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42
A multifractal wavelet model with application to TCP network traffic
 IEEE TRANS. INFORM. THEORY
, 1999
"... In this paper, we develop a new multiscale modeling framework for characterizing positivevalued data with longrangedependent correlations (1=f noise). Using the Haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results, the mo ..."
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Cited by 171 (30 self)
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In this paper, we develop a new multiscale modeling framework for characterizing positivevalued data with longrangedependent correlations (1=f noise). Using the Haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results, the model provides a rapid O(N) cascade algorithm for synthesizing Npoint data sets. We study both the secondorder and multifractal properties of the model, the latter after a tutorial overview of multifractal analysis. We derive a scheme for matching the model to real data observations and, to demonstrate its effectiveness, apply the model to network traffic synthesis. The flexibility and accuracy of the model and fitting procedure result in a close fit to the real data statistics (variancetime plots and moment scaling) and queuing behavior. Although for illustrative purposes we focus on applications in network traffic modeling, the multifractal wavelet model could be useful in a number of other areas involving positive data, including image processing, finance, and geophysics.
LogInfinitely Divisible Multifractal Processes
, 2002
"... We define a large class of multifractal random measures and processes with arbitrary loginfinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined lognormal Multifractal Random Walk processes (MRW) [33, 3] and the logPoisso ..."
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Cited by 37 (5 self)
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We define a large class of multifractal random measures and processes with arbitrary loginfinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined lognormal Multifractal Random Walk processes (MRW) [33, 3] and the logPoisson “product of cynlindrical pulses” [7]. Their construction involves some “continuous stochastic multiplication” [36] from coarse to fine scales. They are obtained as limit processes when the finest scale goes to zero. We prove the existence of these limits and we study their main statistical properties including non degeneracy, convergence of the moments and multifractal scaling.
The multifractal nature of Lévy processes
"... . We show that the sample paths of most L'evy processes are multifractal functions and we determine their spectrum of singularities. Key Words. L'evy processes, multifractals, Holder singularities, Hausdorff dimensions, spectrum of singularities. AMS Classification. 28A80, 60G17, 60G30, 60J30, A L ..."
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Cited by 30 (3 self)
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. We show that the sample paths of most L'evy processes are multifractal functions and we determine their spectrum of singularities. Key Words. L'evy processes, multifractals, Holder singularities, Hausdorff dimensions, spectrum of singularities. AMS Classification. 28A80, 60G17, 60G30, 60J30, A L'evy process X t (t 0) valued in IR d is, by definition, a stochastic process with stationary independent increments: X t+s \Gamma X t is independent of the (X v ) 0vt and has the same law as X s . Brownian motion and Poisson processes are examples of L'evy processes that can be qualified as monofractal; for instance the Holder exponent of the Brownian motion is everywhere 1=2 (the variations of its regularity are only of a logarithmic order of magnitude). These two examples are not typical: we will see that the other L'evy processes are multifractal provided that their L'evy measure is neither too small nor too large near zero. Furthermore their spectrum of singularities depends precise...
Multifractal Processes
, 1999
"... This paper has two main objectives. First, it develops the multifractal formalism in a context suitable for both, measures and functions, deterministic as well as random, thereby emphasizing an intuitive approach. Second, it carefully discusses several examples, such as the binomial cascades and sel ..."
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Cited by 28 (6 self)
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This paper has two main objectives. First, it develops the multifractal formalism in a context suitable for both, measures and functions, deterministic as well as random, thereby emphasizing an intuitive approach. Second, it carefully discusses several examples, such as the binomial cascades and selfsimilar processes with a special eye on the use of wavelets. Particular attention is given to a novel class of multifractal processes which combine the attractive features of cascades and selfsimilar processes. Statistical properties of estimators as well as modelling issues are addressed.
Multifractality in Asset Returns: Theory and Evidence
 REVIEW OF ECONOMICS AND STATISTICS
, 2001
"... This paper investigates the Multifractal Model of Asset Returns, a class of continuoustime processes that incorporate the thick tails and volatility persistence exhibited by many financial time series. The simplest version of the model compounds a Brownian Motion with a multifractal timedeformatio ..."
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Cited by 27 (4 self)
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This paper investigates the Multifractal Model of Asset Returns, a class of continuoustime processes that incorporate the thick tails and volatility persistence exhibited by many financial time series. The simplest version of the model compounds a Brownian Motion with a multifractal timedeformation process. Prices follow a semimartingale, which precludes arbitrage in a standard twoasset economy. Volatility has long memory, and the highest finite moments of returns can take any value greater than two. The local variability of the process is highly heterogeneous, and is usefully characterized by the local Hölder exponent at every instant. In contrast with earlier processes, this exponent takes a continuum of values in any time interval. The model also predicts that the moments of returns vary as a power law of the time horizon. We confirm this property for Deutsche Mark/U.S. Dollar exchange rates and several equity series. We then develop an estimator, and infer a parsimo...
LongRange Dependence and Data Network Traffic
, 2001
"... This is an overview of a relatively recent application of longrange dependence (LRD) to the area of communication networks, in particular to problems concerned with the dynamic nature of packet flows in highspeed data networks such as the Internet. We demonstrate that this new application area off ..."
