Hurst Exponent Research Articles

Fractional Brownian motion as a rough surface

In this study we consider two dimensional fractional Brownian motion (fBM) as a rough surface, tuned by the Hurst exponent H, with a focus on the scaling relations. While this system fulfills partially the requirements for self-similar Gaussian surfaces, (like Gaussian height distribution), it shows deviations for the global features. A thorough examination of the Kondev’s hyperscaling relations is presented. The global features associated with level lines which define contour loop ensemble (CLE) are studied in terms of H. We show that in the thermodynamic limit the fractal dimension of loops (Df), the critical exponent of the distribution of loop length (τl), and the gyration radius (τr) vary linearly with H. The hyperscaling relations between these quantities however challenge this hypothesis. While Df, τl and τr fulfill partially the Kondev relations, the loop correlation exponent xl depends weakly on H, i.e. shows deviations from 12 which is hypothesized by Kondev to be super-universal for the Gaussian surfaces.

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.

Fundamental Dynamics of Popularity-Similarity Trajectories in Real Networks.

Real networks are complex dynamical systems, evolving over time with the addition and deletion of nodes and links. Currently, there exists no principled mathematical theory for their dynamics-a grand-challenge open problem. Here, we show that the popularity and similarity trajectories of nodes in hyperbolic embeddings of different real networks manifest universal self-similar properties with typical Hurst exponents H≪0.5. This means that the trajectories are predictable, displaying antipersistent or "mean-reverting" behavior, and they can be adequately captured by a fractional Brownian motion process. The observed behavior can be qualitatively reproduced in synthetic networks that possess a latent geometric space, but not in networks that lack such space, suggesting that the observed subdiffusive dynamics are inherently linked to the hidden geometry of real networks. These results set the foundations for rigorous mathematical machinery for describing and predicting real network dynamics.

Feynman-Kac formula for general diffusion equations driven by TFBM with Hurst index H ∈ (0,1)

We consider the general diffusion equation driven by tempered fractional Brownian motion (TFBM)∂u(t,x)∂t=Lu(t,x)+u(t,x)dBH,λ(t)dt,(t,x)∈R+×Rd, where L is the infinitesimal generator of some time-homogeneous Markov process {Xt}t≥0 and {BH,λ(t)}t∈R is a tempered fractional Brownian motion with Hurst index H∈(0,12)∪(12,1) and tempering parameter λ>0 which is independent of {Xt}t≥0. Based on approximating TFBM with a family of Gaussian processes possessing absolutely continuous sample paths, a unified framework of the Feynman-Kac formulau(t,x)=Ex[f(Xt)]eBH,λ(t) is established for the general stochastic diffusion equation driven by TFBM with the initial value f, Hurst index H∈(0,12)∪(12,1) and tempering parameter λ>0. The difficulty is that the Hurst parameter H can be allowed to be less than 1/2. The idea is to explore the above simplest form, to utilize the techniques of Malliavin calculus. By using the properties of TFBM and especially that of the modified Bessel function of the second kind, we prove that the process defined by the Feynman-Kac formula is the mild and weak solutions of the general diffusion equation driven by TFBM. From the Feynman-Kac formula, we exhibit the smoothness of the density of the solution and the sharp Hölder regularity. We further show that limt→∞⁡ln⁡‖u(t,⋅)‖pg(t)=0 in the almost surely sense where p∈R and the function g:R+→R+ grows at least as fast as a linear function. The Feynman-Kac formula for the stochastic general diffusion equation in the Skorohod sense is also achieved by exploring the relationship between the Stratonovich and Skorohod integrals.

Maximum Ismail’s Second Entropy(HI q )Formalism of Heavy-Tailed Queues with Hurst Exponent Heuristic Mean Queue Length Combined with Potential Applications of Hurst Exponent to Social Computing and Connected Health

Maximum Ismail’s Second Entropy(HI q )Formalism of Heavy-Tailed Queues with Hurst Exponent Heuristic Mean Queue Length Combined with Potential Applications of Hurst Exponent to Social Computing and Connected Health

Spatiotemporal patterns, sustainability, and primary drivers of NDVI-derived vegetation dynamics (2003-2022) in Nepal.

