Hurst Exponent Research Articles

Synchronization of differential equations driven by linear multiplicative fractional Brownian motion

This paper is devoted to the synchronization of stochastic differential equations driven by the linear multiplicative fractional Brownian motion with Hurst parameter H∈(12,1). We use equivalent transformations to prove that the differential equation has a unique stationary solution, which generates a random dynamical system. Moreover, the system has the pathwise singleton set random attractor. We then establish the synchronization of the coupled differential equations and provide numerical simulation results. At the end, we discuss two specific noise forms and present the corresponding synchronization results.

Effect of correlation time of combustion noise on early warning indicators of thermoacoustic instability.

In this paper, we analyze the effects of finite correlation time (noise color) of combustion noise on noise-induced coherence and early warning indicators (EWIs) via numerical and experimental studies. We consider the Rijke tube as a prototypical combustion system and model combustion noise as an additive Ornstein-Uhlenbeck process while varying noise intensity and correlation time. We numerically investigate corresponding effects on coherence resonance and multi-fractal properties of pressure fluctuations. Subsequently, we experimentally validate results and elucidate the influence of noise color and intensity on trends in coherence resonance and multi-fractal measures that can be expected in a practical scenario using an electroacoustic simulator. We find that the coherence factor, which quantifies the relative contribution of coherent oscillations in a noisy signal, increases as the system approaches the thermoacoustic instability-irrespective of the correlation time. It works at most levels of combustion noise (except for too low and too high noise levels). The Hurst exponent reduces as the system approaches thermoacoustic instability only when the correlation time is small. These results have implications on the prediction and monitoring of thermoacoustic instability in practical combustors.

Approximate controllability of impulsive evolution stochastic functional differential equations driven by a fractional Brownian motion

In this paper, we study the approximate controllability of certain class of impulsive evolution stochastic functional differential equations, with variable delays, driven by a fractional Brownian motion in a separable real Hilbert space. We derive a new set of sufficient conditions for approximate controllability using a stochastic analysis of fractional Brownian motion with Hurst parameter H ∈ (1/2,1) and a Schaefer's fixed point theorem. An example is considered at the end of the paper to illustrate the obtained abstract results.

Diagnosis of pulmonary hypertension associated with congenital heart disease based on statistical features of the second heart sound

Aiming at the problems of obscure clinical auscultation features of pulmonary hypertension associated with congenital heart disease and the complexity of existing machine-aided diagnostic algorithms, an algorithm based on the statistical characteristics of the high-frequency components of the second heart sound signal is proposed. Firstly, an endpoint detection adaptive segmentation method is employed to extract the second heart sounds. Subsequently, the high-frequency component of the heart sound is decomposed using the discrete wavelet transform. Statistical features including the Hurst exponent, Lempel-Ziv information and sample entropy are extracted from this component. Finally, the extracted features are utilized to train an extreme gradient boosting algorithm (XGBoost) classifier, which achieves an accuracy of 80.45% in triple classification. Notably, this method eliminates the need for a noise reduction algorithm, allows for swift feature extraction, and achieves effective multi-classification using only three features. It is promising for early screening of pulmonary hypertension associated with congenital heart disease.

The Influence of Non-Gaussian Roughness and Spectral Properties on Mixed Lubrication for Heavily Loaded Counterformal Contacts

The impact of non-Gaussian height distribution and spectral properties on the lubrication performance of counterformal (point) contacts is quantitatively studied (film parameter, Λ, and asperity load ratio, La) by developing a mixed lubrication model. The Weibull height distribution function and power spectral density (PSD) are used to generate artificial surface topographies (non-Gaussian and Gaussian, isotropic), as these surface topographies are found in many tribological components. The set of variables needed to parametrize and their effect on mixed lubrication is discussed, including the shape parameter, the autocorrelation length, the wavelength ratio, and the Hurst coefficient. It is revealed that a rough surface with a lower shape parameter exhibits higher hydrodynamic lift. The spectral properties (the autocorrelation length and the wavelength ratio) of rough surfaces significantly affect the film parameter and the hydrodynamic and asperity pressures. The film parameter is slightly influenced by the Hurst coefficient.

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.

AN ANALYSIS OF PERFORMANCE PERSISTENCE IN INDIAN MUTUAL FUND USING HURST EXPONENT

We empirically investigated the presence of long memory in financial return series data of selected equity oriented mutual funds in India. The Hurst exponent, a non parametric test was used as the measurement of performance persistence to show presence of long memory. We used Net Asset Value (NAV) of 33 equity oriented growth based Indian mutual funds from different categories which showed a diversified picture of the mutual fund industry in India for the period 2003-2019. We empirically found that the degree of performance persistence is positive in case of selected funds and a few funds showed randomness or Brownian motion. The result suggested that the performance persistence exists in Indian mutual funds, applying the Rescaled Range method of Hurst exponent analysis.

