Hurst Estimation Research Articles

Simulating Rubber Prices under Geometric Fractional Brownian Motion with Different Hurst Estimators

Natural rubber was a vital pillar of Malaysia's export-oriented economy throughout much of the twentieth century, according to the Economic History of Malaya (EHM) website, the worldwide demand for natural rubber is expected to expand at a CAGR of 4.8% in the future (2019–2023). Therefore, it could be concluded that rubber is one of the commodities that should be monitored. Thus, the forecasting of the rubber price should be analysed for the future use. This research focuses on the simulation of rubber prices using geometric Brownian motion and geometric fractional Brownian motion. Moreover, an analysis in the value of hurst exponents' estimator were also done using 3 different methods, Rescaled Range Analysis, Aggravate Variance Method, and Residuals of Regression Method that bases on the slope deviation technique. Finally, an error analysis was done by the calculation of Mean Absolute Percentage Error (MAPE) value to analyse the accuracy of the simulation models.

Estimation of the Hurst Parameter in Spot Volatility

This paper contributes in three stages in a logic of the cognitive process: we firstly propose a new estimation of Hurst exponent by changing frequency method which is purely mathematical. Then we want to check if the new Hurst is efficient, so we prove the advantages of this new Hurst in asymptotic variance in the perspective compared with other two Hurst estimator. However, a purely mathematical game is not enough, a good estimation should be proven by reality, so we apply the new Hurst estimator into truncated and non-truncated spot volatility which fills the gap of previous literatures using 5-min price data (Source: Wind Financial Terminal) of 10 Chinese A-share industry indices from 1 January 2005 until 31 December 2020.

Hurst entropy profiles for Beta Low and Beta High EEG sub-bands Part I: Intragroup comparison.

The non-linear exploration of the entropic structure of the information contained in an EEG allows access to a new source of data that characterizes, in a more global way, the functioning of the brain as a whole. On the understanding that the brain is a hyper-complex system, describing it only through a linear statistical approach, mainly based on the comparison of differences in the magnitude of the power of the electroencephalographic signal between comparable conditions, leaves incomplete the part of the brain phenomenon that obeys non-linear laws. The present work explores the possibility of characterizing the entropic structure of a human brain, through the construction of order/chaos balance profiles. We proceeded to study the entropic content of the EEG signal, by estimating the Hurst exponent in segments of different sizes, for a set of data from 14 EEG electrodes (Emotiv Epoc Research Edition), taken in basal resting conditions, during 3-5 minutes with eyes closed. The EEG recordings of thirteen subjects (N=13) were studied for the low beta (13-21Hz) and high beta (22-30Hz) sub-bands, in four time-processing windows: 125ms, 250ms, 500ms, and 1s. In addition to obtaining the traditional Hurst estimator, the behavior of two derivatives of the Hurst exponent: the moving-Hurst (muHurst), and the mHurst, an estimator interpreted as a global entropic trend of the time series, were also evaluated. The results showed a clear differentiation between the entropic profiles obtained for low beta and high beta, for all individuals, as well as a different entropic structure of both sub-bands, when comparing Hurst exponents and their derivatives with different time processing windows. At the 1s scale (128 EEG frames), the average entropic content of the time series (muHurst) is relatively higher than the entropic load contained in the mid-term estimation of the whole signal (H(trad)). At the same time, estimates of mHurst, a derivative that assesses the overall entropic trend of the signal, indicated mostly lower entropic (more organized) content than estimates at shorter timescales. Our nonlinear exploration of the entropic structure of the brain revealed a wealth of details that can be used to complement the traditional characterization of the brain that linear approaches have provided so far. Our procedure allowed us to discern substantive differences between the entropic characteristics of the low beta and high beta EEG sub-bands. As well as individual differences within the group, potential substrates for the search of neural markers of psychological, pharmacological, or clinical utility.