Understanding the vegetation dynamics and their drivers in Nepal has significant scientific reference value for implementing sustainable ecological policies. This study provides a comprehensive analysis of the spatio-temporal variations in vegetation cover in Nepal from 2003 to 2022 using MODIS NDVI data and explores the effects of climatic factors and anthropogenic activities on vegetation. Mann-Kendall test was used to assess the significant trend in NDVI and was integrated with the Hurst exponent to predict future trends. The driving factors of NDVI dynamics were analyzed using Pearson's correlation, partial derivative, and residual analysis methods. The results indicate that over the last 20 years, Nepal has experienced an increasing trend in NDVI at 0.0013 year-1, with 80% of the surface area (vegetation cover) showing an increasing vegetation trend (~ 28% with a significant increase in vegetation). Temperature influenced vegetation dynamics in the higher elevation areas, while precipitation and human interventions influenced the lower elevation areas. The Hurst exponent analysis predicts an improvement in the vegetation cover (greening) for a larger area compared to vegetation degradation (browning). A significantly increased area of NDVI residuals indicates a positive anthropogenic influence on vegetation cover. Anthropogenic activities have a higher relative contribution to NDVI variation followed by temperature and then precipitation. The results of residual trend and Hurst analysis in different regions of Nepal help identify degraded areas, both in the present and future. This information can assist relevant authorities in implementing appropriate policies for a sustainable ecological environment.

Analysis of the temporal and spatial evolution of turbidity in Tonle Sap Lake and its influencing factors

Turbidity is a crucial indicator of water quality. The European Commission's Copernicus Land Monitoring Service Platform provides free turbidity data for large lakes to monitor the global water quality of lakes. However, the data were missing from April 2012 to April 2016, severely limiting long-term analysis. Based on MODIS and turbidity data, Random Forest and XGBoost models are used to invert Tonle Sap Lake's turbidity. Random Forest outperformed the XGBoost model. Based on Random Forest model, missing data were filled in to construct long-term series data of Tonle Sap Lake turbidity (2004–2021). Trend, persistence and correlation analyses were conducted to reveal spatiotemporal characteristics and driving mechanism of turbidity. The results showed that: (1) spatially, the average annual, monthly, and seasonal turbidity was higher in the north but lower in the south, with regions of higher turbidity exhibiting more significant changes; (2) temporally, the annual turbidity mean was 53.99 NTU and showed an increasing trend. Monthly, turbidity values were higher from March to August and lower from September to February, with the highest and lowest recorded in June and November at 110.06 and 5.82 NTU, respectively. Seasonally, turbidity was higher in spring and summer compared to autumn and winter, with mean turbidity values of 84.16, 93.47, 15.33 and 23.21 NTU, respectively; (3) In terms of sustainability, the Hurst exponent for annual turbidity was 0.23, indicating a reverse trend in the near future; (4) Dam construction's impact on turbidity was not significant. Compared with natural factors (permanent wetlands, grasslands, lake surface water temperature, and remote sensing ecological index), human activities (barren, urban and built-up lands, croplands and population density) had a more significant impact on turbidity. Turbidity was highly correlated with croplands (r = 0.76), followed by population density (r = 0.71), and urban and built-up lands (r = 0.69).

Fractal contours: Order, chaos, and art

Over the recent decades, a variety of indices, such as the fractal dimension, Hurst exponent, or Betti numbers, have been used to characterize structural or topological properties of art via a singular parameter, which could then help to classify artworks. A single fractal dimension, in particular, has been commonly interpreted as characteristic of the entire image, such as an abstract painting, whether binary, gray-scale, or in color, and whether self-similar or not. There is now ample evidence, however, that fractal exponents obtained using the standard box-counting are strongly dependent on the details of the method adopted, and on fitting straight lines to the entire scaling plots, which are typically nonlinear. Here, we propose a more discriminating approach with the aim of obtaining robust scaling plots and extracting relevant information encoded in them without any fitting routines. To this goal, we carefully average over all possible grid locations at each scale, rendering scaling plots independent of any particular choice of grids and, crucially, of the orientation of images. We then calculate the derivatives of the scaling plots, so that an image is described by a continuous function, its fractal contour, rather than a single scaling exponent valid over a limited range of scales. We test this method on synthetic examples, ordered and random, then on images of algorithmically defined fractals, and finally, examine selected abstract paintings and prints by acknowledged masters of modern art.

