R. H. Riedi, "Multifractal processes," Technical Report, ECE Dept. Rice Univ., TR 99-06, submitted for publication 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....An easy consequence of this definition is that (q) is a concave function [22] An alternative approach to multifractal processes, also found in literature, is based on the study of the local erratic behaviour of the process by means of its local Holder exponents. For details on this approach see [29] and references therein. The most obvious examples of multifractals are self similar and multiplicative processes. 3 Approximation for queue tail probabilities We now state our main proposition: Proposition 1 The probabilities for the queue tail asymptotic of a single queueing model with ....

R. H. Riedi. Multifractal processes. In Doukhan, Oppenheim, and Taqqu, editors, Long range dependence : theory and applications, 2001.

....published, the complete understanding of this phenomenon and the application of the models in practice is far from being complete. There are different processes which are candidates for multifractal modeling. Multiplicative cascades were first used as a multifractal model for data traffic [15], 7] This class is the most well known member of the class of multifractal processes. The simplest case of this process is the binomial cascade which can be defined by a binary tree structure [5] 15] Combining this process with the aforementioned fBm we can define a new class called the ....

....modeling. Multiplicative cascades were first used as a multifractal model for data traffic [15] 7] This class is the most well known member of the class of multifractal processes. The simplest case of this process is the binomial cascade which can be defined by a binary tree structure [5] [15]. Combining this process with the aforementioned fBm we can define a new class called the fractional Brownian motions in multifractal time [20] This process has several nice properties, e.g. it is able to capture LRD and multifractal scaling independently. The selfsimilar # stable process [18] ....

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R. H. Riedi. "Multifractal processes," In Doukhan, Oppenheim, and Taqqu, editors, Long range dependence : theory and applications, 2001.

....processes are also called processes with scaling property. Note that an alternative approach to multifractal processes, also found in literature, is based on the study of the local erratic behaviour of the process by means of its local Holder exponents. For details on this approach see [27] and references therein. The most obvious examples of multifractals are self similar and multiplicative processes. The following Proposition provides our queue tail approximation for the presented queueing system: Proposition 1: The probabilities for the queue tail asymptotic of a single ....

R. H. Riedi, "Multifractal processes," in Theory and Applications of Long Range Dependence, P. Doukhan, G. Oppenheim, and M. S. Taqqu, Eds. Birkhauser, Boston, 2002.

.... Multifractal time processes Random processes on IR where time has been distorted using a multifractal measure to de ne a multifractal time have attracted recent attention, particular in modelling share prices and exchange rates, where changes might be supposed to occur at widly varying rates [20]. Let be a locally nite Borel measure on IR with dense support and no atoms, and such that for some ; c 0 we have (B(z; r) cr for all z 2 IR N and r 1. Typically might have the form of a self similar multifractal measure on each unit interval, see [8] We de ne a multifractal ....

Riedi, R.H., Multifractal Processes, in Long range dependence : theory and applications, eds. Doukhan, Oppenheim and Taqqu, 2000. 16

.... non parametric estimates of the covariance function (Istas and Laredo [44] and on the asymptotic behaviour of wavelet coefficients of X (Istas, 42] There are other definitions of Holder indices more suitable to the study of path properties of non Gaussian processes such as multifractals (cf. [72]) Data network applications are discussed in [26] 29] and [73] A process X ff is an ff stable L evy motion if it has stationary independent increments which follow a non normal stable distribution with index ff, 0 ff 2. Clearly Gamma1=ff X ff ( Delta) also has stationary increments, ....

....dependence parameters and Holder exponents. Better understanding of the estimation of Holder exponents would be useful INFINITE SOURCE 37 as well as a clearer understanding of the relationship between treatments using the second order definition (2. 2) and the pathwise treatments in, for example, [72]. Global statistical properties such as heavy tails and long range dependence were amply in evidence in our data as expected and as predicted by the model. Transfer times, file sizes and transfer rates were consistently heavy tailed, usually with 1 ff 2. See Table 2. Traffic rates frequently ....

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R.H. Riedi. Multifractal Processes. Technical report, ECE Dept., Rice University, TR 99-06, 1999.

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R. H. Riedi, "Multifractal processes," Technical Report, ECE Dept. Rice Univ., TR 99-06, submitted for publication 1999.