Analysis of the temporal correlations of TLS range observations from plane fitting residuals

Terrestrial laser scanners (TLS) record a large number of points within a short time. Temporal correlations between observations are unavoidable but often neglected in stochastic modelling. The main consequences are an overestimated precision of the point clouds and potential wrong test decisions when used for deformation analysis with rigorous statistical procedures. Regarding physical considerations, a fractional Gaussian noise, defined by a so-called Hurst exponent, or a combination of fractional Gaussian noises could be used to model the noise of range measurements from a sensor perspective; Temporal correlations are expected to have a long-range dependency due to the high recording rate of the TLS. Scanning settings and configurations can affect the global correlation parameters. These effects can be quantified from the residuals of a least-squares surface approximation from the TLS point cloud. Based on simulation results, real data correlation analysis from indoor and outdoor experiments can be better understood which makes the identification of the dominant correlating noise source possible. Our methodology combines two Hurst-estimators: the Whittle maximum likelihood and the generalised Hurst estimator; It paves the way for a simple and global model for describing the temporal noise of TLS range correlations, usable in point clouds analysis independently of the object under consideration.

Estimation of the Hurst index of the solutions of fractional SDE with locally Lipschitz drift

Strongly consistent and asymptotically normal estimates of the Hurst index H are obtained for stochastic differential equations (SDEs) that have a unique positive solution. A strongly convergent approximation of the considered SDE solution is constructed using the backward Euler scheme. Moreover, it is proved that the Hurst estimator preserves its properties, if we replace the solution with its approximation.

Long memory estimation for complex-valued time series

Long memory has been observed for time series across a multitude of fields, and the accurate estimation of such dependence, for example via the Hurst exponent, is crucial for the modelling and prediction of many dynamic systems of interest. Many physical processes (such as wind data) are more naturally expressed as a complex-valued time series to represent magnitude and phase information (wind speed and direction). With data collection ubiquitously unreliable, irregular sampling or missingness is also commonplace and can cause bias in a range of analysis tasks, including Hurst estimation. This article proposes a new Hurst exponent estimation technique for complex-valued persistent data sampled with potential irregularity. Our approach is justified through establishing attractive theoretical properties of a new complex-valued wavelet lifting transform, also introduced in this paper. We demonstrate the accuracy of the proposed estimation method through simulations across a range of sampling scenarios and complex- and real-valued persistent processes. For wind data, our method highlights that inclusion of the intrinsic correlations between the real and imaginary data, inherent in our complex-valued approach, can produce different persistence estimates than when using real-valued analysis. Such analysis could then support alternative modelling or policy decisions compared with conclusions based on real-valued estimation.

Discriminating Between Explosions and Earthquakes by the SDFA Method

Discriminating Between Explosions and Earthquakes by the SDFA Method Raquel Romes Linhares Abstract The presence of human-made explosions in a seismic catalog leads to errors in statistical analyses of seismicity. We proposed the Hurst estimator by the Smoothed Detrended Fluctuation Analysis method (SDFA) like an additional classification tool to discriminate between natural tectonic earthquakes and human-made explosions. Full Text: PDF DOI: 10.15640/arms.v5n2a1

Hurst Estimation of Scale Invariant Processes with Stationary Increments and Piecewise Linear Drift

The characteristic feature of the discrete scale invariant (DSI) processes is the invariance of their finite dimensional distributions by dilation for certain scaling factor. DSI process with piecewise linear drift and stationary increments inside prescribed scale intervals is introduced and studied. To identify the structure of the process, first, we determine the scale intervals, their linear drifts and eliminate them. Then, a new method for the estimation of the Hurst parameter of such DSI processes is presented and applied to some period of the Dow Jones indices. This method is based on fixed number equally spaced samples inside successive scale intervals. We also present some efficient method for estimating Hurst parameter of self-similar processes with stationary increments. We compare the performance of this method with the celebrated FA, DFA and DMA on the simulated data of fractional Brownian motion (fBm).