Rough Volatility: Fact or Artefact?

We investigate the statistical evidence for the use of ‘rough’ fractional processes with Hurst exponent H<0.5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H< 0.5$$\\end{document} for modeling the volatility of financial assets, using a model-free approach. We introduce a non-parametric method for estimating the roughness of a function based on discrete sample, using the concept of normalized p-th variation along a sequence of partitions. Detailed numerical experiments based on sample paths of fractional Brownian motion and other fractional processes reveal good finite sample performance of our estimator for measuring the roughness of sample paths of stochastic processes. We then apply this method to estimate the roughness of realized volatility signals based on high-frequency observations. Detailed numerical experiments based on stochastic volatility models show that, even when the instantaneous volatility has diffusive dynamics with the same roughness as Brownian motion, the realized volatility exhibits rough behaviour corresponding to a Hurst exponent significantly smaller than 0.5. Comparison of roughness estimates for realized and instantaneous volatility in fractional volatility models with different values of Hurst exponent shows that, irrespective of the roughness of the spot volatility process, realized volatility always exhibits ‘rough’ behaviour with an apparent Hurst index H^<0.5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widehat{H}<0.5$$\\end{document}. These results suggest that the origin of the roughness observed in realized volatility time series lies in the estimation error rather than the volatility process itself.

Space–Time Variations in the Long-Range Dependence of Sea Surface Chlorophyll in the East China Sea and the South China Sea

Gaining insights into the space–time variations in the long-range dependence of sea surface chlorophyll is crucial for the early detection of environmental issues in oceans. To this end, 12 locations were selected along the Yangtze River and Pearl River estuaries, varying in distances from the Chinese coastline. Daily satellite-observed sea surface chlorophyll concentration data at these 12 locations were collected from the Copernicus Marine Service website, spanning from December 1997 to November 2023. The main objective of the current study is to introduce a multi-fractional generalized Cauchy model for calculating the values of Hurst exponents and quantitatively assessing the long-range dependence strength of sea surface chlorophyll at different spatial locations and time instants during the study period. Furthermore, ANOVA was utilized to detect the differences of calculated Hurst exponent values among the locations during various months and seasons. From a spatial perspective, the findings reveal a significantly stronger long-range dependence of sea surface chlorophyll in offshore regions compared to nearshore areas, with Hurst exponent values &gt; 0.5 versus &lt;0.5. It is noteworthy that the values of Hurst exponents at each location exhibit significant differences during various seasons, from a temporal perspective. Specifically, the long-range dependence of sea surface chlorophyll in summer in the nearshore region is weaker than in other seasons, whereas that in the offshore region is stronger than in other seasons. The study concludes that long-range dependence is inversely related to the distance from the coastline, and anthropogenic activity plays a dominant role in shaping the long-range dependence of sea surface chlorophyll in the coastal regions of China.

Clustering analysis on Hurst dynamic

This research focuses on utilizing the Hurst exponent () as a critical measure of relative volatility, with a specific emphasis on its ideal value of . The paper introduces an innovative simulation process for based on an Ornstein-Uhlenbeck stochastic differential equation while simultaneously estimating the parameter that governs the rate at which reverts to its initial efficiency level of . The empirical segment of the study aims to compare the parameters of numerous stocks computed through a stochastic simulation approach. The value represents the speed at which reverts to the value of over extended periods.

Using Nonlinear Features and Logistic Regression for Epilepsy Detection with Linear Complexity

Introduction: In this specific research study, a remarkably accurate and significantly simplified approach has been presented.Material and Metods: this research is encompasses three crucial stages. Firstly, the length of the signal is effectively diminished to an optimal magnitude through the utilization of a technique widely known as windowing. This technique plays a pivotal role in reducing the signal to an ideal size, ensuring the subsequent stages are executed with utmost precision. Secondly, the pertinent features are extracted from the shortened signal, specifically focusing on the Fractal Dimension, the Hurst Exponent, and the Ratio of Determinism to Recurrence Rate. These features are chosen due to the inherent nonlinear nature of the signal, as they provide valuable insights into the complex patterns and structures present within. Lastly, the extracted features undergo Logistic Regression, a widely employed classification algorithm, to effectively categorize and classify them. This step plays a crucial role in providing a clear and concise understanding of the underlying characteristics of the signal.Results: The implementation of the proposed method not only achieves an outstanding accuracy rate of 99.66%, but it also exhibits a linear time complexity, ensuring efficient processing. Additionally, this method leads to a significant reduction in the length of EEG signals, which is of utmost importance in practical applications.Conclusion: The primary objective of this proposed method revolves around the introduction of an online approach that can seamlessly integrate into healthcare systems. This objective is derived from a comprehensive analysis and evaluation of the obtained results, ensuring the method's practicality and effectiveness are thoroughly assessed.

