Generalized Hurst Exponent Research Articles

Ethereum futures and the efficiency of cryptocurrency spot markets

This paper examines the impact of the introduction of Ethereum futures contracts on the market efficiency of major cryptocurrency (Bitcoin, Ethereum, Ripple, Litecoin, and Dogecoin) spot prices. Using a multifractality-based approach and daily data from September 4, 2017 to February 16, 2024, the main results show a slight improvement in market efficiency. Specifically, the degree of multifractality persistence decreases, implying reduced market inefficiencies in major cryptocurrencies. The temporal linear correlation effect and thick tail effect are less pronounced post-launch. The asymmetry of the generalized Hurst exponent increases after the launch of Ethereum futures, with a higher persistence under the downward trend of cryptocurrencies noted. This downward trend emerges after the launch of Ethereum futures, coinciding with the final stage of the COVID-19 pandemic. Additional analysis shows that fat tails and temporal linear correlations are the main sources of multifractality. The results highlight the influence of introducing financial derivatives into the relatively new and volatile cryptocurrency area, which should concern traders, hedgers, investors, and regulators.

Impact of surface-roughness and fractality on electrical conductivity of SnS thin films

Mono- and multi-fractal geometry have been used to explore the surface characteristics of scanning electron microscopy (SEM) micrographs of the SnS films with thicknesses of 100nm (SnS1) to 600nm (SnS4), respectively. For this investigation, the SnS thin films have been grown on fluorine-doped tin oxide (FTO)-coated glass substrate through the thermal evaporation route, and surface morphologies are captured by SEM. Two-dimensional multi-fractal detrended fluctuation analysis (MFDFA) based on the partition function is used to examine whether the surfaces have a multi-fractal nature or not. The partition function is applied to extract the generalized Hurst exponent from the segment size. It has been found that surfaces with higher surface roughness induce substantial nonlinearity and a wider width of the multi-fractal spectrum. The multi-fractal spectrum acquired from the analysis of the geometry and shape of the singularity spectrum is used to quantify the irregularity and complexity of surfaces. Minkowski functionals (MFs) parameters such as volume, boundary, and connectivity were measured for each thin film. Moreover, we tried to correlate the electrical conductivity with the mono- and multi-fractal parameters such as fractal dimension (Df), singularity strength function (Δα), singularity spectrum Δf(α), and it is observed that the conductivity of a thin film decreases with decreasing fractal dimension. The minimum (maximum) resistivity (conductivity) was observed for the surface having a larger fractal dimension. The present investigation suggests that such SnS surfaces, having minimal resistivity and maximum conductivity on the roughest surface, indicate enhanced light trapping capacity and can be utilized as active layers for advanced optoelectronics devices.

Multifractal properties of Covid-19 data

Multifractal detrended moving average analysis (MFDFA) has been performed on time series of three different epidemiological indices, namely daily number of new cases per million population, reproduction rate and positive rate, recorded during the Covid-19 pandemic. We consider India, Japan, USA and UK for this study. For all the series the scaling behavior of MFDMA fluctuation functions has been examined, and the multifractal variables like the generalised Hurst exponent spectra and singularity spectra have been calculated. It is found that the time series of the Covid-19 indices are of multifractal nature. The degree of multifractality of the reproduction rate series is significantly weaker than the other two series for all the countries. To find out the source(s) of multifractality, we generate a set of one hundred shuffled series and Amplitude-Adjusted Fourier Transform surrogate series for each of the original series. Comparing the results obtained from the original series with those from the shuffled and surrogate series we understand that the main source of multifractality in Covid-19 data is long-range temporal correlation among the series values.