Existence and Uniqueness of Strong Solutions of Mixed-Type Stochastic Differential Equations Driven by Fractional Brownian Motions with Hurst Exponents $$H>1/4 $$

Existence and Uniqueness of Strong Solutions of Mixed-Type Stochastic Differential Equations Driven by Fractional Brownian Motions with Hurst Exponents $$H>1/4 $$

Euler scheme for SDEs driven by fractional Brownian motions: Malliavin differentiability and uniform upper-bound estimates

The Malliavin differentiability of a SDE plays a crucial role in the study of density smoothness and ergodicity among others. For Gaussian driven SDEs the differentiability issue is solved essentially in Cass et al., (2013). In this paper, we consider the Malliavin differentiability for the Euler scheme of such SDEs. We will focus on SDEs driven by fractional Brownian motions (fBm), which is a very natural class of Gaussian processes. We derive a uniform (in the step size n) path-wise upper-bound estimate for the Euler scheme for stochastic differential equations driven by fBm with Hurst parameter H>1/3 and its Malliavin derivatives.

Moderate deviations for rough differential equations

AbstractSmall noise problems are quite important for all types of stochastic differential equations. In this paper, we focus on rough differential equations driven by scaled fractional Brownian rough path with Hurst parameter . We prove a moderate deviation principle for this equation as the scale parameter tends to zero.

Multifractal detrended fluctuation analysis on the fracture surface of polycarbonate and acrylonitrile-butadiene-styrene alloy

Multifractal detrended fluctuation analysis (MF-DFA) has been carried out to evaluate the multifractal properties of the fracture morphology for Polycarbonate (PC) and Acrylonitrile-butadiene-styrene (ABS) alloy, and the relationship between the long-range correlation parameter of the samples fracture - Hurst exponent and fractal dimension was explored and analyzed. The results suggest that the existence of multifractal properties within the micrometer scale domain we investigated for the fracture surface of PC/ABS alloy samples. The Hurst exponent and fractal dimension can both be used to measure and characterize the fracture surface morphology of the PC/ABS alloys. As the content of compatibilizer PP-g-MAH increased, the tensile strength of PC/ABS alloy decreased, the fracture surface peaks tend to dominate, the Hurst exponent increased, the fractal dimension decreased, and its strength retention rate increases, displaying persistence.

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.

Permutation entropy and complexity analysis of large-scale solar wind structures and streams

Abstract. In this work, we perform a statistical study of magnetic field fluctuations in the solar wind at 1 au using permutation entropy and complexity analysis and the investigation of the temporal variations of the Hurst exponents. Slow and fast wind, magnetic clouds, interplanetary coronal mass ejection (ICME)-driven sheath regions, and slow–fast stream interaction regions (SIRs) have been investigated separately. Our key finding is that there are significant differences in permutation entropy and complexity values between the solar wind types at larger timescales and little difference at small timescales. Differences become more distinct with increasing timescales, suggesting that smaller-scale turbulent features are more universal. At larger timescales, the analysis method can be used to identify localised spatial structures. We found that, except in magnetic clouds, fluctuations are largely anti-persistent and that the Hurst exponents, in particular in compressive structures (sheaths and SIRs), exhibit a clear locality. Our results shows that, in all cases apart from magnetic clouds at the largest scales, solar wind fluctuations are stochastic, with the fast wind having the highest entropies and low complexities. Magnetic clouds, in turn, exhibit the lowest entropy and highest complexity, consistent with them being coherent structures in which the magnetic field components vary in an ordered manner. SIRs, slow wind and ICME sheaths are intermediate in relation to magnetic clouds and fast wind, reflecting the increasingly ordered structure. Our results also indicate that permutation entropy–complexity analysis is a useful tool for characterising the solar wind and investigating the nature of its fluctuations.

