Generalized Hurst Exponent Research Articles

Generalized Approach to Analysis of Multifractal Properties from Short Time Series

The paper considers a generalized approach to the time series multifractal analysis. The focus of research is on the correct estimation of multifractal characteristics from short time series. Based on numerical modeling and estimating, the main disadvantages and advantages of the sample fractal characteristics obtained by three methods: the multifractal fluctuation detrended analysis, wavelet transform modulus maxima and multifractal analysis using discrete wavelet transform are studied. The generalized Hurst exponent was chosen as the basic characteristic for comparing the accuracy of the methods. A test statistic for determining the monofractal properties of a time series using the multifractal fluctuation detrended analysis is proposed. A generalized approach to estimating the multifractal characteristics of short time series is developed and practical recommendations for its implementation are proposed. A significant part of the study is devoted to practical applications of fractal analysis. The proposed approach is illustrated by the examples of multifractal analysis of various real fractal time series.

Study of the fractal nature of evapotranspiration time series from agricultural regions of Northern Karnataka

Multifractal detrended fluctuation analysis (MFDFA) renders valuable insights pertaining to the randomness, inner regularity and long range correlations in the time series data. We applied this technique to assess the fractal behaviour and inherent correlations of the potential and reference crop evapotranspiration data collected from two regions of Karnataka, viz., Belgaum and Raichur. The annual periodic nature of the evapotranspiration data series was removed using seasonal trend decomposition method and it was seen that all the decomposed series consisted of long-term persistence. The multifractal behaviour in the evapotranspiration series at the two stations were seen owing to the strong dependency of generalised Hurst exponent H(q) on the values of q. We also tested for the reduction in the dependence of h(q) on q by shuffling the evapotranspiration time series and it is indicative of the fact that the multifractality is responsible for the correlation characteristics and also the probability density function of the evapotranspiration data series. The two regions, Belgaum and Raichur vary considerably in climatic and geomorphic conditions and this difference is evident in their corresponding evapotranspiration data as fractal properties. This is depicted by the values of the generalised Hurst exponent H(q) obtained using MFDFA technique.

Study of the fractal nature of evapotranspiration time series from agricultural regions of Northern Karnataka

Multifractal detrended fluctuation analysis (MFDFA) renders valuable insights pertaining to the randomness, inner regularity and long range correlations in the time series data. We applied this technique to assess the fractal behaviour and inherent correlations of the potential and reference crop evapotranspiration data collected from two regions of Karnataka, viz., Belgaum and Raichur. The annual periodic nature of the evapotranspiration data series was removed using seasonal trend decomposition method and it was seen that all the decomposed series consisted of long-term persistence. The multifractal behaviour in the evapotranspiration series at the two stations were seen owing to the strong dependency of generalised Hurst exponent H(q) on the values of q. We also tested for the reduction in the dependence of h(q) on q by shuffling the evapotranspiration time series and it is indicative of the fact that the multifractality is responsible for the correlation characteristics and also the probability density function of the evapotranspiration data series. The two regions, Belgaum and Raichur vary considerably in climatic and geomorphic conditions and this difference is evident in their corresponding evapotranspiration data as fractal properties. This is depicted by the values of the generalised Hurst exponent H(q) obtained using MFDFA technique.

Differentiation of early mild cognitive impairment in brainstem MR images using multifractal detrended moving average singularity spectral features

Brainstem texture analysis can provide valuable information in the diagnosis of Early Mild Cognitive Impairment (EMCI) condition. In this work, 3D brainstem structure is segmented and analysed for texture alterations using multifractal features to differentiate EMCI from other Alzheimer’s disease stages. The images obtained from public domain database are preprocessed for spatial registration, skull stripping and contrast enhancement. White matter volume is segmented from the preprocessed images using fuzzy ‘C’ means clustering algorithm. Midsagittal white matter tissue is used as the initial seed to segment the brainstem volume using sparse field level set method. Multifractal detrended moving average algorithm is used to compute the fluctuation function, generalized Hurst exponent and mass exponent to study the multifractal characteristics of brainstem structure. Features extracted from the multifractal spectrum are analysed to differentiate the images pertaining to EMCI subject group. Results indicate that the proposed technique is able to segment the brainstem structure from all the considered images. The fluctuation function is observed to have linear relationship with scale. The generalized Hurst exponent decreases with order and mass exponent follows a non-linear trend demonstrating the multifractal nature of brainstem. Singularity spectral features namely strength of multifractality, Holder exponent at f(2.8), tangent slope and maximum Holder exponent are found to be most significant in differentiating EMCI from subject groups. As this complex EMCI distinction is clinically important, the proposed approach is useful for early diagnosis of Alzheimer’s condition.