Mindfulness meditation styles differently modulate source-level MEG microstate dynamics and complexity.

The investigation of mindfulness meditation practice, classically divided into focused attention meditation (FAM), and open monitoring meditation (OMM) styles, has seen a long tradition of theoretical, affective, neurophysiological and clinical studies. In particular, the high temporal resolution of magnetoencephalography (MEG) or electroencephalography (EEG) has been exploited to fill the gap between the personal experience of meditation practice and its neural correlates. Mounting evidence, in fact, shows that human brain activity is highly dynamic, transiting between different brain states (microstates). In this study, we aimed at exploring MEG microstates at source-level during FAM, OMM and in the resting state, as well as the complexity and criticality of dynamic transitions between microstates. Ten right-handed Theravada Buddhist monks with a meditative expertise of minimum 2,265 h participated in the experiment. MEG data were acquired during a randomized block design task (6 min FAM, 6 min OMM, with each meditative block preceded and followed by 3 min resting state). Source reconstruction was performed using eLORETA on individual cortical space, and then parcellated according to the Human Connect Project atlas. Microstate analysis was then applied to parcel level signals in order to derive microstate topographies and indices. In addition, from microstate sequences, the Hurst exponent and the Lempel-Ziv complexity (LZC) were computed. Our results show that the coverage and occurrence of specific microstates are modulated either by being in a meditative state or by performing a specific meditation style. Hurst exponent values in both meditation conditions are reduced with respect to the value observed during rest, LZC shows significant differences between OMM, FAM, and REST, with a progressive increase from REST to FAM to OMM. Importantly, we report changes in brain criticality indices during meditation and between meditation styles, in line with a state-like effect of meditation on cognitive performance. In line with previous reports, we suggest that the change in cognitive state experienced in meditation is paralleled by a shift with respect to critical points in brain dynamics.

Feynman–Kac formula for parabolic Anderson model in Gaussian potential and fractional white noise

In this paper, we establish a Feynman–Kac formula for the stochastic parabolic Anderson model with Gaussian potential in space and fractional white noise in time with Hurst parameter H &amp;gt; 1/2. We obtain the necesscary and suffcient condition for the integrability of the Gaussian potential and the exponential integrability of the solution which is defined by Feynman–Kac formula. By the smoothing of the fractional white noise and techniques from Malliavin calculus, we prove that the Feynman–Kac representation is a mild solution of the stochastic parabolic Anderson equation.

Rate of convergence for the Smoluchowski–Kramers approximation for distribution-dependent SDEs driven by fractional Brownian motions

In this paper, we study the rate in the Smoluchowski–Kramers approximation for the solution of the following distribution-dependent SDE driven by fractional Brownian motion [Formula: see text] where [Formula: see text] denotes the law of [Formula: see text], [Formula: see text] is a [Formula: see text]-dimensional fractional Brownian motion with Hurst parameter [Formula: see text]. Based on the techniques of multiple integrals and Malliavin calculus, we provide an explicit bound on total variation distance for the rate of convergence.

Optimal stop-loss rules in markets with long-range dependence

Stop-loss is a common risk management tool for limiting risks and improving trading strategy performance. The effectiveness of stop-loss depends critically on asset price characteristics. This study is the first to analyze stop-loss strategy incorporating long-range dependence of asset prices through a fractional Brownian motion-based market model. It is shown that stop-loss strategy yields a positive return premium over the buy-and-hold return when asset price exhibits long-range dependence. The efficacy of stop-loss strategies and the determining criterions are investigated through both theoretical analysis and simulation studies. The performance of a stop-loss rule depends on the Hurst parameter, mean and volatility of the asset returns. The optimal stop-loss threshold model in a chosen strategy class is fitted by polynomial regression. Empirical analysis demonstrates that the class-specific optimal rules outperform stop-loss rules under alternative asset return-generating models.

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.

