BERAN, J. Statistical methods for data with long-range dependence. Statistical Science 7, 4 (1992), 404--427.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....ARMA model that can capture the longrange dependence of self similar signals. In some sense, these signals are borderline stationary. The AR, MA, ARMA, and ARIMA models are classical time series models described by Box, et al. [3] ARFIMA models are well covered in more recent literature [7, 6, 2]. The RPS technical report [5] also provides an explanation of these models as well as a detailed description of the implementations we use here. The same implementations are used for offline and online analysis in RPS. It is important to point out that we do not follow the Box Jenkins ....

BERAN, J. Statistical methods for data with long-range dependence. Statistical Science 7, 4 (1992), 404--427.

....a more refined data analysis is possible for maximum likelihood type estimates (MLE) and related methods based on the periodogram. Especially, for Gaussian processes ( 0,1, 2, k XXk= # , Whittles approximate MLE has been studied extensively. The Whittle estimator is defined as follows [11, 26, 29, 30, 31]: Let ( f be the spectral density of X, with 3 ( k H = # where is the variance scaling factor, H is the Hurst parameter, and 3 , k # model short range dependence structure of the process. Getting out the variance scaling factor yields: ....

....and the variance are the only parameters to be estimated. Therefore, relation (3.18) reduces to the problem of estimating H that minimizes: QH d (3.20) If X has the length N, integral (3. 20) is converted to a discrete summation over the frequencies 24 , 2 NN = [29]: 1) 2 ( 2 i i QH in (3.21) Finding the value of H minimizing (3.21) can be done by calculating Q(H) on sufficiently fine grid of H values. The spectral density for a FGN process is given by [29, 31] 01, fHAH BH H ## = ....

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Beran, J., Statistical Methods for Data with Long-Range Dependence, Statistical Science, Vol.17, No.4, pp. 404-427, 1992.

....ARFIMA(4, 1,4) model is a fractionally integrated ARMA model that can capture the long range dependence of self similar signals. AR, MA, ARMA, and ARIMA models are classical time series models well covered by Box, et al. [7] ARFIMA models are well covered in more recent literature [20] 18] [5]. The RPS technical report [15] also provides an explanation of these models as well as a detailed description of the implementations we use here. The same implementations are used for offline and online analysis in RPS. For each of the 34 AUCKLAND traces, we performed the analysis described ....

BERAN, J. Statistical methods for data with long-range dependence. Statistical Science 7, 4 (1992), 404--427.

....[6] AR, MA, ARMA, and ARIMA classes might be appropriate for predicting load. On the other hand, the existence of self similarity induced long range dependence suggested that such models might require an impracti cal number of parameters or that the much more com plex ARFIMA model class [20,16,4], which explicitly captures long range dependence, might be more appro priate. Since it is not obvious which model is best, we empirically evaluated the predictive power of the AR, MA, ARMA, ARIMA, and ARFIMA model classes, as well as that of a simple ad hoc windowed mean predic tor called BM ....

....of all frequency components. An important implication is that linear models may be appropriate for predicting load signals. However, the complex frequency domain behavior suggests such models may have to be of unreasonably high order. 5) The traces exhibit self similarity with Hurst Parameters [2,4] ranging from 0.63 to 0.97, with a strong bias toward the top of that range. Hurst parameters in the range of 0.5 to 1.0 indicate self similarity with positive near neighbor correlations. This result tells us that load varies in complex ways on all time scales and has long range dependence. ....

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BERAN, J. Statistical methods for data with long-range dependence. Statistical Science 7, 4 (1992), 404-427.

....the estimator itself. There is a long tradition of hypothesis testing in financial econometrics in which one computes the likelihood of a model to hold given the value of a statistic and rejects accepts the model by comparing the test statistics to a threshold value. With a few exceptions (see [8, 87, 88]) the large majority of statistical tests are based on a central limit theorem for the estimator from which the asymptotic normality is obtained. This central limit theorem can be obtained by assuming that the noise terms (innovations) in the return process are weakly dependent [30] In order ....

Beran J 1992 Statistical methods for data with long range dependence Stat. Sci. 7 404--27

....over many different time scales, often referred to as long range dependence (LRD) or long memory. Some of the better known examples of such systems are found in hydrology [19] However, the phenomenon of LRD occurs in many other systems including chemical, astronomical, and biological systems (see [4] for references) In spite of much statistical evidence, the existence of LRD has often been met with resistance or at least puzzlement. This was caused in large part by the absence of physical ex This work was done while both authors were at INRIA Sophia Antipolis, France. This work was ....

