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In this paper, we propose an improved algorithm based on the original two-dimensional (2D) multifractal detrended fluctuation analysis (2D MF-DFA) that involves increasing the number of cumulative summations in the computational steps of 2D MF-DFA. The proposed method aims to modify the distribution of the generalized Hurst exponent to ensure that skin lesion image features are extracted based on enhanced multifractal features. We calculate the generalized Hurst exponent using 0, 1, or 2 cumulative summation processes. A support vector machine (SVM) is adopted to examine the classification performance under these three conditions. Computation shows that the process involving two cumulative summations achieves an accuracy, sensitivity, and specificity of $95.69\pm 0.1174\%$, $94.25\pm 0.0942\%$, and $97.63\pm 0.1466\%$, respectively, which indicates that its performance is much better than with 0 and 1 cumulative summations.

In this paper, we check for existence of multifractal in volatility of Moroccan family business stock returns and in volatility of Casablanca market index returns based on multifractal detrended fluctuation analysis (MF-DFA) technique. Empirical results show strong evidence of multifractal characteristics in volatility series of both family business stocks and market index. In addition, it is found that small variations in volatility of family business stocks are persistent, whilst small variations in volatility of market index are anti-persistent. However, large variations in family business volatility and market index volatility are both anti-persistent. Furthermore, multifractal spectral analysis based results show strong evidence that volatility in Moroccan family business companies exhibits more multifractality than volatility in the main stock market. These results may provide insightful information for risk managers concerned with family business stocks.

In this paper, automatic electrocardiogram (ECG) recognition and classification algorithms based on multifractal detrended fluctuation analysis (MF-DFA) and multifractal detrended cross-correlation analysis (MF-DXA) were studied. As human heart is a complex, nonlinear, chaotic system, using multifractal analysis to analyze chaotic systems is also a trend. We performed a comparison study of the multifractal nature of the healthy subjects and that of the cardiac dysfunctions ones. To analyze multifractal property quantitatively, the ranges of the Hurst exponent ($\mathrm{\Delta }h$) are computed by MF-DFA and MF-DXA. We found that for MF-DFA, the area of Hurst exponents for atrial premature beat (APB) people was narrower than normal sinus rhythm (NSR) subjects, and for MF-DXA, the difference of $\mathrm{\Delta }h$ ($\mathrm{\Delta }(\mathrm{\Delta }h)$) of NSR and APB subjects was larger than that of MF-DFA. We then regarded the Hurst exponents ($h$) as the input vectors and took them into support vector machine (SVM) for classification. The results showed that $h$ obtained from MF-DXA led to a higher classification accuracy than that of MF-DFA. This is related to the widening of the difference in the values of Hurst exponents in MF-DFA and MF-DXA. The proposed MF-DFA-SVM and MF-DXA-SVM systems achieved classification accuracy of $86.54\%\pm 0.068\%$ and $98.63\%\pm 0.0644\%$, achieved classification sensitivity of $75.03\%\pm 0.1323\%$ and $90.77\%\pm 0.1309\%$, achieved classification specificity of $86.66\%\pm 0.1131\%$ and $96.47\%\pm 0.0891\%$, respectively. In general, the Hurst exponents obtained from MF-DXA played an important role in classifying ECG of the healthy and that of the cardiac dysfunctions subjects. Moreover, MF-DXA was more accurate than MF-DFA in the classification of ECG studied in this paper. The research in automatic medical diagnosis and early warning of major diseases has very important practical value.

In this paper, we develop the multifractal detrending weighted average algorithm of historical volatility (MF-DHV) for one-dimensional multifractal measure based on the classical multifractal detrended fluctuation analysis (MF-DFA). In the calculation process of getting a local trend for MF-DHV, historical volatility is taken to develop an moving average algorithm, which is different from the simple moving average function in multifractal detrended moving average (MF-DMA). We assess the performance of three methods such as MF-DFA, MF-DMA, and MF-DHV based on the p-model multiplicative cascading constructed time series. The computational results show that all the estimated generalized Hurst exponent $H(q)$, the scaling exponent $\tau (q)$, and the singularity spectrum $f(\alpha )$ of MF-DHV are in good agreement with the theoretical values. In addition, we also calculate the standard deviations of $H_{\mathrm{\text{err}}}$ and ${\tau }_{\mathrm{\text{err}}}$ for three methods, and the lowest errors in MF-DHV provides the most accurate estimates. To avoid the accidental selection of parameters, we change the total length of the generated multifractal simulation data and p-value, respectively. It is found that in all the cases, the MF-DHV outperforms the other two methods.