Implied roughness in the term structure of oil market volatility

This paper analyses the attributes and the significance of the roughness of oil market volatility. We employ unspanned stochastic volatility models driven by rough Brownian motions that yield semi-analytical prices for future options entailing efficient calibration applications. By performing a Monte Carlo simulation study, we show that the semi-analytical pricing performs well thus establishing its efficiency for calibration applications. Thus we calibrate option prices written on oil futures and provide empirical evidence of the roughness in oil volatility. Introducing just one additional parameter, the Hurst parameter, indicating the roughness of the volatility improves the calibration by almost a factor of 10. The calibrated option-implied Hurst parameter varies over time, but the entire set of parameters becomes more stable than in the non-rough case corresponding to a fixed Hurst parameter 1/2. These results underscore the importance to model the time dependency of the roughness of oil market volatility.

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.

A Discovery Tour in Random Riemannian Geometry

We study random perturbations of a Riemannian manifold (M,g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\ extsf{M},\ extsf{g})$$\\end{document} by means of so-called Fractional Gaussian Fields, which are defined intrinsically by the given manifold. The fields h∙:ω↦hω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h^\\bullet : \\omega \\mapsto h^\\omega $$\\end{document} will act on the manifold via the conformal transformation g↦gω:=e2hωg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{g}\\mapsto \ extsf{g}^\\omega := e^{2h^\\omega }\\,\ extsf{g}$$\\end{document}. Our focus will be on the regular case with Hurst parameter H>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H>0$$\\end{document}, the critical case H=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H=0$$\\end{document} being the celebrated Liouville geometry in two dimensions. We want to understand how basic geometric and functional-analytic quantities like diameter, volume, heat kernel, Brownian motion, spectral bound, or spectral gap change under the influence of the noise. And if so, is it possible to quantify these dependencies in terms of key parameters of the noise? Another goal is to define and analyze in detail the Fractional Gaussian Fields on a general Riemannian manifold, a fascinating object of independent interest.

Characteristics and Drivers of Vegetation Change in Xinjiang, 2000–2020

Examining the features of vegetation change and analyzing its driving forces across an extensive time series in Xinjiang are pivotal for the ecological environment. This research can offer a crucial point of reference for regional ecological conservation endeavors. We calculated the fractional vegetation cover (FVC) using MOD13Q1 data accessed through the Google Earth Engine (GEE) platform. To discern the characteristics of vegetation changes and forecast future trends, we employed time series analysis, coefficient of variation, and the Hurst exponent. The correlation between climate factors and FVC was investigated through correlation analysis. Simultaneously, to determine the relative impact of meteorological change and anthropogenic actions on FVC, we utilized multiple regression residual analysis. Furthermore, adhering to China’s ecological functional zone classification, Xinjiang was segmented into five ecological zones: R1 Altai Mountains-Junggar West Mountain Forest and Grassland Ecoregion, R2 Junggar Basin Desert Ecoregion, R3 Tianshan Mountains Mountain Forest and Grassland Ecoregion, R4 Tarim Basin-Eastern Frontier Desert Ecoregion, and R5 Pamir-Kunlun Mountains-Altan Mountains Alpine Desert and Grassland Ecoregion. A comparative analysis of these five regions was subsequently conducted. The results showed the following: (1) During the first two decades of the 21st century, the overall FVC in Xinjiang primarily exhibited a trend of growth, exhibiting a rate of increase of 4 × 10−4 y−1. The multi-year average FVC was 0.223. The mean value of the multi-year FVC was 0.223, and the mean values of different ecological zones showed the following order: R1 &gt; R3 &gt; R2 &gt; R5 &gt; R4. (2) The predominant spatial pattern of FVC across Xinjiang’s landscape is characterized by higher coverage in the northwest and lower in the southeast. In this region, 66.63% of the terrain exhibits deteriorating vegetation, while 11% of the region exhibits a notable rise in plant growth. Future changes in FVC will be dominated by a decreasing trend. Regarding the coefficient of variation outcomes, a minor variation, representing 42.12% of the total, is noticeable; the mean coefficient of variation stands at 0.2786. The stability across varied ecological zones follows the order: R1 &gt; R3 &gt; R2 &gt; R4 &gt; R5. (3) Factors that have a facilitating effect on vegetation FVC included relative humidity, daylight hours, and precipitation, with relative humidity having a greater influence, while factors that have a hindering effect on vegetation FVC included air temperature and wind speed, with wind speed having a greater influence. (4) Vegetation alterations are primarily influenced by climate change, while human activities play a secondary role, contributing 56.93% and 43.07%, respectively. This research underscores the necessity for continued surveillance of vegetation dynamics and the enhancement of policies focused on habitat renewal and the safeguarding of vegetation in Xinjiang.