Long Memory Time-series Model (ARFIMA) Based Modelling of Jute Prices in the Samsi Market of Malda District, West Bengal

The objective of this paper is modeling and forecasting the weekly jute prices of Samsi market in the Malda district of West Bengal in the presence of long memory process. The long memory behavior of series is investigated by the ACF plot and Hurst R/S analysis. A fractionally integrated autoregressive moving-average (ARssFIMA) model is fitted using 668 weekly data (January 2009-November 2022). This study shows the efficiencies of the Hurst exponent, GPH, Smoothed periodogram, Local Whittle, and Wavelet methods used to estimate the fractional difference parameter in the ARFIMA model. Furthermore, we compared the forecasting abilities of the ARFIMA and ARIMA models. The results show that long memory is present in the jute price series. The models selected according to each method are ARFIMA (3,0.348,0), ARFIMA (3,0.291,1), ARFIMA (2,0.487,0), ARFIMA (3,0.461,0), ARFIMA (2,0.311,0), and ARIMA (2,1,1) on the basis of minimum AIC and BIC using 534 in-sample data. Finally, the wavelet method based ARFIMA (2,0.311,0) model is found to be the best optimal model in terms of MAE, RMSE, and MAPE criteria using 134 out-of-sample data. A comparative study indicates that the forecasting performance of the ARFIMA model is strongly better than that of the ARIMA model in this regard.

Eigenvalue distributions of high-dimensional matrix processes driven by fractional Brownian motion

In this paper, we study high-dimensional behavior of empirical spectral distributions [Formula: see text] for a class of [Formula: see text] symmetric/Hermitian random matrices, whose entries are generated from the solution of stochastic differential equation driven by fractional Brownian motion with Hurst parameter [Formula: see text]. For Wigner-type matrices, we obtain almost sure relative compactness of [Formula: see text] in [Formula: see text] following the approach in [1]; for Wishart-type matrices, we obtain tightness of [Formula: see text] on [Formula: see text] by tightness criterions provided in Appendix B. The limit of [Formula: see text] as [Formula: see text] is also characterized.

Impact of Subprime and Covid-19 crises on WAEMU stock exchange efficiency: a classic Hurst vs dynamic Hurst approach

Financial markets are alternative sources of financing, above all, in the current context of financial globalization and the development of information and telecommunication technologies. In the WAEMU zone in particular, the size of the economies seemed insufficient to supply national markets without high costs. The union’s authorities therefore set up a regional market that seemed to be a beneficial outcome : the WAEMU stock market. This stock exchange has undergone significant modernizations since its inception, with the aim of achieving optimal resource allocation. The informational efficiency of this stock exchange has become a hot topic of interest among researchers and professionals, giving rise to a growing number of publications on the subject. Our article is set in this context and aims to analyze the WAEMU’s weak-form efficiency. Based on the limitations of traditional methods, we use new approaches. Firstly, the classic Hurst is a tool for characterizing market structure and analyzing efficiency. The results highlight the multifractal nature of the WAEMU’s market and the rejection of the null hypothesis of no memory in the return series. Given the resurgence of crises, it would be interesting to analyze stock market behavior during such events. We also study the impact of crises (subprime and covid-19) on efficiency, using the dynamic Hurst exponent to account for efficient market varying with time. The results show that during periods of crisis, the stock market alternates between efficiency and inefficiency. They also reveal that Covid-19’s impact was faster than subprime crisis. They also reveal that covid impact crisis was faster than that of subprimes. However, despite the big spike linked to the covid context, the composite index picked up and continued in the same vein.

