Hurst Exponent Research Articles

Regional Disparities in Inflation Persistence: Unpacking the Dynamics of Price Growth in Portugal

<p><big>This paper investigates the degree of inflation persistence across regions in Portugal by analyzing the Consumer Price Index (CPI) growth rates for NUTS II regions. The study employs the Augmented Dickey-Fuller (ADF) test to determine whether the CPI data for Portugal is stationary or non-stationary. The results of the ADF test reveal that the IPC data for Portugal is non-stationary, indicating that inflation exhibits persistence in the long run. The study further assesses the persistence of inflation by estimating an autoregressive integrated moving average (ARIMA) model for each region. The Ljung-Box test is used to test for autocorrelation in the time series data, and the Hurst exponent is calculated to evaluate the presence of long-term memory in the time series data. The study finds that there is significant autocorrelation in the time series data for all regions, supporting the presence of persistence in inflation at the regional level in Portugal. The Hurst exponent also shows that the time series data for each region exhibits a high degree of persistence in inflation. Finally, the study applies the ARIMA model to each CPI division's data and uses the Ljung-Box test to test for autocorrelation in the time series data. The results show that some CPI divisions exhibited higher levels of persistence compared to others. For example, the "Housing, water, electricity, gas and other fuels" division exhibited high persistence, while the "Communication" division exhibited low persistence. This study contributes to the existing literature by exploring regional inflation persistence in Portugal and its implications for policymaking. The results provide insights into the inflation persistence patterns across regional levels in Portugal, by emphasizing the need to consider regional differences in inflation dynamics when formulating effective policy interventions. Understanding the persistence of inflation is crucial for policymakers to ensure price stability and sustainable economic growth.</big></p>

The Fama–French Five-Factor Model with Hurst Exponents Compared with Machine Learning Methods

Scholars and investors have been interested in factor models for a long time. This paper builds models using the monthly data of the A-share market. We construct a seven-factor model by adding the Hurst exponent factor and the momentum factor to a Fama–French five-factor model and find that there is a 7% improvement in the average R–squared. Then, we compare five machine learning algorithms with ordinary least squares (OLS) in one representative stock and all A-Share stocks. We find that regularization algorithms, such as lasso and ridge, have worse performance than OLS. SVM and random forests have a good improvement in fitting power, while the neural network is not always better than OLS, depending on the data, frequency, period, etc.

Towards a Better Understanding of Fractional Brownian Motion and Its Application to Finance

The aim of this work is to first build the underlying theory behind fractional Brownian motion and applying fractional Brownian motion to financial market. By incorporating the Hurst parameter into geometric Brownian motion in order to characterize the long memory among disjoint increments, geometric fractional Brownian motion model is constructed to model S &P 500 stock price index. The empirical results show that the fitting effect of fractional Brownian motion model is better than ordinary Brownian motion.

Probabilistic Machine Learning Methods for Fractional Brownian Motion Time Series Forecasting

This paper explores the capabilities of machine learning for the probabilistic forecasting of fractional Brownian motion (fBm). The focus is on predicting the probability of the value of an fBm time series exceeding a certain threshold after a specific number of time steps, given only the knowledge of its Hurst exponent. The study aims to determine if the self-similarity property is preserved in a forecasting time series and which machine learning algorithms are the most effective. Two types of forecasting methods are investigated: methods with a predefined distribution shape and those without. The results show that the self-similar properties of the fBm time series can be reliably reproduced in the continuations of the time series predicted by machine learning methods. The study also provides an experimental comparison of various probabilistic forecasting methods and their potential applications in the analysis and modeling of fractal time series.

Valve Stiction Detection Method Based on Dynamic Slow Feature Analysis and Hurst Exponent

Valve stiction is the most common root of oscillation faults in process control systems, and it can cause the severe deterioration of control performance and system instability, ultimately impacting product quality and process safety. A new method for detecting valve stiction, based on dynamic slow feature analysis (DSFA) and the Hurst exponent, is proposed in this paper. The proposed method first utilizes DSFA to extract slow features (SFs) from the preprocessed and reconstructed data of the controller output and the controlled process variable; then, it calculates the Hurst exponent of the slowest SF to quantify its long-term correlation; and, finally, it defines a new valve detection index to identify valve stiction. The results obtained from simulations and actual process case studies demonstrate that the proposed method, based on a DSFA–Hurst exponent, can effectively detect valve stiction in control loops.

