Browsing by Subject "time delay embedding"

The aim of this thesis was to 1) give an exposition of how topological data analysis (TDA) can be used to look for patterns in periodic data, 2) apply it to financial data and 3) visually explore how a topological analysis of credit data using landscape distances compared to looking directly at the change in credit data in the context of stock market crashes. TDA applies algebraic topology to data. It models data sets as various-dimensional surfaces, or manifolds, and studies their structure to find patterns of interconnectedness. It is a powerful tool for studying large, complex, multi-dimensional and noisy data sets. It is often able to capture subtle patterns in such data sets much better than other methods. It is known that stock market crashes are preceded by periods of credit expansion, but we have no reliable indicator of an imminent crash. Chapter 2 covers the algebraic topological theory needed. Key concepts are simplicial complexes, homology groups and persistent homology. The central theorem is the Nerve Theorem, which establishes an equivalence between the union of a collection of convex sets and the nerve of the collection. Chapter 3 describes the method of time delay embedding to pre-process periodic data. A Vietoris-Rips filtration was applied to sliding windows of credit data. From this persistence diagrams and their corresponding persistence landscapes were obtained. The normalised persistence landscape norms (L1) were plotted to visually explore how well TDA captured the connection between credit expansion and stock market crashes. It was compared to the discrete first derivative of the credit data. Visual inspection of the graphs suggested TDA to be as good, and possibly slightly better, at predicting stock market crashes from bank credit data, than looking at the discrete first derivative directly. No obvious new indicator of an imminent crash was found, however. To unlock the true potential of TDA in analysing large, multivariate data sets, further studies could look to triangulate a better indicator of stock market crashes by combining the credit data with other economic, social and political data. It would also be useful to establish a less subjective, more transparent method for choosing the thresholds used as crash indicators, and to quantify the predictions made by different indicators to better compare them with each other.