässigt möjligt. Syntetiska exempel används för att illustrera processen. En jämförelse mellan topologiska egenskaper hos finansiell tidseries och kvantbrus utförs för att se skillnader mellan dessa. Även komplexitetsberäkningar utförs på dessa data set för att vidare utforska skillnaderna mellan kvantbrus och högfrekventa FX-data. Resultatet visar på att högfrekvent FX-data skiljer sig från kvantbrus och att det finns egenskaper förutom gemensam information hos finansiella tidsserier som topologisk dataanalys visar på.

Place, publisher, year, edition, pages
2017.
Series
TRITA-MAT-E ; 2017:80
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-220355OAI: oai:DiVA.org:kth-220355DiVA, id: diva2:1169943
Subject / course
Financial Mathematics
Educational program
Master of Science - Industrial Engineering and Management
Supervisors
Examiners
Available from: 2017-12-31 Created: 2017-12-31 Last updated: 2017-12-31Bibliographically approved

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CiteExportLink to record
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