Long-range Dependent Processes Research Articles

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.

Long-Range Dependence and Climate Noise Characteristics of Antarctic Temperature Data

Abstract This study examines the long-range dependency, climate noise characteristics, and nonlinear temperature trends of eight Antarctic stations from the Reference Antarctic Data for Environmental Research (READER) dataset. Evidence is shown that Antarctic temperatures are long-range dependent. To identify possible nonlinear trends, the ensemble empirical mode decomposition (EEMD) method is used, and then the question of whether the observed trends can arise from internal atmospheric fluctuations is examined. To answer this question, surrogate data are generated from two paradigmatic null models: a standard first-order autoregressive process representing a short-range dependent process and a fractional integrated process representing a long-range dependent process. It is found that three of the eight stations show statistically significant trends when tested against the short-range dependent process while only the Faraday–Vernadsky station temperature time series shows a significant trend when tested against the long-range dependent null model. All other considered stations show no trends that are statistically significant against the two null models, and thus they can be explained by internal atmospheric variability. These results imply that more attention should be given to assessing the correlation structure of climate time series.

An efficient estimator for locally stationary Gaussian long-memory processes

This paper addresses the estimation of locally stationary long-range dependent processes, a methodology that allows the statistical analysis of time series data exhibiting both nonstationarity and strong dependency. A time-varying parametric formulation of these models is introduced and a Whittle likelihood technique is proposed for estimating the parameters involved. Large sample properties of these Whittle estimates such as consistency, normality and efficiency are established in this work. Furthermore, the finite sample behavior of the estimators is investigated through Monte Carlo experiments. As a result from these simulations, we show that the estimates behave well even for relatively small sample sizes.

Robust estimation of the scale and of the autocovariance function of Gaussian short- and long-range dependent processes

A desirable property of an autocovariance estimator is to be robust to the presence of additive outliers. It is well-known that the sample autocovariance, being based on moments, does not have this property. Hence, the use of an autocovariance estimator which is robust to additive outliers can be very useful for time-series modeling. In this paper, the asymptotic properties of the robust scale and autocovariance estimators proposed by Rousseeuw and Croux (1993) and Genton and Ma (2000) are established for Gaussian processes, with either short-range or long-range dependence. It is shown in the short-range dependence setting that this robust estimator is asymptotically normal at the rate $\sqrt{n}$, where $n$ is the number of observations. An explicit expression of the asymptotic variance is also given and compared to the asymptotic variance of the classical autocovariance estimator. In the long-range dependence setting, the limiting distribution displays the same behavior than that of the classical autocovariance estimator, with a Gaussian limit and rate $\sqrt{n}$ when the Hurst parameter $H$ is less $3/4$ and with a non-Gaussian limit (belonging to the second Wiener chaos) with rate depending on the Hurst parameter when $H \in (3/4,1)$. Some Monte-Carlo experiments are presented to illustrate our claims and the Nile River data is analyzed as an application. The theoretical results and the empirical evidence strongly suggest using the robust estimators as an alternative to estimate the dependence structure of Gaussian processes.

Nonparametric regression for dependent data in the errors-in-variables problem

We consider the nonparametric estimation of the regression functions for dependent data. Suppose that the covariates are observed with additive errors in the data and we employ nonparametric deconvolution kernel techniques to estimate the regression functions in this paper. We investigate how the strength of time dependence affects the asymptotic properties of the local constant and linear estimators. We treat both short-range dependent and long-range dependent linear processes in a unified way and demonstrate that the long-range dependence (LRD) of the covariates affects the asymptotic properties of the nonparametric estimators as well as the LRD of regression errors does.

Fast simulation of self-similar and correlated processes

Simulations with long-range dependent or self-similar input processes are hindered both by the slowness of convergence displayed by the output data and by the high computational complexity of the on-line methods for generating the input process. In this paper, we present optimized algorithms for simulating efficiently the occupancy process of a M/G/ ∞ system, which can be used as a sequential pseudo-random number generator of a broad class of self-similar and correlated sample-paths. We advocate the use of this approach in the simulation toolbox, as a simple method to overcome the drawbacks of other synthetic generators of Gaussian self-similar time series. Our approach to fast simulation of the M/G/ ∞ model is the decomposition of the service time distribution as a linear combination of deterministic and memoryless random variables, plus a residual term. Then, the original M/G/ ∞ system is replaced by a number of parallel, independent, virtual and easier to simulate M/G/ ∞ subsystems, the dynamics of which can be replicated sequentially or in parallel too. We report the results of several experiments demonstrating the substantial improvements attainable with this decomposition.

