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This study establishes the statistical properties of a spectrum-based Whittle parameter estimation procedure for locally stationary long-range dependent processes. Both theoretical and empirical behaviors are investigated. In particular, a central limit theorem for the Whittle likelihood estimation method is derived under mild distributional conditions, extending its application to a wide range of non-Gaussian time series. The finite-sample properties of the estimators are examined using Monte Carlo experiments with gamma and gamma-normal noise distributions. These simulation studies demonstrate that the proposed method behaves properly, even for small to moderate sample sizes. Finally, the practical application of this methodology is illustrated using a well-known real-life data example. [ABSTRACT FROM AUTHOR]

A frequency domain bootstrap (FDB) is a common technique to apply Efron's independent and identically distributed resampling technique (Efron, 1979) to periodogram ordinates – especially normalized periodogram ordinates – by using spectral density estimates. The FDB method is applicable to several classes of statistics, such as estimators of the normalized spectral mean, the autocorrelation (but not autocovariance), the normalized spectral density function, and Whittle parameters. While this FDB method has been extensively studied with respect to short-range dependent time processes, there is a dearth of research on its use with long-range dependent time processes. Therefore, we propose an FDB methodology for ratio statistics under long-range dependence, using semi- and nonparametric spectral density estimates as a normalizing factor. It is shown that the FDB approximation allows for valid distribution estimation for a broad class of stationary, long-range (or short-range) dependent linear processes, without any stringent assumptions on the distribution of the underlying process. The results of a large simulation study show that the FDB approximation using a semi- or nonparametric spectral density estimator is often robust for various values of a long-memory parameter reflecting magnitude of dependence. We apply the proposed procedure to two data examples. [ABSTRACT FROM AUTHOR]

We study the asymptotic behaviour of least squares estimators (LSE) in regression models for long-range dependent random fields observed on spheres. The LSE can be given as a weighted functional of long-range dependent random fields. It is known that in this scenario the limits can be non-Gaussian. We derive the limit distribution and the corresponding rate of convergence for the estimators. The results were obtained under rather general assumptions on the random fields. Simulation studies were conducted to support theoretical findings. [ABSTRACT FROM AUTHOR]

We review recent results on anisotropic scaling limits and the scaling transition for linear and their subordinated nonlinear long-range dependent stationary random fields X on ℤ2. The scaling limits V γ X are taken over rectangles in ℤ2 whose sides increase as O(λ) and O(λγ) as λ→∞for any fixed γ > 0. The scaling transition occurs at γ 0 X > 0 provided that V γ X are different for γ > γ 0 X and γ < γ 0 X and do not depend on γ otherwise. [ABSTRACT FROM AUTHOR]

Let Z denote a Hermite process of order q ≥ 1 and self-similarity parameter H ∈ (1 2 , 1). This process is H -self-similar, has stationary increments and exhibits long-range dependence. When q = 1 , it corresponds to the fractional Brownian motion, whereas it is not Gaussian as soon as q ⩾ 2. In this paper, we deal with a Vasicek-type model driven by Z , of the form d X t = a (b − X t) d t + d Z t. Here, a > 0 and b ∈ R are considered as unknown drift parameters. We provide estimators for a and b based on continuous-time observations. For all possible values of H and q , we prove strong consistency and we analyze the asymptotic fluctuations. [ABSTRACT FROM AUTHOR]

For some practical industrial systems or components, such as blast furnaces and Li-ion batteries, there are two important factors to model the degradation processes. One is the long-range dependence, which can reflect the non-Markovian nature of the degradation processes. The other factor is the existence of multiple modes, because the operating conditions and external environments inevitably change during the whole lifetime of these systems. In this paper, we first propose a fractional Brownian motion (FBM) based degradation model with long-range dependence and multiple modes, and then consider the prediction of remaining useful life. To identify the multiple modes in the degradation process, we propose a two-step method, including change-points detection and linear segments clustering. In each degradation mode, the degradation rate is assumed to be normally distributed. The means and variances of these distributions can be obtained by the maximum likelihood estimation. To describe the switching between different modes, the continuous-time Markov chain is applied, and its transition rate matrix can be estimated by the historical switching time. An approximation of the first passage time with a predefined threshold can be obtained by a weak convergence theorem and a time-space transformation. A numerical simulation and a practical case of a blast furnace wall are provided to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