Investigation of Temperature Multifractrality According to Zugspitze Weather Station Data

The main multifractal properties of time series of mean, maximum and minimum daily temperatures are analyzed using the method of multifractal fluctuation analysis. As initial data, we used the results of instrumental temperature observations made at the Zugspitze meteorological station in the period from August 1, 1900 to January 31, 2023. In general, variations in the mean, maximum and minimum daily temperatures demonstrate multifractal behavior, especially for small time scales, up to about 90 days An analysis of the generalized Hurst exponent found that the considered time series have a long-term positive correlation and that the multifractality is weaker with large fluctuations. The singularity spectrum for all time series is truncated to the left, which means that the time series have a multifractal structure that is insensitive to local fluctuations of large values.

COVID-19 and persistence in the stock market: a study on a leading emerging market

AbstractIn this study, we examine how sectors of the National Stock Exchange from India respond to the uncertainties introduced by the COVID-19 pandemic. By examining the synchronization between the sector-specific and overall market index (NIFTY 50) reaction to COVID-19, we contribute to the inconclusive ongoing academic literature regarding the impact of COVID-19 on the stock market, especially in the context of persistence in an emerging market. To analyze the persistence of sectoral indices, we apply multifractal detrended fluctuation analysis (MFDFA). We use the generalized Hurst exponent and singularity spectrum as indicators for persistence and spectral width as a measure of volatility. Our analysis shows that the sample sectoral indices are persistent before and after the announcement of COVID-19; however, volatility in some sectors reduces post-announcement of COVID-19. The findings will enrich the academic literature on the relationship between sector-specific and overall market indexes. In practice, the paper will guide investors to organize their portfolios, especially during future economic uncertainty.

Modified MF-DFA Model Based on LSSVM Fitting

This paper proposes a multifractal least squares support vector machine detrended fluctuation analysis (MF-LSSVM-DFA) model. The system is an extension of the traditional MF-DFA model. To address potential overfitting or underfitting caused by the fixed-order polynomial fitting in MF-DFA, LSSVM is employed as a superior alternative for fitting. This approach enhances model accuracy and adaptability, ensuring more reliable analysis results. We utilize the p model to construct a multiplicative cascade time series to evaluate the performance of MF-LSSVM-DFA, MF-DFA, and two other models that improve upon MF-DFA from recent studies. The results demonstrate that our proposed modified model yields generalized Hurst exponents h(q) and scaling exponents τ(q) that align more closely with the analytical solutions, indicating superior correction effectiveness. In addition, we explore the sensitivity of MF-LSSVM-DFA to the overlapping window size s. We find that the sensitivity of our proposed model is less than that of MF-DFA. We find that when s exceeds the limited range of the traditional MF-DFA, h(q) and τ(q) are closer than those obtained in MF-DFA when s is in a limited range. Meanwhile, we analyze the performances of the fitting of the two models and the results imply that MF-LSSVM-DFA achieves a better outstanding performance. In addition, we put the proposed MF-LSSVM-DFA into practice for applications in the medical field, and we found that MF-LSSVM-DFA improves the accuracy of ECG signal classification and the stability and robustness of the algorithm compared with MF-DFA. Finally, numerous image segmentation experiments are adopted to verify the effectiveness and robustness of our proposed method.

The Multifractal Phenomenon of Stock Price Caused by “Tesla Rights Defense Event”

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 [Formula: see text] and [Formula: see text] 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 [Formula: see text] and two [Formula: see text], which are regarded as [Formula: see text] and [Formula: see text]. 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 [Formula: see text] and [Formula: see text] 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, [Formula: see text] and [Formula: see text] gradually stabilize, indicating that public memorability will diminish and the market efficiency almost remains unchanged.