No context found.

R. H. Riedi, "Multifractal processes," Technical Report, ECE Dept. Rice Univ., TR 99-06, submitted for publication 1999.

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R. H. Riedi, "Multifractal processes," Stoch. Proc. Appl., preprint, to be submitted 1999.

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R. H. Riedi, "Multifractal processes," IEEE Info. Theory, submitted 1999.

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R. H. Riedi, "Multifractal processes," Technical Report, ECE Dept. Rice Univ., TR 99-06. "Long range dependence: Theory and applications," eds. Doukhan, Oppenheim and Taqqu (2000). Available at www.dsp.rice.edu.

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R. H. Riedi, "Multifractal processes," preprint, 1999.

.... In a nutshell, the multifractal formalism relates N j ( the frequency of bursts of given strength, to the decay rate T (q) of the moments of Y across dyadic scales: IE k j =0 Y ( k j 1)2 ) Y (k j 2 5 2 jT (q) 17) The formalism states that under suitable conditions [41] N j ( 2 jT ( 18) where T is the Legendre transform of T , i.e. inf (q T (q) 19) The Legendre transform T ( gives the smallest distance between the straight line q and the function T (q) If T ( is negative for some , then (18) means that there is no ....

....for some , then (18) means that there is no exponent (t) in the signal. Since the time instances with equal (t) form highly interwoven fractal sets, T ( is called the multifractal spectrum. For a more rigorous, in depth presentation as well as a set of relevant references, see [26, 41]. In practice, traffic traces possess a minimum time resolution and hence a finite range of dyadic time scales. In order to estimate T (q) from such data sets, it is customary to extract the (b) 0.8 0.9 1 1.1 1.2 1.3 0 0.2 0.4 0.6 0.8 1 Holder exponent (a) Multifractal Spectrum WIG ....

[Article contains additional citation context not shown here]

R. H. Riedi, "Multifractal processes," Technical Report, ECE Dept. Rice Univ., TR 99-06. "Long range dependence: Theory and applications," eds. Doukhan, Oppenheim and Taqqu (2000). Available at www.dsp.rice.edu.

.... In a nutshell, the multifractal formalism relates N j ( the frequency of bursts of given strength, to the decay rate T (q) of the moments of Y across dyadic scales: IE k j =0 Y ( k j 1)2 ) Y (k j 2 5 2 jT (q) 17) The formalism states that under suitable conditions [41] N j ( 2 jT ( 18) where T is the Legendre transform of T , i.e. inf (q T (q) 19) The Legendre transform T ( gives the smallest distance between the straight line q and the function T (q) If T ( is negative for some , then (18) means that there is no ....

....for some , then (18) means that there is no exponent (t) in the signal. Since the time instances with equal (t) form highly interwoven fractal sets, T ( is called the multifractal spectrum. For a more rigorous, in depth presentation as well as a set of relevant references, see [26, 41]. In practice, traffic traces possess a minimum time resolution and hence a finite range of dyadic time scales. In order to estimate T (q) from such data sets, it is customary to extract the scaling laws (17) from log log plots. For a tree based model, this reads as T (q) log 2 IE ....

[Article contains additional citation context not shown here]

R. H. Riedi, "Multifractal processes," Technical Report, ECE Dept. Rice Univ., TR 99-06. "Long range dependence: Theory and applications," eds. Doukhan, Oppenheim and Taqqu (2000). Available at www.dsp.rice.edu.

....in general the path of X might not have n derivatives. In the case where the local polynomial P s is constant then h(s) is the largest h such that jX(s ) X(s)j K jj : 12) holds. Note that h(s) may very well be larger than 1, as is the case with all cascades. A simple argument yields [31] the more useful dual statement: if the largest h satisfying (12) is non integer, then the local polynomial P s is necessarily constant and h(s) can be computed using (12) Fig 5 demonstrates the simple scaling structure of fractional Brownian motion; for almost every path and at any time ....

....for almost all paths. In particular, for fBm it consists of only one point: D(H) 1, while it has an inverted U shape for multiplicative cascades. While estimating D from traces is very hard, there exist almost sure upper bounds which are easier to estimate (see Box 11) For an overview see [31]. Here the M k i are independent positive random variables called the multipliers such that siblings add up to one: 2k M 2k 1 = 1. Thus, 11) repartitions the increments of X iteratively. Setting X(0) 0 and X(1) 1) for convenience) de nes the process on [0; 1] This is ....

