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At long times, a fractional Brownian particle in a confining external potential reaches a non-equilibrium (non-Boltzmann) steady state. Here we consider scale-invariant power-law potentials V (x) ∼ | x | m , where m > 0, and employ the optimal fluctuation method (OFM) to determine the large- | x | tails of the steady-state probability distribution P (x) of the particle position. The calculations involve finding the optimal (that is, the most likely) path of the particle, which determines these tails, via a minimization of the exact action functional for this system, which has recently become available. Exploiting dynamical scale invariance of the model in conjunction with the OFM ansatz, we establish the large- | x | tails of ln ⁡ P (x) up to a dimensionless factor α (H , m) , where 0 < H < 1 is the Hurst exponent. We determine α (H , m) analytically (i) in the limits of H → 0 and H → 1, and (ii) for m = 2 and arbitrary H, corresponding to the fractional Ornstein-Uhlenbeck (fOU) process. Our results for the fOU process are in agreement with the previously known exact P (x) and autocovariance. The form of the tails of P (x) yields exact conditions, in terms of H and m, for the particle confinement in the potential. For H ≠ 1 / 2 , the tails encode the non-equilibrium character of the steady state distribution, and we observe violation of time reversibility of the system except for m = 2. To compute the optimal paths and the factor α (H , m) for arbitrary permissible H and m, one needs to solve an (in general nonlinear) integro-differential equation. To this end we develop a specialized numerical iteration algorithm which accounts analytically for an intrinsic cusp singularity of the optimal paths for H < 1 / 2 . [ABSTRACT FROM AUTHOR]

We analyze the error estimates of a fully discrete scheme for solving a semilinear stochastic subdiffusion problem driven by integrated fractional Gaussian noise with a Hurst parameter H ∈ (0 , 1) . The covariance operator Q of the stochastic fractional Wiener process satisfies ∥ A − ρ Q 1 / 2 ∥ H S   <   ∞ for some ρ ∈ [ 0 , 1) , where ∥ · ∥ H S denotes the Hilbert–Schmidt norm. The Caputo fractional derivative and Riemann–Liouville fractional integral are approximated using Lubich's convolution quadrature formulas, while the noise is discretized via the Euler method. For the spatial derivative, we use the spectral Galerkin method. The approximate solution of the fully discrete scheme is represented as a convolution between a piecewise constant function and the inverse Laplace transform of a resolvent-related function. By using this convolution-based representation and applying the Burkholder–Davis–Gundy inequality for fractional Gaussian noise, we derive the optimal convergence rates for the proposed fully discrete scheme. Numerical experiments confirm that the computed results are consistent with the theoretical findings. [ABSTRACT FROM AUTHOR]

While the statistical inference of first-order autoregressive processes driven by independent and identically distributed noises has a long history, the statistical analysis for first-order autoregressive processes driven by dependent noises is more recent. This paper considers the problem of estimating all the unknown parameters in first-order autoregressive processes driven by the fractional Gaussian noise, which is a self-similar stochastic process used to model persistent or anti-persistent dependency structures in observed time series. The estimation procedure is built upon the marriage of the log-periodogram regression and the method of moments. The usual asymptotic properties, including consistency and asymptotic normality, are established under some mild conditions. Monte Carlo simulations are performed to demonstrate the feasibility and effectiveness of the proposed method. Finally, the estimation methods are applied to model some realized volatility time series, where we find that the realized volatility is rough. Moreover, the proposed model is compared with some alternative models, including the first-order autoregressive process driven by the standard Gaussian noise, the heterogeneous autoregression model with lag structure (1,5,21) (HAR(1,5,21)), the autoregressive fractionally integrated moving average model (ARFIMA(1,d,0)) and the scaled fractional Brownian motion, in forecasting the logarithmic realized volatility. [ABSTRACT FROM AUTHOR]

