Melvin Segerman: Visibility Graphs för tidsserier: Matematiska egenskaper och tillämpningar

Abstract: This paper presents the mathematical foundations of visibility-based graph representations of time series, focusing primarily on Natural Visibility Graphs and Horizontal Visibility Graphs, while also introducing Invisibility Graphs as a complementary construction. Key theoretical properties are formulated and proven, including invariance under strictly monotone transformations and the exact degree distribution of Horizontal VIsibility Graphs for i.i.d. stochastic processes. These results are then used to contrast theoretical behaviour of periodic, stochastic and chaotic time series, with empirical focus on stochastic, chaotic and financial data. Building on this theory, the paper develops local visibility- and invisibility-based indicators designed to quantify concave and convex price dynamics within rolling windows. The methodology is applied to daily SPY (SP 500 ETF) price data and compared to simulated reference series generated from white noise and the logistic map in a chaotic regime. The empirical findings show that SPY exhibits graph characteristics that differ from both purely stochastic and low-dimensional chaotic systems, with convex indicators in particular highlighting periods of rapid market stress. The results show that the visibility graphs grasp structural asymmetries in financial time series while being a descriptive tool of trend shape analysis. Possible extensions involve assessing predictability and embedding the proposed indicators within trading or risk-management schemes.