A unifying representation of path integrals for fractional Brownian motions

Fractional Brownian motion (fBm) is an experimentally-relevant, non-Markovian Gaussian stochastic process with long-ranged correlations between the increments, parametrised by the so-called Hurst exponent H; depending on its value the process can be sub-diffusive (0<H<1/2) , diffusive (H=1/2) or super-diffusive (1/2<H<1) . There exist three alternative equally often used definitions of fBm—due to Lévy and due to Mandelbrot and van Ness (MvN), which differ by the interval on which the time variable is formally defined. Respectively, the covariance functions of these fBms have different functional forms. Moreover, the MvN fBms have stationary increments, while for the Lévy fBm this is not the case. One may therefore be tempted to conclude that these are, in fact, different processes which only accidentally bear the same name. Recently determined explicit path integral representations also appear to have very different functional forms, which only reinforces the latter conclusion. Here we develop a unifying equivalent path integral representation of all three fBms in terms of Riemann–Liouville fractional integrals, which links the fBms and proves that they indeed belong to the same family. We show that the action in such a representation involves the fractional integral of the same form and order (dependent on whether H<1/2 or H>1/2 ) for all three cases, and differs only by the integration limits.

Formation of partially embedded Au nanostructures: Ion beam irradiation on thin film.

The Au partially embedded nanostructure (PEN) is synthesized by ion irradiation on an Au thin film deposited on a glass substrate using a 50 keV Ar ion. Scanning electron microscopy results show ion beam-induced restructuring from irregularly shaped nanostructures (NSs) to spherical Au NSs, and further ion irradiation leads to the formation of well-separated spherical nanoparticles. Higuchi's algorithm of surface analysis is utilized to find the evolution of surface morphology with ion irradiation in terms of the Hurst exponent and fractal dimension. The Au PEN is evidenced by Rutherford backscattering spectrometry and optical studies. Also, the depth of the mechanism behind synthesized PEN is explained on the basis of theoretical simulations, namely, a unified thermal spike and a Monte Carlo simulation consisting of dynamic compositional changes (TRIDYN). Another set of plasmonic NSs was formed on the surface by thermal annealing of the Au film on the substrate. Glucose sensing has been studied on the two types of plasmonic layers: nanoparticles on the surface and PEN. The results reveal the sensing responses of both types of plasmonic layers. However, PEN retains its plasmonic behavior as the NSs are still present after washing with water, which demonstrates the potential for reusability. RESEARCH HIGHLIGHTS: Synthesis of PENs by ion irradiation Utilization of Higuchi's algorithm to explore the surface morphology. Unified thermal spike and TRIDYN simulations being used to explain the results. Glucose is only used as a test case for reusability of substrate.

Random and time-persistent depositional processes in turbidite successions: an example from the marine deep-water Aoshima Formation (Neogene, Kyushu Island, southwest Japan)

ABSTRACT The deposits of flood- and earthquake-derived subaqueous sediment gravity flows represent a significant fraction of lacustrine and deep-sea sedimentary successions, thus providing a valuable record of such natural disasters. The magnitude of these events and the thickness of the associated deposits are considered to follow a lognormal or power-law frequency distribution, whilst that of time intervals between subsequent events appear to be best approximated by a Poisson model, indicative of a random, time-independent phenomenon. However, the debate on whether the sedimentary record of these natural disasters is governed by randomness alone or whether there is some underlying stratigraphic ordering is still unsettled and requires detailed time-series analysis. This study consists of a time-series analysis of mudstone- and sandstone-dominated turbidite successions offshore a fan-delta system in the Neogene Aoshima Formation that belongs to the sedimentary fill of the forearc basin of southwest Japan. The formation consists of a monotonous alternation of very fine- to medium-grained sandstones capped by hemipelagic mudstones and, more rarely, by turbidite mudstones. The results show that the autocorrelation function of the time series suggests quasi-periodic variability in the upper sandstone-dominated part, whereas the lower mudstone-dominated part shows a white-noise-like pattern. Rescaled range analysis shows that the number of events per unit time in the lower part is characterized by a random time series, such as Brownian noise with a Hurst exponent of 0.5. In contrast, the thickness of event beds of the lower part and the thickness and the number of events of the upper part are persistent time series with a Hurst exponent &amp;gt; 0.5. These results suggest that the number of turbidite depositional events in the mudstone-dominated part indicates random timing, whereas its thickness time series and the sandstone-dominant part are not governed by simple stochastic processes but are affected by sea-level changes, sediment transport dynamics, and other factors such as, for example, seafloor topography.