J. Beran. Statistical Methods for Data with Long Range Dependence. Statistical Science, 7(4):404--427, 1992.

....of a given time series is presented in Sect. 2.2. 2. 1 Mathematical Characterization Before going into detail, it is worth noting that there exists a considerable amount of non equivalent definitions for self similarity, as also mentioned in [47] and [48] Here, we follow the work of Beran [3] and Cox [10] We consider a discrete time stochastic process X = X(t) t = 0, 1, representing e.g. the number of packet arrivals during the time interval [t, t 1) It is assumed that X is covariance stationary (i.e. the autocovariance function R t (k) ....

....formal definition of self similar stochastic processes. The most important approaches in this area are: Increment processes of exactly self similar processes, which consider the difference of subsequent innovations of their parent process. A common example is fractional Brownian motion (FBM) [3], which is derived from the di#erentiation of fractional Gaussian noise (FGN) 33] Fractional autoregressive integrated moving average (F ARIMA) processes [28] are an extension of standard ARIMA [5] processes. Compared to e.g. FBM, they have a higher flexibility in accounting for short range ....

J. Beran. Statistical methods for data with long-range dependence. Statistical Science, 7:404--427, 1992.

....0:5 H 1. Since we are always dealing with finite data sets, it is in principle not possible to check whether by definition a traffic trace is self similar or not. Instead, we check for different features of self similarity present in actual packet traffic based on the properties listed above [2,7,11]. The following most popular self similarity tests are used to capture some of the listed properties: 3.2. Tests Indices of dispersion A commonly used measure for capturing the variability of traffic over different time scales is provided by the index of dispersion for counts (IDC) 4] i.e. ....

J. Beran. Statistical methods for data with long-range dependence. Statistical Science, 7(4):404--427, 1992.

....improve the model; but I do think that any model which does not account for the features of long range correlations observed in real data is doomed to failure in the long run. Mandelbrot s statistical tools were still rather coarse. They were optimized and expanded in our group (Graf et al. 1984; Beran 1989, 1992), and applied not only to further geophysical series (such as the famous Arosa ozone series) but also to pure measurement series in chemistry (taken from Student 1927) and physics (about 3000 measurements of the velocity of light, taken from Michelson et al. 1935; sets of hundreds of measurements ....

Beran, J. (1992). Statistical methods for data with long--range dependence, Statist. Sci. 7: 404--427, (with discussion).

....the ARMA models fitted previously to these datasets adequately describe the low frequency component. In this revised version of our previous paper we have revised the spectral analysis plot used for checking for long memory. For long memory diagnostic checking we now recommend, as is suggested in Beran (1992), the use of a log scale for the frequency axis as well as for the spectrum. This scaling emphasizes the low frequencies and at the low frequencies a long memory spectrum is indicated by a straight line. It is also more convenient to use normalized spectra obtained by dividing the spectrum by the ....

....(1969) A more flexible approach to long memory models was initiated by Granger and Joyeux (1980) and Hosking (1981) who suggested what is now referred to as the fractional ARMA model. This provides a comprehensive family of stationary and ergodic models which generalize the usual ARMA model. Beran (1992) gives a recent review of long memory time series models and several other researchers have enthusiastically recommended long memory models for various types of geophysical data. A.I. McLeod K.W. Hipel 13 4.1 Exploratory Spectral Analysis of Hydrological Time Series Hipel and McLeod (1978) ....

Beran, J. (1992), Statistical methods for data with long-range dependence, Statistical Science 7(4) 404--427.

....as 0:82 [57] 3.7. 2 Generating Self Similar Traffic The literature gives number of traffic models that generate long range dependent traffic (see [45, 57, 80] and the references therein) In developing M, we model a particular class of self similar traffic called Fractional Gaussian Noise (FGN) [12] using the Fast Fourier Transform (FFT) method developed by Paxson [79] The advantage to a FGN process is that the degree of self similarity can be expressed solely by the Hurst parameter. We choose the FFT method (given in Appendix A.1) because an approximate FGN sample path of length P can be ....

....of FGN, the sample path is guaranteed to have the autocorrelation properties of an FGN process. The steps to the FFT technique are given below [79] 1. Construct a sequence of values f 1 ; f n , where f i = f(2i=n; H) The power spectrum of FGN, developed by Whittle, is given below by Beran [12]: f( H) A( H) j j Gamma2H Gamma1 B( H) A.1) A( H) 2 sin(H ) Gamma(2H 1) 1 Gamma cos ) A.2) B( H) 1 X j=1 [ 2j ) Gamma2H Gamma1 (2j Gamma ) Gamma2H Gamma1 ] A.3) 2. Multiply each f i by an independent exponential random variable with mean 1, creating a ....

