Topology Data Analysis Using Mean Persistence Landscapes in Financial Crashes

Alejandro Aguilar,Katherine Ensor

doi:10.4236/jmf.2020.104038

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.

Highlights

Topological data analysis (TDA) extracts topological features by examining the shape of the data through persistent homology to produce topological summaries
Using Equations (43), (44), (47), (50), (53), and (56), the permutation is performed with a significance level α = 0.05 when comparing all the stock indices and all the ETF sectors in the same sliding windows between July 1, 2019 and July 1, 2020, which results in 199 p-values of 0.0000 and 2 p-values of 0.001 for homology in degree 1
From reviewing the norms of the persistence landscape, the daily log returns, persistence diagrams, persistence landscapes, and mean landscapes for all of the selected dates, it is clear that the number of the loops in the relevant point clouds are more pronounced resulting in more persistence, which signifies that the stock market is transitioning from a stable state to a more unpredictable, volatile state

Summary

Introduction

Topological data analysis (TDA) extracts topological features by examining the shape of the data through persistent homology to produce topological summaries. These topological summaries lack geometric properties and do not have a unique (Fréchet) mean [4], which makes it difficult to conduct statistical analysis and machine learning. This study presents a topological data analysis of financial time series data. We provide background material about four relevant areas: algebraic topology, homology, topological summaries, and norms for persistent landscapes. We apply topological data analysis to a sequence of point cloud data sets to examine their topological properties within a point cloud matrix of d 1-dimensional time series.

Objectives

Methods

Results

Discussion

Conclusion