Results 1  10
of
26
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.
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.
Exceptions to the Multifractal Formalism for Discontinuous Measures
, 1997
"... In an earlier paper [MR] the authors introduced the inverse measure y (dt) of a given measure (dt) on [0; 1] and presented the `inversion formula' f y (ff) = fff(1=ff) which was argued to link the respective multifractal spectra of and y . A second paper [RM2] established the formula under ..."
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Cited by 18 (9 self)
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In an earlier paper [MR] the authors introduced the inverse measure y (dt) of a given measure (dt) on [0; 1] and presented the `inversion formula' f y (ff) = fff(1=ff) which was argued to link the respective multifractal spectra of and y . A second paper [RM2] established the formula under the assumption that and y are continuous measures. Here, we investigate the general case which reveals telling details of interest to the full understanding of multifractals. Subjecting selfsimilar measures to the operation 7! y creates a new class of discontinuous multifractals. Calculating explicitly we find that the inversion formula holds only for the `fine multifractal spectra' and not for the `coarse' ones. As a consequence, the multifractal formalism fails for this class of measures. A natural explanation is found when drawing parallels to equilibrium measures. In the context of our work it becomes natural to consider the degenerate Holder exponents 0 and 1.
Iterated function systems with overlaps and selfsimilar measures
 J. London Math. Soc
, 2001
"... The paper considers the iterated function systems of similitudes which satisfy a separation condition weaker than the open set condition, in that it allows overlaps in the iteration. Such systems include the wellknown Bernoulli convolutions associated with the PV numbers, and the contractive simili ..."
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Cited by 12 (8 self)
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The paper considers the iterated function systems of similitudes which satisfy a separation condition weaker than the open set condition, in that it allows overlaps in the iteration. Such systems include the wellknown Bernoulli convolutions associated with the PV numbers, and the contractive similitudes associated with integral matrices. The latter appears frequently in wavelet analysis and the theory of tilings. One of the basic questions is studied: the absolute continuity and singularity of the selfsimilar measures generated by such systems. Various conditions to determine the dichotomy are given. 1.
Inverse Measures, the Inversion Formula and Discontinuous Multifractals
"... The present paper is part I of a series of three closely related papers in which the inverse measure ..."
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Cited by 9 (6 self)
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The present paper is part I of a series of three closely related papers in which the inverse measure
Inversion Formula for Continuous Multifractals
, 1997
"... In a previous paper [MR] the authors introduced the inverse measure y of a probability measure on [0; 1]. It was argued that the respective multifractal spectra are linked by the `inversion formula' f y (ff) = fff(1=ff). Here, the statements of [MR] are put in more mathematical terms and proof ..."
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Cited by 9 (5 self)
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In a previous paper [MR] the authors introduced the inverse measure y of a probability measure on [0; 1]. It was argued that the respective multifractal spectra are linked by the `inversion formula' f y (ff) = fff(1=ff). Here, the statements of [MR] are put in more mathematical terms and proofs are given for the inversion formula in the case of continuous measures. Thereby, f may stand for the Hausdorff spectrum, the packing spectrum, or the coarse grained spectrum. With a closer look at the special case of selfsimilar measures we offer a motivation of the inversion formula as well as a discussion of possible generalizations. Doing so we find a natural extension of the scope of the notion `selfsimilar' and a failure of the usual multifractal formalism.
The pressure function for products of nonnegative matrices
 Math. Research Letter
"... Abstract. Let (ΣA,σ) be a subshift of finite type and let M(x) be a continuous function on ΣA takingvalues in the set of nonnegative matrices. We extend the classical scalar pressure function to this new settingand prove the existence of the Gibbs measure and the differentiability of the pressure f ..."
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Cited by 7 (4 self)
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Abstract. Let (ΣA,σ) be a subshift of finite type and let M(x) be a continuous function on ΣA takingvalues in the set of nonnegative matrices. We extend the classical scalar pressure function to this new settingand prove the existence of the Gibbs measure and the differentiability of the pressure function. We are especially interested on the case where M(x) takes finite values M1, ·· ·,Mm. The pressure function reduces to P (q): = limn→ ∞ 1 log n J∈Σ ‖MJ ‖ A,n q. The expression is important when we consider the multifractal formalism for certain iterated function systems with overlaps. 1.
Hausdorff Dimension Of SelfSimilar Sets With Overlaps
 J. LONDON MATH. SOC
, 2000
"... We introduce the notion of "finite type" iterated function systems of contractive similitudes, and describe a scheme for computing the exact Hausdorff dimension of their attractors in the absence of the open set condition. This method extends a previous one by Lalley and applies not only to the clas ..."
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Cited by 6 (2 self)
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We introduce the notion of "finite type" iterated function systems of contractive similitudes, and describe a scheme for computing the exact Hausdorff dimension of their attractors in the absence of the open set condition. This method extends a previous one by Lalley and applies not only to the classes of selfsimilar sets studied by Edgar, Lalley, Rao and Wen, and others, but also to some new classes that are not covered by the previous ones.
Multifractal analysis of complex random cascades
 COMMUN MATH. PHYS
, 2009
"... We achieve the multifractal analysis of a class of complex valued statistically selfsimilar continuous functions. For we use multifractal formalisms associated with pointwise oscillation exponents of all orders. Our study exhibits new phenomena in multifractal analysis of continuous functions. In ..."
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Cited by 5 (2 self)
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We achieve the multifractal analysis of a class of complex valued statistically selfsimilar continuous functions. For we use multifractal formalisms associated with pointwise oscillation exponents of all orders. Our study exhibits new phenomena in multifractal analysis of continuous functions. In particular, we find examples of statistically selfsimilar such functions obeying the multifractal formalism and for which the support of the singularity spectrum is the whole interval [0, ∞].
Secondorder Selfsimilar Identities and Multifractal Decompositions
"... ABSTRACT. Motivated by the study of convolutions of the Cantor measure, we set up a framework for computing the multifractal Lqspectrum τ(q), q>0, for certain overlapping selfsimilar measures which satisfy a family of secondorder identities introduced by Strichartz et al. We apply our results to t ..."
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Cited by 4 (2 self)
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ABSTRACT. Motivated by the study of convolutions of the Cantor measure, we set up a framework for computing the multifractal Lqspectrum τ(q), q>0, for certain overlapping selfsimilar measures which satisfy a family of secondorder identities introduced by Strichartz et al. We apply our results to the family of iterated function systems Sjx = (1/m)x + [(m − 1)m]/j, j = 0, 1,..., m, where m is an odd integer, and obtain closed formulas defining τ(q), q>0, for the associated selfsimilar measures. As a result, we can show that τ(q) is differentiable on (0, ∞) and justify the multifractal formalism in the region q>0. Furthermore, expressions for the Hausdorff and entropy dimensions of these measures can also be derived. By letting m = 3, we obtain all these results for the 3fold convolution of the standard Cantor measure. 1.