Multiscale Volatility Analysis for Noisy High-Frequency Prices

We present a multiscale analysis of the volatility of intraday prices from high-frequency data. Our multiscale framework includes a fractional Brownian motion and microstructure noise as the building blocks. The proposed noisy fractional Brownian motion model is shown to possess a variety of volatility behaviors suitable for intraday price processes. Algorithms for numerical estimation from time series observations are then introduced, with a new Hurst exponent estimator proposed for the noisy fractional Brownian motion model. Using real-world high-frequency price data for a collection of U.S. stocks and ETFs, we estimate the parameters in the noisy fractional Brownian motion and illustrate how the volatility varies over different timescales. The Hurst exponent and noise level also exhibit an intraday pattern whereby the the noise ratio tends to be higher near market close.

Distance to criticality undergoes critical transition before epileptic seizure attacks

Epilepsy is a common neurological disorder characterized by recurring seizures, but its underlying mechanisms remain poorly understood. Despite extensive research, there are still gaps in our knowledge about the relationship between brain dynamics and seizures. In this study, our aim is to address these gaps by proposing a novel approach to assess the role of brain network dynamics in the onset of seizures. Specifically, we investigate the relationship between brain dynamics and seizures by tracking the distance to criticality. Our hypothesis is that this distance plays a crucial role in brain state changes and that seizures may be related to critical transitions of this distance. To test this hypothesis, we develop a method to measure the evolution of the brain network’s distance to the critical dynamic systems (i.e., the distance to the tipping point, DTP) using dynamic network biomarker theory and random matrix theory. The results show that the DTP of the brain decreases significantly immediately after onset of an epileptic seizure, suggesting that the brain loses its well-defined quasi-critical state during seizures. We refer to this phenomenon as the “criticality of the criticality” (COC). Furthermore, we observe that DTP exhibits a shape transition before and after the onset of the seizures. This phenomenon suggests the possibility of early warning signal (EWS) identification in the dynamic sequence of DTP, which could be utilized for seizure prediction. Our results show that the Hurst exponent, skewness, kurtosis, autocorrelation, and variance of the DTP sequence are potential EWS features. This study advances our understanding of the relationship between brain dynamics and seizures and highlights the potential for using criticality-based measures to predict and prevent seizures.

Spatiotemporal Characteristics of Urbanization in the Taiwan Strait Based on Nighttime Light Data from 1992 to 2020

Urbanization is a crucial indicator which reflects the socio-economic development of a country or region. The regions across the Taiwan Strait (TSR) have garnered attention worldwide as being representative of typical urbanization development along the southeastern coast. Currently, research in the TSR predominantly focuses on individual regions, with limited academic achievements comparing urbanization paths across the strait. In particular, the domain of comparative analysis of the spatiotemporal characteristics of urbanization dynamics in TSR by using long time series of nighttime light data remains largely underexplored. Therefore, this study focused on comparing the urbanization paths in the TSR and analyzing the spatiotemporal characteristics of urbanization by using the long-term nighttime light data from 1992 to 2020. Additionally, some methods such as Theil–Sen median trend analysis, Mann–Kendall significance test, Hurst exponent, spatial statistics, and time series were used to quantitatively analyze the spatial distribution patterns and temporal trends of nighttime lights in the TSR since 1992. The results were as follows: (1) From 1992 to 2020, the spatial distribution of nighttime light data in TSR exhibited significant spatial heterogeneity, with high-value areas mainly located in southeastern Fujian and northwestern Taiwan, while low-value areas were concentrated in Fujian’s inland regions; (2) During this period, nighttime lighting data increased from 729,863 in 1992 to 2,729,052 in 2020, and the percentage of its high-value (40–063) increased from 2.59% in 1992 to 12.22% in 2020; (3) Comparison of nighttime light data across representative cities from Taiwan (Taipei, Hsinchu) and Fujian (Xiamen, Fuzhou) uncovered distinct growth patterns—while Taiwanese cities had a high initial brightness value (the lowest value in the last 30 years was 518,379.4), their growth was relatively slow (average growth rate of 17%); Fujian cities, on the other hand, started from lower initial brightness value (the lowest value in the last 30 years was 35,123.1), but displayed substantial growth vigor (average growth rate of 222%); (4) During the study period, the nighttime light data of the vast majority of cities in the TSR demonstrated a significant increasing trend, particularly in coastal areas and urban centers; (5) Predictions of future trend variation suggest that the significantly increasing trend of cities surrounding Taiwan’s primary metropolitan areas will intensify, whereas metropolitan regions such as Keelung may witness a decline in future trend variations. However, only a mere 0.03% of the nighttime light data show a significant decreasing trend. Additionally, there are distinct differences in the urbanization development stages of the TSR. Fujian is currently undergoing rapid urbanization, while Taiwan’s urbanization has entered a stable stage. The study reveals that factors such as geographical location, natural resources, transportation infrastructure, population size, and industrial structure collectively influence the urbanization characteristics of the TSR. This research bears substantial significance for deepening the comprehension of the patterns and processes of urbanization development in the TSR and provides valuable insights for urban construction and development across the strait.