Particle systems with quasi-homogeneous initial states and their occupation time fluctuations

Occupation time fluctuation limits of particle systems in R^d with independent motions (symmetric stable Levy process, with or without critical branching) have been studied assuming initial distributions given by Poisson random measures (homogeneous and some inhomogeneous cases). In this paper, with d=1 for simplicity, we extend previous results to a wide class of initial measures obeying a quasi-homogeneity property, which includes as special cases homogeneous Poisson measures and many deterministic measures (simple example: one atom at each point of Z), by means of a new unified approach. In previous papers, in the homogeneous Poisson case, for the branching system in "low" dimensions, the limit was characterized by a long-range dependent Gaussian process called sub-fractional Brownian motion (sub-fBm), and this effect was attributed to the branching because it had appeared only in that case. An unexpected finding in this paper is that sub-fBm is more prevalent than previously thought. Namely, it is a natural ingredient of the limit process in the non-branching case (for "low" dimension), as well. On the other hand, fractional Brownian motion is not only related to systems in equilibrium (e.g., non-branching system with initial homogeneous Poisson measure), but it also appears here for a wider class of initial measures of quasi-homogeneous type.

Steady-State Simulation of Long-Range Dependent Queuing Processes

Long-range dependence was first noted in hydrology by H. E. Hurst from a study of the water levels of the Nile river which showed a tendency for a flood year to be followed by another flood year. Long-range dependent processes are relevant not only in telecommunication networks but also in such areas of scientific activity as econophysics, climatology, economics, environmental sciences, geology, geophysics, hydrology, computer science, and computer engineering. They provide good models of packet trac in telecommunication networks, for example, in local area networks and video trac, and good models of persistence analysis in econophysics, especially for short-term interest rates. In this paper, the eects of long-range dependence on the behaviors of queuing systems have been investigated. We have also investigated how the long-range dependence in arrival processes aects the length of sequential steady-state simulations being executed to obtain simulation results with the required level of statistical error. Our results show that the finite buer overflow probability of a queuing system with a long-range dependent input is much greater than the equivalent queuing system with a Poisson or a short-range dependent input process and that the overflow probability increases as the Hurst parameter approaches one.

The effect of long-range dependence on modelling extremes with the generalised extreme value distribution

Two effects arise for the modelling of block maxima from dependent time series: a reduced rate of convergence for the block maxima probability distribution towards the generalised extreme value distribution, and an increase in uncertainty of the parameter estimates compared to independent or short range dependent records. These effects are exemplified with a simulation study using a white noise, a short-range and a long-range dependent process. The two issues raised turned out to be relatively unproblematic for short-range dependent processes. For long-range dependent processes, especially the increased parameter uncertainty poses a problem. Incautious use of standard procedures would lead to a severe underestimation of the parameter uncertainty which implies a misconception of accuracy for derived quantities, such as return levels which are frequently used for risk assessment and dimensioning of hydraulic structures.

Modelling profit series: nonstationarity and long memory

The dynamic structure of profit rates for 156 US manufacturing companies is analysed by means of fractional integration techniques as an alternative to the commonly used ARIMA models with respect to the ‘persistence of profits’. Thereby the pseudo-spectral density aproach of Velasco and Robinson together with model selection criteria is applied. The results show–despite the short lengths of the series and tests for the integer degrees of integration (d = 0, 1)–that 35.5% of the series may be well-approximated by long-range dependent processes, and 54% are nonstationary. This is a confirmation of the strong challenge to the competitive environment hypothesis obtained by previous studies.

On Berry–Esseen bounds for non-instantaneous filters of linear processes

Let $X_n=∑^∞_{i=1}a_iɛ_{n−i}$, where the $ɛ_i$ are i.i.d. with mean 0 and at least finite second moment, and the $a_i$ are assumed to satisfy $|a_i|=O(i^{−β})$ with $β>1/2$. When $1/2<β<1, X_n$ is usually called a long-range dependent or long-memory process. For a certain class of Borel functions $K(x_1, …, x_{d+1}), d≥0$, from ${\mathcal{R}}^{d+1}$ to $\mathcal{R}$, which includes indicator functions and polynomials, the stationary sequence $K(X_n, X_{n+1}, …, X_{n+d})$ is considered. By developing a finite orthogonal expansion of $K(X_n, …, X_{n+d})$, the Berry–Esseen type bounds for the normalized sum $Q_{N}/\sqrt{N}, Q_N=∑^N_{n=1}(K(X_n, …, X_{n+d})−\mathrm{E}K(X_n, …, X_{n+d}))$ are obtained when $Q_{N}/\sqrt{N}$ obeys the central limit theorem with positive limiting variance.