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Cited by 23 (1 self)
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This is an overview of a relatively recent application of longrange dependence (LRD) to the area of communication networks, in particular to problems concerned with the dynamic nature of packet flows in highspeed data networks such as the Internet. We demonstrate that this new application area offers unique opportunities for significantly advancing our understanding of LRD and related phenomena. These advances are made possible by moving beyond the conventional approaches associated with the widespread "blackbox" perspective of traditional time series analysis and exploiting instead the physical mechanisms that exist in the networking context and that are intimately tied to the observed characteristics of measured network traffic. In order to describe this complexity we provide a basic understanding of the design, architecture and operations of data networks, including a description of the TCP/IP protocols used in today's Internet. LRD is observed in the large scale behavior of the data traffic and we provide a physical explanation for its presence. LRD tends to be caused by user and application characteristics and has little to do with the network itself. The network affects mostly small time scales, and this is why a rudimentary understanding of the main protocols is important. We illustrate why multifractals may be relevant for describing some aspects of the highly irregular traffic behavior over small time scales. We distinguish between a timedomain and waveletdomain approach to analyzing the small time scale dynamics and discuss why the waveletdomain approach appears to be better suited than the timedomain approach for identifying features in measured traffic (e.g., relatively regular traffic patterns over certain time scales) that have a direct networking interpretation (e....
A Multifractal Model of Asset Returns
, 1997
"... This paper presents the multifractal model of asset returns (“MMAR”), based upon ..."
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Cited by 23 (2 self)
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This paper presents the multifractal model of asset returns (“MMAR”), based upon
Multidimensional infinitely divisible cascades. application to the modelling of intermittency in turbulence
 European Physical J. B
, 2005
"... Abstract—We propose to model the statistics of natural images, thanks to the large class of stochastic processes called Infinitely Divisible Cascades (IDCs). IDCs were first introduced in one dimension to provide multifractal time series to model the socalled intermittency phenomenon in hydrodynami ..."
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Cited by 21 (4 self)
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Abstract—We propose to model the statistics of natural images, thanks to the large class of stochastic processes called Infinitely Divisible Cascades (IDCs). IDCs were first introduced in one dimension to provide multifractal time series to model the socalled intermittency phenomenon in hydrodynamical turbulence. We have extended the definition of scalar IDCs from one to N dimensions and commented on the relevance of such a model in fully developed turbulence in [1]. In this paper, we focus on the particular 2D case. IDCs appear as good candidates to model the statistics of natural images. They share most of their usual properties and appear to be consistent with several independent theoretical and experimental approaches of the literature. We point out the interest of IDCs for applications to procedural texture synthesis. Index Terms—Stochastic processes, picture/image generation, fractals, image processing and computer vision, statistical, image models. 1
Combining multifractal additive and multiplicative chaos
 COMMUN. MATH. PHYS
, 2005
"... In this work, we study the new class of multifractal measures, which combines additive and multiplicative chaos, defined by νγ,σ = X b j≥1 −jγ j2 X 0 ≤ k ≤bj µ([kb −1 −j, (k + 1)b −j)) σ δkb−j (γ ≥ 0, σ ≥ 1), where µ is any positive Borel measure on [0, 1] and b is an integer ≥ 2. The singularities ..."
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Cited by 20 (12 self)
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In this work, we study the new class of multifractal measures, which combines additive and multiplicative chaos, defined by νγ,σ = X b j≥1 −jγ j2 X 0 ≤ k ≤bj µ([kb −1 −j, (k + 1)b −j)) σ δkb−j (γ ≥ 0, σ ≥ 1), where µ is any positive Borel measure on [0, 1] and b is an integer ≥ 2. The singularities analysis of the measures νγ,σ involves new results on the mass distribution of µ when µ describes large classes of multifractal measures. These results generalize ubiquity theorems associated with the Lebesgue measure. Under suitable assumptions on µ, the multifractal spectrum of νγ,σ is linear on [0, hγ,σ] for some critical value hγ,σ, and then it is strictly concave on the right of hγ,σ, and deduced from the one of µ by an affine transformation. This untypical shape is the result of the combination between Dirac masses and atomless multifractal measures. These measures satisfy multifractal formalisms. These measures open interesting perspectives in modeling discontinuous phenomena.
From multifractal measures to multifractal wavelet series
 THE JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS
, 2004
"... Given a positive locally finite Borel measure µ on R, a natural way to construct multifractal wavelet series Fµ(x) = ∑ j≥0,k∈Z dj,kψj,k(x) is to set dj,k  =2 −j(s0−1/p0) −j −j 1/p0 µ([k2, (k + 1)2)) , where s0,p0 ≥ 0, s0 − 1/p0> 0. Indeed, under suitable conditions, it is shown that the function ..."
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Cited by 12 (7 self)
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Given a positive locally finite Borel measure µ on R, a natural way to construct multifractal wavelet series Fµ(x) = ∑ j≥0,k∈Z dj,kψj,k(x) is to set dj,k  =2 −j(s0−1/p0) −j −j 1/p0 µ([k2, (k + 1)2)) , where s0,p0 ≥ 0, s0 − 1/p0> 0. Indeed, under suitable conditions, it is shown that the function Fµ inherits the multifractal properties of µ. The transposition of multifractal properties works with many classes of statistically selfsimilar multifractal measures, enlarging the class of processes which have selfsimilarity properties and controlled multifractal behaviors. Several perturbations of the wavelet coefficients and their impact on the multifractal nature of Fµ are studied. As an application, multifractal Gaussian processes associated with Fµ are created. We obtain results for the multifractal spectrum of the socalled Wcascades introduced by Arnéodo et al.