The fractional Brownian motion (fBm) process, governed by a fractional parameter H∈(0,1), is a continuous‐time Gaussian process with its increment being the fractional Gaussian noise (fGn). This article first provides a computationally feasible expression for the spectral density of fGn. This expression enables us to assess the accuracy of a range of approximation methods, including the truncation method, Paxson's approximation, and the Taylor series expansion at the near‐zero frequency. Next, we conduct an extensive Monte Carlo study comparing the finite sample performance and computational cost of alternative estimation methods for H under the fGn specification. These methods include two semi‐parametric methods (based on the Taylor series expansion), two versions of the Whittle method (utilising either the computationally feasible expression or Paxson's approximation of the spectral density), a time‐domain maximum likelihood (ML) method (employing a recursive approach for its likelihood calculation), and a change‐of‐frequency method. Special attention is paid to highly anti‐persistent processes with H close to zero, which are of empirical relevance to financial volatility modelling. Considering the trade‐off between statistical and computational efficiency, we recommend using either the Whittle ML method based on Paxson's approximation or the time‐domain ML method. We model the log realized volatility dynamics of 40 financial assets in the US market from 2012 to 2019 with fBm. Although all estimation methods suggest rough volatility, the implied degree of roughness varies substantially with the estimation methods, highlighting the importance of understanding the finite sample performance of various estimation methods. [ABSTRACT FROM AUTHOR]

We propose a new time series model called Rational Gaussian Noise (rGn) to describe a pattern of long-range dependence. The rGn model is shown to be an extension of the traditional fractional Gaussian noise (fGn). Theoretical formulas such as autocorrelation function, and some properties of rGn are derived and compared to that of fGn. Transformed S&P500 daily excessive return data is used as a case study where parameters for both the rGn and fGn models are estimated. [ABSTRACT FROM AUTHOR]

The assessment of physiological complexity via the estimation of monofractal exponents or multifractal spectra of biological signals is a recent field of research that allows detection of relevant and original information for health, learning, or autonomy preservation. This tutorial aims at introducing Whittle’s maximum likelihood estimator (MLE) that estimates the monofractal exponent of time series. After introducing Whittle’s maximum likelihood estimator and presenting each of the steps leading to the construction of the algorithm, this tutorial discusses the performance of this estimator by comparing it to the widely used detrended fluctuation analysis (DFA). The objective of this tutorial is to propose to the reader an alternative monofractal estimation method, which has the advantage of being simple to implement, and whose high accuracy allows the analysis of shorter time series than those classically used with other monofractal analysis methods.

This article is devoted to investigating the evolution of the probability density function for power system excited by fractional stochastic noise. First, the single-machine-infinite-bus (SMIB) power system model excited by fractional Gaussian noise (FGN) is established. Second, we derive the Fokker–Planck–Kolmogorov (FPK) equation for the proposed model and solve the FPK equation using the finite difference method. Finally, the numerical results verify that the addition of FGN would influence dynamical stability of the SMIB power system under certain conditions. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the properties of entropy as a function of the Hurst index, which corresponds to the fractional Gaussian noise. Since the entropy of the Gaussian vector depends on the determinant of the covariance matrix, and the behavior of this determinant as a function of the Hurst index is rather difficult to study analytically at high dimensions, we also consider simple alternative entropy functionals, whose behavior, on the one hand, mimics the behavior of entropy and, on the other hand, is not difficult to study. Asymptotic behavior of the normalized entropy (so called entropy rate) is also studied for the entropy and for the alternative functionals. [ABSTRACT FROM AUTHOR]

This paper explores the effectiveness of using ordinal pattern probabilities to evaluate antipersistency in the sign decomposition of long-range anti-correlated Gaussian fluctuations. It is numerically shown that ordinal patterns are able to effectively measure both persistent and antipersistent dynamics by analyzing the sign decomposition derived from fractional Gaussian noise. These findings are crucial given that traditional methods such as Detrended Fluctuation Analysis are unsuccessful in detecting anti-correlations in such sequences. The numerical results are supported by physiological and environmental data, illustrating its applicability in real-world situations. • Ordinal patterns can successfully quantify antipersistency in binary sequences. • Sign decomposition of fractional Gaussian noise is considered as a numerical test. • Performance of the present approach in real world applications is also evidenced. [ABSTRACT FROM AUTHOR]

This paper deals with the maximum likelihood estimator for the parameter of first-order autoregressive models driven by the stationary Gaussian noises (Colored noise) together with an input. First, we will find the optimal input that maximizes the Fisher information, and then, with the method of the Laplace transform, both the asymptotic properties and the asymptotic design problem of the maximum likelihood estimator will be investigated. The results of the numerical simulation confirm the theoretical analysis and show that the proposed maximum likelihood estimator performs well in finite samples. [ABSTRACT FROM AUTHOR]

