Multiscale Multifractal Analysis Research Articles

Multifractal Multiscale Analysis of Human Movements during Cognitive Tasks.

Continuous adaptations of the movement system to changing environments or task demands rely on superposed fractal processes exhibiting power laws, that is, multifractality. The estimators of the multifractal spectrum potentially reflect the adaptive use of perception, cognition, and action. To observe time-specific behavior in multifractal dynamics, a multiscale multifractal analysis based on DFA (MFMS-DFA) has been recently proposed and applied to cardiovascular dynamics. Here we aimed at evaluating whether MFMS-DFA allows identifying multiscale structures in the dynamics of human movements. Thirty-six (12 females) participants pedaled freely, after a metronomic initiation of the cadence at 60 rpm, against a light workload for 10 min: in reference to cycling (C), cycling while playing "Tetris" on a computer, alone (CT) or collaboratively (CTC) with another pedaling participant. Pedal revolution periods (PRP) series were examined with MFMS-DFA and compared to linearized surrogates, which attested to a presence of multifractality at almost all scales. A marked alteration in multifractality when playing Tetris was evidenced at two scales, τ ≈ 16 and τ ≈ 64 s, yet less marked at τ ≈ 16 s when playing collaboratively. Playing Tetris in collaboration attenuated these alterations, especially in the best Tetris players. This observation suggests the high sensitivity to cognitive demand of MFMS-DFA estimators, extending to the assessment of skill/demand interplay from individual behavior. So, by identifying scale-dependent multifractal structures in movement dynamics, MFMS-DFA has obvious potential for examining brain-movement coordinative structures, likely with sufficient sensitivity to find echo in diagnosing disorders and monitoring the progress of diseases that affect cognition and movement control.

Multiscale adaptive multifractal cross-correlation analysis of multivariate time series

In the real world, most time series generated from complex systems are nonlinear. To effectively study their fractal properties, in this work, we first generalize the adaptive fractal analysis (AFA) to the adaptive multifractal cross-correlation analysis (AMFCCA), which can be used to study the multifractal cross-correlation between two time series. Considering the complexity of time series from complex systems, we extend AMFCCA to the case of multivariate time series, namely multivariate adaptive multifractal cross-correlation analysis (MV-AMFCCA). In order to detect multiscale multifractality, we propose multiscale multivariate adaptive multifractal cross-correlation analysis (MMV-AMFCCA). By the numerical simulation on synthetic multivariate processes, our method shows the theoretical validity. Furthermore, we applied these methods to the multifractal analysis of three pollutants in urban and suburban areas in Beijing. After removing the seasonal trend, we find that the urban and suburban systems both have multifractality, especially the multifractality of the urban system are more evident than the suburban systems, and the degree of multifractality in spring and winter stronger than that in summer and autumn. Therefore, our methods are suitable for systems with multiple outputs and provide more comprehensive methods for characterizing multivariate autocorrelation and cross-correlation.

Detection of EEG signals in normal and epileptic seizures with multiscale multifractal analysis approach via weighted horizontal visibility graph

Electroencephalogram (EEG) signals contain important information about the regulation of brain system. Thus, automatic detection of epilepsy by analyzing the characteristics obtained from EEG signals has important research implications in the field of clinical medicine. In this paper, the horizontal visibility graph (HVG) algorithm is used to map multifractal EEG signals into complex networks. Then, we study the structure of the networks and explore the nonlinear dynamics properties of the EEG signals inherited from these networks. In order to better describe complex brain behaviors, we use the angle between two connected nodes as the edge weight of the network and construct the weighted horizontal visibility graph (WHVG). In our studies, fractality and multifractality of WHVG are innovatively used to analyze the structure of related networks. However, these methods only analyze the reconstructed dynamical system in general characterizations, they are not sufficient to describe the complex behavior and cannot provide a comprehensive picture of the system. To this effect, we propose an improved multiscale multifractal analysis (MMA) for network, which extends the description of the network dynamics features by focusing on the relationship between the multifractality and the measured scale-free intervals. Furthermore, neural networks are applied to train the above-mentioned parameters for the classification and identification of three kinds of EEG signals, i.e., health, interictal phase, and ictal phase. By evaluating our experimental results, the classification accuracy is 99.0%, reflecting the effectiveness of the WHVG algorithm in extracting the potential dynamic characteristics of EEG signals.