Detection of long-range correlation causing multifractality in H time series of geomagnetic field over the Northern Hemisphere during quiet geomagnetic conditions

The generation of geomagnetic time series frequently involves intricate spatio-temporal dynamics, wherein nonlinearity and scaling emerge as pivotal mechanisms. The present work investigates the fractal properties of geomagnetic field, and its sources during quiet geomagnetic conditions, by analyzing the horizontal component data of 1 min resolution (H time series) from different latitude regions. Analysis using Auto Correlation Function (ACF), Power spectrum analysis (PSA), Rescaled range analysis (R/S) and Detrended Fluctuation Analysis (DFA) gave indications of the presence of long-range correlations, fractal properties and scaling behaviour of quiet period H time series data. The strong q dependence of generalized Hurst exponent h(q) and scaling exponent Ʈ(q), obtained from the Multifractal Detrended Fluctuation Analysis (MFDFA) method, confirmed the multifractal behaviour of quiet period H time series. The multifractal source analysis of the quiet period H time series with its shuffled and surrogate time series data revealed that, the multifractal characteristics are caused almost entirely by the long-range correlations between small and large fluctuations. The analysis of the latitudinal variations in the multifractal characteristics is also performed as a part of the work, by considering H time series belonging to various latitudes under almost same longitude. The observed decrease in the generalized Hurst exponent values with increase in latitude indicates the change in the multifractal characteristics of the time series with latitude. Also the work revealed that, the H time series from high latitude region shows anti-persistent behaviour, even when the H time series from observatories located at the low and mid latitude regions exhibiting persistent behaviour. The change in the persistence, anti-persistence behaviour can be related to the strength of Earth’s magnetic field.

The multifractality of the internal geomagnetic field

It is well established that the external geomagnetic field exhibits distinct multifractal behaviour with multiple characteristic timescales that are a manifest of the complex solar-driven dynamics of the magnetosphere or ionosphere. The internal magnetic field on the other hand is characterized by much longer timescales. Consequently, in order to observe any multifractal behaviour, there is needed a time series of magnetic dipolar moment spanning thousands or even millions of years of the past geomagnetic field of the core. Here, we analyse two series that describe the dipolar ingredient of the geomagnetic field for the last 4 and 2 million years, respectively. The first series is constructed from a continuous streak of sedimentary samples while the other series is a composite of a myriad of shorter time series contained within the 2 million years frame. We analysed the Generalized Hurst Exponent through the Multi Fractal-Detrended Fluctuation Analysis method and in each case, we observed typical multifractal structures. These results may be the manifestation of different phenomena evolving in the liquid outer core, possibly providing hints and insights into the details of the corresponding mechanisms. Analysis of the time evolution of the Generalized Hurst Exponent for each series displays a complicated behaviour. Typically the anti-persistent character of the series becomes more evident when close to e dipole reversal. Special efforts need to be dedicated in identifying connections between geomagnetic reversals and time dependence of the Generalized Hurst Exponent.

Analysis of the structural complexity of Crab Nebula observed at radio frequency using a multifractal approach

The Crab Nebula is an astrophysical system that exhibits complex morphological patterns at different observing frequencies. We carry out a systematic investigation of the structural complexity of the nebula using publicly available imaging data at radio frequency. For the analysis, we use the well-known multifractal detrended fluctuation analysis in two dimensions. We find that radio data exhibit long-range correlations, as expected from the underlying physics of the supernova explosion and evolution. The correlations follow a power-law scaling with length scales. The structural complexity is found to be multifractal in nature, as evidenced by the dependence of the generalized Hurst exponent on the order of the moments of the detrended fluctuation function. By repeating the analysis on shuffled data, we further probe the origin of the multifractality in the radio imaging data. For the radio data, we find that the probability density function is close to a Gaussian form. Hence, the multifractal behavior is due to the differing nature of long-range correlations of the large and small detrended fluctuation field values. We investigate the multifractal parameters across different partitions of the radio image and find that the structures across the image are highly heterogeneous, making the Crab Nebula a structurally complex astrophysical system. Our analysis thus provides a fresh perspective on the morphology of the Crab Nebula from a complexity science viewpoint.

