B. B. Mandelbrot and J. R. Wallis. Computer experiments with fractional gaussian noises. Water Resources Research, 5:228--267, 1969.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....traffic [20] and NSFNET [21] has suggested that the auto correlation decays to zero at a slower rate than exponential. This slow decay in correlation has been observed before in other statistical applications, e.g. hydrology [22] Mathematical models have been developed to capture this behavior [23, 24]. BACKGROUND ON LONG RANGE DEPENDENCE Let X t , t = 0, 1, 2, be a wide sense stationary stochastic process, i.e. a process with a stationary mean = E[X t ] a stationary and finite variance v = E[ X t ) 2 ] and a stationary auto covariance function g k = E[ X t ) X t k ) ....

....(1 B) d = d k k= 0 ( 1) k B k , n Figure 7. Examples of auto correlation structures of a) shortrange dependent process, b) long range dependent process. IEEE Communications Magazine . July 1997 88 The fractional Brownian motion fB t can be deduced from Brownian motion B t [24, 25] by forming the integral: As Eq. 28 shows, the interdependence between the increments of the fractional Brownian motion can said to be infinite. In the discrete case [25] the auto correlation of the increment sequence is obtained by replacing t 1 , t 2 , t 3 , and t 4 in Eq. 27 by n, n 1, n ....

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B. Mandelbrot and J. Wallis, "Computer Experiments with Fractional Gaussian Noises," Water Resources Res., vol. 5, Feb. 1969, pp. 228--67.

.... parsimoniously described via the notion of long range dependence (LRD) which was brought to the attention of statisticians and probabilists by Mandelbrot and his co workers [51, 53, 54] mainly through applications in such areas as hydrology (e.g. annual river ow data) 56, 55, 57] geophysics [58, 59], and nance (e.g. stock prices) 49] As far as the networking application area is concerned, the last decade has seen an enormous increase in empirical studies of high quality and highvolume data sets of trac measurements from a variety of di erent data networks, but especially from di erent ....

B. B. Mandelbrot and J. R. Wallis. Computer experiments with fractional Gaussian noises. Water Resources Research, 5:228-267, 1969.

....Brownian motion (FBM) random process. This process is a (discrete time) stationary Gaussian process with mean ,variance# 2 and 2 # k = 1 2 ( k 1 2H 2 k 2H k 1 2H ) k 0. A FBM process, which is the sum of FGN increments, is characterised by three properties [16]: i) it is a continuous zero mean Gaussian process X t = X s : s # 0and 0 H 1 with ACF given by # s,t = 1 2 (s 2H t 2H s t 2H )wheres is time lag and t is time; ii) its increments X t X t 1 form a stationary random process; iii) it is self similar with ....

Mandelbrot, B., and Wallis, J. Computer Experiments with Fractional Gaussian Noises. Water Resources Research 5, 1 (1969), 228--267.

....Brownian motion (FBM) random process. This process is a (discrete time) stationary Gaussian process with mean ,variance# 2 and # k = 1 2 ( k 1 2H 2 k 2H k 1 2H ) k 0. A FBM process, which is the sum of FGN increments, is characterised by three properties [19]: i) it is a continuous zero mean Gaussian process X t = X s : s # 0and 0 H 1 with ACF given by # s,t = 1 2 (s 2H t 2H s t 2H )wheres is time lag and t is time; ii) its increments X t X t 1 form a stationary random process; iii) it is self similar with ....

Mandelbrot, B., and Wallis, J. Computer Experiments with Fractional Gaussian Noises. Water Resources Research 5, 1 (1969), 228--267.

....Brownian motion (FBM) random process. This process is a (discrete time) stationary Gaussian process with mean ,variance# 2 and # k = 1 2 ( k 1 2H 2 k 2H k 1 2H ) k 0. A FBM process, which is the sum of FGN increments, is characterised by three properties [15]: i) it is a continuous zero mean Gaussian process X t = X s : s # 0and 0 H 1 with ACF given by # s,t = 1 2 (s 2H t 2H s t 2H )wheres is time lag and t is time; ii) its increments X t X t 1 form a stationary random process; iii) it is 2 self similar ....

Mandelbrot, B., and Wallis, J. Computer Experiments with Fractional Gaussian Noises. Water Resources Research 5, 1 (1969), 228--267.

....map (running on a 70MHz SPARC 5) generated 20 times more samples on line in the same amount of time. 4. Experimental Results Two types of statistical checks for self similarity were carried out. The rescaled adjusted range (R S) statistic and variance analysis. The reader is referred to refs. [9 11] for outlines of how to perform these tests. In the case of the R S analysis the Hurst parameter can be obtained directly from the slope of the doubly logarithmic plot of R S and the sample lag, whereas in the variance analysis the Hurst parameter is obtained via the relationship H = 1 b 2 where b ....