Comparison of various estimators in simulated FGN

The Hurst parameter is the simplest numerical characteristic of self-similar long-range dependent stochastic processes. Such processes have been identified in many natural and man-made systems. In particular, since they were discovered in the Internet and other multimedia telecommunication networks a decade ago, they have been the subject of numerous investigations. Typical quantitative assessment of self-similarity and long-range dependency, begins with the estimation of the Hurst parameter H. There have been a number of techniques proposed for this. This paper reports results of a comparative analysis of the six most frequently used estimators of H. To set up a credible framework for this, the minimal acceptable sample size is first determined. The Hurst parameter estimators are then compared for bias and variance. Our experimental results have confirmed that the Abry–Veitch Daubechies Wavelet-Based (DWB) and the Whittle ML (Maximum Likelihood) estimators of H are the least biased. However, the latter has significantly smaller variance and can be applied to shorter data samples than the Abry–Veitch DWB estimator. On the other hand, the Abry–Veitch DWB estimator is computationally simpler and faster than the Whittle ML estimator.

Decorrelation of Wavelet Coefficients for Long-Range Dependent Processes

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> We consider a discrete-time stationary long-range dependent process <emphasis><formula formulatype="inline"><tex>$(X_k)_{k\in Z}$</tex></formula></emphasis> such that its spectral density equals <emphasis><formula formulatype="inline"> <tex>${\varphi}(\vert {\lambda}\vert)^{-2d}$</tex></formula></emphasis>, where <emphasis><formula formulatype="inline"><tex>${\varphi}$</tex></formula></emphasis> is a smooth function such that <emphasis><formula formulatype="inline"><tex>${\varphi}(0)={\varphi}^{\prime\prime}(0)=0$</tex> </formula></emphasis> and <emphasis><formula formulatype="inline"><tex>${\varphi}({\lambda})\geq c{\lambda}$</tex></formula></emphasis> for <emphasis><formula formulatype="inline"> <tex>${\lambda}\in [0,\pi]$</tex></formula></emphasis>. Then for any wavelet <emphasis><formula formulatype="inline"><tex>$\psi$</tex></formula></emphasis> with <emphasis><formula formulatype="inline"><tex>$N$</tex></formula></emphasis> vanishing moments, the lag <emphasis><formula formulatype="inline"><tex>$k$</tex></formula></emphasis> within-level covariance of wavelet coefficients decays as <emphasis><formula formulatype="inline"> <tex>${\cal O}(k^{2d-2N-1})$</tex></formula></emphasis> when <emphasis><formula formulatype="inline"><tex>$k\to\infty$</tex></formula></emphasis>. The result applies to fractionally integrated autoregressive moving average (ARMA) processes as well as to fractional Gaussian noise. </para>

Minimizing Large Deviation Paths for a Family of Long-Range Dependent Processes and Their Fractional Brownian Approximations

The long-range dependent cumulative traffic processes generated by the superposition of constant rate fluid sources having exponentially distributed inter-start times and Pareto-distributed durations with finite mean and infinite variance satisfy a large deviation principle when the session intensity is increased and the processes are centered and appropriately scaled. We show that close to its zero point the rate function in this large deviation principle behaves approximately like the one appearing in a sample path large deviation principle for fractional Brownian motion. This allows for an approximation of large deviation rates and minimizing paths of the traffic processes by those of fractional Brownian motions. We explicitly calculate certain minimizing paths representing the asymptotically most probable behavior of the traffic processes leading to a fixed positive or negative increase and compare them to the corresponding behavior of fractional Brownian motions. We develop analogous results for the minimizing paths of the queue length processes obtained by sending the traffic through a buffer with infinite capacity and constant service rate. These paths are approximated by minimizing paths of reflected fractional Brownian motion.

Long-range dependence of stationary processes in single-server queues

The stationary processes of waiting times {W n } n = 1,2,? in a GI/G/1 queue and queue sizes at successive departure epochs {Q n}n = 1,2,? in an M/G/1 queue are long-range dependent when 3 < ? S < 4, where ? S is the moment index of the independent identically distributed (i.i.d.) sequence of service times. When the tail of the service time is regularly varying at infinity the stationary long-range dependent process {W n } has Hurst index ½(5?? S ), i.e. $${\rm sup} \left\{h : {\rm lim sup}_{n\to\infty}\, \frac{{\rm var}(W_1+\cdots+W_n)}{n^{2h}} = \infty \right\} = \frac{5-\kappa_S} {2}\,.$$ If this assumption does not hold but the sequence of serial correlation coefficients {? n } of the stationary process {W n } behaves asymptotically as cn ?? for some finite positive c and ? ? (0,1), where ? = ? S ? 3, then {W n } has Hurst index ½(5?? S ). If this condition also holds for the sequence of serial correlation coefficients {r n } of the stationary process {Q n } then it also has Hurst index ½(5? S )

