Narrow Results

Much attention has been given to the long-range dependence and fractal properties in network traffic engineering, and these properties are also widely observed in many fields of science and technologies. Traffic time series is conventionally characterized by its fractal dimension D, which is a measure for roughness, and by the Hurst parameter H, which is a measure for long-range dependence, see for examples (Refs. 10–12). Each property has been traditionally modeled and explained by self-affine random functions, such as fractional Gaussian noise (FGN)^{1,10–13,18,22–28} and fractional Brownian motion (FBM),^6,7 where a linear relationship between D and H, say D = 2 - H for one-dimensional series, links the two properties. The limitation of single parameter models (e.g., FGN) in long-range dependent (LRD) traffic modeling has been noticed as can be seen from Refs. 1, 18 and 25. Hence, models which can provide good fitting of LRD traffic for both short-term lags and long-term ones are worth studying due to the importance of accurate models of traffic in network communications.¹³ This letter utilizes a statistical model called the Cauchy correlation model to model LRD traffic. This model characterizes D and H separately, and it allows any combination of two within the constraint of LRD condition. It is a new power-law correlation model for LRD traffic modeling with its local and global behavior decoupling. Its flexibility in data modeling in comparison with a single parameter model of FGN is briefly discussed, and applications to LRD traffic modeling demonstrated.

Electroencephalogram (EEG), the measures and records of the electrical activity of the brain, exhibits evidently nonlinear, nonstationary, chaotic and complex dynamic properties. Based on these properties, many nonlinear dynamical analysis techniques have emerged, and much valuable information has been extracted from complex EEG signals using these nonlinear analysis techniques. Among these techniques, the Hurst exponent estimation was widely used to characterize the fractional or scaling property of the EEG signals. However, the constant Hurst exponent H cannot capture the detailed information of dynamic EEG signals. In this research, the multifractional property of the normal human sleep EEG signals is investigated and characterized using local Hölder exponent H(t). The comparison of the analysis results for human sleep EEG signals in different stages using constant Hurst exponent H and the local Hölder exponent H(t) are summarized with tables and figures in the paper. The results of the analysis show that local Hölder exponent provides a novel and valid tool for dynamic assessment of brain activities in different sleep stages.

Hospital emergency arrivals are often modeled as Poisson processes because of the similarity of the problem to a telephone exchange model, i.e. applications that involve counting the number of times a random event occurs in a given time, where the interval between individual counts follows the exponential distribution. In this paper, we propose a statistical modeling of hospital emergency ward arrivals, using self-similar processes. This modeling takes into account the fact that it is known empirically that the emergency time series consists of periods marked by "bursts" of high level demand peaks, followed by periods of lower demand. This is explained neither by a Poisson during fixed-lengths intervals nor by a lognormal distribution. We show that a self-similar distribution can model this phenomenon. We also show that the commonly used Poisson models seriously underestimate the "burstiness" of emergency arrivals over a wide range of time scales, and that the emergency time series cannot be modeled by a lognormal distribution. The self-similar distribution was tested by the Hurst parameter, which we have calculated using five different methods, all of which agree on the value of the parameter.

We present efficient methods for simulation, using the Fast Fourier Transform (FFT) algorithm, of two classes of processes with symmetric α-stable (SαS) distributions. Namely, (i) the linear fractional stable motion (LFSM) process and (ii) the fractional autoregressive moving average (FARIMA) time series with SαS innovations. These two types of heavy-tailed processes have infinite variances and long-range dependence and they can be used in modeling the traffic of modern computer telecommunication networks.

We generate paths of the LFSM process by using Riemann-sum approximations of its SαS stochastic integral representation and paths of the FARIMA time series by truncating their moving average representation. In both the LFSM and FARIMA cases, we compute the involved sums efficiently by using the Fast Fourier Transform algorithm and provide bounds and/or estimates of the approximation error.

We discuss different choices of the discretization and truncation parameters involved in our algorithms and illustrate our method. We include MATLAB implementations of these simulation algorithms and indicate how the practitioner can use them.

A self-similar process has power spectrum with power law depending on its self-similarity parameter H and we use this property for its generation by the method of surrogate data. The surrogates are a set of random data with the same distribution as that of the increments of the process. These are iteratively rearranged according to the rank order of a time series obtained with the same power law spectrum. The method is fast, and reliable as shown by the characteristics reproduced. It is also possible to extend this method to prediction of a self-similar process, the success of which will depend on the number of available data.

A stationary time series is said to be long-range dependent (LRD) if its autocovariance function decays as a power of the lag, in such a way that the sum (over all lags) of the autocovariances diverges. The asymptotic rate of decay is determined by a parameter H, called the Hurst parameter. The time series is said to be short-range dependent (H = 1/2) if the sum converges. It is commonly believed that a random permutation of a sequence maintains the marginal distribution of each element but destroys the dependence, and in particular, that a random permutation of an LRD sequence creates a new sequence whose estimate of Hurst parameter H is close to 1/2. This paper provides a theoretical basis for investigating these claims. In reality, a complete random permutation does not destroy the covariances, but merely equalizes them. The common value of the equalized covariances depends on the length N of the original sequence and it decreases to 0 as N → ∞. Using the periodogram method, we explain why one is led to think, mistakenly, that the randomized sequence yields an "estimated H close to 1/2.".