J. Beran. Statistical Methods for Data with Long-Range Dependence. In Statistical Science, 7(4), pages 404--427, 1992.

....Box Jenkins [6] AR, MA, ARMA, and ARIMA classes might be appropriate for predicting load. On the other hand, the existence of selfsimilarity induced long range dependence suggested that such models might require an impractical number of parameters or that the much more expensive ARFIMA model class [18, 14, 4], which explicitly captures long range dependence, might be more appropriate. Since it is not obvious which model is best, we empirically evaluated the predictive power of the AR, MA, ARMA, ARIMA, and ARFIMA model classes, as well as a simple ad hoc windowed mean predictor called BM and a ....

....40000 60000 80000 0.04 0.02 0.0 0.02 0.04 ) 0 ( 2 a WhiteNoise a s 2 t z s 2 2 a s s Unpredictable Random Sequence Fixed Linear Filter Partially Predictable Load Sequence Figure 3. Linear time series models. 5) The traces exhibit self similarity with Hurst Parameters [2, 4] ranging from 0.63 to 0.97, with a strong bias toward the top of that range. Hurst parameters in the range of 0.5 to 1.0 indicate self similarity with positive near neighbor correlations. This result tells us that load varies in complex ways on all time scales and has long range dependence. ....

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BERAN, J. Statistical methods for data with long-range dependence. Statistical Science 7, 4 (1992), 404--427.

....by computers. The use of traditional models of teletra#c can result in overly optimistic estimates of performance of telecommunication networks, insu#cient allocation of communication and data processing resources, and di#culties in ensuring the quality of service expected by network users [1], 17] 20] On the other hand, if the strongly correlated character of teletra#c is explicitly taken into account, this can also lead to more e#cient tra#c control mechanisms. # Technical Report TR COSC 03 98 1 Several methods for generating pseudo random self similar sequences have been ....

....be required that Var[X( 1 2 ) X(0) 1 4 Var[X(1) X(0) S 2 1 = 1 2 ) 2H # 2 0 . Thus S 2 1 = 1 2 1 ) 2H (1 2 2H 2 )# 2 0 . 0 1 1 A self similar sequence Add o ffs e s to midpoints Gaussian random numbers Uniformly distributed random numbers Subdivide interval [0,1] recursively Normalisation Figure 2: RMD method X t d1 d2,1 d2,2 0.25 0.50 0.75 1.00 d3,1 d3,2 d3,3 d3,4 Figure 3: The first three steps in the RMD method 5 Step.3 Reduce the scaling factor by # 2, that is, now assume 1 # 8 , and divide the two intervals from 0 and 1 2 and from 1 2 to ....

Beran, J. Statistical Methods for Data with Long Range Dependence. Statistical Science 7,4 (1992), 404--427.

....tra#c generated by computers. The use of traditional models of teletra#c can result in overly optimistic estimates of performance of computer networks, insu#cient allocation of communication and data processing resources, and di#culties in ensuring the quality of service expected by network users [2], 21] 23] On the other hand, if the strongly correlated character of teletra#c is explicitly taken into account, this can also lead to more e#cient tra#c control mechanisms. Several methods for generating pseudo random self similar sequences have been proposed. They include methods based on ....

Beran, J. Statistical Methods for Data with Long Range Dependence. Statistical Science 7, 4 (1992), 404--427.

....tra#c generated by computers. The use of traditional models of teletra#c can result in overly optimistic estimates of performance of computer networks, insu#cient allocation of communication and data processing resources, and di#culties in ensuring the quality of service expected by network users [1], 16] 19] On the other hand, if the strongly correlated character of teletra#c is explicitly taken into account, this can also lead to more e#cient tra#c control mechanisms. Several methods for generating pseudo random self similar sequences have been proposed. They include methods based on ....

Beran, J. Statistical Methods for Data with Long Range Dependence. Statistical Science 7,4 (1992), 404--427.

....12 of values of A n were rejected to exclude the initial transient. In particular we are interested in calculating the Hurst parameter H, which is a measure of the degree to which a time series is self similar. The calculations are based on [Leland 1994] Garrett 1994] Mandelbrot 1969] [Beran 1992]. 6.1 The rescaled adjusted range statistic The rescaled adjusted range statistic, R(n; k) S(n; k) is calculated for a selection of subsets of the time series A i , starting at n and of size k 1 [Mandelbrot 1969] The adjusted range R(n; k) has the following physical interpretation. Suppose ....