Non-Gaussian and Long Memory Statistical Characterizations for Internet Traffic with Anomalies

The goals of the present contribution are twofold. First, we propose the use of a non-Gaussian long-range dependent process to model Internet traffic aggregated time series. We give the definitions and intuition behind the use of this model. We detail numerical procedures that can be used to synthesize artificial traffic exactly following the model prescription. We also propose original and practically effective procedures to estimate the corresponding parameters from empirical data. We show that this empirical model relevantly describes a large variety of Internet traffic, including both regular traffic obtained from public reference repositories and traffic containing legitimate (flash crowd) or illegitimate (DDoS attack) anomalies. We observe that the proposed model accurately fits the data for a wide range of aggregation levels. The model provides us with a meaningful multiresolution (i.e., aggregation level dependent) statistics to characterize the traffic: the evolution of the estimated parameters with respect to the aggregation level. It opens the track to the second goal of the paper: anomaly detection. We propose the use of a quadratic distance computed on these statistics to detect the occurrences of DDoS attack and study the statistical performance of these detection procedures. Traffic with anomalies was produced and collected by us so as to create a controlled and reproducible database, allowing for a relevant assessment of the statistical performance of the proposed (modeling and detection) procedures

Ambiguities in estimates of critical exponents for long-range dependent processes

In this article, we show that the Hurst parameter and a well-known wavelet estimator fail to distinguish on/off models with substantially different distributions of on-times. We identify the problem as arising when standard power spectrum techniques, which are well-defined for commonly considered processes, are applied to processes with extremely heavy tailed on-times. We provide an elementary mechanism whereby such extremely heavy-tailed on-time processes arise in a simple queuing model.

Bootstrap approaches and confidence intervals for stationary and non-stationary long-range dependence processes

This paper deals with different bootstrap approaches and bootstrap confidence intervals in the fractionally autoregressive moving average ( ARFIMA ( p , d , q ) ) process [J. Hosking, Fractional differencing, Biometrika 68(1) (1981) 165–175] using parametric and semi-parametric estimation techniques for the memory parameter d. The bootstrap procedures considered are: the classical bootstrap in the residuals of the fitted model [B. Efron, R. Tibshirani, An Introduction to the Bootstrap, Chapman and Hall, New York, 1993], the bootstrap in the spectral density function [E. Paparoditis, D.N Politis, The local bootstrap for periodogram statistics. J. Time Ser. Anal. 20(2) (1999) 193–222], the bootstrap in the residuals resulting from the regression equation of the semi-parametric estimators [G.C Franco, V.A Reisen, Bootstrap techniques in semiparametric estimation methods for ARFIMA models: a comparison study, Comput. Statist. 19 (2004) 243–259] and the Sieve bootstrap [P. Bühlmann, Sieve bootstrap for time series, Bernoulli 3 (1997) 123–148]. The performance of these procedures and confidence intervals for d in the stationary and non-stationary ranges are empirically obtained through Monte Carlo experiments. The bootstrap confidence intervals here proposed are alternative procedures with some accuracy to obtain confidence intervals for d.

Estimation of Hurst exponent revisited

In order to estimate the Hurst exponent of long-range dependent time series numerous estimators such as based e.g. on rescaled range statistic ( R / S ) or detrended fluctuation analysis (DFA) are traditionally employed. Motivated by empirical behaviour of the bias of R / S estimator, its bias-corrected version is proposed. It has smaller mean squared error than DFA and behaves comparably to wavelet estimator for traces of size as large as 2 15 drawn from some commonly considered long-range dependent processes. It is also shown that several variants of R / S and DFA estimators are possible depending on the way they are defined and that they differ greatly in their performance.

GPS scheduling

We consider a system with two service classes with heterogeneous traffic characteristics and Quality-of-Service requirements. The available bandwidth is shared between the two traffic classes in accordance with the Generalized Processor Sharing (GPS) discipline. GPS-based scheduling algorithms, such as Weighted Fair Queueing, provide a popular mechanism for service differentiation among heterogeneous traffic classes. While the performance of GPS for given weights has been thoroughly examined, the problem of selecting weight values that maximize the traffic-carrying capacity, has only received limited attention so far. In the present paper, we address the latter problem for the case of general Gaussian traffic sources. Gaussian models cover a wide variety of both long-range dependent and short-range dependent processes, and are especially suitable at relatively high levels of aggregation. In particular, we determine the realizable region, i.e., the combinations of traffic sources that can be supported for given Quality-of-Service requirements in terms of loss and delay metrics. The results yield the remarkable observation that simple priority scheduling strategies achieve nearly the full realizable region. 1 .