Evaluation of non-linear heart rate variability using multi-scale multi-fractal detrended fluctuation analysis in mice: Roles of the autonomic nervous system and sinoatrial node.

Nonlinear analyses of heart rate variability (HRV) can be used to quantify the unpredictability, fractal properties and complexity of heart rate. Fractality and its analysis provides valuable information about cardiovascular health. Multi-Scale Multi-Fractal Detrended Fluctuation Analysis (MSMFDFA) is a complexity-based algorithm that can be used to quantify the multi-fractal dynamics of the HRV time series through investigating characteristic exponents at different time scales. This method is applicable to short time series and it is robust to noise and nonstationarity. We have used MSMFDFA, which enables assessment of HRV in the frequency ranges encompassing the very-low frequency and ultra-low frequency bands, to jointly assess multi-scale and multi-fractal dynamics of HRV signals obtained from telemetric ECG recordings in wildtype mice at baseline and after autonomic nervous system (ANS) blockade, from electrograms recorded from isolated atrial preparations and from spontaneous action potential recordings in isolated sinoatrial node myocytes. Data demonstrate that the fractal profile of the intrinsic heart rate is significantly different from the baseline heart rate in vivo, and it is also altered after ANS blockade at specific scales and fractal order domains. For beating rate in isolated atrial preparations and intrinsic heart rate in vivo, the average fractal structure of the HRV increased and multi-fractality strength decreased. These data demonstrate that fractal properties of the HRV depend on both ANS activity and intrinsic sinoatrial node function and that assessing multi-fractality at different time scales is an effective approach for HRV assessment.

Multifractal Analysis of the Impact of COVID-19 on NASDAQ, CIOPI, and WTI Crude Oil Market

In this paper, we explore the impact of COVID-19 on auto-correlations and cross-correlations among NASDAQ stock index of the USA, China iron ore price index (CIOPI), and West Texas Intermediate Crude Oil price (WTI). To find out the effect of COVID-19 on financial markets, we divide the investigated data series into two sub-periods, i.e., pre-COVID19 period and post-COVID19 period. First, multifractal detrended fluctuation analysis (MF-DFA) of those series shows a general trend of strong multifractality after COVID-19, indicating lower market efficiency after the pandemic shock. Second, multifractal detrended cross-correlation analysis (MF-DCCA) method is employed to examine cross-correlations among NASDAQ, CIOPI, and WTI. The three cross-correlations all increase in post-COVID19. The correlation between NASDAQ and CIOPI increases the most, becoming the strongest correlation among the three cross-correlations in post-COVID19. The surrogate procedure shows that the post-COVID19 cross-correlation multifractalities are mostly due to fat-tail distribution. Third, we use multi-scale multifractal analysis (MMA) to visualize the dynamic behaviors of correlations among the series. The Hurst surfaces of the three cross-correlations have more fluctuation, both at small and large scale in post-COVID19 than that of pre-COVID19. Particularly, the Hurst surface of cross-correlation between NASDAQ and CIOPI exhibits stronger multifractality during the outbreak of COVID-19 than that in both pre-COVID19 and post-COVID19. The above investigations provide helpful insights of relevant market trends.

