B. B. Mandelbrot and J. Wallis. Some long-run properties of geophysical records. Water Resources Research, 5:321--340, 1969.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....turn implies the maximum possible degree of self similarity. xxiv The rescaled adjusted range statistic pox plots of R S: The rescale adjusted range statistic, R(n, k) S(n, k) is calculated for a selection of subsets of the arrival times for the packets # , starting at n and of size k 1 [8]. The adjusted range R(n, k) has the following physical interpretation. Suppose that the times series # represents the amount of water per time unit flowing into a reservoir. Furthermore, water flows out of the reservoir at a constant rate, this rate being such that the reservoir contains the ....

B.B. Mandelbrot & J.R. Wallis, "Some long-run properties of geophysical records", 5, 2, 1969, pp/ 321-340

....decay. A recent analysis of traffic measurements of Ethernet LAN 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 ] ....

B. B. Mandelbrot and J. R. Wallis, "Some Long-Run Properties of Geophysical Records," Water Resources Res., vol. 5, 1969, pp. 321--40.

....1 Hurst acknowledged that standard forecasting methods fail for this data. Instead of independence or weak correlations between data points far away from each other, he observed strong dependencies. The phenomenon of long range dependence in water flow data was observed in many other rivers by Mandelbrot Wallis (1969). Also the Rhine River exhibits long range dependence (see Lohre Sibbertsen (2001) and references therein) Additional geophysical applications of long memory are for instance the temperature data of the northern hemisphere. Other domains of application are Computer Science and Economics. Many ....

Mandelbrot, B. B., Wallis, J. R. (1969): Some long--run properties of geophysical records. Water resources research 5, 321--340.

.... 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. Some long-run properties of geophysical records. Water Resources Research, 5:321-340, 1969. Long-Range Dependence and Data Network Trac 37

....invented much earlier in an abstract setting by Kolmogorov (1940) moreover he showed that not only hydrological, but virtually all geophysical processes he studied (river flows, geological layers, tree ring indices, annual precipitations, earthquake frequencies . are long range correlated (Mandelbrot and Wallis 1969). There was a big debate, particularly in hydrology, pro and against Mandelbrot s model. Now I do not think that the model of self similar processes is the only reasonable model (e.g. fractional ARIMA processes, invented later, behave very similarly) and if reasons for nonstationarity are ....

Mandelbrot, B. B. and Wallis, J. R. (1969). Some long--run properties of geophysical records, Water Resources Research 5: 321--340.

....large at very low frequencies as the sample size increases although these models are non stationary as well. The first stationary model suggested was Mandelbrot s fractional Gaussian noise (FGN) model described by Mandelbrot and Van Ness (1968) and advocated for hydrological time series by Mandelbrot and Wallis (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 ....

Mandelbrot, B.B.and Wallis, J.R. (1969), Some long-run properties of geophysical records. Water Resources Research 5(2), 321--340.

....response given in Equation (19) is developed from the parameters of the signal and noise as well as the knowledge of the wavelets. As a consequence, more information is utilized in the signal estimation. 4 Simulation Results In the simulation process, the discrete version of the fBm is given by [25] B ff (t) Gamma B ff (t Gamma 1) n Gammaff Gamma(ff 0:5) f n X i=1 (i) ff Gamma0:5 (1 n(M t) Gammai) n(M Gamma1) X i=1 ( n i) ff Gamma0:5 Gamma (i) ff Gamma0:5 (1 n(M Gamma1 t) Gammai) g; 20) where f i g is the set of Gaussian random variables with unit ....

B. Mandelbrot and J. Wallis, "Some long-run properties of geophysical records," in Water Resources Res., vol. 5, No. 2, 1969.

....passes and the first 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 ....

....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 the time series A i represents the amounts of water per time unit flowing into a reservoir. Furthermore, water flows out of the reservoir at a constant rate, 8 0.0 0.2 0.4 0.6 0.8 1.0 Time (1000 token passes) ....