Comparison of the Scaling Properties of EUV Intensity Fluctuations in Coronal Hole and Quiet-Sun Regions

Abstract Using detrended fluctuation analysis and rescaled range analysis, we investigate the scaling properties of extreme ultraviolet (EUV) intensity fluctuations of low-latitude coronal holes (CHs) and neighboring quiet-Sun (QS) regions in signals obtained with the Solar Dynamics Observatory (SDO)/Atmospheric Imaging Assembly instrument. Contemporaneous line-of-sight SDO/Helioseismic and Magnetic Imager magnetic fields provide a context for the physical environment. We find that the intensity fluctuations in the time series of EUV images present at each spatial point a scaling symmetry over the range ∼20 minute to ∼1 hr. Thus we are able to calculate a generalized Hurst exponent and produce image maps, not of physical quantities like intensity or temperature, but of a single dynamical parameter that sums up the statistical nature of the intensity fluctuations at each pixel. In QS regions and in CHs with magnetic bipoles, the scaling exponent (1.0 < α ≤ 1.5) corresponds to anticorrelated turbulent-like processes. In CHs, and in QS regions primarily associated with (open) magnetic field of dominant polarity, the generalized exponent (0.5 < α < 1) corresponds to positively correlated (persistent) processes. We identify a tendency for α ∼ 1 near CH boundaries and in other regions in which open and closed magnetic fields are in proximity. This is a signature of an underlying 1/f type process that is characteristic for self-organized criticality and shot-noise models.

Rough volatility of Bitcoin

Recent studies have found that the log-volatility of asset returns exhibits roughness. This study investigates roughness or the anti-persistence of Bitcoin volatility. Using multifractal detrended fluctuation analysis, we obtain the generalized Hurst exponent of the log-volatility increments and find that the generalized Hurst exponent is less than 1/2, which indicates rough log-volatility increments. Furthermore, we find that the generalized Hurst exponent is not constant. This observation indicates that the log-volatility has a multifractal property. Using shuffled time series of the log-volatility increments, we infer that the source of multifractality partly derives from the distributional property.

Multifractal analysis of the impact of US–China trade friction on US and China soy futures markets

We investigate the impact of the 2018 US–China trade friction on the multifractality of soy futures markets. Using DMCA and MFDMA methods, we study the cross-correlation coefficient of soy futures markets, and analyze the multifractality and origins of multifractality before and during this trade friction. We find: (a) The cross-correlations coefficients decrease significantly during the trade friction, and different trends of cross-correlations are found. (b) The singularity widths of all soy futures markets and the cross-correlations decrease during trade friction; whilst the generalized Hurst exponents of all series experience significant increases, especially for large fluctuations. (c) The finite-size effect and the fat-tailed probability distributions can explain the major decrease in multifractality. Comparing the period before, soy futures markets became less correlated, less multifractal and more persistent during this trade friction. The evidences suggest that as the tariff on soy was put into effect, investigators in two countries may take different hedging behaviors and be more risk averse.

Market Efficiency, Liquidity, and Multifractality of Bitcoin: A Dynamic Study

This letter investigates the dynamic relationship between market efficiency, liquidity, and multifractality of Bitcoin. We find that before 2013 liquidity is low and the Hurst exponent is less than 0.5, indicating that the Bitcoin time series is anti-persistent. After 2013, as liquidity increased, the Hurst exponent rose to approximately 0.5, improving market efficiency. For several periods, however, the Hurst exponent was found to be significantly less than 0.5, making the time series anti-persistent during those periods. We also investigate the multifractal degree of the Bitcoin time series using the generalized Hurst exponent and find that the multifractal degree is related to market efficiency in a non-linear manner.

Multi-fractal Behaviors of long term daily relative humidity and temperature observed over Benin synoptic stations (West Africa)