Asymmetric multiscale multifractal analysis (AMMA) of heart rate variability

Objective. The physiological activity of the heart is controlled and modulated mostly by the parasympathetic and sympathetic nervous systems. Heart rate variability (HRV) analysis is therefore used to observe fluctuations that reflect changes in the activity in these two branches. Knowing that acceleration and deceleration patterns in heart rate fluctuations are asymmetrically distributed, the ability to analyze HRV asymmetry was introduced into MMA. Approach. The new method is called asymmetric multiscale multifractal analysis (AMMA) and the analysis involved six groups: 36 healthy persons, 103 cases with aortic valve stenosis, 36 with hypertrophic cardiomyopathy, 32 with atrial fibrillation, 59 patients with coronary artery disease (CAD) and 13 with congestive heart failure. Main results. Analyzing the results obtained for the 6 groups of patients based on the AMMA method, i.e. comparing the Hurst surfaces for heart rate decelerations and accelerations, it was noticed that these surfaces differ significantly. And the differences occur in most groups for large fluctuations (multifractal parameter q > 0). In addition, a similarity was found for all groups for the AMMA Hurst surface for decelerations to the MMA Hurst surface—heart rate decelerations (lengthening of the RR intervals) appears to be the main factor determining the shape of the complete Hurst surface and so the multifractal properties of HRV. The differences between the groups, especially for CAD, hypertrophic cardiomyopathy and aortic valve stenosis, are more visible if the Hurst surfaces are analyzed separately for accelerations and decelerations. Significance. The AMMA results presented here may provide additional input for HRV analysis and create a new paradigm for future medical screening. Note that the HRV analysis using MMA (without distinguishing accelerations from decelerations) gave satisfactory screening statistics in our previous studies.

TIME-DELAY MULTISCALE MULTIFRACTAL DETRENDED PARTIAL CROSS-CORRELATION ANALYSIS OF HIGH-FREQUENCY STOCK SERIES

To study the time-delay multiscale partial cross-correlation between the high-frequency stock series SSEC and SZSE in the Chinese stock market, this paper proposes a new partial cross-correlation exponent, namely the [Formula: see text]-order time-delay multiscale partial cross-correlation exponent (denoted by [Formula: see text]). We integrate the time-delay factor and the multiscale range fluctuation function into partial cross-correlation analysis. Numerical experiments show that the newly proposed exponent [Formula: see text] can be used to detect the partial cross-correlation of two time series with time delay. After excluding the influence of other stock indexes, we found that the persistence and multifractal characteristics between SSEC and SZSE have become stronger. Among them, the American high-frequency stock series has a greater impact on the two Chinese stocks than the Asian high-frequency stock series. Comparing the methods of time-delay multiscale multifractal detrended partial cross-correlation analysis method (time-delay MM-DPXA) and time-delay multiscale multifractal detrended cross-correlation analysis method (time-delay MM-DCCA), we find that cross-correlation and partial cross-correlation have different properties on different time scales and different time delays.

Identification of Electroencephalogram Signals in Alzheimer's Disease by Multifractal and Multiscale Entropy Analysis.

Alzheimer's disease (AD) is the most common form of dementia and is a progressive neurodegenerative disease that primarily develops in old age. In recent years, it has been reported that early diagnosis of AD and early intervention significantly delays disease progression. Hence, early diagnosis and intervention are emphasized. As a diagnostic index for AD patients, evaluating the complexity of the dependence of the electroencephalography (EEG) signal on the temporal scale of Alzheimer's disease (AD) patients is effective. Multiscale entropy analysis and multifractal analysis have been performed individually, and their usefulness as diagnostic indicators has been confirmed, but the complemental relationship between these analyses, which may enhance diagnostic accuracy, has not been investigated. We hypothesize that combining multiscale entropy and fractal analyses may add another dimension to understanding the alteration of EEG dynamics in AD. In this study, we performed both multiscale entropy and multifractal analyses on EEGs from AD patients and healthy subjects. We found that the classification accuracy was improved using both techniques. These findings suggest that the use of multiscale entropy analysis and multifractal analysis may lead to the development of AD diagnostic tools.

