Wei Shao

In this paper, we use multifractal detrend fluctuation analysis to calculate the multifractal characteristics of particulate matter with particle size less than or equal to 2.5 microns in five representative cities in China in the south and north from 2015 to 2019. The T-test shows that the average annual PM2.5 concentration in northern cities is significantly higher than that in southern cities. In terms of average annual temperature, northern cities are significantly less than southern cities. By observing the local Q-order wave function image and Q-order Hurst index image, it is confirmed that the PM2.5 concentration time series has multi-fractal characteristics and long memory. We observe a clear north–south difference in [ ]. To examine whether temperature is statistically associated with this contrast, we compute Pearson correlation coefficients between temperature and PM2.5 concentration in the observed cities from 2015 to 2019, and the correlation is found to be moderate-to-weak negative correlation.

In this paper, we explore the periodicity of the multifractal phenomenon of stock price caused by public relations events using the multifractal detrended fluctuation analysis (MF-DFA) method. The Tesla stock closing price in the context of the Tesla rights defense event is used to obtain the generalized Hurst exponent and multifractal spectrum. We find that the stock price exhibits multifractal characteristics. Afterwards, we conduct a multifractal time-varying analysis using high-frequency data for 12 time periods before and after the Tesla rights defense event, which happened on 19 April 2021. We calculate the [ ] and [ ] before and after the rights defense event to measure the degree of multifractal of Tesla stock price. In addition, we also compute the difference between the two [ ] and two [ ], which are regarded as [ ] and [ ]. The results show that the Tesla stock market efficiency degree raised as the period time increases until the eighth period. And after the eighth period, both [ ] and [ ] tend to be stable, and the degree of market efficiency caused by the Tesla rights defense event remains almost unchanged. In addition, we use three broad market indices in the context of the Russia–Ukraine conflict to test the universality. The three broad market indices are Dow Jones Industrial Average (DJIA), Shanghai Stock Exchange Composite Index (SSEC) and Nikkei 225 ETF (N225). The results demonstrate that with the extending of time period, [ ] and [ ] gradually stabilize, indicating that public memorability will diminish and the market efficiency almost remains unchanged.

In this paper, we use the finite difference methods to explore step-down Equity Linked Securities (ELS) options under the fractional Black-Scholes model. We establish Crank-Nicolson scheme under one asset and study the impact of Hurst exponent (H) on return of repayment under fixed stock price. We also explore the impact of stock price on return of repayment under different H. Through numerical experiments, it is found that the return of repayment of options is related to H, and the result of difference scheme will increase with the increase of H. In the case of two assets, we establish implicit scheme, and in the case of three assets, we use operator splitting method (OSM) method to establish semi-implicit scheme. We get the result that the H also influences the return of repayment in two and three assets. We also conduct Greeks analysis. Through Greeks analysis, we find that the long-term correlation of stocks has a huge impact on investment gains or losses. Therefore, we take historical volatility (fractal exponents) into account which can significantly reduce risk and increase revenue for investors.

In this paper, we explore whether the activity of SARS-CoV-2 was associated with seasonality. MF-DFA model is utilized to calculate multifractal strength and multifractal complexity to evaluate the change state of SARS-CoV-2 activity. We select 10 countries with serious epidemic in the world, which are distributed in different latitudes of the northern and southern hemispheres. The study utilized the time series data of daily new cases and daily new deaths recorded in these countries. We regard May to October as the “high temperature season” for countries in the northern hemisphere, November to April as the “low temperature season”, and the southern hemisphere is just the opposite. By comparing the multifractal intensity [ ] and multifractal complexity [ ] of the two time series in the two seasons, we draw a conclusion that, for both the sequence of the daily newly diagnosed persons and the daily newly increased number of deaths, in the countries of both the northern and southern hemispheres, [ ] and [ ] are weaker in the “low temperature season”. That is, in the low temperature environment, SARS-CoV-2 can survive for a long time and be more infectious. In addition, we also observe that in the northern hemisphere, Iran is at a lower latitude, and although the SARS-CoV-2 activity in the low temperature season is higher than that in the high temperature season, the difference is not significant. Therefore, the lower latitude may resist this phenomenon. However, most of the countries in the southern hemisphere are within 30° of south latitude, with low latitude, and other meteorological characteristics such as humidity in the countries in the southern hemisphere are also relatively unique. Although SARS-CoV-2 is characterized by high activity in low temperature seasons, no direct evidence related to the characteristics of latitude distribution has been found.

In this paper, we propose the multifractal detrending fluctuation analysis based on [ ]-norm constrain (MF-DFN). We assess the performance of the proposed algorithm by constructing a [ ]-model based multiplicative cascades time series. We calculate the generalized [ ] exponent [ ], [ ] exponent [ ], singularity exponent [ ] and multifractal spectrum [ ] for multifractal detrended fluctuation analysis (MF-DFA) and MF-DFN, respectively. We notice that under different norm constraints, the distribution of multifractal characteristics is quite different. Appropriate norm constraints can make the multifractal characteristics of time series described more accurately. Based on the analytical solution curve, the distribution of different multifractal numerical curves determines the correct selection of norm constraint value. In this study, using a combination of norms [ ] and [ ], the depicted numerical curves almost overlap the theoretical curve, which shows that the proposed MF-DFN is superior to MF-DFA. To eliminate the influence of specific time series, we reset parameters [ ] and [ ] in [ ]-model. Various experimental results also show the effectiveness of the proposed MF-DFN. In addition, we also verify the efficiency of the proposed MF-DFN by using the practical application. The experimental results show that our proposed method plays a significant role in ECG classification.