A Hurst‐based diffusion model using time series characteristics for influence maximization in social networks

AbstractOnline social networks have grown exponentially in the recent years while finding applications in real life like marketing, recommendation systems, and social awareness campaigns. An important research area in this field is Influence Maximization, which pertains to finding methods for maximizing the spread of information (influence) across an OSN. Existing works in IM widely use a pre‐defined edge propagation probability for node activation. Hurst exponent (H), which depicts the self‐similarity in the time series depicting a user's past interaction behaviour, has also been used as activation criteria. In this work, we propose a Time Series Characteristic based Hurst‐based Diffusion Model (TSC‐HDM), which calculates H based on the stationary or non‐stationary characteristic of the time series. TSC‐HDM selects a handful of seed nodes and activates a seed node's inactive successor only if H > 0.5. The proposed model has been tested on four real‐world OSN datasets. The results have been compared against four other IM models – Independent Cascade, Weighted Cascade, Trivalency, and Hurst‐based Influence Maximization. TSC‐HDM is found to have achieved as much as 590% higher expected influence spread as compared to the other models. Moreover, TSC‐HDM has attained 344% better average influence spread than other state‐of‐the‐art models namely LIR, A‐Greedy, LPIMA, Genetic Algorithm with Dynamic Probabilities, NeighborsRemove, DegreeDecrease, IGIM, IRR, and PHG.

Strong convergence of a fractional exponential integrator scheme for finite element discretization of time-fractional SPDE driven by fractional and standard Brownian motions

The aim of this work is to provide the first strong convergence result of a numerical approximation of a general time-fractional second order stochastic partial differential equation involving a Caputo derivative in time of order α∈(12,1) and driven simultaneously by a multiplicative standard Brownian motion and additive fBm with Hurst parameter H∈(12,1), more realistic to model the random effects on transport of particles in medium with thermal memory. We prove the existence and uniqueness results, and perform the spatial discretization using the standard finite element and the temporal discretization based on a generalized exponential time differencing method (GETD). We provide the temporal and spatial convergence proofs for our fully discrete scheme and the result shows that the convergence orders depend on the regularity of the initial data, the power of the fractional derivative, and the Hurst parameter H. Numerical results are provided to illustrate our theoretical results.

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.

A Fractal and Comparative View of the Memory of Bitcoin and S&P 500 Returns

The majority of previous studies used autocorrelation-based methodologies to explore the dependency structure for Bitcoin, but this paper follows Benoit Mandelbrot in taking a fractal point of view. This perspective showed that Bitcoin and S&P 500 returns exhibit fractal-like behavior. Additional evidence suggested that the infinite variance hypothesis cannot be rejected for either asset supporting Mandelbrot’s (1963) early study on cotton price changes. This result held across non-overlapping subsamples. Following Mandelbrot (2008), Hurst exponents were estimated using rescaled/range analysis. The key findings are that (a) Bitcoin returns exhibit a higher level of persistence than S&P 500 returns across various subsamples, (b) the level of persistence in Bitcoin returns did not change over time, (c) the S&P 500 moved from efficiency in the first subsample to inefficiency in the ex-post June 17, 2018, period, (d) even if it was assumed that the variance of S&P 500 returns was finite, the kurtosis remained statistically undefined. The study concluded that the correlation-based methods used to explore the S&P 500 universe result in misleading answers.

Statistical Analysis of Plasma Dynamics in Gyrokinetic Simulations of Stellarator Turbulence.