Multiscale multifractal detrended partial cross-correlation analysis of Chinese and American stock markets

• We analyzed multifractal binomial measures contaminated with strong white noises. • We investigated the cross correlations between stock markets using MM DPXA methods. • New conclusions regarding multiscale correlation structures were obtained. • We studied the effect of financial crisis on the cross correlations of stock markets. In this paper, a new method called multiscale multifractal detrended partial cross-correlation analysis (MM-DPXA) method is proposed, which combines multifractal detrended partial cross-correlation analysis (MF-DPXA) with multiscale multifractal analysis (MMA). To demonstrate the advantages of this method, we analyze multifractal binomial measures contaminated with strong white noises and compare the performance of MM-DPXA method to traditional cross-correlation techniques. It is found that MM-DPXA method can not only eliminate the influence of other variables, but also provide more valuable information from multiscale perspective. Moreover, this method is able to characterize monofractality or multifractality of the time series in a wide range of scales simultaneously and without assuming any presumed time scale. To further show the utility of MM-DPXA method in complex systems, we provide new evidence on the financial time series. By comparing Hurst surfaces before and after removing common influences, we conclude that cross-correlations and intrinsic cross-correlations show different properties in different scales. Furthermore, we also study the effect of financial crisis on the cross-correlations between Chinese and American stock markets using MF-DCCA and MF-DPXA methods.

Multiscale adaptive multifractal analysis and its applications.

To precisely analyze the fractal nature of a short-term time series under the multiscale framework, this study introduces multiscale adaptive multifractal analysis (MAMFA) combining the adaptive fractal analysis method with the multiscale multifractal analysis (MMA). MAMFA and MMA are both applied to the two kinds of simulation sequences, and the results show that the MAMFA method achieves better performances than MMA. MAMFA is also applied to the Chinese and American stock indexes and the R-R interval of heart rate data. It is found that the multifractal characteristics of stock sequences are related to the selection of the scale range s. There is a big difference in the Hurst surface's shape of Chinese and American stock indexes and Chinese stock indexes have more obvious multifractal characteristics. For the R-R interval sequence, we find that the subjects with abnormal heart rate have significant shape changes in three areas of Hurst surface compared with healthy subjects, thereby patients can be effectively distinguished from healthy subjects.

The effect of persistent U-shaped patterns in RR night-time series on the heart rate variability complexity in healthy humans

Objective: U-shaped patterns, characteristic periods of time observed in tachograms, are a specific subgroup among very low frequency components characterized by relatively short periods of smooth accelerations followed by decelerations of heart rhythm. In this study, we characterize this phenomenon and its effect on heart rate variability (HRV) parameters. Approach: We calculated linear (the mean and standard deviation of RR intervals, RMSSD, pNN50 and the power of the frequency components) and nonlinear (V0, V1 and V2 Porta’s symbolic analysis, Shannon and Sample entropy, Guzik’s and Porta’s asymmetry indexes, the exponents α1 and α2 of detrended fluctuation analysis and the Hurst surface h(q,s) of multiscale multifractal analysis (MMA)) HRV parameters for 65 RR interval night-time series (39 females, 37.5(11.3) years old and 26 males, 41.7(16.5) years old; all without organic heart diseases). All parameters were calculated for original data and for the three kinds of test data in which the following parts of the time series were replaced by 1/f noise: (A) the U-shape patterns annotated in a given data set, (B) randomly chosen windows of similar size as the U-shaped patterns, (C) acceleration-deceleration events shorter than U-shaped patterns. Main results: We found that the U-shaped patterns, as the most persistent structures in RR night-time intervals series, affect the long-range correlation properties (measured by α2). We also found that the U-shaped patterns importantly strictly affect the shape of h(q,s) surface at different scales s. Removing the U-shaped patterns results in the shape of the h(q,s) surface losing the properties characteristic for healthy heart rhythm. The largest quantitative effect of U-shaped patterns was obtained for the power of the VLF component. The mean percentage difference of the VLF component between the original data and the A to C type test data were 19.4%, −4.3% and 5.3%, respectively. Significance: Although percentage contribution of U-shaped patterns is small compared to the whole night-time series (on the average 3.1%(1.7%) with a standard deviation of 1.7%), these patterns have a considerable impact on the HRV parameters describing the VLF, persistency, nonlinear correlations and multifractal properties.