MULTIFRACTAL CHARACTERISTICS OF THE MITOCHONDRIAL BK CHANNEL ACTIVITY IN THE PRESENCE OF THE CHANNEL-ACTIVATING FLAVONOIDS

Over the years, the (multi)fractal nature of signals describing the activity of ion channels has been observed both at the level of single-channel currents and the temporal scaling of the corresponding sojourns in conducting and non-conducting states. The recognized nonlinear and self-similar properties were interpreted regarding the underlying channel gating machinery. Nevertheless, the literature lacks in (multi)fractal description of the biochemical stimulation of ion channels. Therefore, in this work, we provide a multifractal description of the effects exerted by two different flavonoids — naringenin (Nar) and quercetin (Que) — being the regulators of the large-conductance voltage- and Ca[Formula: see text]-activated K[Formula: see text] channels from the inner mitochondrial membrane (mitoBK). Toward this aim, the focus-based multifractal detrended fluctuation analysis (MFDFA) was applied to the patch-clamp data obtained in the presence of Nar and Que in micromolar concentrations and the corresponding controls. Our results show that mitoBK channel activity has well-pronounced multifractal characteristics and long-range memory (measured by the generalized Hurst exponent) under biochemical stimulation by Nar and Que and in the absence of these substances. The spectral properties significantly change after the randomization of the analyzed series. Therefore, multifractality is an inherent feature of the mitoBK channel currents, which stems from their statistical distribution and temporal organization. Both flavonoids have similar structures and exert channel-activating effects, but they differently affect the parameters of the multifractal spectra of the corresponding patch-clamp signals. Coordination of Que leads to more prominent changes in signal complexity than Nar at membrane hyper- and depolarization. It suggests that this flavonoid has a stronger impact on the conformational dynamics of the mitoBK channels in comparison with naringenin.

Multifractality approach of a generalized Shannon index in financial time series.

Multifractality is a concept that extends locally the usual ideas of fractality in a system. Nevertheless, the multifractal approaches used lack a multifractal dimension tied to an entropy index like the Shannon index. This paper introduces a generalized Shannon index (GSI) and demonstrates its application in understanding system fluctuations. To this end, traditional multifractality approaches are explained. Then, using the temporal Theil scaling and the diffusive trajectory algorithm, the GSI and its partition function are defined. Next, the multifractal exponent of the GSI is derived from the partition function, establishing a connection between the temporal Theil scaling exponent and the generalized Hurst exponent. Finally, this relationship is verified in a fractional Brownian motion and applied to financial time series. In fact, this leads us to proposing an approximation called local fractional Brownian motion approximation, where multifractal systems are viewed as a local superposition of distinct fractional Brownian motions with varying monofractal exponents. Also, we furnish an algorithm for identifying the optimal q-th moment of the probability distribution associated with an empirical time series to enhance the accuracy of generalized Hurst exponent estimation.

THE (IN)EFFICIENCY OF USA EDUCATION GROUP STOCKS: BEFORE, DURING AND AFTER COVID-19

This paper represents a pioneering effort to investigate multifractal dynamics that exclusively encompass the return time series of USA Education Group Stocks concerning two non-overlapping periods (before, during, and after COVID-19). Given this, we employ the Multifractal Detrended Fluctuations Analysis (MF-DFA). In this sense, we investigate the generalized Hurst exponent [Formula: see text] and the Rényi exponent [Formula: see text] for each asset and quantify their statistical properties, which allowed us to observe separately the contributing small scale (primarily via the negative moments [Formula: see text]) and the large scale (via the positive moments [Formula: see text]). We perform a fourth-degree polynomial regression fit to estimate the complexity parameters that describe the degree of multifractality of the underlying process. Also, we shall apply the inefficiency multifractal index to assess the COVID-19 shock for both periods. Our findings show that for both periods, the majority of these assets are marked by multifractal dynamics associated with persistent behavior [Formula: see text], a higher degree of multifractality and the dominance of large fluctuations. At the same time, most of these assets show asymmetry parameter [Formula: see text] for both periods, indicating that large fluctuations contributed more to multifractality in the time series of returns.