J. Beran, "Statistical methods for data with long-range dependence", Statistical Science, 7, 4, 1992, pp. 404--427

....of underlying self similarity. y = 5.7637x 0.4354 R 2 = 0.9986 0.1 1 10 1 10 100 1000 10000 100000 Ana lysis in terval (m in) Figure 1. Aggregate Variance Plot Showing Self Similarity Over 4 Orders of Magnitude. Normally, self similarity is estimated using an aggregate variance method [7,8]. However this technique is only accurate for stationary data sets, and it is known that cache hit rates exhibit a diurnal variation. The data sets were therefore normalised (by subtracting a suitable moving average) and also analysed using a wavelet estimator [9] The benefit of wavelet ....

....the EduWeb cache. Fig 1 shows clearly that significant selfsimilarity can be observed over at least 4 orders of magnitude at the NLANR LJ cache. The exponent of 0.435 equates to a Hurst parameter of 0. 78, which is high enough to suggest a large amount of self similarity is influencing the data [8]. This was confirmed by wavelet estimation giving H = 0.805 0.012. These results were further validated by using daily hit rate statistics for an 18 month period at the NLANR caches (available at http: ircache.nlanr.net Cache Statistics Reports ) to generate aggregate variance plots. This ....

J Beran. 'Statistical methods for data with long-range dependence'. Statistical Sciences 1992, Vol. 7, No. 4, 404-427.

.... Similarity of Network Traffic The concept of self similarity in the communications field was first introduced by Mandelbrot [9] He proposed a fractal like model of random error perturbations that appeared to come in bursts not unlike those observed in the early hydrological data studied by Hurst [1]. In 1994, Leland, Taqqu, Willinger and Wilson demonstrated the presence of slowly decaying packet count bursts across all time scales in traffic on an operational corporate Ethernet LAN [8] Time series with these characteristics are considered to exhibit LRD and are termed self similar . Figure ....

....(size = 0.01 second) Time interval (size = 0.1 second) Time interval (size = 0.01 second) Figure 2 Illustration of self similarity (Poisson with = 1000 versus inverse Pareto with = 1.1 and = 0. 0002) The R S statistic is one of several methods to estimate the Hurst parameter (see also [1] and [6] Hurst discovered that many naturally occurring time series are well represented by the relation, H cn n S n R E (3) where as n c is an integer constant and H is a Hurst parameter ( 0 . 1 50 . 0 H ) The estimation of H values can be based on a heuristic ....

J. Beran, "Statistical Methods for Data with LongRange Dependence", Statistical Science, Vol. 7, No. 4., pp. 404 - 416, 1992.

....Control Protocol (TCP) 14 16] Motion Pictures Experts Group (MPEG) video [6] World Wide Web [4, 12] and Signaling System 7 [5] traffic. An important characteristic of self similar traffic is its longrange dependence, i.e. the existence of correlations over a broad range of time scales [2, 3, 17]. Until now, most network traffic measurements were performed on wired networks. The question arises whether the traffic in wireless data networks exhibits self similar behavior as well. If so, it is important to determine if this traffic characteristic affects the provisioning and design of ....

J. Beran, "Statistical methods for data with longrange dependence," Statistical Science, vol. 7, no. 4, pp. 404 - 427, 1992.

....of a given time series is presented in Sect. 2.2. 2. 1 Mathematical Characterization Before going into detail, it is worth noting that there exists a considerable amount of non equivalent de nitions for self similarity, as also mentioned in [47] and [48] Here, we follow the work of Beran [3] and Cox [10] We consider a discrete time stochastic process X = fX(t) t = 0; 1; g, representing e.g. the number of packet arrivals during the time interval [t; t 1) It is assumed that X is covariance stationary (i.e. the autocovariance function R t (k) E[ X(t) E[X] X(t k) E[X] ....

....the formal de nition of self similar stochastic processes. The most important approaches in this area are: Increment processes of exactly self similar processes, which consider the difference of subsequent innovations of their parent process. A common example is fractional Brownian motion (FBM) [3], which is derived from the di erentiation of fractional Gaussian noise (FGN) 33] Fractional autoregressive integrated moving average (F ARIMA) processes [28] are an extension of standard ARIMA [5] processes. Compared to e.g. FBM, they have a higher exibility in accounting for short range ....

J. Beran. Statistical methods for data with long-range dependence. Statistical Science, 7:404-427, 1992.

....sixties onwards, when Mandelbrot suggested the appropriateness of longmemory models for economic time series, there has been a steady growth in the literature on the subject. Robinson (1994) and Baillie (1996) provided useful surveys of the developments in the econometric modelling of long memory; Beran (1992) reviewed developments in longmemory modelling in other areas. Beran s monograph, Beran (1994) discusses most of the central issues, including forecasting. The arfima(p; d; q) model is used here for the statistical analysis of a univariate time series y t with long memory. We write it as (L) ....