The multifractal structure of daily temperature and relative humidity is investigated in this study. Multifractal Detrended Fluctuation Analysis (MFDFA) method has been applied on data observed from 1967 to 2012 at the six synoptic stations of Benin (Cotonou, Bohicon, Parakou, Save, Natitingou and Kandi). We estimate the generalized Hurst exponent, the Renyi exponent, and the singularity spectrum from the data to quantify the multi-fractal behaviors. The results show that multi-fractality exists in both daily humidity and temperature record at Benin synoptic stations. It shows multi-fractality with the curves of h (q), τ (q) and D (q), depending on the values of q. The comparison of the multifractal properties shows that, at all the synoptic stations, the multifractal strength of the temperature is significantly different from the feature the humidity.For the temperature, among the six study sites, the multifractal strength at Natitingou is largest (∆α = 0.6917). This means that Natitingou is the city in which the multifractal property is strongly observed for temperature. At Parakou the multifractal strength is smallest (∆α = 0.5252), meaning that Parakou is the city in which the multifractal property is weakly observed. At all synoptic stations the multifractal strength are superior to 0.5 (Δα> 0.5) indicating the degree of multifractal in temperature time series.For the relative humidity, multifractal strength is smallest Kandi (∆α = 0.3031). This means that Kandi is the city in which the multifractal property is weakly observed. Furthermore, the multifractal strength of Parakou is largest (∆α = 0.7691) meaning that for the relative humidity, Parakou is the city in which the multifractal property is strongly observed. The geographic distribution of the multifractal strength reflects the role of climate dynamic processes on the multi-fractal behavior of humidity and the distinctiveness of physical processes in Benin.

Trends of Global Seismic Noise Properties in Connection to Irregularity of Earth’s Rotation

The properties of continuous records of low-frequency seismic noise on a global network consisting of 229 broadband seismic stations located around the world are considered. Changes in the properties of seismic noise, estimated daily for the time interval from the beginning of 1997 to the end of February 2019, are investigated. We consider the generalized Hurst exponent, the singularity spectrum support width, the entropy of the wavelet coefficients, and wavelet-based Donoho–Johnstone index. For the centers of 50 clusters of seismic stations, the average values of these 4 statistics from the 5 nearest operational stations are calculated daily. As a result, 4 multidimensional time series with a dimension of 50 are obtained with a time step of 1 day for more than 22 years of measurements. Average daily values of the noise properties studied, calculated over all cluster centers, have piecewise linear trends, the break point of which is estimated by the principal component method as mid-2003. After the break point, the average values of generalized Hurst exponent, singularity spectrum support width and Donoho–Johnstone index the parameters decrease whereas the entropy increases. This is interpreted as a simplification of the noise structure which is typical for areas of high seismic hazard. Trends in average noise properties after 2003 are accompanied by a linear increase with imposed 3-years quasi-periodic fluctuations in the average value of pairwise correlation coefficients between the values in cluster centers when evaluated in a sliding time window with a length of 1 year. It is hypothesized that the simultaneous simplification of the structure of global seismic noise, an increase in its spatial correlation and an increase in the intensity of the strongest earthquakes in the world after the end of 2004 is a single process associated with the irregularity of the Earth’s rotation. To confirm this hypothesis, a change in the coherence spectrum between the first principal component of the seismic noise properties and the time series of the length of day is estimated.

Analysis of uterine EMG signals in term and preterm conditions using generalised Hurst exponent features

An attempt has been made in this Letter to analyse term (week of gestation (WOG) >37) and preterm (WOG ≤ 37) conditions using uterine electromyography (uEMG) signals and generalised Hurst exponent (GHE) features. For this analysis, public database signals recorded from the surface of abdomen are considered. Multifractal detrended fluctuation analysis is performed on the signals and the GHE is calculated. From the exponent, seven features are extracted and data-balancing based on synthetic minority over-sampling technique is used to retain a balanced feature contribution by the term and preterm records. Two classification algorithms namely, Naive Bayes and logistic regression (LR) are employed to classify the signals. Ten-fold cross validation approach is executed and the performance is validated using accuracy, precision and recall. The results show the uEMG signals exhibit multifractal characteristics and five GHE features are significant in distinguishing the term and preterm uEMG signals. The LR classifier gives the highest accuracy of 97.8%. Therefore, it appears that the multifractal Hurst exponent features in combination with LR classifier can be used as biomarkers for predicting the preterm or term delivery during the early stage of gestation.

Multifractality and market efficiency of carbon emission trading market: Analysis using the multifractal detrended fluctuation technique

An efficient carbon market is important for mitigating carbon emissions and many other energy issues. In this study we investigate the multifractality and efficiency of carbon market using Multifractal Detrended Fluctuation Analysis, and the data are based on the carbon price returns of seven pilots in the carbon market of China. First, we find the price return series is multifractal for both the short term and the long term. Asymmetric distributions of carbon prices and the general Hurst exponents in the return series are detected, which suggest a lack of efficiency in the pilot markets under study. Then, we measure the efficiencies of these carbon market pilots using an indicator that we develop based on the generalized Hurst exponents, and the results suggest ubiquitous inefficiencies among them. But it is also found that the efficiencies of most of the pilots have been improving over time. The ranking of the pilot markets’ efficiencies depends on the time window under study. For time scales shorter than a month, Chongqing pilot generally ranks the highest and Hubei pilot the lowest. Beijing pilot performs the best while Tianjin the worst when the time scale is longer. Lastly, regression results show that the carbon market pilot’s inefficiency value is positively correlated with market activity in the short run while negatively in the long run. Based on the empirical results, effect of the inefficiency on low-carbon development and the policy implications are discussed.

