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

Processes with long-range correlations or called long-range dependent (LRD) processes are all around us in nature. The presence and nature of LRD is characterized by the Hurst parameter (0 < H < 1). The aim of this paper is to make a practical analysis of the robustness of the Hurst parameter estimators. A simple model of exactly self-similar process-Fractional Gaussian noise (FGN) with parameter H ∈ (0, 1) is applied to evaluate Hurst parameter estimators. The white Gaussian noise or the Symmetric α-stable (SαS) noise is superimposed in order to evaluate the reliability and the robustness of different estimators. In this paper, six statistic analysis methods, R/S statistic, Aggregated Variance method, Absolute Value method, Residuals of Regression method, Periodogram method, and Whittle method are analyzed. It follows from the comparison that the Variance of Residuals method is almost unbiased for non-noise LRD processes. And the Whittle method has best robustness to Symmetric α-stable (SαS) noisy LRD processes. The robustness analysis has practical value for analyzing noisy LRD time series, especially for the economic data, under water signal, biomedical signal and the communication signal which are corrupted by impulsive noise.

Many recent studies on a wide range of networks by means of high quality, high time-resolution measurements have indicated that traffic burstiness and correlations among inter-arrival intervals are present over many time scales, which can be modelled using the long-range dependent (LRD) process. A LRD process is characterised by a hyperbolically decaying correlation function. However, interconnection networks used in current multicomputers have been mainly designed and analysed under the assumption that traffic follows the traditional short-range dependent process (e.g., the Poisson arrival process). In an effort to reevaluate network performance, this paper presents a new analytical model for circuit-switched interconnection networks under LRD traffic. Simulation experiments reveal that the model exhibits a good degree of accuracy under various operating conditions.

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.

Mean value estimation of processes exhibiting long range dependence (LRD) requires a different approach than the techniques applied to those exhibiting short range dependence (SRD), except for the independent replication method. We describe a nonoverlapping batch means method able to deal with LRD processes, the LRD Batch Means method. This method exploits the behavior of asymptotically second-order self-similar processes: their aggregated processes become well approximated by fractional Gaussian noise (FGN) processes for large aggregation levels. Once tested positively for this similarity, the method produces a correlation-adjusted confidence interval from an empirical approximation of the distribution of the standardized average for the particular case of FGN processes. Afterwards, we measure its performance over both LRD and SRD processes.

Frequency domain bootstrap for ratio statistics under long-range dependence

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.

A frequency domain bootstrap for Whittle estimation under long-range dependence

Recent researches have highlighted the long-range dependence and impulsive nature of the aggregated network traffic, and consequently long-range dependent α-stable processes model has been widely used to capture the network traffic. This paper presents an analytical demonstration of aggregated network traffic converging to long-range dependent α-stable processes, which provides inherent mechanism explanation for the widely used long-range dependent α-stable processes model. Simulations validate the accuracy of the presented theoretical analysis.

The North Atlantic Oscillation (NAO) is one of the most important climatic patterns in the Northern Hemisphere. Indices based on the normalised pressure difference between Iceland and a southern station, such as Lisbon or Gibraltar, have been defined in order to describe NAO temporal evolution. Although exhibiting interannual and decadal variability, the signals are statistically rather featureless and therefore it is difficult to discriminate between different types of stochastic models. In this study, Lisbon and Gibraltar NAO winter indices are analysed using the discrete wavelet transform discrete wavelet transform(DWT). A multi-resolution analysis (MRA) is carried out for a scale-based description of the indices and the wavelet spectrum is used to identify and estimate long-range dependence. The degree of association of the two NAO indices is assessed by estimating the wavelet covariance for the two signals. The scale-based approach inherent to the discrete wavelet methodology allows a scale-by-scale comparison of the signals and shows that although the short-term temporal pattern is very similar for both indices, the long-term temporal structure is distinct. Furthermore, the degree of persistence or ‘memory’ is also distinct: the Lisbon index is best described by a long-range dependent (LRD) process, while the Gibraltar index is adequately described by a short-range process. Therefore, while trend features in the Lisbon NAO index may be explainable by long-range dependence alone, with no need to invoke external factors, for the Gibraltar index such features cannot be interpreted as resulting only from internal variability through long-range dependence. Copyright © 2006 Royal Meteorological Society.