Bucket random permutations (shuffling) are used to modify the dependence structure of a time series, and this may destroy long-range dependence, when it is present. Three types of bucket permutations are considered here: external, internal and two-level permutations. It is commonly believed that (1) an external random permutation destroys the long-range dependence and keeps the short-range dependence, (2) an internal permutation destroys the short-range dependence and keeps the long-range dependence, and (3) a two-level permutation distorts the medium-range dependence while keeping both the long-range and short-range dependence. This paper provides a theoretical basis for investigating these claims. It extends the study started in Ref. 1 and analyze the effects that these random permutations have on a long-range dependent finite variance stationary sequence both in the time domain and in the frequency domain.

Stoev and Taqqu¹ presented a fast simulation method for the linear fractional stable motion (lfsm) process, using the fast Fourier transform. In this paper, we extend this method for fast simulation of the log-fractional stable motion (log-fsm) process. These two heavy-tailed processes, i.e. lfsm and log-fsm, can be used for modeling the modern telecommunication networks. In addition, a necessary and sufficient condition is given for the efficiency of this method. Finally, an upper bound for approximation error is presented.

The FARIMA models, which have long-range-dependence (LRD), are widely used in many areas. Through the derivation of a precise characterization of the spectrum and variance time function, we show that this family is very atypical among LRD processes, being extremely close to the fractional Gaussian noise in a precise sense which results in ultra-fast convergence to fGn under rescaling. Furthermore, we show that this closeness property is not robust to additive noise. We argue that the use of FARIMA, and more generally fractionally differenced time series, should be reassessed in some contexts, in particular when convergence rate under rescaling is important and noise is expected.

This paper proposes the shifted Legendre polynomial approximations-based stochastic operational matrix of integration method to solve persistent processes-based fractional stochastic differential equations. The operational matrix of integration, stochastic operation matrix and fractional stochastic operational matrix of the shifted Legendre polynomials are derived. The stochastic differential equation is transformed into an algebraic system of $(N+1)$ equations by the operational matrices. For the proposed approach, a thorough discussion of the error analysis in $L^2$ norm is provided. The proposed method’s applicability, correctness, and accuracy are examined using a few numerical examples. Comparing the numerical examples to the other methods discussed in the literature demonstrates the solution’s effectiveness and attests to the solution’s high quality. The error analysis also reveals the method’s superiority. A more accurate solution is obtained, thus maintaining a minimum error.

The joint presence of heavy-tailed distribution and long memory in time series always leads to certain trouble in correctly obtaining the statistical characteristics for time series modeling. These two properties i.e., heavy-tailed distribution and long memory, cannot be neglected in time series analysis, because the tail thickness of the distribution and long memory property of the time series are critical in characterizing the essence of the resulting natural or man-made phenomenon of the time series. Meanwhile, the fluctuation of the varying local long memory parameter may be used to capture the internal changes which underlie the externally observed phenomenon. Therefore, in this paper, we proposed to use the variance trend, heavy-tailed distribution, long memory, and local long memory characteristics to analyze a time series recorded as in [1] from tracking the jumps of individual molecules on cell membranes. The tracked molecules are Class I major histocompatibility complex (MHCI) expressed on rat hepatoma cells. The analysis results show that the jump time series of molecular motion on the cell membrane obviously has both heavy-tailed distribution and local long memory characteristics. The tail heaviness parameters, long memory parameters, and the local long memory parameters of ten MHCI molecular jump time series are all summarized with tables and figures in the paper. These reported tables and figures are not only interesting but also important in terms of additional novel insights and characterization of the time series under investigation.

This is an extended version of the lecture notes to a mini-course devoted to fractional Brownian motion and delivered to the participants of the 7th Jagna International Workshop.

In this work, we will study the Hurst parameter embedded in the discrete-time fraction Gaussian noise (DFGN) and derive the entropy with this parameter. It will be applied to the electromyogram (EMG) of external urethral sphincter (EUS). These signals come from intact rats and the injured ones from spinal cord injury (SCI). Analysis indicates that we can discriminate between intact and SCI rats from this information.

It is generally accepted that self-similar (or fractal) processes may provide better models for teletraffic in modern telecommunication networks than Poisson processes. Neglecting this fact can lead to inaccurate conclusions about performance of telecommunication networks. Thus we need an efficient generator of long synthetic stochastic self-similar sequences for conducting simulation studies of telecommunication networks.

Five generators of pseudo-random self-similar sequences, based on the fast Fourier transform (FFT), random midpoint displacement (RMD), successive random addition (SRA), fractional autoregressive integrated moving-average (F-ARIMA), and fractional Gaussian noise and Daubechies wavelets (FGN-DW) methods, are considered in this paper. We discuss their properties in the sense of their statistical accuracy required to produce sequences of a given (long) length.

The statistical estimation of the Hurst index is one of the fundamental problems in the literature of long-range dependent and self-similar processes. In this article, the Hurst index estimation problem is addressed for a special class of self-similar processes that exhibit long-memory, the Hermite processes. These processes generalize the fractional Brownian motion, in the sense that they share its covariance function, but are non-Gaussian. Existing estimators such as the R/S statistic, the variogram, the maximum likelihood and the wavelet-based estimators are reviewed and compared with a class of consistent estimators which are constructed based on the discrete variations of the process. Convergence theorems (asymptotic distributions) of the latter are derived using multiple Wiener-Itô integrals and Malliavin calculus techniques. Based on these results, it is shown that the latter are asymptotically more efficient than the former.