In this paper, we develop a new multiscale modeling framework for characterizing positive-valued data with longrange-dependent 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 N-point data sets. We study both the second-order 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 (variance-time 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.

For describing the local structure of a random self-similar measure we use the multifractal decomposition of its support into sets of points of different local dimension. Under the strong open set condition we compute the Hausdorff dimensions of these sets and the generalized dimensions of the random self-similar measure. Furthermore, the tangential distribution of the random self--similar measure is investigated.

We generalize the definition of the fractional Brownian motion of exponent H to the case where H is no longer a constant, but a function of the time index of the process. This allows us to model non stationary continuous processes, and we show that H(t) and 2 \Gamma H(t) are indeed respectively the local Holder exponent and the local box and Hausdorff dimension at point t. Finally, we propose a simulation method and an estimation procedure for H(t) for our model.

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 self-similar 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 self-similar processes. Statistical properties of estimators as well as modelling issues are addressed.

This is an overview of a relatively recent application of long-range dependence (LRD) to the area of communication networks, in particular to problems concerned with the dynamic nature of packet flows in high-speed 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 wide-spread "black-box" 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 time-domain and wavelet-domain approach to analyzing the small time scale dynamics and discuss why the wavelet-domain approach appears to be better suited than the time-domain approach for identifying features in measured traffic (e.g., relatively regular traffic patterns over certain time scales) that have a direct networking interpretation (e....

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.

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 self-similar 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.

some basic properties

The interest for multifractal stochastic processes is mainly motivated by the need for accurate models in the study of the variability of wild signals. These locally irregular signals come from physical phenomena such as fully developed turbulence, TCP Internet traffic, variations of financial prices, or heart beats.

This article investigates the natural problem of performing the multifractal analysis of heterogeneous sums of Dirac masses ν = ∑ n≥0 wn δxn, where (xn)n≥0 is a sequence of points in [0, 1] d and (wn)n≥0 is a positive sequence of weights such that ∑ n≥0 wn < ∞. We consider the case where the points xn are roughly uniformly distributed in [0, 1] d, and the weights wn depend on a random self-similar measure µ, a parameter ρ ∈ (0, 1], and a sequence of positive radii (λn)n≥1 converging to 0 in the following way wn = λ d(1−ρ) n µ ( B(xn, λ ρ n) ) | log λn | −2. The measure ν has a rich multiscale structure. The computation of its multifractal spectrum is related to heterogeneous ubiquity properties of the system {(xn, λn)}n with respect to µ.