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.

Fractal behavior and long-range dependence have been observed in an astonishing number of physical, biological, geological, and socio-economic systems. Time series, profiles, and surfaces have been characterized by their fractal dimension, a measure of roughness, and by the Hurst coefficient, a measure of long-memory dependence. Either phenomenon has been modeled and explained by self-affine random functions, such as fractional Gaussian noise and fractional Brownian motion. The assumption of statistical self-affinity implies a linear relationship between fractal dimension and Hurst coe#cient and thereby links the two phenomena. This article introduces stochastic models that allow for any combination of fractal dimension and Hurst coefficient. Associated software for the synthesis of images with arbitrary, pre-specified fractal properties and power-law correlations is available. The new models suggest a test for self-affinity that assesses coupling and decoupling of local and global behavior.

This thesis describes describes a small number of problems arising from the applied study of networks in various contexts. The work can be split into two main areas: telecommunications networks (particularly the Internet) and road networks.

The Hurst parameter H characterizes the degree of long-range dependence (and asymptotic self-similarity) in stationary time series. Many methods have been developed for the estimation of H from data. In practice, however, the classical estimation techniques can be severely affected by non-stationary artifacts in the time series. In fact, the assumption that the data can be modeled by a stationary process with a single Hurst exponent H may be unrealistic. We focus on practical issues associated with the detection of long-range dependence in Internet traffic data and develop two tools designed to address some of these issues. The first is an animation tool which is used to visualize the local dependence structure. The second is a statistical tool for the local analysis of self-similarity (LASS). The LASS tool is designed to handle time series that have long-range dependence and are long enough that some parts are essentially stationary, while others exhibit non-stationarity, which are either deterministic or stochastic in nature. The tool uses wavelets to analyze the local dependence structure in the data over a set of windows. It can be used to visualize local deviations from self-similar, long-range dependence scaling and to provide reliable local estimates of the Hurst exponents. The tool, which is illustrated by using a trace of Internet traffic measurements, can also be applied to economic time series. We also develop a median-based wavelet spectrum which can be used to obtain robust local or global estimates of the the Hurst parameter that are less susceptible to local nonstationarity. We make the software tools freely available and describe their use in an appendix.

DMS-0112069. Any opinions, findings, and conclusions or recommendations expressed in this material are

In this paper we propose a novel methodology to generate realistic traffic traces to be used for performance evaluation of switches. Indeed, real Internet traffic shows long and short range dependency characteristics, difficult to be captured by flexible, yet simple, synthetic models. One option is to use real traffic traces, which however are difficult to obtain, as requires to capture traffic in different places with synchronization and management problems. We therefore present a methodology to generate several synthetic traffic traces from a single real trace of packets, by carefully grouping packets belonging to the same flow to guarantee to keep the same statistical properties of the original trace. After formalizing the problem, we solve it and apply the results to assess the performance of scheduling algorithms in high performance switches, comparing the results to other simpler traffic models traditionally adopted in the switching community. Our results show that realistic traffic degrades the performance of the switch by more than one order of magnitude with respect to the traditional traffic models.

Abstract—We introduce an extended family of continuous-domain stochastic models for sparse, piecewise-smooth signals. These are specified as solutions of stochastic differential equations, or, equivalently, in terms of a suitable innovation model; the latter is analogous conceptually to the classical interpretation of a Gaussian stationary process as filtered white noise. The two specific features of our approach are 1) signal generation is driven by a random stream of Dirac impulses (Poisson noise) instead of Gaussian white noise, and 2) the class of admissible whitening operators is considerably larger than what is allowed in the conventional theory of stationary processes. We provide a complete characterization of these finite-rate-of-innovation signals within Gelfand’s framework of generalized stochastic processes. We then focus on the class of scale-invariant whitening operators which correspond to unstable systems. We show that these can be solved by introducing proper boundary conditions, which leads to the specification of random, spline-type signals that are piecewise-smooth. These processes are the Poisson counterpart of fractional Brownian motion; they are nonstationary and have the same-type spectral signature. We prove that the generalized Poisson processes have a sparse representation in a wavelet-like basis subject to some mild matching condition. We also present a limit example of sparse process that yields a MAP signal estimator that is equivalent to the popular TV-denoising algorithm. Index Terms—Fractals, innovation models, Poisson processes, sparsity, splines, stochastic differential equations, stochastic processes,

Abstract: In this paper the whole family of fractional Brownian motions is constructed as a single Gaussian field indexed by time and the Hurst index simultaneously. The field has a simple covariance structure and it is related to two generalizations of fractional Brownian motion known as multifractional Brownian motions. A mistake common to the existing literature regarding multifractional Brownian motions is pointed out and corrected. The Gaussian field, due to inherited “duality”, reveals a new way of constructing martingales associated with the odd and even part of a fractional Brownian motion and therefore of the fractional Brownian motion. The existence of those martingales and their stochastic representations is the first step to the study of natural wavelet expansions associated to those processes in the spirit of our earlier work on a construction of natural wavelets associated to Gaussian-Markov processes. 1.

The central theme of this pair of papers (Parts I and II in this issue) is self-similarity, which is used as a bridge for connecting splines and fractals. The first part of the investigation is deterministic, and the context is that of L-splines; these are defined in the following terms: @ A is a cardinal L-spline iff v @ A a ‘ “ @ A, where L is a suitable pseudodifferential operator. Our starting point for the construction of “self-similar” splines is the identification of the class of differential operators L that are both translation and scale invariant. This results into a two-parameter family of generalized fractional derivatives,, where is the order of the derivative and is an additional phase factor. We specify the corresponding L-splines, which yield an extended class of fractional splines. The operator is used to define a scale-invariant energy measure—the squared P-norm of the th derivative of the signal—which provides a regularization functional for interpolating or fitting the noisy samples of a signal. We prove that the corresponding variational (or smoothing) spline estimator is a cardinal fractional spline of order P, which admits a stable representation in a B-spline basis. We characterize the equivalent frequency response of the estimator and show that it closely matches that of a classical Butterworth filter of order P. We also establish a formal link between the regularization parameter and the cutoff frequency of the smoothing spline filter: H P. Finally, we present an efficient computational solution to the fractional smoothing spline problem: It uses the fast Fourier transform and takes advantage of the multiresolution properties of the underlying basis functions.

Computer network use is becoming increasingly widespread, both in terms of number of users and variety of applications. In order to provide consistently high quality service, network engineers and other professionals must monitor several aspects of the network, including the traffic intensity on the links that comprise the network. As networks grow, this type of monitoring has potential to become burdensome in terms of resources required. Motivated by the prospect of monitoring only a small subset of links, this paper explores the problem of using observed traffic measurements on selected links to predict the traffic on other, unobserved links. The characteristics of such unobserved links are learned through auxiliary data. Although more expensive to obtain, this extra data set provides the necessary information to represent important structure in the network, and can significantly improve the results of prediction as compared with more naive approaches. In addition, we introduce an adjusted control chart methodology that shows possible applications of our prediction results in situations where all links may be observed. 1. Introduction. Modern