Asymmetric Multifractal Analysis of Rebar Futures and Spot Market in China

In this study, a wavelet denoising method is first used to eliminate the influence of noise. Then, an overlapping smooth window technique is introduced into the asymmetric multifractal detrended cross-correlation analysis method, which was combined with the multiscale multifractal analysis method, resulting in the proposed asymmetric multiscale multifractal detrended cross-correlation analysis method. This method not only remedies the pseudo-fluctuation defect of the traditional method, but also explores the asymmetric multifractal cross-correlation between China’s rebar futures and spot markets at different scales. The results show the existence of an asymmetric multifractal cross-correlation between rebar futures and spot markets with upward and downward trends at different scales. This cross-correlation is highly complex at the small-scale, and more pronounced when the futures market is in an uptrend.

Multifractal analysis of the WTI crude oil market, US stock market and EPU

This paper adopts multifractal methods to analyze the nonlinear correlations among economic policy uncertainty (EPU), the crude oil market and the stock market. First, using multifractal detrended fluctuation analysis (MF-DFA), we find that each of the three series shows multifractality, and the sources of multifractality are all from long-range correlations and fat-tailed distributions. However, for the stock and oil price series, the long-range correlation has a greater impact, while the fat-tailed distribution contributes more to EPU multifractality. Furthermore, we apply multifractal detrended cross-correlation analysis (MF-DCCA) to analyze the cross-correlation multifractal features among the three series. The generalized Hurst exponents of any two series are significantly greater than 0.5, and the stock and oil price series have the strongest cross-correlation values. Coupling detrended fluctuation (CDFA) analysis of the three series shows that the coupling correlation among the three sequences is also multifractal. The chi-squared statistic reveals that the contribution of the stock market to the multifractality of the coupling correlation is greater than that of other series, and EPU has the smallest influence on the coupling correlation. Finally, we employ multiscale multifractal analysis (MMA) to visualize the dynamic behaviors of the correlations among the series. The results show that the cross-correlation of stock with the oil price has some symmetry at small- and large-scale fluctuations. Additionally, the influence of EPU on the overall coupling features is mainly reflected under small-scale fluctuations, and at large scales, the coupling correlation of the three series remains similar to the Hurst surface of stock with the oil price. These results are useful for building predictive models.

Multiscale multifractal analysis on air traffic flow time series: A single airport departure flight case

Multiscale multifractality of air traffic flow volume time series characterizes the operation and the evolution characteristics of the air traffic system in the temporal dimension, which are helpful in gaining a better understanding of the dynamical evolution mechanism of this system. We identify the airport departure air traffic flow volume time series is of multiscale multifractality via the investigation on the Hurst surface obtained by the multiscale multifractal analysis. The resulting Hurst surface is of distinct local fluctuation characteristics and shows distinct fractal characteristics on different scales. We confirm that the two types of multifractality both present in the time series, with the long-range correlation being the main cause of the multifractality. We extend multifractal spectra to multifractal spectrum surfaces to provide a common multiscale framework for exploring the multifractal strength, the dominant fluctuation type and the insensitivity of the multifractality. Furthermore, we find that we can use the time series of only 101 days, saving us 116 days, to obtain a quite similar Hurst surface whose difference to the original one is only 2.11662−7.