Arrhythmia detection from heart rate variability by SDFA method

The smoothed detrended fluctuation analysis (SDFA) based on DFA and the principal of wavelet shrinkage procedures is a scaling analysis method to represent the correlation properties of a time series. Since there is not a specific rule for the choice of the numbers of regressors in SDFA Method, we present here an asymptotic optimal choice. We carried out some Monte Carlo simulations on fractional Gaussian noise (FGN) models, to investigate the effect of the numbers of regressors in SDFA Method. We analyze the long dependence property in view of the SDFA method to compare 10 healthy and 10 unhealthy (with cardiac arrhythimia) RR time series randomly selected from databases of the PhysioBank. It is proposed that utilizing Hurst estimator by SDFA method, as an additional diagnostic tool may provide an indication of cardiac arrhythmia.

Tracking performance and robustness analysis of Hurst estimators for multifractional processes

In this study, the authors focus on the tracking performance and the robustness of 12 sliding-windowed Hurst estimators for multifractional processes with linear trend local Hölder exponent, noisy multifractional processes and multifractional processes with infinite second-order statistics. Four types of multifractional processes are synthesised to test the tracking performance and robustness of these 12 sliding-windowed Hurst estimators. They are (i) noise-free multifractional process; (ii) multifractional process corrupted by 30-dB signal-to-noise ratio (SNR) white Gaussian noise; (iii) multifractional process corrupted by 30-dB SNR impulse noise; and (iv) multifractional stable process, which has no finite second-order statistics. Furthermore, the standard error of different sliding-windowed Hurst estimators are calculated in order to quantify the accuracy and robustness. This study provides a guideline and principle in the selection of Hurst estimators for noise-free multifractional process, noise-corrupted multifractional process and multifractional process with infinite second-order statistics. The results of this analysis show that the sliding-windowed Kettani and Gubner's method provides the best-tracking performance for multifractional processes with linear trend local Hölder exponent and good robustness to noise.

On the robustness of Hurst estimators

The presence and the nature of long-range dependent (LRD) are usually characterised by the Hurst parameter. In order to meet the requirements of analysing the LRD processes, a number of practical estimation methods have been proposed in the literature. Furthermore, some efforts have been made to evaluate the accuracy and validity of the Hurst estimators for LRD processes. In practice, however, many signals measured are corrupted with various types of noises, and sometimes even the concerned signal itself has infinite variance. In such cases, which estimator has the best robustness to the LRD property of the signal and its noise involved, and how robust it is are still unresolved. The aim of this paper is to make a quantitative analysis of the robustness of twelve commonly used Hurst parameter estimators. In this paper, we considered four types of LRD signals with possible noises. They are 1) LRD process alone; 2) LRD process corrupted by 30 dB signal to noise ratio (SNR) white Gaussian noise; 3) LRD process corrupted by 30 dB SNR stable noise; 4) fractional autoregressive moving average (FARIMA) time series with stable innovations. Moreover, the standard errors of each estimator are provided.

Comparison of daubechies wavelets for hurst parameter estimation

Time scale dependence on the working nature of wavelet analysis makes it a valuable tool for Hurst parameter estimation. Similar to other wavelet-based signal processing applications, the selection of a particular wavelet type and vanishing moment in wavelet based Hurst estimation is a challenging problem. In this paper, we investigate the best Daubechies wavelet in wavelet based Hurst estimation for an exact self similar process, fractional Gaussian noise and how Daubechies vanishing moment affects the Hurst estimation accuracy. Daubechies wavelets are preferred in analysis because increasing vanishing moment does not cause excessive increase of time support of Daubechies wavelets. Thus, limited time support of wavelets reduces the border effects. Results show that Daubechies wavelets with one vanishing moment (Daubechies 1) gives the best estimation result for short range dependent fractional Gaussian noise. Daubechies 2 is the best preference for long range dependent fractional Gaussian noise.