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R. Riedi, \Multifractal Processes", in: \Long range dependence : theory and applications", eds. Doukhan, Oppenheim and Taqqu, to appear 2001.

.... Gammanqff 2 Gamman inf ff (qff Gammaf (ff) 39) We conclude that we must expect T (q) to equal inf ff (qff Gamma f(ff) the so called Legendre transform of f(ff) For the special case of an MWM process, i.e. Y = D (see Section 2. 2 for the definition of D) it can be shown (see [45]) that the inverse relation holds, called the multifractal formalism f(ff) T (ff) inf (qff Gamma T (q) 40) In order to estimate T (q) from a data set, it is customary to use the approximation GammanT (q) S n (q) For the MWM this is equivalent to Gammaj T (q) k=0 j2 ....

R. H. Riedi, "Multifractal processes," Technical Report, ECE Dept. Rice Univ., TR 99-06, submitted for publication 1999.

.... and using (27) we get Sn (q) ff nff (2 Gammanff ff 2 nfG (ff) Gammanqff 2 Gamman inf ff (qff Gammaf G (ff) 29) We conclude that we must expect T (q) to equal inf ff (qff Gamma fG (ff) For the special case of an MWM process, i.e. Y = D, it can be shown (see [42]) that the dual relation holds. This relation is called the multifractal formalism and reads f(ff) T (ff) inf (qff Gamma T (q) 30) Simple calculus shows that T (ff) qff Gamma T (q) at ff = T (q) provided T 00 (q) 0. This relation via the Legendre transform is typical ....

R. H. Riedi, "Multifractal processes," IEEE Info. Theory, submitted 1999.

....is finite, we have: If N H g (t 0 ) then N , i.e. there is a constant C such that jW (t 0 ; s)j C jsj : 19) We note that wavelet coefficients may decay significantly faster than the order of the wavelet. For examples we refer the interested reader to the study of cascades [27]. Let us assume t 0 = 0 for simplicity. To establish the first statement we proceed indirectly and assume that N . Then, 15) holds for any r N and any r r. Letting r become arbitrarily close to N proves that H N and we are done. Note that lemma 7 makes the dual statement: if ....

....if its multifractal partition function is non linear. It is known from multifractal theory that an honestly concave (non linear) partition function is indicative of the presence of a rich range of different degrees of Holder regularity in the signal which occur in a highly erratic way (see, e.g. [27]) To the contrary, a (mono )fractal process which spots a smoothly varying degree of regularity, An earlier version of (q) relies on the q th order moments of the increment process Ys (t) X(t st 0 ) X(t) instead of the wavelet coefficients W [X] t; s) 30 possesses a piecewise linear ....

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R. H. Riedi. Multifractal processes. in: "Long range dependence : theory and applications", eds. Doukhan, Oppenheim and Taqqu, to appear 2002.

....i.e. the behavior of the moments IE[A (t) as t 0. Indeed, while Kolmogorov s well known criterion for continuity [22] permits to deduce the global degree of Holder continuity, a much more refined assessment of pathwise variability is provided through the multifractal formalism (see [21] for a large body of references) Scale invariance and related phenomena have received considerable attention in the past from the point of view of both, analysis and estimation as well as modelling and synthesis. Various kinds of scaling form an indisputable component of empirical data observed ....

....phenomena call for both, appropriate tools of analysis with known accuracy as well as novel models with controllable parameters leading to deeper understanding and allowing for physical interpretations. However, the development of advanced tools of multifractal analysis, notably using wavelets [1, 19, 11, 21], has only partly been balanced by equally satisfactory models. Missing are processes with rich scaling properties, yet appealing statistical properties such as stationary increments. The disadvantages of Binomial cascades, e.g. are only too obvious. The increment process is not even second order ....

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R. H. Riedi. Multifractal processes. in: "Long range dependence : theory and applications", eds. Doukhan, Oppenheim and Taqqu, to appear 2002.