Exploring the dynamic relationship between crude oil price and implied volatility indices: A MF-DCCA approach

This paper investigates the cross-correlation between crude oil prices and implied volatility indices — theinvestor’s fear gauges, using a cross-correlation statistical test and multifractal detrended cross-correlation analysis (MF-DCCA). The results show that the cross-correlations between crude oil prices and three different implied volatility indices are multifractal. By finding the “crossover”, we separate the three pairs of series into the short- and long-term, respectively, and find that the cross-correlations are strongly anti-persistent in both short- and long-term. Moreover, cross-correlations of small and large fluctuations are anti-persistent in the short- and long-term, suggesting that crude oil prices and implied volatility indices are susceptible to each other. We also find that the cross-correlation exponents are less than the average generalized Hurst exponent when q<0 and more than the average generalized Hurst exponent when q>0 in the short-term and that the opposite results are true in the long-term.

A Modified Multifractal Detrended Fluctuation Analysis (MFDFA) Approach for Multifractal Analysis of Precipitation in Dongting Lake Basin, China

Multifractal detrended fluctuation analysis (MFDFA) method can examine higher-dimensional fractal and multifractal characteristics hidden in time series. However, removal of local trends in MFDFA is based on discontinuous polynomial fitting, resulting in pseudo-fluctuation errors. In this paper, we propose a two-stage modified MFDFA for multifractal analysis. First, an overlap moving window (OMW) algorithm is introduced to divide time series of the classic MFDFA method. Second, detrending by polynomial fitting local trend in traditional MFDFA is replaced by ensemble empirical mode decomposition (EEMD)-based local trends. The modified MFDFA is named OMW-EEMD-MFDFA. Then, the performance of the OMW-EEMD-MFDFA method is assessed by extensive numeric simulation experiments based on a p-model of multiplicative cascading process. The results show that the modified OMW-EEMD-MFDFA method performs better than conventional MFDFA and OMW-MFDFA methods. Lastly, the modified OMW-EEMD-MFDFA method is applied to explore multifractal characteristics and multifractal sources of daily precipitation time series data at the Mapoling and Zhijiang stations in Dongting Lake Basin. Our results showed that the scaling properties of the daily precipitation time series at the two stations presented a long-range correlation, showing a long-term persistence of the previous state. The strong q-dependence of H ( q ) and τ ( q ) indicated strong multifractal characteristics in daily precipitation time series data at the two stations. Positive Δ f values demonstrate that precipitation may have a local increasing trend. Comparing the generalized Hurst exponent and the multifractal strength of the original precipitation time series data with its shuffled and surrogate time series data, we found that the multifractal characteristics of the daily precipitation time series data were caused by both long-range correlations between small and large fluctuations and broad probability density function, but the broad probability density function was dominant. This study may be of practical and scientific importance in regional precipitation forecasting, extreme precipitation regulation, and water resource management in Dongting Lake Basin.

Chaos and order in the bitcoin market

The bitcoin price has surged in recent years and it has also exhibited phases of rapid decay. In this paper we address the question to what extent this novel cryptocurrency market can be viewed as a classic or semi-efficient market. Novel and robust tools for estimation of multi-fractal properties are used to show that the bitcoin price exhibits a very interesting multi-scale correlation structure. This structure can be described by a power-law behavior of the variances of the returns as functions of time increments and it can be characterized by two parameters, the volatility and the Hurst exponent. These power-law parameters, however, vary in time. A new notion of generalized Hurst exponent is introduced which allows us to check if the multi-fractal character of the underlying signal is well captured. It is moreover shown how the monitoring of the power-law parameters can be used to identify regime shifts for the bitcoin price. A novel technique for identifying the regimes switches based on a goodness of fit of the local power-law parameters is presented. It automatically detects dates that can be associated with some known events in the bitcoin market place. A very surprising result is moreover that, despite the wild ride of the bitcoin price in recent years and its multi-fractal and non-stationary character, this price has both local power-law behaviors and a very orderly correlation structure when it is observed on its entire period of existence.