A technological demand made the integration of multi core chips into a single chip which leads to the advancement of processing elements and memory units. But this leads to communication between cores more challenging. On the downside, simulation with real traffic becomes inflexible and time-consuming. This paper analyzes a novel method for generation of the self-similar process exhibited by the bursty HVEC MPEG-4 video applications for the on-chip architecture with the algorithm running cost at most O (nlogn). Statistical properties of the relevant generated traces from the video clips show the existence of the self-similarity in the traffic of high MPEG-4 video. The multicore architecture helps us in highlighting the inference of the findings on the size of memory (buffer) and present required quantitative analysis for various video streams running on on-chip architecture. We have generated traces from the statistical properties of videos using the Circular Embedding Technique (CIET) resemblance with the bursty nature. Parameters are analyzed under different traffics and loads which results in the reduction of power and latency. Our methodology is to figure the buffer loss probability for the fixed-sized buffer framework utilizing Maximum Variance Asymptotic (MVA) approximations open new bearings and information on research with better ramifications on comparable crucial issues for mixed media applications for on-chip network structure. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the long-range correlations and trends between consecutive earthquakes by means of the scaling parameter so-called locally Hurst parameter, H(t), and examine its variations in time, to find a specific pattern that exists between Earthquakes. The long-range correlations are usaully detected by calculating a constant Hurst parameter. However, the multi-fractal structure of earthquakes caused that more than one scaling exponent is needed to account for the scaling properties of such processes. Thus, in this paper, we consider the time-dependent Hurst exponent to realize scale variations in trend and correlations between consecutive seismic activities, for all times. We apply the Hilbert-Huang transform to estimate H(t) for the time series extracted from seismic activities occurred in California during 12 years, from 2/24/2007 to 9/29/2017. The superiority of the method is discovering some specific hidden patterns that exist between consecutive earthquakes, by studying the trend and variations of H(t). Estimationg H(t) only as a measure of dependency, may lead to misleading results, but using this method, the trend and variations of the parameter is studying to discover hidden dependencies between consecutive earthquakes. Recognizing such dependency patterns can help us in prediction of future main shocks. [ABSTRACT FROM AUTHOR]

Fractional Gaussian noise (fGn) is a stationary time series model with long-memory properties applied in various fields like econometrics, hydrology and climatology. The computational cost in fitting an fGn model of length n using a likelihood-based approach is O (n 2) , exploiting the Toeplitz structure of the covariance matrix. In most realistic cases, we do not observe the fGn process directly but only through indirect Gaussian observations, so the Toeplitz structure is easily lost and the computational cost increases to O (n 3) . This paper presents an approximate fGn model of O (n) computational cost, both with direct and indirect Gaussian observations, with or without conditioning. This is achieved by approximating fGn with a weighted sum of independent first-order autoregressive (AR) processes, fitting the parameters of the approximation to match the autocorrelation function of the fGn model. The resulting approximation is stationary despite being Markov and gives a remarkably accurate fit using only four AR components. Specifically, the given approximate fGn model is incorporated within the class of latent Gaussian models in which Bayesian inference is obtained using the methodology of integrated nested Laplace approximation. The performance of the approximate fGn model is demonstrated in simulations and two real data examples. [ABSTRACT FROM AUTHOR]