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B.B. Mandelbrot & J.R. Wallis, "Some long-run properties of geophysical records", Water Resources Research, 5, 2, 1969, pp. 321--340

.... zero mean Gaussian process, BH = fBH (t)g 1 t=0 , with 0 H 1 and autocorrelation function ae X (s; t) 1=2(jsj 2H jtj 2H Gamma jt Gamma sj 2H ) Fractional Gaussian noise and fractional Brownian motion have been particularly popular in hydrological modeling (see, e.g. [72]) Despite its rigid autocorrelation structure, fractional Gaussian noise is often a reasonable first approximation of more complex structures due to the fact that certain long range dependent processes yield fractional Gaussian noise as limits under a special type of central limit theorem. ....

Mandelbrot, B.B. and Wallis, J.R., "Some Long-Run Properties of Geophysical Records", Water Resources Research 5 (1969), 321--340.

....of ff can be made from time series that are much shorter than generally presumed. 4.1 Introduction Long term correlations have been observed in many types of time series from physical, biological, physiological, economic, technological and sociological systems. Examples include geophysical data [32, 36, 18] such as rainfall, temperature measurements, sun spot numbers, earthquake frequencies, and river flows, frequency fluctuations in electrical oscillators [65] rate of traffic flow [32] voltage or current fluctuations in metal films and semiconductor devices [65] loudness fluctuations in speech ....

....2 (f) 3) and the regression was performed on ln S(k) versus ln fi k . The slope of the regressed line is 1 Gamma ff. 4.3. 3 Method 3: Hurst exponent Perhaps the most famous quantifier of long range dependence is that introduced by Hurst [19] and expounded and modified by Mandelbrot and Wallis [36, 27]. The method is nicely described in Bassingthwaighte et al. 5] Hurst was initially motivated by the problem of how large to make a Nile river dam in order to have acceptably low levels of fluctuations. He analyzed river flow data with an eye toward the functioning of dams. For each segment of ....

B.B. Mandelbrot and J.R. Wallis (1969) Some longrun properties of geophysical records, Water Resources Res. 5(2):321-340.

....are) and can never be fully validated from finite data sets. As we will show, however, these idealizations offer simplification and clarity and capture in a parsimonious manner important characteristics of the data at hand. The terms Joseph Effect and Noah Effect were coined by Mandelbrot [33]. 2 Self Similarity Through High Variability In [53] we presented an idealized ON OFF source model which allows for long packet trains ( ON periods, i.e. periods during which packets arrive at regular intervals) and long inter train distances ( OFF periods, i.e. periods with no packet ....

B. B. Mandelbrot and J. R. Wallis. Some Long-Run Properties of Geophysical Records. Water Resources Research, 5:321--340, 1969.

....to investigate how such LRD might affect networks mean self similar teletraffic models are required. Since self similarity had been investigated in other areas for some time the framework for such models is available. The best known self similar process is fractional Brownian motion (fBm) 34] [36] from which Norros developed a teletraffic model [53] We discuss both fBm and the Norros model in the next section. Another form of self similar model that we consider is the auto regressive integrated moving average (ARIMA) process [45] The third model that we consider is based on the marginal ....

B. B. Mandelbrot and J. R. Wallis, "Some long-run properties of geophysical records," Water Resources and Reservoirs, vol. 5, pp. 321--340, 1969.

....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, "Some long-run properties of geophysical records," Water Resources Research, vol. 5, pp. 321-40, 1969.

.... 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] biophysics [143, 267 269] and biology [339, 340] An area of application where self similarity and long range dependence continue to play 4 Willinger, Taqqu and Erramilli a significant role and where ....

....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, ....

B. B. Mandelbrot and J. R. Wallis. Some long-run properties of geophysical records. Water Resources Research, 5:321--340, 1969.

....0:78. The slope of the dotted line in Fig. 11 is fi = Gamma1:0, corresponding to an H value of 0:5. Another graphical method for estimating the Hurst parameter from a given empirical record is called the R=S analysis . This method has been used to model a wide variety of geophysical phenomena [MAND69b] and is based on the log10(m) 0 1 2 3 4 Figure 11: Variance time plot for VBR video trace. rescaled adjusted range statistics R=S, originally introduced by H. E. Hurst [HURS51] The heuristic behind the R=S statistic is to capture the fluctuations in a given time series in order to size, for ....