.... (27) we get Sn (q) P ff P ff nff (2 Gammanff ) q P ff 2 nfG (ff) 2 Gammanqff 2 Gamman inf ff (qff Gammaf G (ff) 29) We conclude that we must expect T (q) to equal inf ff (qff Gammaf G (ff) For the special case of an MWM process, i.e. Y = D, it can be shown (see [42]) that the dual relation holds. This relation is called the multifractal formalism and reads f(ff) T (ff) inf q (qff Gamma T (q) 30) Simple calculus shows that T (ff) qff Gamma T (q) at ff = T 0 (q) provided T 00 (q) 0. This relation via the Legendre transform T is typical ....

R. H. Riedi, "Multifractal processes," IEEE Info. Theory, submitted 1999.

....[30] we will discuss later in this section some de nitive advantages that the former seems to have over the latter when analyzing measured TCP IP traces. Multifractal analysis (MFA) What follows is an intuitive way of understanding multifractals. For a mathematical account of the subject, see [74]. The aim of multifractal analysis (MFA) is to provide information about the singularity exponents in a given signal (as de ned in equations (4.2) and (4.4) resp. and to come up with compact geometrical or statistical descriptions of the signal s overall singularity. Conceptually, the time ....

....for FBM with = H; but we also have f( 1 if only a certain constant fraction of n s equals , as is the case with the concatenation of FBMs described earlier [76] Only if certain values of n are considerably more spurious than others will we observe f( 1. In fact, it can be shown [77, 74] that the rate function f( relates to the Hausdor dimension dim(K ) and that we have dim(K ) f( 4.9) It is in this sense that f provides information on the occurrence of the various fractal exponents and has been termed multifractal spectrum. Also, 28 Walter Willinger, Vern ....

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R. H. Riedi. Multifractal processes. In P. Doukhan, G. Oppenheim, and M. S. Taqqu, editors, Long-range Dependence: Theory and Applications. Birkhauser, 2001. Appears in this volume.

.... well suited to modeling non homogeneous phenomena [17, 18] More recently, the multifractal nature of network traffic has been demonstrated convincingly, first in [19] and subsequently in [20, 21] The beauty of the multifractal formalism has motivated considerable research effort in mathematics [22 32]; however, few multifractal data models have been developed to date. In the most simple terms, multifractals possess a local smoothness H t that depends on t in an erratic way. Equivalently, multifractals have moments that scale non linearly. By matching the multifractal properties of training ....

.... from the accumulated theoretical and practical knowledge of the field of multifractals, including a precise understanding of the convergence of the algorithm, properties of the marginal distributions, advantages over monofractal fGn models, and a range of possible refinements and extensions [15,16,22 32,49 57]. The theory of cascades comes with a dedicated set of tools for analysis, both theoretical and numerical, that we will outline in the next two sections (see Appendices A and B for more details) At this point, our discussion will become decidedly more technical, mainly because we wish to extend ....

[Article contains additional citation context not shown here]

R. H. Riedi, "Multifractal processes," Stoch. Proc. Appl., preprint, to be submitted 1999.

.... well suited to modeling non homogeneous phenomena [17, 18] More recently, the multifractal nature of network traffic has been demonstrated convincingly, first in [19] and subsequently in [20, 21] The beauty of the multifractal formalism has motivated considerable research effort in mathematics [15,22 32]; however, few multifractal data models have been developed to date. In the most simple terms, multifractals possess a local smoothness H t that depends on t in an erratic way. Equivalently, multifractals have moments that scale non linearly. By matching the multifractal properties of training ....

....Analysis of the MWM So far we have noted two attractive properties of cascades: their increment processes are spiky and have nonGaussian marginals. Surprisingly, these two properties are strongly related, and much effort has been expended connecting them rigorously under various assumptions [23 32]. The scaling of moments, which is captured with the simple and efficient partition function T (q) acts as the bridge. This function can be viewed as a concise way of describing various features of cascades and of processes in general. After introducing the various multifractal spectra f(ff) ....

[Article contains additional citation context not shown here]

R. H. Riedi, "Multifractal processes," Stoch. Proc. Appl., preprint 1999.

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R. H. Riedi. Multifractal processes. In P. Doukhan, G. Oppenheim, and M. S. Taqqu, editors, Theory And Applications Of Long-Range Dependence, pages 625--716. Birkhauser, 2003.

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R. H. Riedi. Multifractal processes. 1999. preprint.

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