Abstract In this work we introduce the class of beta autoregressive fractionally integrated moving average models for continuous random variables taking values in the continuous unit interval (0 , 1). The proposed model accommodates a set of regressors and a long-range dependent time series structure. We derive the partial likelihood estimator for the parameters of the proposed model, obtain the associated score vector and Fisher information matrix. We also prove the consistency and asymptotic normality of the estimator under mild conditions. Hypotheses testing, diagnostic tools and forecasting are also proposed. A Monte Carlo simulation is considered to evaluate the finite sample performance of the partial likelihood estimators and to study some of the proposed tests. An empirical application is also presented and discussed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract We provide a complete description of anisotropic scaling limits of stationary linear random field (RF) on Z 3 with long-range dependence and moving average coefficients decaying as O (| t i | − q i ) in the i th direction, i = 1 , 2 , 3. The scaling limits are taken over rectangles in Z 3 whose sides increase as O (λ γ i ) , i = 1 , 2 , 3 when λ → ∞ , for any fixed γ i > 0 , i = 1 , 2 , 3. We prove that all these limits are Gaussian RFs whose covariance structure is determined by the fulfillment or violation of the balance conditions γ i q i = γ j q j , 1 ≤ i < j ≤ 3. The paper extends recent results in Puplinskaitė and Surgailis (2015, 2016) [27,28] , Pilipauskaitė and Surgailis (2016, 2017) [31,29] on anisotropic scaling of long-range dependent RFs from dimension 2 to dimension 3. [ABSTRACT FROM AUTHOR]

Various trading strategies have been proposed that use estimates of the Hurst coefficient, which is an indicator of long-range dependence, for the calculation of buy and sell signals. This paper introduces frequency-domain tests for longrange dependence which do, in contrast to conventional procedures, not assume that the number of used periodogram ordinates grow with the length of the time series. These tests are applied to series of gold price returns and stock index returns in a rolling analysis. The results suggest that there is no long-range dependence, indicating that trading strategies based on fractal dynamics have no sound statistical basis. [ABSTRACT FROM AUTHOR]

Statistical process control (SPC) has natural applications in data network surveillance. However, network data are commonly autocorrelated, which presents challenges to the basic SPC methods. Most existing SPC methods for correlated data assume parametric models to account for the correlation structure within the data. Those model assumptions can be difficult to justify in practice. In this paper, we propose a nonparametric cumulative sum (CUSUM) control chart for autocorrelated processes. In our proposed approach, we incorporate a wavelet decomposition and a nonparametric multivariate CUSUM control chart to obtain a robust procedure for autocorrelated processes without distribution assumptions. Extensive simulations show that the procedure appropriately controls the in‐control average run length and also has good sensitivity for detecting location shifts. [ABSTRACT FROM AUTHOR]

Abstract The Hermite rank appears in limit theorems involving long memory. We show that a Hermite rank higher than one is unstable when the data is slightly perturbed by transformations such as shift and scaling. We carry out a "near higher order rank analysis" to illustrate how the limit theorems are affected by a shift perturbation that is decreasing in size. We also consider the case where the deterministic shift is replaced by centering with respect to the sample mean. The paper is a companion of Bai and Taqqu (2017) which discusses the instability of the Hermite rank in the statistical context. [ABSTRACT FROM AUTHOR]

This study finds that the scaling properties of India's nominal and real Treasury rates are time varying, as is their multiscaling behaviour. We observe an association between the scaling behaviour of interest rates and the stages of development of the bill market. Interest rate behaviour is influenced by structural reforms, microstructure changes, and improvement in the operational efficiency of the Treasury market. Our findings suggest that monetary policy shocks have a persistent effect, but rates eventually revert to the mean. We show that the adaptive market hypothesis helps to delineate the dynamics of an emerging market undergoing a series of institutional and structural changes. [ABSTRACT FROM AUTHOR]

Ordinal pattern dependence is a multivariate dependence measure based on the co-movement of two time series. In strong connection to ordinal time series analysis, the ordinal information is taken into account to derive robust results on the dependence between the two processes. This article deals with ordinal pattern dependence for a long-range dependent time series including mixed cases of short- and long-range dependence. We investigate the limit distributions for estimators of ordinal pattern dependence. In doing so, we point out the differences that arise for the underlying time series having different dependence structures. Depending on these assumptions, central and non-central limit theorems are proven. The limit distributions for the latter ones can be included in the class of multivariate Rosenblatt processes. Finally, a simulation study is provided to illustrate our theoretical findings.

