Analysis of financial time series using TDA: theoretical and empirical results

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2019, Director: Carles Casacuberta i Josep Vives i Santa Eulàlia[en] Topological Data Analysis (TDA) is a recently developed tool designed to study the geometry of finite data sets. In these notes, we describe the theory of persistent homology, which is the background underlying the application of TDA. Our work is both practical and theoretical. We describe in detail persistence landscape functions, which are a means of visualizing persistent homology, and study some of their properties while deriving a few novel results. From a statistical approach, our theoretical work corroborates the use of TDA to measure changes in the underlying distribution of a data set. We employ TDA to analyze the log returns of four main financial European indices throughout 2005–2015, comparing our results with the ones in the paper Topological Data Analysis of Financial Time Series: Landscapes of Crashes [19]. As in this article, we observe that the norms of persistence landscapes show strong growth prior to substantial financial instability