This study explores persistent homology to detect early warning signals of the 2017 and 2019 major financial crashes in Bitcoin. Sliding window is used to obtain point cloud datasets from a multidimensional time series (Bitcoin, Ethereum, Litecoin and Ripple). We apply persistent homology to quantify transient loops that appear in multiscale topological spaces, which associated on each point cloud dataset and encode the quantified information in a persistence landscape. Temporal changes in persistence landscapes are measured via their L1-norms. Consequently, a new representative is attained, called L1-norms time series. The L1-norms is associated with indicators: autocorrelation function at lag 1, variance and mean power spectrum at low frequencies to detect the signals. By using Kendall’s tau correlation and significance test, significant rising trend events that occur before major financial crashes in Bitcoin are defined as the signals. A threshold is determined to scan entire data and record all the significant rising trend events. Lastly, we compare L1-norms with residuals time series, which is another representative obtained from de-trending approach. Our result portrays that autocorrelation function at lag 1 and variance of the L1-norms successfully detect early warning signals before the 2017 and 2019 major financial crashes. However, variance of the L1-norms is better since it able to signal another 2018 major financial crash. For the residuals, no early warning signals are detected. Hence, persistent homology provides a better representative than de-trending approach. Overall, persistent homology is a promising method to detect early warning signals of major financial crashes in Bitcoin.

In this study, a new market representation from persistence homology, known as the L¹-norm time series, is used and applied independently with three critical slowing down indicators [autocorrelation function at lag 1, variance, and mean for power spectrum (MPS)] to examine two historical financial crises (Dotcom crash and Lehman Brothers bankruptcy) in the US market. The captured signal is the rising trend in the indicator time series, which can be determined by Kendall's tau correlation test. Furthermore, we examined Pearson's and Spearman's rho correlation tests as potential substitutes for Kendall's tau correlation. After that, we determined a correlation threshold and predicted the whole available date. The point of comparison between these correlation tests is to determine which test is significant and consistent in classifying the rising trend. The results of such a comparison will suggest the best test that can classify the observed rising trend and detect early warning signals (EWSs) of impending financial crises. Our outcome shows that the L¹-norm time series is more likely to increase before the two financial crises. Kendall's tau, Pearson's, and Spearman's rho correlation tests consistently indicate a significant rising trend in the MPS time series before the two financial crises. Based on the two evaluation scores (the probability of successful anticipation and probability of erroneous anticipation), by using the L¹-norm time series with MPS, our result in the whole prediction demonstrated that Spearman's rho correlation (46.15 and 53.85%) obtains the best score as compared to Kendall's tau (42.31 and 57.69%) and Pearson's (40 and 60%) correlations. Therefore, by using Spearman's rho correlation test, L¹-norm time series with MPS is shown to be a better way to detect EWSs of US financial crises.

Investigating the dependence structures among the characteristics of the current unhealthy air pollution events is a valuable endeavor to understand the pollution behavior more clearly and determine the potential future risks. This study determined the characteristics of air pollution events based on their duration, severity, and intensity. It focused on modeling the dependence structures for all the possible pairs of characteristics, which were (duration, intensity), (severity, intensity), and (duration, severity), using various parametric copula models. The appropriate copula models for describing the behavior of the relationship pairs of the (duration, intensity), (severity, intensity), and (duration, severity) were found to be the Tawn type 1, 180°-rotated Tawn type 1, and Joe, respectively. This result showed that the dependence structures for the pairs were skewed and asymmetric. Therefore, the obtained copulas were appropriate models for such non-elliptical structures. These obtained models can be further extended in future work through the vine copula approach to provide a more comprehensive insight into the tri-variate relationship of the duration–intensity–severity characteristics.

Contaminated air is unhealthy for people to breathe and live in. To maintain the sustainability of clean air, air pollution must be analyzed and controlled, especially after unhealthy events. To do so, the characteristics of unhealthy events, namely intensity, duration, and severity are studied using multivariate modeling. In this study, the vine copula approach is selected to study the characteristics data. Vine copula is chosen here because it is more potent than the standard multivariate distributions, and multivariate copulas, especially in modeling the tails related to extreme events. Here, all nine different vine copulas are analyzed and compared based on model fitting and the comparison of models. In model fitting, the best model obtained is Rv123-Joint-MLE, a model with a root nodes sequence of 123, and optimized using the joint maximum likelihood. The components for the best model are the Tawn type 1 and Rotated Tawn type 1 180 degrees representing the pair copulas of (intensity, duration), and (intensity, severity), respectively, with the Survival Gumbel for the conditional pair copula of (duration, severity; intensity). Based on the best model, the tri-variate dependence structure of the intensity, duration, and severity relationship is positively correlated, skewed, and follows an asymmetric distribution. This indicates that the characteristic’s, including intensity, duration, and severity, tend to increase together. Using comparison tests, the best model is significantly different from others, whereas only two models are quite similar. This shows that the best model is well-fitted, compared to most models. Overall, this paper highlights the capability of vine copula in modeling the asymmetric dependence structure of air pollution characteristics, where the obtained model has a better potential to become a tool to assess the risks of extreme events in future work.

A dynamical system is a system that evolves with time. Research in the field of dynamical system is largely focused on the nature of chaos on that system. Nowadays, there are various definitions of chaotic dynamical systems. However, the most well-known definition of chaos is Devaney chaos that states three chaotic conditions in its definition; sensitivity dependence on initial conditions, transitivity and density of periodic points. In this paper, we are investigating the presence of chaotic behavior in a discrete space, even shift. The even shift space is a space of all infinite sequences over symbols 0 and 1 such that between any two 1’s there are an even number of 0’s. By the end of this investigation, we prove that even shift space is not only Devaney chaos but also satisfies some other stronger chaotic conditions i.e. totally transitive, topologically mixing, blending, and locally everywhere onto.