Topological data analysis of financial time series: Landscapes of crashes

• Published in: Physica A Vol. 491; pp. 820 - 834
• Main Authors: Gidea, Marian, Katz, Yuri
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.02.2018
• Subjects: Early warning signals; Financial time-series; Topological data analysis; Topological data analysis; Financial time-series; Early warning signals
• ISSN: 0378-4371; 1873-2119
• DOI: 10.1016/j.physa.2017.09.028

We explore the evolution of daily returns of four major US stock market indices during the technology crash of 2000, and the financial crisis of 2007–2009. Our methodology is based on topological data analysis (TDA). We use persistence homology to detect and quantify topological patterns that appear in multidimensional time series. Using a sliding window, we extract time-dependent point cloud data sets, to which we associate a topological space. We detect transient loops that appear in this space, and we measure their persistence. This is encoded in real-valued functions referred to as a ’persistence landscapes’. We quantify the temporal changes in persistence landscapes via their Lp-norms. We test this procedure on multidimensional time series generated by various non-linear and non-equilibrium models. We find that, in the vicinity of financial meltdowns, the Lp-norms exhibit strong growth prior to the primary peak, which ascends during a crash. Remarkably, the average spectral density at low frequencies of the time series of Lp-norms of the persistence landscapes demonstrates a strong rising trend for 250 trading days prior to either dotcom crash on 03/10/2000, or to the Lehman bankruptcy on 09/15/2008. Our study suggests that TDA provides a new type of econometric analysis, which complements the standard statistical measures. The method can be used to detect early warning signals of imminent market crashes. We believe that this approach can be used beyond the analysis of financial time series presented here. •We introduce a new method, based on topological data analysis (TDA), to analyze financial time series, and detect possible early signs prior to financial crashes.•We analyze the time-series of daily log-returns of four major US stock market indices: S&P 500, DJIA, NASDAQ, and Russell 2000.•We use persistence homology to detect and quantify topological patterns that appear in the multidimensional time series.•We find that, in the vicinity of financial meltdowns, the Lp-norms of persistence landscapes exhibit strong growth prior to the primary peak, which ascends during a crash.•Our method is very general and can be applied to any asset-types and mixtures of time series.