Beran, J. (1992). Statistical methods for data with long-range dependence. Statistical Science 7, 404-416.

....over many different time scales, often referred to as long range dependence (LRD) or long memory. Some of the better known examples of such systems are found in hydrology [20] However, the phenomenon of LRD occurs in many other systems including chemical, astronomical, and biological systems (see [4] for references) In spite of much statistical evidence, the existence of LRD has often been met with resistance or at least puzzlement. This was caused in large part by the absence of physical explanations for the observed phenomenon. Hydrologists for example wondered By what sort of physical ....

J. Beran. Statistical Methods for Data with Long Range Dependence. Statistical Science, 7(4):404--427, 1992.

.... (k) as k 1; 9) where 0 fi 1 and L 1 is slowly varying at infinity, that is, lim t 1 L 1 (tx) L 1 (t) 1, for all x 0; examples of such slowly varying functions are L 1 (t) const and L 1 (t) log(t) A stochastic process satisfying relation (9) is said to exhibit long range dependence [6, 16, 93]. In Mandelbrot s terminology [74] long range dependence is also referred to as the Joseph Effect, in reference to the Old Testament figure who had interpreted Pharaoh s dream of the seven lean cows and the seven fat cows to mean the seven fat years and seven lean years that ancient Egypt was ....

....2 g 1 m=1 satisfies (oe (m) X ) 2 b m Gamma1 ; as m 1; where b is a finite positive constant independent of m. The consequences of the slowlydecaying variances, oe (m) X ) 2 , can be disastrous for classical statistical tests and confidence or prediction intervals (see, e.g. [6]) since the usual standard errors (derived for conventional models) are wrong by a factor that tends to infinity in the sample size. The statistical properties of slowly decaying variances, long range dependence, and a power law spectral density are thus seen to be different manifestations of ....

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Beran, J., "Statistical Methods for Data with Long-Range Dependence", Statistical Science 7:4 (1992), 404--427.

....series is that a process fXng has long range dependence if 1 X j=1 jcorr(X 0 ; X j )j = 1 (cf. Brockwell and Davis, 1991) Here corr( Delta; Delta) is the correlation coefficient, whose use is legitimate in the finite variance case. A very useful recent reference on long range dependence is Beran, 1992. In contexts outside traditional times series where correlations may not exist or be difficult to compute or to interpret, other measures of dependence may be more meaningful. In this case, long range dependence may refer to a slowly decreasing dependence between blocks of random variables as the ....

....range dependence in a stationary process fTng is that the probability P [T i r; i = 1; n] decreases slowly to 0 as n 1 for a suitable choice of constant r. Statistical evidence is mounting that traffic on certain types of data networks may exhibit long range dependence. For instance, Beran et. al (1992) report on the analysis of several data sets representing the traffic seen as a result of transmitting video conference scenes. The data sets are large, sometimes in excess of 50,000 data points and consequently one may expect estimates of the autocorrelation function to be accurate to very large ....

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Beran, J., Statistical methods for data with long-range dependence, Statistical Science 7 (1992), 404--427.

....y = 0.0334x 0.6322 0.001 0.01 0.1 1 10 100 X Days Hurst Parameter =0.68 Figure 5. Aggregate variance of fitted exponents over time. The Hurst parameter was calculated using the formula Slope = 2H 2 The H of 0. 68 indicates that there is some self similarity in best fit exponents over time [5], this implies that large sample sizes will be required for statistics to be safe . It is worth noting that the self similarity revealed in figure 5 is over 3 orders of magnitude of timescale from hours to weeks. It is thus quite different from the long range dependency reported elsewhere [13] at ....

J Beran. 'Statistical methods for data with long-range dependence'. Statistical Sciences 1992, Vol. 7, No. 4, 404-427.

....of the predictor. ARFIMA(p,d,q) models: ARFIMA(p,d,q) autoregressive fractionally integrated moving average ) models implement Equation 1 for fractional values of d, 0 d 0:5. It can be shown that this fractional integration can model long range dependence such as arises from self similarity [4, 17, 14]. In addition, the ARMA part of the model models the short range dependence in the signal. To fit ARFIMA models, we use Fraley s Fortran 77 code [12] which does maximum likelihood estimation of ARFIMA models assuming a normally distributed white noise source following Haslett and Raftery [16] ....

BERAN, J. Statistical methods for data with long-range dependence. Statistical Science 7, 4 (1992), 404--427.

