A comparative study of financial time series PointCloud construction methods based on TakensEmbedding and distance matrix

In recent years, as financial market fluctuations have become increasingly prevalent, a variety of novel analysis methods and tools have emerged. Among these, Topological Data Analysis (TDA) has garnered attention and application in the financial field due to its capacity to profoundly comprehend the nonlinear relationships and intricate network structures among stocks in the market. Furthermore, TDA is well-suited for nonlinear, high-dimensional, and complex financial data. However, a comparative analysis of the merits and drawbacks of constructing point clouds using TakensEmbedding and the distance matrix in topological data analysis is lacking. In this study, we aim to address this knowledge gap by systematically comparing and analyzing the performance of constructing PointClouds based on TakensEmbedding and the distance matrix for daily time series of stock market indices. By replicating the same steps and parameter settings, we obtain Normalized L1-norm of Persistence Landscapes result plots, which demonstrate that the L1-norm generated by the distance matrix-based approach to building PointCloud exhibits a significant upward trend in the absence of market crashes, leading to an excessive number of false alarms. Finally, the experimental results indicate that the TakensEmbedding-based performance surpasses that of PointCloud construction using the distance matrix. This study offers an interpretable guideline for selecting a point cloud construction method.