Table of Links I. Introduction I. Introduction II. Methodology II. Methodology III. TDA Approach to analyzing multiple time series III. TDA Approach to analyzing multiple time series IV. Data Analyzed IV. Data Analyzed V. Results and Discussion V. Results and Discussion A. Obtaining point cloud from stock price time-series A. Obtaining point cloud from stock price time-series B. EE due to the 2008 Financial crisis B. EE due to the 2008 Financial crisis C. EE due to COVID-19 pandemic C. EE due to COVID-19 pandemic D. Impact of COVID-19 on different Indian sectors D. Impact of COVID-19 on different Indian sectors VI. Conclusion VI. Conclusion VII. Acknowledgments and References VII. Acknowledgments and References Extreme events (EEs) are rare, unexpected occurrences that stand out significantly from normal happenings. These can include events like floods, heart attacks, power blackouts, and stock market crashes. For instance, the 2008 financial crisis and the COVID-19 pandemic caused massive stock market crashes worldwide, leading to significant financial losses for investors. Identifying and analyzing these extreme events is essential. One effective method for detecting EEs is topological data analysis (TDA), which can analyze multiple time-series data simultaneously. Using TDA-based norms and Wasserstein distance, we can identify EEs in stock markets across different continents. Additionally, we can conduct a sector-wise analysis to understand how various sectors were differently impacted during the COVID-19 pandemic based on their future outlook. Extreme events (EEs) are rare, unexpected occurrences that stand out significantly from normal happenings. These can include events like floods, heart attacks, power blackouts, and stock market crashes. For instance, the 2008 financial crisis and the COVID-19 pandemic caused massive stock market crashes worldwide, leading to significant financial losses for investors. Identifying and analyzing these extreme events is essential. One effective method for detecting EEs is topological data analysis (TDA), which can analyze multiple time-series data simultaneously. Using TDA-based norms and Wasserstein distance, we can identify EEs in stock markets across different continents. Additionally, we can conduct a sector-wise analysis to understand how various sectors were differently impacted during the COVID-19 pandemic based on their future outlook. I. INTRODUCTION The stock market has witnessed various crashes starting from the great depression of 1929, the 1987 crash, the tech bubble of 2000, the 2008 financial crisis, and the latest COVID-19 pandemic[1–5]. During these crashes, investors lost significant capital due to the panic sell-off caused by irrational decisions[6–8]. However, the same crashes have provided opportunities for investors to make huge profits[7]. Due to the huge risk and opportunity, the study of crash dynamics in the stock market is significant. An extreme event (EE) is identified by the abrupt occurrence of unusual events[9,10]. EEs can happen in technological, social, and natural contexts[9]. They could originate naturally or due to human activity[11]. Previously, floods, power blackouts, earthquakes, heart attacks, stock market crashes and upsurges were referred to as EEs.[10,12,13]. The study of EEs in different fields is important as the impact of these events is severe. There has been very limited work in the stock market in terms of the identification of EEs[10,13]. extreme event Mahata et al.[10] recognized the stock price crash caused by the COVID-19 pandemic as an EE in different companies and indices. The analysis of each time series is carried out separately. The author applied the Empirical Mode Decomposition (EMD) based Hilbert–Huang transform (HHT) to identify EEs. The EMD-based HHT has been widely used to understand the stock market dynamics[14,15]. Further, Rai et al.[13] also identified the different factors leading to positive and negative EEs in different stocks using the EMD-based HHT. In these analyses, the authors have identified EE considering each stock price time series individually. However, when dealing with multiple time series, this method becomes impractical. Hence, we need a more convenient approach to analyze several time series simultaneously. That’s where Topological Data Analysis (TDA) comes into play. TDA attempts to solve data science problems using tools from algebraic topology and geometry[16,17]. The idea behind employing TDA in our investigation is to extract the “shape” of a multi-dimensional time-series dataset. Using a fixed window of time, we first convert the slice of the time series into a Euclidean point cloud. Further, a topological signature—known as persistent homology—of the point cloud is computed. Now, as the slice moves along the time series, the evolution of the signature is then studied and analyzed[18]. One of the key advantages of using persistent homology is its robustness under small, random perturbations of the data points. In contrast to other techniques that consider each time series separately, TDA allows us to analyze multiple time series at once in a complete and insightful manner—hence providing a more efficient and accurate identification of EEs. TDA has been successfully applied to problems in a growing number of fields, such as material sciences[19,20], 3D shape analysis[21], multivariate time-series analysis22, biology[23],medicine[24], chemistry[25], network sensors[26], early warning systems for flood[27]. Recently, TDA has also lent itself to analyzing the stock market crash dynamics. Some of the pivotal and relevant developments in this direction are discussed below. • Further, the persistence landscape was applied to perform enhanced indexing[34]. Moreover, TDA was applied to cluster time-series models by their topological similarity and classification of the simulated dataset. This work provided evidence that stock price movements are sector-dependent[35]. In addition, a method that combines TDA with machine learning was used to understand the dynamics before a critical transition[36]. The authors applied Taken’s theorem to estimate the persistence landscapes from Bitcoin time-series data. The so-called ‘Topological Tail Dependence Theory’ was also proposed to bridge the gap between the mathematical theory of persistent homology and the financial stock market theory[37]. The study found that incorporating persistent homology information systematically improves the forecasting accuracy of models, especially during turbulent periods. It also showed that the 2D Wasserstein distance is statistically significant for linear models. The structure of our paper is as follows. Section II explains the methodology employed in the paper and Section III contains the TDA approach to analyzing time series. Section IV contains the data analyzed. In section V we have discussed our results and Section VI contains the conclusion of the work. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv Authors: (1) Anish Rai, Department of Physics, National Institute of Technology Sikkim, Sikkim, India-737139; (2) Buddha Nath Sharma, Department of Physics, National Institute of Technology Sikkim, Sikkim, India-737139; (3) Salam Rabindrajit Luwang, Department of Physics, National Institute of Technology Sikkim, Sikkim, India-737139; (4) Md.Nurujjaman, Department of Physics, National Institute of Technology Sikkim, Sikkim, India-737139; (5) Sushovan Majhi, Data Science Program, George Washington University, USA, 20052. Authors: Authors: (1) Anish Rai, Department of Physics, National Institute of Technology Sikkim, Sikkim, India-737139; (2) Buddha Nath Sharma, Department of Physics, National Institute of Technology Sikkim, Sikkim, India-737139; (3) Salam Rabindrajit Luwang, Department of Physics, National Institute of Technology Sikkim, Sikkim, India-737139; (4) Md.Nurujjaman, Department of Physics, National Institute of Technology Sikkim, Sikkim, India-737139; (5) Sushovan Majhi, Data Science Program, George Washington University, USA, 20052.