A geometrical method for assessing stochastic processes in plasma turbulence is investigated in this study. The thermodynamic length methodology allows using a Riemannian metric on the phase space; thus, distances between thermodynamic states can be computed. It constitutes a geometric methodology to understand stochastic processes involved in, e.g., order-disorder transitions, where a sudden increase in distance is expected. We consider gyrokinetic simulations of ion-temperature-gradient (ITG)-mode-driven turbulence in the core region of the stellarator W7-X with realistic quasi-isodynamic topologies. In gyrokinetic plasma turbulence simulations, avalanches, e.g., of heat and particles, are often found, and in this work, a novel method for detection is investigated. This new method combines the singular spectrum analysis algorithm with a hierarchical clustering method such that the time series is decomposed into two parts: useful physical information and noise. The informative component of the time series is used for the calculation of the Hurst exponent, the information length, and the dynamic time. Based on these measures, the physical properties of the time series are revealed.

On the Analysis of Ait-Sahalia-Type Model for Rough Volatility Modelling

Fractional Brownian motion with Hurst parameter H<12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H<\\frac{1}{2}$$\\end{document} is used widely, for instance, to describe ‘rough’ volatility data in finance. In this paper, we examine a generalised Ait-Sahalia-type model driven by a fractional Brownian motion with H<12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H<\\frac{1}{2}$$\\end{document} and establish theoretical properties such as an existence-and-uniqueness theorem, regularity in the sense of Malliavin differentiability and higher moments of the strong solutions.

Noise Cancellation Method Based on TVF-EMD with Bayesian Parameter Optimization

To separate the noise and important signal features of the indoor carbon dioxide (CO2) concentration signal, we proposed a noise cancellation method, based on time-varying, filtering-based empirical mode decomposition (TVF-EMD) with Bayesian optimization (BO). The adaptive parameters of TVF-EMD, that is, bandwidth threshold ξ and B-spline order n, were determined by the BO algorithm, and the correlation coefficient for the kurtosis index (CCKur) constituted the objective function. Initially, the objective function CCKur was introduced to systematically identify anomalous signals while preserving signal feature extraction between the modes and the input signal. Subsequently, the proposed signal noise cancellation model based on TVF-EMD and the BO algorithm were employed, along with the Hurst exponent, to extract the sensitive mode. An examination of the optimization indices of the decomposed intrinsic mode functions (IMFs), namely CC, Kur, MI, EE, EEMI, and CCKur, revealed that the synthetic measurement index CCKur and objective function fitness were reasonable and effective. The proposed method exhibited better signal cancellation performance, compared to that of TVF-EMD with the default values, EMD, the moving average method, and the exponential smoothing method.

Asymmetric market efficiency of the Eurozone using the MF-DFA: a comparison between global financial crisis and COVID-19 era

PurposeThis research paper aims to explore asymmetric market efficiency of the 13 Euro countries, i.e. Austria, Belgium, Finland, France, Germany, Greece, Ireland, Italy, Netherland, Portugal, Slovakia, Slovenia and Spain, concerning the period before global financial crisis (GFC), after GFC and period of COVID-19 pandemic.Design/methodology/approachMultifractal detrended fluctuation analysis (MF-DFA) is applied to examine the persistence and anti-persistency. It also discusses the random walk behavior hypothesis of these 13 countries non-stationary time series. Additionally, generalized Hurst exponents are applied to estimate the relative efficiency between short- and long-run horizons and small and large fluctuations.FindingsThe current study results suggest that most countries' markets are multifractal and exhibit long-term persistence in the short and long run. Moreover, the results with respect to full sample confirm that Portugal is the most efficient country in short run and Austria is the least efficient country. However, in long run, Austria appeared to be highly efficient, and Slovakia is the least efficient. In the pre-GFC period, Greece is said to be the relatively most efficient market in the short run, whereas Austria is the most efficient market in the long run. In the case of Post-GFC, Netherland and Ireland are the most efficient markets in short and long run, respectively. Lastly, COVID-19 results indicate that Finland's stock market is the most efficient in short run. Whereas, in the long run, the high efficiency is illustrated by Germany. In contrast, the most affected stock market due to COVID-19 is Belgium.Originality/valueThis study will add value to the present knowledge on efficient market hypothesis (EMH) with the MF-DFA approach. Also, with the MF-DFA approach, potential investors will be capable of ranking the stock markets of Eurozone countries based on their efficiency in the period before and after GFC and then specifically in the period of COVID-19.