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 $\mathrm{\Delta }H$ and multifractal complexity $\mathrm{\Delta }\alpha$ 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, $\mathrm{\Delta }H$ and $\mathrm{\Delta }\alpha$ 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.

This paper investigates the impact of Chinese Treasury bond (CTB) futures on the information content of interest rate swap (IRS) from a multifractality perspective. We first use multifractal detrended fluctuation analysis (MF-DFA) method and show that the swap rate and the CTB yield exhibit strong multifractality. In addition, employing multifractal detrended cross-correlation analysis (MF-DCCA) method, we find that cross-correlations between the swap rate and the CTB yield are multifractally persistent. Moreover, after the reintroduction of Treasury bond futures, the persistence of cross-correlation between the series is weaker. Our results indicate that the information content of IRS decreased after the re-launch of CTB futures.

This study proposes a novel approach to investigating the multifractality of time series using the multifractal cross-correlation detrended moving average analysis (MF-X-DMA). The study demonstrates the behavioral differences of MF-X-DMA in coherent and non-coherent time periods. Due to the lack of a mechanism to capture the dynamical cross-correlation in time series, correlated time series with multifractal structure present a barrier for analysis. The study shows that when the wavelet coherence method is applied to time series, co-movement between time series can be easily captured in certain time intervals, providing an efficient way to find time intervals to apply MF-X-DMA. The study applies the wavelet coherence method to the daily spot prices of gold and platinum from January 1987. It shows that the wavelet coherence method is an excellent engine to extract designated time series in certain frequency and time intervals, eliminating the need for windowing or shuffling methods. Additionally, the study observes a long-term power law cross-correlation using detrended cross-correlation analysis coefficients of inversed series for both low-correlated and high-correlated series. Finally, the findings indicate that MF-X-DMA leads to superior results compared to MF-DFA when provided with highly correlated data.

In this paper, we adopt multifractal detrended fluctuation analysis (MF-DFA) to explore relationships between the Russia–Ukraine conflict and defense stock markets. Specifically, we analyze the behaviors of 20 U.S. defense stock markets confronting with the Russia–Ukraine conflict. By using the stock price charts, combined with multifractal spectra and singularity exponents calculated by MF-DFA, we explore how the conflict affects the defense stock markets in perspectives of closing price, market efficiency and stability. In addition, the obtained results reveal high level of consistency while each type shows distinct features. According to singularity exponents, we observe that all 20 stocks can be divided into three types which we note as $Types$ A, B and C. We infer from the singularity spectra that the $Type$ A stock price will experience plummet after the conflict, instead, stock price of $Type$ A increases based on stock price charts, while stock price of $Types$ B and C rises as predicted. For $Type$ A stocks, their market efficiency and stability show increments where we draw completely opposite conclusion for $Type$ B stocks. Furthermore, we also note that $Type$ C stocks include two defense stocks having a special phenomenon, and their multifractal spectra indicate the increase in stock price which behave like $Type$ B stocks. However, their singularity exponents reduce during the conflict, meaning the slump in their market efficiency, which share the same characteristic as $Type$ A stocks. Hence, we treat $Type$ C stocks as a unique type. To mitigate the influence of stochastic elements in the experimental process, three comparative analyses are undertaken. We humbly believe that the induced implications are aroused by the Russia–Ukraine conflict.

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 $\mathrm{\Delta }h$ and $\mathrm{\Delta }\alpha$ 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 $\mathrm{\Delta }h$ and two $\mathrm{\Delta }\alpha$, which are regarded as $D1$ and $D2$. 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 $D1$ and $D2$ 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, $D1$ and $D2$ gradually stabilize, indicating that public memorability will diminish and the market efficiency almost remains unchanged.

We introduce a new approach to improve the forecasting performance by investigating the multifractal features and the dynamic correlations of return on spot prices of precious metals, namely, gold and platinum. The Hölder exponent of multifractal time series is employed to detect the critical fluctuations during the financial crises through measuring the multifractal behavior. We also consider co-movement of Hölder exponents and forecast the Hölder exponents of multifractal precious metal time series on coherent time periods. The results indicate that forecasting of multiple wavelet coherence of Hölder exponents of multifractal precious metal time series is efficiently improved by using Vector FARIMA and VARIMA models.