....Box Jenkins [6] AR, MA, ARMA, and ARIMA classes might be appropriate for predicting load. On the other hand, the existence of selfsimilarity induced long range dependence suggested that such models might require an impractical number of parameters or that the much more expensive ARFIMA model class [18, 14, 4], which explicitly captures long range dependence, might be more appropriate. Since it is not obvious which model is best, we empirically evaluated the predictive power of the AR, MA, ARMA, ARIMA, and ARFIMA model classes, as well as a simple ad hoc windowed mean predictor called BM and a ....

....Time 20000 40000 60000 80000 0.04 0.02 0.0 0.02 0.04 ) 0 ( 2 a WhiteNoise a s 2 z s 2 2 z a s s Unpredictable Random Sequence Fixed Linear Filter Partially Predictable Load Sequence Figure 3. Linear time series models. 5) The traces exhibit self similarity with Hurst Parameters [2, 4] ranging from 0.63 to 0.97, with a strong bias toward the top of that range. Hurst parameters in the range of 0.5 to 1.0 indicate self similarity with positive near neighbor correlations. This result tells us that load varies in complex ways on all time scales and has long range dependence. ....

[Article contains additional citation context not shown here]

BERAN, J. Statistical methods for data with long-range dependence. Statistical Science 7, 4 (1992), 404--427.

....in the investigation of self similar queuing problems and these are discussed in section 2.2. ffl Self similar processes have a strong dependency over time (i.e. slowly decaying autocorrelation functions) which makes non parametric estimates such as mean and variance very slow to converge [8]. This suggests that more sophisticated, parametric network monitoring techniques are required, adding complexity to the network. It is the second of the above points that is the focus of this paper which continues by defining and explaining self similarity and discussing why it can have such a ....

J. Beran. Statistical methods for data with long-range dependence. Statistical Science, 7(4):404--427, 1992.

....estimated the Hurst parameter as high as 0:82 [9] In the literature, there are a number of traffic models which generate long range dependent traffic, see [6, 9, 12] and references therein. In developing M we model a particular class of self similar traffic called Fractional Gaussian Noise (FGN) [2] using the Fast Fourier Transform (FFT) method developed by Paxson [11] The advantage to a FGN process is that the degree of self similarity can be expressed solely by the Hurst parameter. We choose the FFT method because an approximate FGN sample path of length P can be efficiently generated in ....

J. Beran. Statistical Methods for Data with Long-Range Dependence. In Statistical Science, Lucas: (M;P ; S) -- An Efficient Background Traffic Model for Wide-Area Network Simulation 12 7(4), pages 404--427, 1992.

.... Gamma1 N ( X: 36) The search for the set of parameters that jointly maximizes (34) requires evaluation of the inverse of the covariance matrix RN ( This operation is computationally intensive and in some cases may be numerically unstable. Whittle s method uses the following approximation [13]: ffl Approximation of log(det(R N ( lim N 1 1 N log(det(R N ( 1 2 Z Gamma log fS 2X ( g d : 37) ffl Approximation of X T R Gamma1 N ( X: The matrix R Gamma1 N ( is replaced by A( ff(j Gamma l) j;l=1; Delta Delta Delta;N (38) with ff(j Gamma l) ....

J. Beran. Statistical Methods for Data with Long-Range Dependence. Statistical Science, 7(4):404427, 1992.

....in terms of a non stationary trend are influenced by a priori assumptions regarding the causal influence of increasing greenhouse gases on global climate. The observed behavior may however be equally well described in terms of stationary, purely stochastic models with long range dependence (Beran, 1992; Bloomfield and Nychka, 1992; Kunsch, 1986; Smith, 1993) This possibility is investigated in more detail. The IPCC record is analyzed in the context of fractional Gaussian noise as a model for long range dependence. Specifically, one assumes a spectral density function, which depends on ....

Beran, J., 1992: Statistical methods for data with long-range dependence. Statistical Science, 7, 404--427.

....value of n, there may be more than one set of flog[R(n) S(n) log[n]g corresponding to non overlapping intervals in the data. The R S statistic has good robustness properties, in particular, with respect to long tailed distributions. However, its drawback is that it can lead to biased estimates [4]. The VT plots and PG plots are, similarly, useful starting points in data analysis. However, neither of these techniques are suitable for precise estimation [23] More refined statistical analyses based on Maximum Likelihood Estimate (MLE) methods are used to estimate H when fX t g comes from a ....