Brownian motion (BM) has been widely used for degradation modeling and remaining useful life (RUL) prediction, but it is essentially Markovian. This implies that the future state in a BM-based degradation process relies only on its current state, independent of the past states. However, some practical industrial devices such as Li-ion batteries, ball bearings, turbofans, and blast furnace walls show degradations with long-range dependence (LRD), where the future degradation states depend on both the current and past degradation states. This type of degradation naturally brings two interesting problems, that is, how to model the degradations and how to predict their RULs. Recently, in contrast to the work that uses only BM, fractional Brownian motion (FBM) is introduced to model practical degradations. The most important feature of the FBM-based degradation models is the ability to characterize the non-Markovian degradations with LRD. Although FBM is an extension of BM, it is neither a Markovian process nor a semimartingale. Therefore, how to obtain the first passage time of an FBM-based degradation process has become a challenging task. In this paper, a review of the transition of RUL prediction from BM to FBM is provided. The peculiarities of FBM when addressing the LRD inherent in some practical degradations are discussed. We first review the BM-based degradation models of the past few decades and then give details regarding the evolution of FBM-based research. Interestingly, the existing BM-based models scarcely consider the effect of LRD on the prediction of RULs. Two practical cases illustrate that the newly developed FBM-based models are more generalized and suitable for predicting RULs than the BM-based models, especially for degradations with LRD. Along with the direction of FBM-based RUL prediction, we also introduce some important and interesting problems that require further study. [ABSTRACT FROM AUTHOR]

Hypergraph neural networks have recently drawn widespread attention and have succeeded in many fields. However, existing hypergraph-based neural network approaches are limited to learning local neighborhoods and ignore the representation of long-range dependence. Since the lack of necessary long-range interactions tends to lose critical signals from distant nodes, this paper proposes a new mask-guided hypergraph structure learning (MHSL) method. Specifically, MHSL is comprised of two learning components, i.e., the structure learning component and the initial hypergraph learning component. We design a flexible hypergraph structure learning module in the structure learning component to generate a hypergraph representing the global structure by node stream and hyperedge stream. The initial hypergraph learning component contains the hypergraph convolution that retains the local topological information. Additionally, we design a bidirectional node and hyperedge contrastive learning to learn the consistency between local and global structures. The node classification experiments on the hypergraph datasets demonstrate that the proposed MHSL method achieves competitive performance. • We propose MHSL which combines long-range dependence modeling for model performance. • MHSL includes a mask-guided structure learning module to extract high-level features. • MHSL incorporates bidirectional node and hyperedge contrastive learning. [ABSTRACT FROM AUTHOR]

This short article exhibits that there exists critical point of the power for the generalized function t-a for a > 0. The present results show that it is long-range dependent if 0 1. My motivation of studying that dependence issue comes from the power-law type functions in fractal time series. The present results may yet be useful to investigate fractal behavior of fractal time series from a new point of view.

For typical sample sizes occurring in economic and financial applications, the squared bias of estimators for the memory parameter is small relative to the variance. Smoothing is therefore a suitable way to improve the performance in terms of the mean squared error. However, in an analysis of financial high-frequency data, where the estimates are obtained separately for each day and then combined by averaging, the variance decreases with the sample size but the bias remains fixed. This paper proposes a method of smoothing that does not entail an increase in the bias. This method is based on the simultaneous examination of different partitions of the data. An extensive simulation study is carried out to compare it with conventional estimation methods. In this study, the new method outperforms its unsmoothed competitors with respect to the variance and its smoothed competitors with respect to the bias. Using the results of the simulation study for the proper interpretation of the empirical results obtained from a financial high-frequency dataset, we conclude that significant long-range dependencies are present only in the intraday volatility but not in the intraday returns. Finally, the robustness of these findings against daily and weekly periodic patterns is established.