We study the projection of an element of fractional Gaussian noise onto its neighbouring elements. We prove some analytic results for the coefficients of this projection. In particular, we obtain recurrence relations for them. We also make several conjectures concerning the behaviour of these coefficients, provide numerical evidence supporting these conjectures, and study them theoretically in particular cases. As an auxiliary result of independent interest, we investigate the covariance function of fractional Gaussian noise, prove that it is completely monotone for H > 1 / 2 , and, in particular, monotone, convex, log-convex along with further useful properties. [ABSTRACT FROM AUTHOR]

This document is an editorial published in the journal Frontiers in Physics. It discusses the theory and applications of long-range dependent (LRD) processes, which are of interest to researchers in various fields, including physics and computer science. LRD processes differ from conventional random processes in terms of their autocorrelation function, power spectrum density, and probability density function. The editorial highlights two models of LRD processes: fractional Gaussian noise (fGn) and the generalized Cauchy process. It also mentions several papers included in the research topic, which explore the use of LRD processes in precision time protocol, dynamic routing algorithms, identification of cryptomining behavior, and modeling global sea surface chlorophyll concentration. [Extracted from the article]

Fractional spline wavelets act as fractional differentiators for essentially low-pass signals, which is a useful property to analyze time series with fractal behavior. Moreover, under suitable scaling of wavelet coefficients, one can whiten a fractional Gaussian noise inputted in the wavelet channels. In this article, we demonstrate the two former statements and, by employing them, propose a method based on maximum likelihood to estimate the long-memory parameter using fractional spline wavelets. The proposed estimator exhibits competitive behavior in terms of mean squared error in the simulation studies we conducted, dominating all other methods as the sample size increases. In empirical applications, the proposed method outperformed other approaches in 10 of 12 evaluations for three different time series. Moreover, the proposed method allows for estimation in a continuous search grid, even when the Hurst coefficient lies outside of stationarity boundaries, which is a clear advantage over many algorithms: in such cases, the proposed method displays a stable behavior and an increasing outperformance margin. We also propose an identification procedure for the persistence or antipersistence of a given signal, based on the energy concentration of the coefficients of wavelet decomposition and its theoretical properties. • Novel estimator for long-memory parameter with fractional spline wavelets. • We demonstrate whitening & differentiation properties of fractional spline wavelets. • Allows estimation in a continuous grid, even beyond stationarity boundaries. • The estimator outperforms 7 well-established methods in simulations in terms of MSE. • With real datasets, the method had the best performance in 10 out of 12 evaluations. [ABSTRACT FROM AUTHOR]

Fractional Gaussian noise models the time series with long-range dependence; when the Hurst index H(1/2, 1), it has positive correlation reflecting a persistent autocorrelation structure. This paper studies the numerical method for solving the stochastic fractional diffusion equation driven by fractional Gaussian noise. Using the operator theoretical approach, we present the regularity estimate of the mild solution and the fully discrete scheme with finite element approximation in space and backward Euler convolution quadrature in time. The O(H-pα) convergence rate in time in space are obtained, showing the relationship between the regularity of noise and convergence rates, where p is a parameter to measure the regularity of noise and α (0, 1). Finally, numerical experiments are performed to support the theoretical results. [ABSTRACT FROM AUTHOR]

A Non-parametric Ensemble Empirical Mode Decomposition (NCEEMD) method is a novel technique for nonlinear and non-stationary signal analysis to detect a gearbox fault. The NCEEMD method was based on the CEEMD, but the Gaussian white noise was replaced by the fractional Gaussian noise. The NCEEMD method does not need to choose the appropriate SNR and the number of ensemble trials before signal processing, which makes it a non-parametric method. This new approach was evaluated using a simulated malfunction signal representing two typical faults in gearbox systems: modulation and rub-impact. Its performance was evaluated in terms of MSE and computation time. A comparative study between the EMD, EEMD, CEEMD, and NCEEMD methods showed that the latter performed better by improving the computation time and accuracy of CEEMD. The proposed method is a non-parametric method that provides a powerful tool in extracting the modulation and the rub-impact features from a vibration signal. The NCEEMD method helps to track down the gearbox faults and resolve this crucial problem in mechanical machines. [ABSTRACT FROM AUTHOR]