Mandelbrot B., and Wallis J. R., Computer Experiments with Fractional Gaussian Noise Part2 , Rescaled Ranges and Spectra. Water Resources Research, Vol. 5, No 1, February 1968, pp 242- 259.

....map (running on a 70MHz SPARC 5) generated 20 times more samples on line in the same amount of time. 4. Experimental Results Two types of statistical checks for self similarity were carried out. The rescaled adjusted range (R S) statistic and variance analysis. The reader is referred to refs. [9 11] for outlines of how to perform these tests. In the case of the R S analysis the Hurst parameter can be obtained directly from the slope of the doubly logarithmic plot of R S and the sample lag, whereas in the variance analysis the Hurst parameter is obtained via the relationship H = 1 b 2 where b ....

Mandelbrot B., and Wallis J. R., Computer Experiments with Fractional Gaussian Noise Part2 , Rescaled Ranges and Spectra. Water Resources Research, Vol. 5, No 1, February 1968, pp 242- 259.

....(but not in a statistically significant manner) suggesting the possibility that this parameter is to some degree independent of the type of file system measured. 4. 3 R S Analysis and Pox plots A second estimate of the Hurst parameter can be obtained through R S analysis (originally presented in [16], and fully explained in [3] Given a set of observations (X k : k = 1; 2; N ) that set is subdivided into K disjoint, contiguous subsets of length (N=K) The R S statistic R(t i ; n) S(t i ; n) equation 11) is then computed for the starting points t i of the K disjoint subsets and for ....

Mandelbrot, B. B., and Wallis, J. R. Computer experiments with fractional gaussian noises. Water Resources Research 5 (1969), 228--267.

....= 2 Gamma D f . An increment in yH (x) is related to the change in x by: Deltay H (x) Deltax H ; 3:23) 49 where H = 0:5 corresponds to the classical Brownian motion trace. A special property of fBm is that the increments Deltay H (x) are stationary, while yH (x) itself is nonstationary [103]. Equation 3.23 represents a spatial relationship between different points in the profile, and provides one basis for synthesizing a profile with a given fractal dimension. A similar synthesis procedure has been used in fractal terrain generation [57] A second approach to profile synthesis using ....

B. B. Mandelbrot and J. R. Wallis. Computer experiments with fractional gaussian noises: Parts 1-3. Water Resources Research, 5(1):229--267, February 1969.

....and (3) detection of nonperiodic cycles. However, there are also two deficiencies associated with rescaled range analysis and the estimation of Hurst exponents: 1) estimation errors exist when the time scale is very small or very large relative to the number of observations in the time series, (Mandelbrot Wallis 1969, Wallis Matalas 1970, Feder 1988, Ambrose, Ancel Griffiths 1993, Moody Wu 1995a, Muller, Dacorogna Pictet 1995) and (2) the rescaled range is sensitive to short term dependence (McLeod Hipel 1978, Hipel McLeod 1978, Lo 1991) The second shortcoming will sometimes lead to completely ....

Mandelbrot, B. & Wallis, J. (1969), `Computer experiments with fractional Gaussian noises. Part 3, mathematical appendix', Water Resources Research 3(1), 260--267.

....all m and all k; i.e. the correlation structure is preserved across different time scales. Such a process is frequently characterized by the Hurst coefficient, H; where H = 1 Gamma fi=2: Notice that, for 0 fi 1; we have 1=2 H 1: It was observed by Hurst [20] and by Mandelbrot and Wallis [27, 28] that many naturally occurring time series 1 exhibit H 1=2: Examples include annual water flows of the Nile and several other rivers, sun spot numbers, tree ring indices, etc. The Leland et al. study [23] on traffic traces from the Bellcore Ethernet al..so shows second order self similarity ....

B. B. Mandelbrot and J. R. Wallis, "Computer experiments with fractional gaussian noises," Water Resources Research, vol. 5, pp. 228-267, 1969.

....on chaos, probability and statistics, see [29, 46, 47] An overview of statistical inference methods for self similar models and random processes with long range dependence can be found in [22, 24] the papers [392 394] listing additional techniques. More specifically, R S analysis is discussed in [18, 24, 26, 28, 130, 200, 258, 272, 273, 286, 288, 290 292, 302, 310, 394] (see also [10,131] variance time analysis in [24,26,77,258,310,331,394,399] and for spectral domain methods using periodograms, see [24,26,48,84,140,149,157, 159,183,203,204,206,249,253,353,357 366,393,407, 418] Examples of new statistical techniques in this area include [3, 7, 20,21,25,27, ....

....practice. There exist numerous methods to date for generating self similar traffic traces. Exact methods, which are based on the Durbin Levinson algorithm [37, 394] are discussed in [24, 146, 196, 198, 394] They are generally impractical for long time series. Approximate methods are described in [30, 74, 120, 215, 245, 252, 258, 285, 290, 331, 334, 337, 356, 374, 376, 404,412]; some of these methods rely on earlier results reported in [77, 165,391] derived for a different purpose and re interpreted here in the context of synthetic traffic generation. These methods are generally very fast and feasible for even A Bibliographical Guide 7 very long time series. However, ....