Multiscale Multifractal Detrended Cross-Correlation Analysis of High-Frequency Financial Time Series

In order to obtain richer information on the cross-correlation properties between two time series, we introduce a method called multiscale multifractal detrended cross-correlation analysis (MM-DCCA). This method is based on the Hurst surface and can be used to study the non-linear relationship between two time series. By sweeping through all the scale ranges of the multifractal structure of the complex system, it can present more information than the multifractal detrended cross-correlation analysis (MF-DCCA). In this paper, we use the MM-DCCA method to study the cross-correlations between two sets of artificial data and two sets of 5[Formula: see text]min high-frequency stock data from home and abroad. They are SZSE and SSEC in the Chinese market, and DJI and NASDAQ in the US market. We use Hurst surface and Hurst exponential distribution histogram to analyze the research objects and find that SSEC, SZSE and DJI, NASDAQ all show multifractal properties and long-range cross-correlations. We find that the fluctuation of the Hurst surface is related to the positive and negative of [Formula: see text], the change of scale range, the difference of national system, and the length of time series. The results show that the MM-DCCA method can give more abundant information and more detailed dynamic processes.

Multiscale Experiments and Multifractal Analysis of Bituminous Coals: Dominant Pore Properties for Gas Permeability

Abstract The pore characteristics of coal are highly correlated with gas permeability. In this study, multifractal analysis based on mercury intrusion porosimetry (MIP) was used to investigate the pore spatial distributions and heterogeneity of pore size distributions (PSDs) of three sister coal samples collected from the Xin Zhouyao mine in China. The mercury intrusion data, under pressures of over 10 MPa, were corrected in combination with the gas adsorption results. All CT images were processed by using the Image-Pro Plus® (IPP) software (Media Cybernetics, Inc., Rockville, MD). The pore volume percentage information obtained from the MIP was found to be coincident with the 3D reconstructions. Seepage experiments were also carried out under different gas pressures to analyze the effect of the pore parameters on permeability. The results from these experiments showed that for the three coal samples, the degree of clustering of the pores is the dominant factor for permeability and should be the first factor considered. Variations of μ (q, ϵ) versus ϵ indicated that the PSDs of coals have multifractal behavior. Multifractal analysis was suggested by the extracted parameters from the PSDs, including the capacity dimension D0, the information dimension D1, the correlation dimension D2, the widths of the right side D0–D10, and of the left side D−10–D0 of Dq spectra, and the widths of left side α0–αq+ and right side α0–αq− of the f (α) spectrum. These parameters were further discussed to evaluate the permeability.

Multiscale multifractal detrended fluctuation analysis of multivariate time series

This work extends the multivariate multifractal detrended fluctuation analysis(MV-MFDFA) method to multiscale case, named multiscale multivariate multifractal detrended fluctuation analysis (MMV-MFDFA). The benefits of the proposed approach are illustrated by numerical simulations on synthetic multivariate processes. Furthermore, the proposed MMV-MFDFA method is applied to the fractal auto-correlation analysis of six pollutants’ (PM2.5, PM10, SO2, NO2, CO and O3) hourly data in different seasons. The results show that the seasonal periodicity has robust impact on the auto-correlation of pollutants in spring and summer. Besides, we also find that the pollutants in the four seasons possess strong multifractal auto-correlation nature, even after the removal of the seasonal pattern. Finally, the source of multifractality among more than two series is also discussed, and some interesting results are obtained. PM2.5 not only dominates the underlying evolution process in fall and winter, but also is more correlated to the other pollutants than the other ones to each other except in spring. The proposed MMV-MFDFA methodology can provide reliable ways of measuring the fractal auto-correlation properties of multivariate series, and it can be applied to any system with multiple data channels.

Analyzing Complexity and Fractality of Glucose Dynamics in a Pregnant Woman with Type 2 Diabetes under Treatment.