Multifractal characterization of meteorological drought in India using detrended fluctuation analysis

This study presents multifractal detrended fluctuation analysis (MF‐DFA) to describe the multifractality of Standardized Precipitation Index (SPI) series from 30 meteorological subdivisions of India estimated at different aggregation timescales (3, 6 and 12 months) based on long‐term monthly rainfall data sets of 1871–2016 period. The plots of fluctuation function and generalized Hurst exponents confirmed that multifractality is evident in most of the SPI series; however, it is found that its strength and persistency differ with timescale and space. The Hurst exponents of long‐term (SPI‐12) and medium‐term (SPI‐6) drought series of all subdivisions showed long‐term persistence irrespective of the climatic conditions of the region, while the short‐term droughts (SPI‐3) of six subdivisions displayed short‐term persistence. Further, the analysis by partitioning the data to four time spells: 1871–1905, 1906–1940, 1941–1975 and 1976–2016 showed that the persistency of long‐ and medium‐term drought is rather permanent. The analysis on changes in multifractal properties with respect to the climatic shift of 1976/1977 period showed that there is a clear increase in multifractal properties of all types of drought series in the recent past (post‐1976 period). This study also finds that the Hurst exponents and degree of multifractality of all types of droughts increase with increase in aggregation timescales. The multifractality of all the long‐term drought series (SPI‐12) and most of the SPI‐6 series are found to be due to correlation properties, while that of SPI‐3 is because of joint effect of correlation properties and the broadness of probability distribution.

Experimental and numerical study of multicomponent nitride coatings based on TiAlSiY fabricated by CA-PVD method

We report technical parameters of the deposition of multielement nitride protective coatings based on TiAlSiY by CA-PVD method and fractal analysis of their surfaces that are presented by the scanning electronic microscopy (SEM) images. Numerical analysis of the digital images was performed within two-dimensional multifractal detrended fluctuation analysis. Numerical models of surfaces of studied samples were built from their SEM images and generalized Hurst exponents and multifractal spectrums were calculated for all obtained structures. The performed calculations showed the widest range of Hurst exponent for TiAlSiY coating, which can indicate the highest relative roughness of its surface.

Accurate Classification of Seizure and Seizure-Free Intervals of Intracranial EEG Signals From Epileptic Patients

Electroencephalogram (EEG) signals are widely used to detect epileptic seizures in a patient’s neuronal activity. Since visual inspection and interpretation of EEG signal are time-consuming and prone to errors, various computer-aided diagnostic (CAD) tools have been proposed. In this paper, we present a novel automated detection system to distinguish between intracranial EEG time courses with seizures and those that are seizure-free based on complexity measures. Specifically, the features used to characterize the EEG signals are estimates of multiscaling properties over a large spectrum measured by using the generalized Hurst exponent. We tested the capacity of these estimates to correctly classify seizure intervals using a publicly available data set. Using the k-nearest neighbor classifier and testing with tenfold cross validation, we achieved 100% accurate classification. Our proposed CAD system outperformed the existing state-of-the-art models. Moreover, our CAD system is not only accurate but also fast and simple to implement. Therefore, it can be used as an expert system to support a decision in clinical applications.

Multifractal detrended fluctuation analysis on high-frequency SZSE in Chinese stock market

We use multifractal detrended fluctuation analysis (MF-DFA) method to investigate the multifractal behavior of Shenzhen Component Index (SZSE) 5-minute high-frequency stock data from 2017.6.15 - 2018.4.11. We determine generalized Hurst exponent and singularity spectrum and find that these fluctuations have multifractal nature. In order to maintain the long-term memory of the stock , all 9696 data are divided into 6 units. According to the multifractal spectrum, the main parameters of the six units are obtained. Comparing the MF-DFA results for the original SZSE high-frequency time series with those for shuffled series, we conclude that the origin of multifractality is due to both the broadness of probability density function and long-range correlation. The generalized Hurst exponent obviously depend on the order of fluctuation function and change with it. The curve of scaling function clearly departs from a straight line, i.e. it shows clearly nonlinear property, and the multifractal spectrum displays the commonly observed bell shape. This will provide an important and theoretical foundation for researching the forecasting of finance markets.

Memory Property of Grey Accumulation Generation Sequence

The influence of grey accumulation generation operator on the memory of time sequence is discussed. Generalized Hurst exponent (GHE) approach is used to calculate the memory in three cases, namely, the M3‐Competition data, the air quality index in Beijing, Xingtai, and Handan, and the power generating capacity, and car production index in Hebei province. The result indicates that one order accumulation generation operator (1‐AGO) can weaken the volatility and strengthen the memory of time sequence. It also explains the reason that one‐order accumulation can be used in grey prediction.