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WAVELETS FOR THE ANALYSIS, ESTIMATION, AND SYNTHESIS OF SCALING DATA
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based estimators are essentially optimal computationally speaking, as they have a complexity of only O n , and a direct, nonproblematic implementation that can even be performed in real time [62]. Simple estimators of scaling such as the variogram [17] also have excellent computational properties; however, they suffer from signi cant bias and high variance [68]. At the other end of the spectrum, fully parametric maximum likelihood estimators require the inversion of an n n autocovariance matrix, an O n3 operation, which is unsuitable for anything but small data sets. Even approximate forms such as the Whittle estimator, or discrete versions of it [5, 17], involve numerical minimization and are prohibitively slow for the large n > 214 data sets now routinely encountered in teletraf c studies. Statistically, the best performing estimators are fully parametric, such as those based on maximum likelihood, which offer zero bias and optimal variance provided that the data t the chosen parametric model. As mentioned above, to avoid extreme computational dif culties encountered for all but small data sets, approximate forms are used in practice, which retain these desirable statistical properties asymptotically [5, 17]. Taqqu et al. [68, 71], give a comparative discussion of the statistical properties of a variety of estimators of long-range dependence. It is shown that approximate maximum likelihood-based estimators such as the Whittle, aggregated Whittle, and local Whittle methods still offer the best statistical performance when compared against alternatives such as the absolute value method, the variance method, the variance of residuals method, the R=S method, and the periodogram method (see also Teverovsky et al. [66] and Taqqu and Teverovsky [69, 72]). We therefore compare against such parametric alternatives. Being parametric based, such estimators can make full use of the data and will therefore outperform logscale diagram based estimators, which are constrained to use only those scales where the scaling is both present and apparent. Although it is possible that all the data may be accessible to a logscale diagram based estimator that is, that j1 ; j2 can be chosen correctly as j1 ; j2 1; log2 n , this is unlikely in general. It is even possible that a data set is too short to contain scales in the scaling range, in which case the logscale diagram based estimators, or indeed any semiparametric estimator of scaling, will be useless. On the other hand, in practice typically one cannot know the ``true'' model for the data, and parametric estimators based on the wrong model can yield meaningless results. In contrast, logscale diagram based estimation is not sensitive to nonscaling details of the data, provided the scaling range is correctly identi ed, and is also more robust in other ways, as discussed below. Another key advantage is the ability to measure in a uniform framework both stationary and nonstationary forms of scaling. ^ A detailed comparison of the performance of a against that of the discrete Whittle estimator for the FGN and Gaussian FARIMA(0; d; 0) processes is given by Abry and Veitch [5]. Veitch and Abry [75] compared ^ ; cf against a joint discrete Whittle a ^ estimator for the Gaussian FARIMA(0; d; 0) process, and (^ ; cf C) against a joint a d maximum likelihood estimator for a ``pure'' scaling process de ned in the wavelet domain [79]. The main conclusion is that the logscale diagram based estimators offer almost unbiased estimates, even for data of small length, with far greater robustness, for the price of a small to moderate increase in variance compared to that of the
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2.3 WAVELETS AND SCALING: ESTIMATION
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parametric alternatives. They can also be used to treat data of arbitrary length both from the computational complexity and memory requirement points of view [5, 62, 75]. Comparison against parametric estimators for processes that are further from such ``ideal'' processes will appear elsewhere. 2.3.3 The Multiscale Diagram and Multifractals
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2.3.3.1 The Need for Statistics Other than Second Order It is natural and straightforward to generalize the logscale diagram to the study of statistics other than second order, by replacing Eq. (2.19) with m q 1=nj k jdX j; k jq , q P R. The j resulting qth-order logscale diagrams are, naturally enough, of interest in situations where information relevant to the analysis of scaling is beyond the reach of secondorder statistics. Let us concentrate on two important examples: self-similarity and multifractality. Self-Similarity From the de nition of self-similarity, the moments of X t satisfy EjX t jq EjX 1 jq jtjqH , Vt. As for the wavelet coef cients, it follows from Eq. (2.10) that EjdX j; k jq EjdX 0; k jq 2j qH q=2 , implying that Em q j Cq 2j z q q=2 , Vj, with z q qH. This relation suggests that self-similarity can be detected by testing the linearity of z q with q. Multifractal Processes For the class of multifractal processes, assuming that jTX o a; t jq dt % az q q=2 , a 3 0, can be related to the multifractal properties of the process, m q is expected to behave according to m q % 2j z q q=2 for small j. j j From these relations one can measure z q in practice and therefore estimate the Legendre multifractal spectrum. A particularly interesting question in the multi fractal formalism is to test whether, in the range of q where jTX o a; t jq dt is nite, z q takes a simple linear form: z q qh, or not. Clearly in such cases the Legendre spectrum is somewhat degenerate as it is entirely determined by h and the range of q where z q is de ned. For instance, self-similar processes, for which m q % 2j qH q=2 for all scales, satisfy z q qH and are therefore fractal processes j with h H. More speci cally the self-similar Levy processes have in nite variance and are multifractal [41], yet their spectra are parameterized by H and are therefore derivative of the strong self-similar property. The FBM is another, even simpler example, often referred to as ``monofractal,'' con rming the intuition that a single scaling parameter controls all of its scaling properties. 2.3.3.2 The Multiscale Diagram and Its Use In both the self-similar and multifractal cases, therefore, accurately measuring the deviation of z q from a simple linear form is a crucial issue. To test this, and to investigate the form of z q in general, the qth-order scaling exponent aq z q q=2 can be estimated in the qth-order logscale diagram for a variety of q values, and then the q dependence examined in the following tool:
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