We analyze the asymptotic tail distribution of stationary waiting times and stationary virtual waiting times in a single-server queue with long-range dependent arrival process and subexponential service times. We investigate the joint impact of the long range dependency of the arrival process and of the tail distribution of the service times. We consider two traffic models that have been widely used to characterize the long-range dependence structure, namely, the M/G/8 input model and the Fractional Gaussian Noise (FGN) model. We focus on the response times of the customers in a First-Come First-Serve (FCFS) queueing system, although the results carry through to the backlog distribution of the system with any arbitrary queueing discipline. When the arrival process is driven by an M/G/8 input model we show that if the residual service time tail distribution Fe is lighter than the residual session duration Ge, then the stationary waiting time is dominated by the long-range dependence structure, which is determined by the residual session duration Ge. If the residual service time distribution Fe is heavier than the residual session duration Ge, then the tail distribution of the stationary waiting time is dominated by that of the residual service time. When the arrival process is modeled by an FGN, we show that the waiting time tail distribution is asymptotically equal to the tail distribution of the residual service time if the latter is asymptotically heavier than Weibull distribution with shape parameter 2-2H, where H is the Hurst parameter of the FGN. If, however, this residual service time is asymptotically lighter than Weibull distribution with shape parameter 2-2H, then the waiting time tail distribution is dominated by the dependence structure of the arrival process so that it is asymptotically equal to Weibull distribution with shape parameter 2-2H.

We analyze the asymptotic tail distribution of stationary waiting times and stationary virtual waiting times in a single-server queue with long-range dependent arrival process and subexponential service times. We investigate the joint impact of the long range dependency of the arrival process and of the tail distribution of the service times. We consider two traffic models that have been widely used to characterize the long-range dependence structure, namely, the M/G/ 8 input model and the Fractional Gaussian Noise (FGN) model. We focus on the response times of the customers in a First-Come First-Serve (FCFS) queueing system, although the results carry through to the backlog distribution of the system with any arbitrary queueing discipline. When the arrival process is driven by an M/G/ 8 input model we show that if the residual service time tail distribution F e is lighter than the residual session duration G e , then the stationary waiting time is dominated by the long-range dependence structure, which is determined by the residual session duration G e . If the residual service time distribution F e is heavier than the residual session duration G e , then the tail distribution of the stationary waiting time is dominated by that of the residual service time. When the arrival process is modeled by an FGN, we show that the waiting time tail distribution is asymptotically equal to the tail distribution of the residual service time if the latter is asymptotically heavier than Weibull distribution with shape parameter 2-2H, where H is the Hurst parameter of the FGN. If, however, this residual service time is asymptotically lighter than Weibull distribution with shape parameter 2-2H, then the waiting time tail distribution is dominated by the dependence structure of the arrival process so that it is asymptotically equal to Weibull distribution with shape parameter 2-2H.

We estimate the marginal density function of a long-range dependent linear process by the kernel estimator. We assume the innovations are i.i.d. Then it is known that the term of the sample mean is dominant in the MISE of the kernel density estimator when the dependence is beyond some level which depends on the bandwidth and that the MISE has asymptotically the same form as for i.i.d. observations when the dependence is below the level. We call the latter the case where the dependence is not very strong and focus on it in this paper. We show that the asymptotic distribution of the kernel density estimator is the same as for i.i.d. observations and the effect of long-range dependence does not appear. In addition we describe some results for weakly dependent linear processes.

Fractional order fuzzy control of nuclear reactor power with thermal-hydraulic effects in the presence of random network induced delay and sensor noise having long range dependence

ABSTRACTWe establish the existence of multivariate stationary processes with arbitrary marginal copula distributions and long-range dependence. The effect of long-range dependence on extreme value copula estimation is illustrated in the case of known marginals, by deriving functional limit theorems for a standard non parametric estimator of the Pickands dependence function and related parametric projection estimators. The asymptotic properties turn out to be very different from the case of iid or short-range dependent observations. Simulated and real data examples illustrate the results.

Long-range dependence (LRD) is discovered in time series arising from different fields, especially in network traffic and econometrics. Detecting the presence and the intensity of LRD plays a crucial role in time-series analysis and fractional system identification. The existence of LRD is usually indicated by the Hurst parameters. Up to now, many Hurst parameter estimators have been proposed in order to identify the LRD property involved in a time series. Since different estimators have different accuracy and robustness performances, in this study, 13 most popular Hurst parameter estimators are summarised and their estimation performances are investigated. LRD processes with known Hurst parameters are generated as the control data set for the robustness evaluation. In addition, three types of LRD processes are also obtained as the test signals by adding noises in terms of means, trends and seasonalities to the control data set. All 13 Hurst parameter estimators are applied to these LRD processes to estimate the existing Hurst parameters. The estimation results are documented and quantified by the standard errors. Conclusions of the accuracy and robustness performances of the estimators are drawn by comparing the estimation results.