Characterising the variability in ocean data using fractal and correlation dimension analysis

Understanding the variability of ocean data is essential for predicting its behaviour in time domain. This study aims to examine the usability of several nonlinear time series analysis techniques for diagnosing the complexity of the underlying dynamics of various ocean data. For such purpose, one-year data from an offshore platform and two-year data from a weather buoy were collected and analysed. The results indicate that all the ocean data considered here are chaotic in nature, reflected by the unfolded structure in the reconstructed phase space. In general, the surface currents speed possess the largest fractal dimension (or lowest Hurst exponent) and correlation dimension, while the significant wave height has the lowest values. This implies that the dynamics associated with surface currents speed tend to be the most complex, which can lead to more random and unpredictable temporal behaviour. The fractal analysis shows that the fluctuation of these ocean data are mostly persistent, and the dependence of fractal properties on window length appears to vary amongst different ocean data. Overall, the results derived from various analysis techniques are consistent, which confirms the potential of using chaotic time series analysis technique for characterizing ocean data variability.

Discerning Xylella fastidiosa-Infected Olive Orchards in the Time Series of MODIS Terra Satellite Evapotranspiration Data by Using the Fisher–Shannon Analysis and the Multifractal Detrended Fluctuation Analysis

Xylella fastidiosa is a phytobacterium able to provoke severe diseases in many species. When it infects olive trees, it induces the olive quick decline syndrome that leads the tree to a rapid desiccation and then to the death. This phytobacterium has been recently detected in olive groves in southern Italy, representing an important threat to the olive growing of the area. In this paper, in order to identify patterns revealing the presence of Xylella fastidiosa, several hundreds pixels of MODIS satellite evapostranspiration covering infected and healthy olive groves in southern Italy were analyzed by means of the Fisher–Shannon method and the multifractal detrended fluctuation analysis. The analysis of the receiver operating characteric curve indicates that the two informational quantities (the Fisher information measure and the Shannon entropy) and the three multifractal parameters (the range of generalized Hurst exponents and the width and the maximum of the multifractal spectrum) are well suited to discriminate between infected and healthy sites, although the maximum of the multifractal spectrum performs better than the others. These results could suggest the use of both the methods as an operational tool for early detection of plant diseases.

Fractal dimension used as a proxy to understand the spatial distribution for carlin-type gold deposits

The spatial distribution of ore-forming element grades typically exhibits strong nonlinear characteristics as a result of continuous geological processes. The trend of change in ore grade sequences is a crucial indicator of the degree of enrichment of an ore deposit. Therefore, quantitative evaluation of the spatial distribution of ore-forming elements holds great significance for understanding ore-forming processes as well as for ore exploration. In this work, based on the principles of fractal theory, a MATLAB-based program to calculate fractal dimensions was developed, and fractal analysis was conducted on the gold concentration values (g/t) of 25 drill holes in the Lannigou Gold Mining Area, Guizhou, China. Rescaled range analysis and the fractal correlation integral were utilized to obtain the spatial distribution of gold concentrations in the drill holes. Rescaled range analysis revealed that the Hurst exponent (HR/S) values vary in the range of 0.58–0.97, which reflected the persistent behavior of vertical variation in gold tonnage at the studied wells. Furthermore, the relationship between gold grade distribution and associated fluctuations in the fractal dimension derived from the correlation dimension (DC) was discussed. The DC value was found to vary between 1.01 and 1.59, indicating that high-grade gold deposits were distributed along steep fractal gradients and that fractal dimension values could be used as a reliable proxy to efficiently understand the migration pathways of auriferous fluid. Studies have shown that the Hurst exponent of the Au grade sequence exhibited a strong dependence on the grade of mineralization, with gold mineralized veins persisting throughout the studied drill holes. Thus, fractal dimension gradients can serve as an important proxy for identifying the locations of potential economic gold deposits in unexplored areas. Moreover, this approach can be employed to map the distribution patterns of other types of ore deposits.

Memory-enhanced momentum in commodity futures markets

Motivated by the deteriorating performance of traditional cross-sectional momentum strategies in commodity futures markets, we propose to resurrect momentum by incorporating autocorrelation information into the asset selection process. Put differently, we introduce measures of short and long memory (variance ratios and Hurst coefficients, respectively) telling us whether past winners and losers are likely to persist or not. Our empirical findings suggest that a memory-enhanced momentum strategy based on variance ratios significantly outperforms traditional momentum in terms of reward and risk, effectively prevents momentum crashes and is not bound to the movement of the overall commodity market. The strategy returns cannot be explained by typical factor portfolios and macroeconomic variables. They are also robust to alternative data sets, transaction costs and data snooping. In comparison, Hurst coefficients carry less investment-relevant information and cannot outperform variance ratios in terms of risk premia and investment alpha.