Using Topological Data Analysis (TDA) and Persistent Homology to Analyze the Stock Markets in Singapore and Taiwan

Abstract

In recent years, persistent homology (PH) and topological data analysis (TDA) have gained increasing attention in the fields of shape recognition, image analysis, data analysis, machine learning, computer vision, computational biology, brain functional networks, financial networks, haze detection, etc. In this article, we will focus on stock markets and demonstrate how TDA can be useful in this regard. We first explain signatures that can be detected using TDA, for three toy models of topological changes. We then showed how to go beyond network concepts like nodes (0-simplex) and links (1-simplex), and the standard minimal spanning tree or planar maximally filtered graph picture of the cross correlations in stock markets, to work with faces (2-simplex) or any k-dim simplex in TDA. By scanning through a full range of correlation thresholds in a procedure called filtration, we were able to examine robust topological features (i.e. less susceptible to random noise) in higher dimensions. To demonstrate the advantages of TDA, we collected time-series data from the Straits Times Index and Taiwan Capitalization Weighted Stock Index (TAIEX), and then computed barcodes, persistence diagrams, persistent entropy, the bottleneck distance, Betti numbers, and Euler characteristic. We found that during the periods of market crashes, the homology groups become less persistent as we vary the characteristic correlation. For both markets, we found consistent signatures associated with market crashes in the Betti numbers, Euler characteristics, and persistent entropy, in agreement with our theoretical expectations.

Understanding Changes in the Topology and Geometry of Financial Market Correlations during a Market Crash

“…To put it simply, our analysis of the Betti numbers suggests that the cross correlations in the first six time windows were more or less similar topologically, whereas for (15 September 2019, 15 March 2020) and subsequent time windows, the Betti numbers became extremely sensitive to over a broad range of , suggesting a non-trivial topological transition over the last three time windows. Another signature of this topological transition is the persistence weakening phenomenon that we observed in our earlier paper [59], where we found first a slow increase in the number of simplices, and then a rapid increase in the number of simplices after some threshold.…”

“…Recently, we published a TDA paper in the Frontier in Physics Special Issue “From Physics to Econophysics back to Physics: Methods and Insights” [ 59 ]. In this paper, we worked out the TDA signatures for (1) coalescing spheres, (2) torus to horn torus to spindle torus to sphere, and (3) sphere to ellipsoids, and used these toy models to develop a hypothesis on market crashes corresponding to the fragmentation of a multiply connected manifold with a non-zero genus.…”

“…In a recent paper [ 59 ], we demonstrated how TDA can be used to understand the topological changes that accompany market crashes. For such extreme events in financial markets, one of the key questions not well answered through the use of MSTs or planar maximally filtered graphs (PMFGs) is how the hierarchy of cross correlations between stocks re-organizes itself.…”

Machine learning techniques on homological persistence features for prostate cancer diagnosis

“…Note that the maximum persistent entropy corresponds to the situation in which all of the lines in the barcode are of equal length. Many articles described the use of these statistical features derived from persistent homology especially persistent entropy [45][46][47]. These methods are mathematically robust methods of turning a persistence diagram into a stable feature vector.…”

Early Warning Signals of Financial Crises Using Persistent Homology and Critical Slowing Down: Evidence From Different Correlation Tests

“…In addition, dynamics of financial market correlations based on topology and geometry are examined using PH in Yen et al [31]. Moreover, Yen and Cheong [32]also tested PH to analyze Singapore and Taiwan markets. The extreme event called flash crash was also explored in Kim et al [33] by applying PH and dynamic time series analysis.…”