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Topology Data Analysis Using Mean Persistence Landscapes in Financial Crashes

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Document pages: 31 pages

Abstract: Topologicalfeatures in high dimensional time series are used to characterize changes instock market dynamics over time. We explored the daily log returns of fourmajor US stock market indices and 10 ETF sectors between January 2010-June 2020. Topological data analysis andpersistence homology were used on two sequences of point cloud data sets thestock indices and the ETF sectors,respectively. Using these sequences, the daily log returns, persistence diagrams,persistence landscapes, and mean landscapes were used to quantify topologicalpatterns in the multidimensional time series. For example, norms of thepersistence landscapes were generated to detect critical transitions in thedaily log returns. To measure statistical significance, we implemented threepermutation tests with a significance level α = 0.05 to determine iftopological features change within a particular time frame by comparing slidingwindows 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 stockmarket. Critical transitions are identified by the statistical properties ofthe norms of the persistence landscape between contiguous daily sliding windowsof the stock indices and ETF sectorseries.

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