B .B. Mandelbrot and J. R. Wallis. Computer experiments with fractional Gaussian noises, Parts 1,2,3Water Resources Research, 5:228--267, 1969.

....times k H for some H 1 2 . This behaviour, found also in many geophysical and climatological data sets, and now known as Hurst effect, is in contradiction to results holding for short memory processes. Mandelbrot and his co workers (Mandelbrot and van Ness, 1968; Mandelbrot and Wallis, 1968; Mandelbrot and Wallis, 1969) introduced the so called fractional Gaussian noise (see Section 1.3) and were able to show that the Hurst effect can be explained using that model. The use of standard short memory models, such as the ARMA family, to analyse strongly correlated data generally results in underestimating the ....

Mandelbrot, B. B. and Wallis, J. R. (1969). Computer experiments with fractional gaussian noises, Water Resources Research 5: 228--267.

....in the presence of strong low frequency components, a property which is clearly visible for our data (see Fig. 1) An LRD process plotted on a graph of any time scale, appears to have a dominant periodicity with a few cycles fitting on the plot. If more data is plotted, new dominant modes appear [MAND69a]. The increase of apparent amplitude withincreased time scale can be understood intuitively for video, especially movies. Within each scene there is random movement and variation of bandwidth. Changes of camera angle can alter the general level of complexity more than the changes within the scene, ....

....0.7. Others have since shown that (2) holds for short range dependent processes with H = 0:5 [FELL51, MAND68] and for increment processes of selfsimilar models, where 0:5 H 1 [MAND79] When working with empirical data, a practical implementation of the R=S analysis has been proposed by [MAND69a] (see also [MAND79] and consists of plotting R(n) S(n) versus n, for different lags n and for different partitions of the observations. For our video trace, the resulting data is plotted in a Pox diagram of R=S in Fig. 12. H can be measured ideally as the asymptotic slope of a straight line. ....

B. B. Mandelbrot and J. R . Wallis, "Computer Experiments with Fractional Gaussian Noises", Water Resources Res., Vol. 5, pp. 228--267, 1969.

....with the theoretical slope 2(d 1=ff) 2H are included. The innovations in the infinite variance series (stable, Pareto) have ff = 1:5. 16 Taqqu and Teverovsky 4. 4 R S Method The R=S method is one of the oldest methods for estimating H and is discussed in detail in Mandelbrot and Wallis [MW69] Mandelbrot [Man75] and Mandelbrot and Taqqu [MT79] Given the partial sums of a time series Y (t) P t i=1 X(i) and its sample variance S 2 (n) the R=S statistic is defined: R S (n) 1 S(n) max 0tn Y (t) Gamma t n Y (n) Gamma min 0tn Y (t) Gamma t n Y (n) ....

B .B. Mandelbrot and J. R. Wallis. Computer experiments with fractional Gaussian noises, Parts 1,2,3 Water Resources Research, 5:228--267, 1969.

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B. B. Mandelbrot and J. R. Wallis. Computer experiments with fractional gaussian noises. Water Resources Research, 5:228--267, 1969.

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B. B. Mandelbrot and J. R. Wallis, Computer experiments with fractional Gaussian noises, Parts 1,2,3 Water Resources Research, 5:228-267, 1969.

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B.B. Mandelbrot and J.R. Wallis. Computer Experiments with Fractional Gaussian Noises. In Water Resources Research, Vol. 5, pp. 228-267, 1969.

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Mandelbrot, B.B. and Wallis, J.R. : Computer experiments with fractional Gaussian noises, Parts 1,2,3, Water Resources Research 5, 228-267 (1969).

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MANDELBROT, B.B. & J.M. WALLIS (1969): \Computer Experiments with Fractional Gaussian Noises", Water Resources Research, #, 228-267.

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B. B. Mandelbrot, "Computer experiments with fractional Gaussian Noise," Water Resource Research, Vol. 5, No 1, pp. 228-267, Feb. 1969.

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B. Mandelbrot and J. R. Wallis, \Computer Experiments with Fractional Gaussian Noise," Water Resources Research Vol. 5, No 1, pp. 228 - 267, February 1968.

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Mandelbrot B., and Wallis J. R., 1968, "Computer Experiments with Fractional Gaussian Noise" Water Resources Research, Vol. 5, No 1, 228 - 267.

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B. Mandelbrot and J. R. Wallis. Computer Experiments with Fractional Gaussian Noise. Water Resources Research, Vol. 5, No 1:228 - 267, February 1968.

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B. B. Mandelbrot and J. R. Wallis, "Computer Experiments with Fractional Gaussian Noises", Water Resources Research, Vol. 5, pp. 228-267, 1969.

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