The signals in numerous fields usually have scaling behaviors (long-range dependence and self-similarity) which is characterized by the Hurst parameter H. Fractal Brownian motion (FBM) plays an important role in modeling signals with self-similarity and long-range dependence. Wavelet analysis is a common method for signal processing, and has been used for estimation of Hurst parameter. This paper conducts a detailed numerical simulation study in the case of FBM on the selection of parameters and the empirical bias in the wavelet-based estimator which have not been studied comprehensively in previous studies, especially for the empirical bias. The results show that the empirical bias is due to the initialization errors caused by discrete sampling, and is not related to simulation methods. When choosing an appropriate orthogonal compact supported wavelet, the empirical bias is almost not related to the inaccurate bias correction caused by correlations of wavelet coefficients. The latter two causes are studied via comparison of estimators and comparison of simulation methods. These results could be a reference for future studies and applications in the scaling behavior of signals. Some preliminary results of this study have provided a reference for my previous studies.

In this paper we discuss the details, limitations, and difficulties of the implementation in hardware of a memristor-based random number generator that exhibits monofractal/multifractal behavior. To do so, the components and selection criteria of a reference memristor and one proposed by the authors, the chaotic circuit leveraging them, and the processing that is performed on the chaotic signals to achieve the random discrete sequences are described. After applying the estimation tools, findings indicate that more than 60% of the proposed combinations allow generating random discrete sequences, with long-range dependence, and that both monofractal and multifractal behaviors can also be obtained. Consequently, a hardware system was achieved that can be used as a source of entropy in future synthetic biological signal generators., Competing Interests: The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:Cesar Hernandez reports administrative support and equipment, drugs, or supplies were provided by District University Francisco José de Caldas., (© 2024 The Authors.)

This paper gives the quantitative relationship between prediction error and given past sample size in our research of sea level time series. The present result exhibits that the prediction error of sea level time series in terms of given past sample size follows decayed power functions, providing a quantitative guideline for the quality control of sea level prediction.

Minimax-rate adaptive nonparametric regression has been intensively studied under the assumption of independent or uncorrelated errors in the literature. In many applications, however, the errors are dependent, including both short- and long-range dependent situations. In such a case, adaptation with respect to the unknown dependence is important. We present a general result in this direction under Gaussian errors. It is assumed that the covariance matrix of the errors is known to be in a list of specifications possibly including independence, short-range dependence and long-range dependence as well. The regression function is known to be in a countable (or uncountable but well-structured) collection of function classes. Adaptive estimators are constructed to attain the minimax rate of convergence automatically for each function class under each correlation specification in the corresponding lists. [ABSTRACT FROM AUTHOR]

When analyzing time series which are supposed to exhibit long-range dependence (LRD), a basic issue is the estimation of the LRD parameter, for example the Hurst parameter H∈(1/2,1). Conventional estimators of H easily lead to spurious detection of long memory if the time series includes a shift in the mean. This defect has fatal consequences in change-point problems: Tests for a level shift rely on H, which needs to be estimated before, but this estimation is distorted by the level shift. We investigate two blocks approaches to adapt estimators of H to the case that the time series includes a jump and compare them with other natural techniques as well as with estimators based on the trimming idea via simulations. These techniques improve the estimation of H if there is indeed a change in the mean. In the absence of such a change, the methods little affect the usual estimation. As adaption, we recommend an overlapping blocks approach: If one uses a consistent estimator, the adaption will preserve this property and it performs well in simulations. [ABSTRACT FROM AUTHOR]

For data-driven remaining useful life (RUL) prediction, an appropriate degradation model is critically important to achieve accurate prediction. The degradation processes in some practical systems are not only related to the age but also related to the current degradation state, and the degradation processes may be non-Markovian processes. However, most existing stochastic process-based degradation models only depend on the age, and simply assume that the increments are independent. In this paper, an age- and state-dependent degradation model with long-range dependence is developed, which is more general than most of the existing models based on either Brownian motions (BMs) or fractional Brownian motions (FBMs). The Radon-Nikodym derivative is utilized to obtain a likelihood ratio function of unknown parameters, and the estimates are obtained by maximizing the likelihood ratio function. A weak convergence theorem is introduced to approximate the FBM by a BM with a time-varying coefficient. A time-space transformation is further utilized to obtain an approximate explicit solution of the RUL. At last, numerical simulations and two real case studies of blast furnace walls and ball bearings are adopted to verify the effectiveness of the proposed model. [ABSTRACT FROM AUTHOR]