....refined statistical analyses based on Maximum Likelihood Estimate (MLE) methods are used to estimate H when fX t g comes from a Gaussian distribution. These methods and their approximations are based on the spectral density of X t and properties of the MLEs are discussed by a number of authors [4, 10, 37]. One specific method that has been used extensively is Whittle s approximate MLE (e.g. see [23] Most of the approximate MLEs are defined via quadratic forms which make them quite sensitive to deviations from normality. However, transformation of the data may sometimes alleviate these types of ....

J. Beran, "Statistical methods for data with long-range dependence," Statistical Science, vol. 7, pp. 404-427, 1992.

.... models are short range dependent (i.e. have exponentially decaying autocorrelations) measured packet traffic data are consistent with long range dependence (i.e. hyperbolically decaying autocorrelations) The probability theory of self similarity and long range dependence is discussed in [22, 24, 77, 171, 177, 287, 389, 390, 402]. The books [130, 178, 337, 338] also contain large sections on self similar processes, and extensive bibliographies can be found in [14, 22, 24, 287, 374, 389] Self similar stochastic processes were introduced by Kolmogorov [239] in a theoretical context and brought to the attention of ....

.... decaying autocorrelations) The probability theory of self similarity and long range dependence is discussed in [22, 24, 77, 171, 177, 287, 389, 390, 402] The books [130, 178, 337, 338] also contain large sections on self similar processes, and extensive bibliographies can be found in [14, 22, 24, 287, 374, 389]. Self similar stochastic processes were introduced by Kolmogorov [239] in a theoretical context and brought to the attention of probabilists and statisticians by Mandelbrot and his co workers [287 292] They have been used in hydrology [200 202, 214, 254, 302, 310 312] geophysics [40, 322] ....

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J. Beran. Statistical methods for data with long-range dependence. Statistical Science, 7(4):404--416, 1992. With discussions and rejoinder, pages 404-427.

....with a slowly varying function of . A slowly varying function of , as goes to zero, is a function L such that lim 0 L(t ) L( 1; for every t 0: 1.12) Thus, the condition expressed by (1.10) becomes: lim 0 f( L( Gammad = 1; for some slowly varying function L; 1. 13) see Beran (1992). We will not make use of this generalisation. Two classic examples of long memory processes are the fractionally differenced white noise and the fractional Gaussian noise. Let us start with the latter. The idea is to consider the increments of a continuous time self similar stochastic process. ....

Beran, J. (1992). Statistical methods for data with long-range dependence, Statistical Science 7: 404--427.

....generally regarded as prototypes of i.i.d. observations, are not independent but long range correlated. A number of results have been obtained for parametric estimation under long range dependences and on the estimation of dependence parameters. For a review of work in that direction, see Beran [8], 9] We here focus on the case with nonparametric estimation of f . Results on minimax rates of convergence are obtained for long range dependent errors with a one dimensional equally spaced fixed design in Hall and Hart [29] Wang [71] and Johnstone and Silverman [39] for some classical ....

J. Beran. Statistical methods for data with long-range dependence. Statistical Science, 7:404--416, 1992.

....selfsimilar processes, and for generating synthetic network traffic that reflects the salient characteristics of these processes. In this paper we present a fast algorithm for generating approximate sample paths for a type of self similar process known as fractional Gaussian noise (FGN) [B92b]. The algorithm is based on synthesizing sample paths that have the same power spectrum as FGN. These sample paths can then be used in simulations as traces of self similar network traffic. The key to the algorithm is a fast approximation of the power spectrum of an FGN process; this approximation ....

.... From these definitions it is not obvious at first glance that self similar processes actually exist, but in fact a number of families of self similar processes are known [ST94] The most widely studied self similar processes are fractional Gaussian noise (FGN) and fractional ARIMA processes [B92b, ST94, GW94]. Associated with FGN is fractional Brownian motion (FBM) which is simply the integrated version of FGN (that is, an FBM process is simply the sum of FGN increments) In this paper we are concerned with synthesizing FGN. There are several existing methods 1 For a slowly varying function L, lim ....

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J. Beran, "Statistical Methods for Data with LongRange Dependence", with discussion, Statistical Science, 7(4), pp. 404-427, 1992.

....many dioeerent time scales, often referred to as long range dependence (LRD) or long memory. Some of the bet ter known examples of such systems are found in hydrology [16] However, the phenomenon of LRD occurs in many other systems including chemical, astronomical, and biolog ical systems (see [3] for references) In spite of much statistical evidence, the existence of LRD has often been met with resistance or at least puzzlement. This was caused in large part by the absence of physical ex planations for the observed phenomenon. Hydrologists for example wondered iBy what sort of physical ....