B. B. Mandelbrot and J. R . Wallis, "Some LongRun Properties of Geophysical Records", Water Resources Res., Vol. 5, pp. 321--40, 1969.

....73:50:Td, 74:40: k Keywords: noise, noise parameter estimation, noise generation 1 Introduction Long term correlations have been observed in many types of time series from physical, biological, physiological, economic, technological and sociological systems. Examples include geophysical data [1, 2, 3] such as rainfall, temperature measurements, sun spot numbers, earthquake frequencies, and river flows, frequency fluctuations in electrical oscillators [4] rate of traffic flow [1] voltage or current fluctuations in metal films and semiconductor devices [4] loudness fluctuations in speech and ....

....2 (f) 1) and the regression was performed on ln S(k) versus ln fi k . The slope of the regressed line is 1 Gamma ff. 2. 3 Method 3: Hurst exponent Perhaps the most famous quantifier of long range dependence is that introduced by Hurst [17] and expounded and modified by Mandelbrot and Wallis [2]. The method is nicely described in Bassingthwaighte et al. 23] Hurst was initially motivated by the problem of how large to make a Nile river dam in order to have acceptably low levels of fluctuations. He analyzed river flow data with an eye toward the functioning of dams. For each segment of ....

B.B. Mandelbrot and J.R. Wallis (1969) Some long-run properties of geophysical records, Water Resources Res. 5(2):321-340.

....g decays in proportion to m H ; instead of m 1=2 even for large m: The Central Limit Theorem for independent random variables leads to H = 1=2: Mandelbrot and Van Ness [21] has shown that H = 1=2; even for short memory processes. It was observed by Hurst [13] and by Mandelbrot and Wallis [22, 23] that many naturally occurring time series show H 1=2: Examples include annual water flows of the Nile and many other rivers, sun spot numbers, etc. One river with H equal to 1=2 is the Rhine. It is also known from Leland et al. s study [20] that Ethernet traffic traces show second order ....

Mandelbrot, B.B., and J.R. Wallis, "Some long-run properties of geophysical records" Water Resoures Reseach, 5; pp. 321-40, 1969.

....log R=S should be proportional to k 1 2 , for k large. The discovery of slopes proportional to k H , with H 1 2 , was in direct contradiction to the theory of such processes at the time. This discovery is known as the Hurst effect. Mandelbrot and co workers (Mandelbrot and van Ness 1968; Mandelbrot and Wallis 1969) showed that the Hurst effect can be modeled by fractional Gaussian noise with self similarity parameter 0 H 1 (H being for Hurst) More information about the history of long memory processes can be found in Beran (1994) Examples of such behavior can be found in a variety of disciplines, such ....

Mandelbrot, B. B. and J. R. Wallis (1969). Some long-run properties of geophysical records. Water Resources Research 5 (2), 321--340.

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B. B. Mandelbrot and J. Wallis. Some long-run properties of geophysical records. Water Resources Research, 5:321--340, 1969.

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B. B. Mandelbrot and J. R. Wallis, Some long-run properties of geophysical records, Water Resources Research 5, 422-437 (1969).

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B. B. Mandelbrot, J. R. Wallis, "Some Long-Run Properties of Geophysical Records", Water Resources Research 5, 321-340, 1969.

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B. Mandelbrot, J.R. Wallis, "Some Long-Run Properties of Geophysical Records," Water Resources Research 5, 228-267, 1969. References 37

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B. B. Mandelbrot and J. R. Wallis, "Some long-run properties of geophysical records," Water Resources Research, vol. 5, pp. 321-40, 1969.

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B. B. Mandelbrot, J. R. Wallis, "Some Long-Run Properties of Geophysical Records", Water Resources Research 5, 321-340, 1969.

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Mandelbrot, B.B., and J.R. Wallis, "Some long-run properties of geophysical records" Water Resources Research, 5; pp. 321-40, 1969.

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