  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 $L^p$-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 $L^p$-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 $L^p$-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 goes beyond 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.