Random noise arising from physical processes is an inherent characteristic of measurements and a limiting factor for most signal processing and data analysis tasks. Given the recent interest in generative adversarial networks (GANs) for data-driven modeling, it is important to determine to what extent GANs can faithfully reproduce noise in target data sets. In this paper, we present an empirical investigation that aims to shed light on this issue for time series. Namely, we assess two general-purpose GANs for time series that are based on the popular deep convolutional GAN architecture, a direct time-series model and an image-based model that uses a short-time Fourier transform data representation. The GAN models are trained and quantitatively evaluated using distributions of simulated noise time series with known ground-truth parameters. Target time series distributions include a broad range of noise types commonly encountered in physical measurements, electronics, and communication systems: band-limited thermal noise, power law noise, shot noise, and impulsive noise. We find that GANs are capable of learning many noise types, although they predictably struggle when the GAN architecture is not well suited to some aspects of the noise, e.g. impulsive time-series with extreme outliers. Our findings provide insights into the capabilities and potential limitations of current approaches to time-series GANs and highlight areas for further research. In addition, our battery of tests provides a useful benchmark to aid the development of deep generative models for time series.

The main difficulty of analyzing the response of nonlinear dynamical systems to fractional Gaussian noise (fGn) is the non-Markov property and non-usefulness of diffusion process theory. Currently, only numerical simulation can be applied to obtain the response of nonlinear systems to fGn. In the present paper, noting the rather flat property of the fGn power spectral density (PSD) in most part of frequency band, the stochastic averaging method for quasi-integrable Hamiltonian systems under wide-band noise excitation is applied to predict the response of quasi-integrable and non-resonant Hamiltonian systems to fGn. By using this method, the averaged Itô stochastic differential equations (SDEs) are established and the probability density function (PDF) of system response can be obtained from solving the corresponding Fokker-Planck-Kolmogorov (FPK) equation. All of the statistics of system response are then obtained from the PDF analytically and verified through the comparison with the simulation results.

Phytoplankton movement patterns and swimming behavior are important and basic topics in aquatic biology. Heavy tail distribution exists in diverse taxa and shows theoretical advantages in environments. The fat tails in the movement patterns and swimming behavior of phytoplankton in response to the food supply were studied. The log-normal distribution was used for fitting the probability density values of the movement data of Oxyrrhis marina. Results showed that obvious fat tails exist in the movement patterns of O. marina without and with positive stimulations of food supply. The algal cells tended to show a more chaotic and disorderly movement, with shorter and neat steps after adding the food source. At the same time, the randomness of turning rate, path curvature and swimming speed increased in O. marina cells with food supply. Generally, the responses of phytoplankton movement were stronger when supplied with direct prey cells rather than the cellfree filtrate. The scale-free random movements are considered to benefit the adaption of the entire phytoplankton population to varied environmental conditions. Inferentially, the movement pattern of O. marina should also have the characteristics of long-range dependence, local self-similarity and a system of fractional order. [ABSTRACT FROM AUTHOR]

Given a sequence of independent standard Gaussian variables (Zn), the classical Pisier algebra P is the class of all continuous functions f on the unit circle T such that for each t ∈ 𝕋, the random Fourier series ∑ n ∈ ℤ Z n f ̂ n × exp 2 π i nt converges in L2 and the corresponding sums constitute a Gaussian process that admits a continuous version. It was constructed by Pisier in 1979 to answer a long-standing question raised by Katznelson. In this paper, we consider the general random Fourier series ∑ n ∈ ℤ Z n f ̂ n × exp 2 π i nt where ξ = (ξn) is a discrete Gaussian process of standard Gaussian random variables but with the restriction of independence relaxed and study the corresponding class P(ξ) of continuous functions f on 𝕋. We obtain sufficient conditions (based on some spectral properties of the covariance matrix of (ξn)) for each of the relations 𝒫 ⊂ (ξ), 𝒫(ξ) ⊂ 𝒫, and 𝒫 = 𝒫(ξ). We illustrate these results by the classical fractional Gaussian noise. Whether in general (ξ) is also a Banach algebra is an open problem. [ABSTRACT FROM AUTHOR]

