Topology Data Analysis Using Mean Persistence Landscapes in Financial Crashes

Abstract

Topological features in high dimensional time series are used to characterize changes in stock market dynamics over time. We explored the daily log returns of four major US stock market indices and 10 ETF sectors between January 2010-June 2020. Topological data analysis and persistence homology were used on two sequences of point cloud data sets the stock indices and the ETF sectors, respectively. Using these sequences, the daily log returns, persistence diagrams, persistence landscapes, and mean landscapes were used to quantify topological patterns in the multidimensional time series. For example, norms of the persistence landscapes were generated to detect critical transitions in the daily log returns. To measure statistical significance, we implemented three permutation tests with a significance level α = 0.05 to determine if topological features change within a particular time frame by comparing sliding windows in the sequence of point cloud data sets. We found that between July 1, 2019 and July 1, 2020, there is evidence of changing structure in the US stock market. Critical transitions are identified by the statistical properties of the norms of the persistence landscape between contiguous daily sliding windows of the stock indices and ETF sector series.