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2.1 THE SCALING PHENOMENA
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than in the classical case. Noting that the ratio of the size of the LRD-based variance to the classical one grows to in nity with n, it becomes apparent that con dence intervals based on traditional assumptions, even for a quantity as simple as the sample mean, can lead to serious errors when in fact the data are LRD. We focus here on how a wavelet-based approach allows the threefold objective of the detection, identi cation, and measurement of scaling to be ef ciently achieved. Fundamentally, this is due to the nontrivial fact that the analyzing wavelet family itself possesses a scale-invariant feature, a property not shared by other analysis methods. A key advantage is that quite different kinds of scaling can be analyzed by the same technique, indeed by the same set of computations. The semiparametric estimators of the scaling parameters that follow from the approach have excellent properties negligible bias and low variance and in many cases compare well even against parametric alternatives. The computational advantages, based on the use of the discrete wavelet transform (DWT), are very substantial and allow the analysis of data of arbitrary length. Finally, there are very valuable robustness advantages inherent in the method, particularly with respect to the elimination of superposed smooth trends (deterministic functions). Another important issue connected with modeling and performance studies concerns the generation of time series for use in simulations. Such simulations can be particularly time consuming for long memory processes where the past exerts a strong in uence on the future, disallowing simple approximations based on truncation. Wavelets offer in principle a parsimonious and natural way to generate good approximations to sample paths of scaling processes, which bene t from the same DWT-based computational advantages enjoyed by the analysis method. This area is less well developed than is the case for analysis, however. 2.1.2 Mapping the Land of Scaling and Wavelets
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The remainder of the chapter is organized as follows. Section 2.2, Wavelets and Scaling: Theory, discusses in detail the key properties of the wavelet coef cients of scaling processes. It starts with a brief, yet precise, introduction to the continuous and discrete wavelet transforms, to the multiresolution analysis theory underlying the latter, and the low complexity decomposition algorithm made possible by it. It recalls concisely the de nitions of two of the main paradigms of scaling self-similarity and long-range dependence. The properties of the wavelet coef cients of self-similar, long-range-dependent, and fractal processes are then given, and it is shown how the analysis of these various kinds of scaling can be gathered into a single framework within the wavelet representation. Extensions to more general classes of scaling processes requiring a collection of scaling exponents, such as multifractals, are also discussed. The aim of Section 2.3, Wavelets and Scaling: Estimation, is to indicate how and why this wavelet framework enables the ef cient analysis of scaling processes. This is achieved through the introduction of the logscale diagram, where the key analysis tasks of the detection of scaling interpretation of the nature of scaling and estimation of scaling parameters can be performed. Practical issues in the use of
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WAVELETS FOR THE ANALYSIS, ESTIMATION, AND SYNTHESIS OF SCALING DATA
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the logscale diagram are addressed, with references to examples from real traf c data and arti cially generated traces. De nitions, statistical performance, and pertinent features of the estimators for scaling parameters are then studied in detail. The logscale diagram, rst de ned with respect to second-order statistical quantities, is then extended to statistics of other orders. It is also indicated how the tool allows for and deals with situations=processes departing from pure scaling, such as superimposed deterministic nonstationarities. Finally, clear connections between the wavelet tool and a number of more classical statistical tools dedicated to the analysis of scaling are drawn, showing how the latter can be pro tably generalized in their wavelet incarnations. Section 2.4, Wavelet and Scaling: Synthesis, proposes a wavelet-based synthesis of the fractional Brownian motion. It shows how this process can be naturally and ef ciently expanded in a wavelet basis, allowing, provided that the wavelets are suitably designed, its accurate and computationally ef cient implementation. Finally, in Section 2.5, Wavelets and Scaling: Perspectives, a brief indication is given of what may lay ahead in the broad land of scaling and wavelets. 2.2 2.2.1 WAVELET AND SCALING: THEORY Wavelet Analysis: A Brief Introduction
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2.2.1.1 The (Continuous) Wavelet Transform The continuous wavelet decomposition (CWT) consists of the collection of coef cients fTX a; t hX ; ca;t i; a P R ; t P Rg that compares (by means of inner products) the signal X to be analyzed with a set of analyzing functions & ' u t  1 ca;t u  p c0 ;a P R ;t P R : a a
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This set of analyzing functions is constructed from a reference pattern c0 , called the mother wavelet, by the action of a time-shift operator tt c0 t  c0 t t and a dilation (change of scale) operator p da c0 t  1= ac0 t=a : c0 is chosen such that both its spread in time and frequency are relatively limited. It consists of a small wave de ned on a support, which is almost limited in time and having most of its energy within a limited frequency band. While the time support and frequency band cannot both be nite, there is an interval on which they are effectively limited. The time-shift operator enables the selection of the time instant around which one wishes to analyze the signal, while the dilation operator de nes
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