However, all clustering algorithms assume that a stock can be a member-only of one cluster. There are also standard clustering algorithms like k-means and hierarchical clustering that can be used to study the correlation structure in a market. Information filtering methods like the MST and PMFG are not clustering algorithms, but there are clustering algorithms based off them. There is also a second gap in our understanding of these dynamic correlations, and that is the problem of overlapping communities. It can change with window length over the same period, and it can also depend on which market we are looking at. This best genus can change with time for the same window length. ![]() Therefore, the determination of the optimum genus represents a gap in our understanding of correlations in the stock market. However, it is possible that the pattern of dynamic correlations may be explainable more naturally in terms of some nontrivial geometrical structure with g > 0. ![]() This is a good starting point for understanding the correlated price movements between different stocks. In the PMFG method, important correlations are projected onto a sphere, which has genus g = 0. These represent some of the methodological contributions by econophysicists. Physicists also love to strip problems down to their simplest essence, using information filtering approaches such as the minimal spanning tree (MST), planar maximally filtered graph (PMFG), triangular maximally filtered graph (TMFG), etc. Also, their dynamical properties can be described in terms of fractals (in terms for example, of the Hurst exponent) and multifractals instead of the random walk proposed by Bachelier to model price movements in the stock market. The next significant milestone in econophysics is the realization that stock returns follow heavy-tailed Levy distributions instead of a normal distribution. RMT thus allows physicists to discriminate between noise and signal. In RMT, one treats noise as a kind of symmetry, and thus information represents some form of symmetry breaking. The earliest success of econophysics is the application of random matrix theory (RMT, which is a theory combining nuclear physics and statistical mechanics) to the stock market. 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. We found that during the periods of market crashes, the homology groups become less persistent as we vary the characteristic correlation. 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. less susceptible to random noise) in higher dimensions. By scanning through a full range of correlation thresholds in a procedure called filtration, we were able to examine robust topological features (i.e. 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. We first explain signatures that can be detected using TDA, for three toy models of topological changes. In this article, we will focus on stock markets and demonstrate how TDA can be useful in this regard. 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. 3Complexity Institute, Nanyang Technological University, Singapore, Singapore. ![]() ![]() 2Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore.1Energy Research Institute NTU Singapore, Singapore.Peter Tsung-Wen Yen 1 Siew Ann Cheong 2,3*
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