Spatiotemporal Variation and Long-Range Correlation of Groundwater Levels in Odessa, Ukraine

Increasing groundwater levels (GWLs) may become one of the most serious issues for the city of Odessa, Ukraine. This study investigated the spatial distribution characteristics and multifractal scaling behaviour of the groundwater-level/-depth fluctuations for a Quaternary aquifer in the city of Odessa using a geostatistical approach and multifractal detrended fluctuation analysis (MF-DFA). These two methods were applied to monthly GWL fluctuation time series from 1970 to 2020 to monitor 72 hydrogeological wells situated in different parts of the city of Odessa. The spatial distribution of the GWLs revealed an overall trend of decline and recovery from 1970 to 2020 in the study area, except for most of the southern region, where a persistent recovery of the groundwater depth was observed. The MF-DFA results suggest that the dynamics of the GWL fluctuations have multifractal characteristics in the Odessa area. In addition, both long-range correlations and fat-tail probability distribution contribute to the multifractality. However, long-range correlations among the fluctuations made a major contribution to the observed multifractality of the GWL fluctuation time series. The generalised Hurst exponents show a wide range of change (0.20 < h(q) < 2.85), indicating the sensitivity of the GWL fluctuations to changes in small-scale factors and large-scale factors. Regarding the long-range correlations of the GWL depths, the Hurst exponents (q = 2) demonstrated the positive persistence of groundwater-depth recovery in the southern region and the persistence of groundwater-depth variation in the other regions of the study area. The dynamic changes in the GWL depths in the Odessa area may be influenced by both natural and anthropogenic factors.

Regional Multifractal Variability of the Overall Seismic Activity in Pakistan from 1820 to 2020 via the Application of MDFA on Earthquake Catalogs

The overall seismicity of Pakistan from 1820 to 2020 is analysed in terms of its multifractal behaviour. Seismic events of magnitude ML = 3.0 and above are spatially clustered into four distinct groups, each one corresponding to a different region of high seismic activity. The Multifractal Detrended Fluctuation Analysis (MFDA) method applied on each cluster reveals pronounced inter-cluster heterogeneity in terms of the resulting generalised Hurst exponent and fractality spectrum, possibly due to the particular tectonic characteristics of the regions under investigation. Additional results on the variability of the Gutenberg–Richter b-value across the defined clusters further corroborate the uniqueness of the seismic profile of each region.

Santa Ana Winds: Multifractal Measures and Singularity Spectrum

A multifractal analysis based on the time series of temperature, pressure, relative humidity, wind speed, and wind direction was performed for 16 weather stations located in the hydrographic basin of the Guadalupe River in Baja California, Mexico. Our analysis included a 38-year dataset from MERRA-2 database, we investigated the multifractal nature of daily time series data for climatic variables associated with the Santa Ana Winds. We employed the Multifractal Detrended Fluctuation Analysis (MFDFA) method to extract multifractal complexity parameters (α0, ∆α, and r). This was adequate to evaluate the multifractality of the time series that represented the conditions of the phenomenon’s occurrence. From the estimation of the generalized Hurst exponent (hq), it was possible to characterize the time series of the meteorological variables in terms of the characteristics of persistence, anti-persistence, or randomness. Finally, the values corresponding to the parameters and characteristics of the multifractal spectrum or singularities can be used as quantitative and qualitative indicators to describe the dynamics of meteorological processes during the occurrence of the Santa Ana winds in the Guadalupe basin.