This article presents a hybrid wavelet-based algorithm to suppress the ocular artifacts from electroencephalography (EEG) signals. The hybrid wavelet transform (HWT) method is designed by the combination of discrete wavelet decomposition and wavelet packet transform. The artifact suppression is performed by the selection of sub-bands obtained by HWT. Fractional Gaussian noise (fGn) is used as the reference signal to select the sub-bands containing the artifacts. The multichannel EEG signal is decomposed HWT into a finite set of sub-bands. The energies of the sub-bands are compared to that of the fGn to the desired sub-band signals. The EEG signal is reconstructed by the selected sub-bands consisting of EEG. The experiments are conducted for both simulated and real EEG signals to study the performance of the proposed algorithm. The results are compared with recently developed algorithms of artifact suppression. It is found that the proposed method performs better than the methods compared in terms of performance metrics and computational cost. [ABSTRACT FROM AUTHOR]

The reliability of quasi integrable and non-resonant Hamiltonian system under fractional Gaussian noise (fGn) excitation is studied. Noting rather flat fGn power spectral density (PSD) in most part of frequency band, the fGn is innovatively regarded as a wide-band process. Then, the stochastic averaging method for quasi integrable Hamiltonian systems under wide-band noise excitation is applied to reduce 2n-dimensional original system into n-dimensional averaged Itô stochastic differential equations (SDEs). Reliability function and mean first passage time are obtained by solving the associated backward Kolmogorov equation and Pontryagin equation. The validity of the proposed procedure is tested by applying it to an example and comparing the numerical results with those from Monte Carlo simulation. [ABSTRACT FROM AUTHOR]

The likelihood ratio test for a change in the mean-reverting parameter of a first order autoregressive model with stationary Gaussian noise is considered. The test statistic converges in distribution to the Gumbel extreme value distribution under the null hypothesis of no change-point for a large class of covariance structures including long-memory processes as the fractional Gaussian noise. [ABSTRACT FROM AUTHOR]

Summary: The paper analyses temperature data from 96 selected weather stations world wide, and from reconstructed northern hemisphere temperature data over the last two millennia. Using a non‐parametric test, we find that the stationarity hypothesis is not rejected by the data. Subsequently, we investigate further properties of the data by means of a statistical model known as the fractional Gaussian noise (FGN) model. Under stationarity FGN follows from the fact that the observed data are obtained as temporal aggregates of data generated at a finer (basic) timescale where temporal aggregation is taken over a 'large' number of basic units. The FGN process exhibits long‐range dependence. Several tests show that both the reconstructed and most of the observed data are consistent with the FGN model. [ABSTRACT FROM AUTHOR]

Temporal persistence in unforced climate variability makes detection of trends in surface temperature difficult. Part of the challenge is methodological since standard techniques assume a separation of time scales between trend and noise. In this work we present a novel Bayesian approach to trend detection under the assumption of long-range dependent natural variability, and we use estimates of historical forcing to test if the method correctly discriminates trends from low-frequency natural variability. As an application we analyze 2° × 2° gridded data from the GISS Surface Temperature Analysis. In the time period from 1900 to 2015 we find positive trends for 99% of the grid points. For 84% of the grid points we are confident that the trend is positive, meaning that the 95% credibility interval for the temperature trend contained only positive values. This number increased to 89% when we used estimates of historical forcing to specify the noise model. For the time period from 1900 to 1985 the corresponding ratios were 42 and 52%. Our findings demonstrate that positive trends since 1900 are now detectable locally over most of Earth's surface.

Abstract The fractional Brownian motion and its increment process, the fractional Gaussian noise (fGn), are highly popular models for data exhibiting anomalous diffusion. In this paper, an explicit formula for the dynamical functional, a tool for testing ε-ergodicity breaking and a statistic helpful in the process identification, is provided for the fractional Gaussian noise. Its basic characteristics are derived and the distribution of its single trajectory estimator is studied. Additionally, the sensibility of the convergence of the dynamical functional to the Hurst parameter H is analysed. [ABSTRACT FROM AUTHOR]

Fractal investigation of time series is very complex for several reasons. Due to the existence of fully continuous model, on which the majority of conventional methods are based, the quality of Hurst exponent estimate is often influenced by the number of input data and its sampling rate. In this work, we present a novel approach of unbiased Hurst exponent estimate that is suitable especially for short time series. The crucial idea is deriving the discrete fractional Brownian bridge and its statistical properties that can be subsequently used for model parameter estimation. For the verification and demonstration of efficiency of the method, several generators of fractional Gaussian noise are presented and tested. [ABSTRACT FROM AUTHOR]