Currently, the rapid development of continuous glucose monitoring (CGM) device brings new insights into the treatment of diabetic patients including those during pregnancy. Complexity and fractality have recently under fast development for extracting information embodied in glucose dynamics measured using CGM. Although scientists have investigated the difference of complexity in glucose dynamics between diabetes and non-diabetes in order to discover better approaches for diabetes care, no one has analyzed the complexity and fractality of glucose dynamics during the process of adopting CGM to successfully treat pregnant women with type 2 diabetes. Thus, we analyzed the complexity and fractality using power spectral density (PSD), multi-scale sample entropy (MSE) and multifractal detrended fluctuation analysis (MF-DFA) in a clinical case. Our results show that (i) there exists multifractal behavior in blood glucose dynamics; (ii) the alpha stable distribution fits to the glucose increment data better than the Gaussian distribution; and (iii) the “global” complexity indicated by multiscale entropy, spectrum exponent and Hurst exponent increase and the “local” complexity indicated by multifractal spectrum decrease after the successful therapy. Our results offer findings that may bring value to health care providers for managing glucose levels of pregnant women with type 2 diabetes as well as provide scientists a reference on applying complexity and fractality in the clinical practice of treating diabetes.

Heart rate variability, multifractal multiscale patterns and their assessment criteria

Objective: Both the central nervous system and the autonomic nervous system are complex physiological networks which modulate the heart rate. They are spatially extended, have built-in delays and work on many time scales simultaneously—nonhomogeneous networks with multifractal dynamics. The object of our research was the analysis of human heart rate variability (HRV) using the nonlinear multiscale multifractal analysis (MMA) method for several cardiovascular diseases. The analysis of HRV (night-time recordings) involved six groups of patients: 61 healthy persons, 104 cases with aortic valve stenosis, 42 with hypertrophic cardiomyopathy, 36 with atrial fibrillation, 70 patients with coronary artery disease and 19 with congestive heart failure. 85% of patients formed a training data set (282 subjects) and 15% formed a test data set (50 subjects). Approach: Multiscale multifractal analysis allows one to analyze the complexity of HRV and find the scaling properties of its fluctuations. The main result of MMA is the Hurst surface, the shape of which changes depending on the medical case analyzed. We prepared six criteria to distinguish a multifractal pattern for healthy subjects. We also prepared additional criteria, enabling one to recognize atrial fibrillation. Main results: For the training data set, we obtained the following accuracy statistics in distinguishing the patients from the healthy: 68% for coronary artery disease, 67% for hypertrophic cardiomyopathy, 88% for atrial fibrillation, 74% for aortic valve stenosis and 83% for congestive heart failure. For the complete training data set we obtained an accuracy of 73%, and 80% for the test data set (mean for ten random selections of the test data set). Significance: The results of MMA presented here provide an additional input into the diagnostic process and may help to create a paradigm for future studies on medical screening methods, especially in that MMA focuses on very low frequencies of HRV not easily accessible by standard medical techniques. Satisfactory statistics for screening using both MMA and the unfiltered version of LF/HF indicate that the nature of the complete network moderating heart rhythm needs to be studied and that sinus rhythm in clinical patients may not always be separated from arrhythmia when its incidence is large.

Multifractal analysis of the Chinese stock, bond and fund markets

The stock, bond and fund markets are three important components of a financial market, and the volatility of the markets and correlations between the markets have been paid extensive attention by researchers and investors. In this paper, we devote our efforts to studying the Shanghai financial market, while the return series of Shanghai Composite Index, Shanghai Bond Index and Shanghai Fund Index are considered. Statistical tests are used to detect the nonlinear auto-correlated structures and long-range cross-correlations of the three time series. The multifractal detrended fluctuation analysis and multifractal spectrum analysis methods are applied, by which the existence of multifractality in these three return series are revealed and the sources of multifractality are explored. In particular, the multiscale multifractal detrended cross-correlation analysis method is employed for the first time to generate the Hurst surfaces, which can be used to visualize the dynamic behaviors of cross-correlations among the markets. Empirical results show that the cross-correlations among the markets present different fractal features at different time scales. Further, our study finds that the correlation between the stock and fund markets is stronger than that of the other two groups, and the correlation between the stock and bond markets is unstable. These findings can help to better understand the dynamic mechanisms that govern the volatility of security markets and aid in performing better financial risk assessment and management.