In time series analysis one of the main goals is to model time dependent phenomena with stochastic processes. Some phenomena exhibit strong dependence so that even the distant past is significantly related to the future. In this paper we will describe some models of longrange dependence, their properties and methods for detecting longrange dependence in data. The application is illustrated on the yearly water levels of Nile river. [ABSTRACT FROM AUTHOR]

This work defines two classes of processes, that we term tempered fractional multistable motion and tempered multifractional stable motion. They are extensions of fractional multistable motion and multifractional stable motion, respectively, obtained by adding an exponential tempering to the integrands. We investigate certain basic features of these processes, including scaling property, tail probabilities, absolute moment, sample path properties, pointwise Hölder exponent, Hölder continuity of quasi norm, (strong) localisability and semi-long-range dependence structure. These processes may provide useful models for data that exhibit both dependence and varying local regularity/intensity of jumps. [ABSTRACT FROM AUTHOR]

It is well-known that the transition function of the Ornstein-Uhlenbeck process solves the Fokker-Planck equation. This standard setting has been recently generalized in different directions, for example, by considering the so-called α-stable driven Ornstein-Uhlenbeck, or by time-changing the original process with an inverse stable subordinator. In both cases, the corresponding partial differential equations involve fractional derivatives (of Riesz and Riemann-Liouville types, respectively) and the solution is not Gaussian. We consider here a new model, which cannot be expressed by a random time-change of the original process: we start by a Fokker-Planck equation (in Fourier space) with the time-derivative replaced by a new fractional differential operator. The resulting process is Gaussian and, in the stationary case, exhibits long-range dependence. Moreover, we consider further extensions, by means of the so-called convolution-type derivative. [ABSTRACT FROM AUTHOR]

We study persistence probabilities of Hermite processes. As a tool, we derive a general decorrelation inequality for the Rosenblatt process, which is reminiscent of Slepian's lemma for Gaussian processes or the FKG inequality and which may be of independent interest. This allows us to compute the persistence exponent for the Rosenblatt process. For general Hermite processes, we derive upper and lower bounds for the persistence probabilities with the conjectured persistence exponent, but with nonmatching boundaries. [ABSTRACT FROM AUTHOR]

Abstract This paper addresses an important problem of modeling and prediction of phenomena with antipersistent behavior and variance changing in time. As a proper stochastic model we propose an autoregressive fractionally integrated moving average (ARFIMA) process with generalized autoregressive conditional heteroskedasticity (GARCH) noise. First, we introduce a simple identification and validation algorithm for such model. Second, we apply the algorithm to weekday data of hot water usage at urban residential blocks. We extract the deterministic sinusoidal component from the data and fit successfully the ARFIMA–GARCH model to the stochastic part. The goodness of fit is checked by examining model errors and prediction performance. All analyses are performed by the rigorous statistical procedure. The proposed model allows for real-time accurate predictions and when implemented at a hot water supply level will lead to a better optimization of the control system and energy efficiency use. Highlights • A problem of modeling and prediction of hot water usage is addressed. • Hot water usage data depicts antipersistent behavior and time-dependent variance. • A simple identification and validation algorithm for ARFIMA–GARCH is introduced. • The presented scheme can be also applied to other phenomena. [ABSTRACT FROM AUTHOR]

We investigate the problem of estimating a function f based on observations from its noisy convolution when the noise exhibits long-range dependence (LRD). We consider both Gaussian and sub-Gaussian errors. We construct an adaptive estimator based on the kernel method, with the optimal selection of the bandwidths performed via Lepski's Method. We derive a minimax lower bound for the -risk when f belongs to a Sobolev ball and show that such estimator attains optimal or near-optimal rates that deteriorate as the LRD worsens. We carry out a limited simulations study which confirms our conclusions from theoretical results. [ABSTRACT FROM AUTHOR]