Hotel Comment Emotion Classification Based on the MF-DFA and Partial Differential Equation Classifier

Due to the significant value that hotel reviews hold for both consumers and businesses, the development of an accurate sentiment classification method is crucial. By effectively distinguishing the authenticity of reviews, consumers can make informed decisions, and businesses can gain insights into customer feedback to improve their services and enhance overall competitiveness. In this paper, we propose a partial differential equation model based on phase-field for sentiment analysis in the field of hotel comment texts. The comment texts are converted into word vectors using the Word2Vec tool, and then we utilize the multifractal detrended fluctuation analysis (MF-DFA) model to extract the generalized Hurst exponent of the word vector time series to achieve dimensionality reduction of the word vector data. The dimensionality reduced data are represented in a two-dimensional computational domain, and the modified Allen–Cahn (AC) function is used to evolve the phase values of the data to obtain a stable nonlinear boundary, thereby achieving automatic classification of hotel comment texts. The experimental results show that the proposed method can effectively classify positive and negative samples and achieve excellent results in classification indicators. We compared our proposed classifier with traditional machine learning models and the results indicate that our method possesses a better performance.

A modified multifractal detrended fluctuation analysis to study the precipitation across northeast India

Using multifractal detrended fluctuation analysis (MF-DFA), the pre-monsoon rainfall and Indian summer monsoon rainfall (ISMR) from 1871 to 2016 are examined in this work for northeast India. According to observations, rainfall during the pre-monsoon and summer monsoons both exhibit persistence and quasi-randomness. The generalized Hurst Exponent value indicates that the pre-monsoon rainfall time series and the summer monsoon rainfall time series have long-term positive auto-correlation. The fluctuation function obtained through MF-DFA by means of the slopes of ln(Fq(s)) against ln(s) plots is dominated by segments with significant variance, which leads us to believe that the rainfall time series related to the pre- and summer monsoon exhibit multifractal behaviour. The presence of multifractality in the pre- and summer monsoon rainfall time series is further substantiated through the association between the qth-order variance and the length scale.

Performance degradation assessment for mechanical system based on semi-analytical solution of self-similar stable distribution process

To more accurately predict remaining useful life (RUL) and quantitatively evaluate the uncertainty of the predicted results, a performance degradation assessment framework based on semi-analytical solution of self-similar stable distribution process is proposed. The established performance degradation model based on adaptive fractional Lévy stable motion (AFLSM) is more flexible in revealing the long-range dependence, non-Gaussian, and heavy-tailed distribution properties of the incremental behavior. The corresponding stable distribution parameters are estimated through characteristic function method, and Hurst exponent is calculated based on the generalized Hurst exponent approach with narrower confidence interval. Aiming at the difficulties in solving the exact analytical solution and the excessive computation of the numerical solution in the whole process, based on Mellin-Stieltjes transform and direct integration, a semi-analytical solution of RUL distribution function is proposed, which can be readily implemented in practical equipment operations. The proposed performance degradation assessment framework is validated by the novel truck transmission dataset and the benchmark rolling bearing dataset. Experimental results indicate that the developed framework is more effective and superior than other state-of-the-art approaches in terms of RUL prediction.

Correlating the Nonlinear Roughening and Optical Properties of Anatase Thin Films—A Fractal Geometric Approach

AbstractThe anatase phase of TiO2 is highly suitable for photocatalytic application consequently, prerequisite an unpretentious scale‐independent understanding of its roughening‐facilitated surface topography dynamics. Herein, the nonlinear roughening in anatase thin films and its correlation with the nonlinear trend of optical properties in the framework of fractal geometry are investigated. The self‐affine nature of analyzed surfaces is confirmed from the autocorrelation function. The dynamic scaling exponents indicate the existence of Kardar–Parisi–Zhang scaling in surface growth. In addition, the trend of generalized Hurst exponent and mass exponent indicates insignificant multifractal characteristics in the analyzed surfaces. Moreover, fractal analysis explains and aids to description of roughening trend from stereometric and Minkowski functionals analysis. Furthermore, the significance of fractal dimension for probing the surface roughening is validated from the principal component analysis. Consequently, variation in optical bandgap and linear refractive index are investigated in regards to the fractal dimension and root mean‐squared surface slope and regression equations are proposed for tuning of bandgap.