We consider the problem of estimating the response function in a deconvolution model under fractional Gaussian noise and noisy kernel. A preliminary threshold is used to stabilize the inversion. Our estimator is adaptive and attains minimax optimal or near-optimal rates in a wide range of Besov balls. [ABSTRACT FROM AUTHOR]

The paper examines the ability of neural networks to classify Internet traffic data in terms of self-similarity expressed by the Hurst exponent. Fractional Gaussian noise is used for the generation of synthetic data for modeling the genuine ones. It is presented that the trained model is capable of classifying the synthetic data obtained from the Pareto distribution and the real traffic data. We present the results of training for different optimizers of the cost function and a different number of convolutional layers in the neural network.

Delta function is a widely used generalized function in various fields, ranging from physics to mathematics. How to express its fractional derivative with integral representation is a tough problem. In this paper, we present an integral representation of the fractional derivative of the delta function. Moreover, we provide its application in representing the fractional Gaussian noise.

Many signals appear fractal and have self-similarity over a large range of their power spectral densities. They can be described by so-called Hermite processes, among which the first order one is called fractional Brownian motion (fBm), and has a wide range of applications. The fractional Gaussian noise (fGn) series is the successive differences between elements of a fBm series; they are stationary and completely characterized by two parameters: the variance, and the Hurst coefficient (H). From physical considerations, the fGn could be used to model the noise of observations coming from sensors working with, e.g., phase differences: due to the high recording rate, temporal correlations are expected to have long range dependency (LRD), decaying hyperbolically rather than exponentially. For the rigorous testing of deformations detected with terrestrial laser scanners (TLS), the correct determination of the correlation structure of the observations is mandatory. In this study, we show that the residuals from surface approximations with regression B-splines from simulated TLS data allow the estimation of the Hurst parameter of a known correlated input noise. We derive a simple procedure to filter the residuals in the presence of additional white noise or low frequencies. Our methodology can be applied to any kind of residuals, where the presence of additional noise and/or biases due to short samples or inaccurate functional modeling make the estimation of the Hurst coefficient with usual methods, such as maximum likelihood estimators, imprecise. We demonstrate the feasibility of our proposal with real observations from a white plate scanned by a TLS.

Long-memory processes, in particular fractional Gaussian noise processes, have been applied as models for many phenomena occurring in nature. Non-stationarities, such as trends, mean level-shifts, etc., impact the accuracy of long-memory parameter estimators, giving rise to biases and misinterpretations of the phenomena. In this article, a novel methodology for the detection and location of mean level-shifts in stationary long-memory fractional Gaussian noise (fGn) signals is proposed. It is based on a joint application of the wavelet-Tsallis q-entropy as a preprocessing technique and a peak detection methodology. Extensive simulation experiments in synthesized fGn signals with mean level-shifts confirm that the proposed methodology not only detects, but also locates level-shifts with high accuracy. A comparative study against standard techniques of level-shift detection and location shows that the technique based on wavelet-Tsallis q-entropy outperforms the one based on trees and the Bai and Perron procedure, as well.

Abstract The contributions given in this article are mainly in two folds with regard to the fractional Gaussian noise of the Weyl type. The first one is to present a complete form of fGn, which is taken as the output of a fractional-order filter driven by white noise. The second is to propose the analytic expressions of the impulse response function and the frequency transfer function of that fractional-order filter. We also give an expression of the fractional Brownian motion of the Weyl type as a consequence of the present fGn. Besides, we suggest that the statistical dependences of fGn may be described based on the frequency transfer function or the impulse response function of the fractional-order filter. Highlights • The article gives the expressions of the complete form of the fractional Gaussian noise and the fractional Brownian motion of the Weyl type (fGn). • The article proposes the analytic expressions of the impulse response function and the frequency transfer function of the fractional-order filter that produces the fGn under the excitation of white noise. • The article suggests that the statistical dependences of fGn may be described from the point of view of the frequency transfer function or the impulse response function of the fractional-order filter. [ABSTRACT FROM AUTHOR]