J. Beran. Statistical Methods for Data with Long Range Dependence. Statistical Science, 7(4):404427, 1992.

....one value per octave frequency band and is particularly useful when the spectrum is relatively featureless within each octave band. For example, a model that commonly arises in the physical sciences is that the spectrum obeys the power law S Y (f) # f # over a certain interval of frequencies (Beran, 1992), which, using the above approximation, translates into the statement that # 2 Y (#) # # # 1 over a corresponding set of scales. A region of linear variation on a plot of log # 2 Y (#) versus log # indicates the existence of a power law behavior, and the slope of the line can be used to ....

....yields a rather imperfect high pass filter. We can evaluate E analytically for certain choices of S Y . As a simple example, suppose that S Y (f) # sin(#f) # so that S Y varies as f # for frequencies close to zero. Processes with such spectra occur in a wide range of applications (Beran, 1992). Note that the process Y t corresponding to S Y is stationary if # 1; if in addition # 1, then Y t corresponds to a stationary and invertible fractional di#erence process (Granger Joyeux, 1980; Hosking, 1981) Table 1 shows E for three blue noise processes # = 1, 1 2 and 1 4 , a ....

BERAN, J. (1992). Statistical methods for data with long-range dependence. Statist.

....predictions are close to that average. In a previous paper [11] we evaluated the performance of linear time series models for predicting our load traces using the criterion of consistent predictability discussed above. Although load exhibits statistical properties such as self similarity (Beran [5] provides a good introduction to self similarity and long range dependence) and epochal behavior [9] that suggest that complex, expensive models such as ARFIMA [14] models might be necessary to ensure consistent predictability, we found that relatively simple models actually performed just about ....

BERAN, J. Statistical methods for data with long-range dependence. Statistical Science 7, 4 (1992), 404--427.

....average. In a previous paper [12] we evaluated the performance of linear time series models for predicting our load traces using the criterion of consistent predictability discussed above. Although load exhibits statistical properties such as self similarity (Bassingwaighte, et al. 3] and Beran [6] provide good introductions to self similarity and long range dependence) and epochal behavior [11] that suggest that complex, expensive models such as ARFIMA [16, 14] or TAR [27] models might be necessary to ensure consistent predictability, we found that relatively simple models actually ....

BERAN, J. Statistical methods for data with long-range dependence. Statistical Science 7, 4 (1992), 404--427.

....of H since the periodogram is not appropriate to estimate the spectral density [10] More sophisticated methods have to be applied to obtain useful estimates of H. Several periodogram based estimators can be found in the literature. In this paper we will focus on an MLE as presented in [1, 13] which is based on Whittle s approximate MLE for Gaussian processes [12] For Gaussian sequences this estimator is asymptotically normal and efficient [4, 3] The spectral density of the self similar process is denoted by f( where the parameter vector of the process = 1 ; M ) ....

J. Beran. Statistical methods for data with long-range dependence. Statistical Science, 7(4):404--427, 1992.

....the qth Hermite polynomial. Moreover, it is assumed that the spectral density f Z is continuous in [ 0) 0; and diverges to in nity at the origin at the rate f Z ( j j 0 c f;Z j j 2d (2) for some 1 2 1 2m d 1 2 ; m 2 N; and 0 c f;Z 1: This condition implies (see e.g. Beran 1992, 1994) 1. Z (k) c ;Z jkj 2d 1 (0 c ;Z 1) as jkj 1; 2. the process i = fG(Z i )g has long memory in the sense that its spectral density f has a pole at zero of the form c f; j j 2dm with 0 dm = 1 2 m(d 1 2 ) 1 2 ; 3. k) cov( i ; i k ) m c 2 m c m ....

Beran, J. (1992). Statistical methods for data with long-range dependence.

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BERAN, J. Statistical methods for data with long-range dependence. Statistical Science 7, 4 (1992), 404--427.

No context found.

BERAN, J. Statistical methods for data with long-range dependence. Statistical Science 7, 4 (1992), 404--427.

No context found.

Jan Beran, "Statistical Methods for Data with Long-Range Dependence," Statistical Science, 1992, vol.7, No.4, pp.404-427

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J. Beran, \Statistical Methods for Data with Long-Range Dependence," Statistical Science, vol. 7, no. 4, pp. 404-416, 1992.

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Beran, J. (1992) "Statistical Methods for Data with Long-Range Dependence," Statistical Science, 7, 404 -- 416.

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Beran J. (1992) Statistical methods for data with long-range dependence. Statistical Science, 7, 404-427.

No context found.

Beran J. (1992), "Statistical Methods for Data with Long-Range Dependence," Statistical Sciences, 7, 404-427.

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J. Beran, "Statistical Methods for Data with Long-Range Dependence", Statistical Science 7, No. 4, 1992.

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