WAVELETS FOR THE ANALYSIS, ESTIMATION, AND SYNTHESIS OF SCALING DATA in .NET framework

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WAVELETS FOR THE ANALYSIS, ESTIMATION, AND SYNTHESIS OF SCALING DATA
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Fig. 2.7 Eliminating mean level shifts. Top: Ethernet byte data aggregated over 10 s intervals. A marked, but smooth, level shift seems to occur around 1050 seconds. Bottom: Logscale diagram of the byte data aggregated over 10 ms intervals. With N 2, estimates to the left and the right of the shift, and over the whole series, are consistent, showing that smooth level shifts can effectively be eliminated in practice.
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model inference problem, to which an optimal (uniformly most powerful invariant) test exists [77] (see also the discussion in Abry and Veitch [5]). 2.3.5 Relations to Other Tools
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2.3.5.1 Aggregation Procedure Let X t be a centered second-order stationary process with nite variance and form the discrete-time process X T n , n P Z, de ned by X
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Such a quantity, referred to as the aggregated process of X with aggregation level T , turns out to play a key role with respect to self-similarity and long-range dependence. In fact, if we assume that X is LRD in the sense that its autocovariance
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2.3 WAVELETS AND SCALING: ESTIMATION
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decreases asymptotically as in Eq. (2.7), it can be shown [23] that, in the limit of arbitrarily large aggregation levels T , the normalized autocovariance r T k EX T n X T n k ; Var X T n k 1; 2; . . .
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of the aggregated process tends to a well-de ned limiting form that coincides with that of an FGN with unit variance (see Eq. (2.9)), while its variance satis es Var X T n % T a 1 ; T 3 I: 2:31
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Aggregation appears therefore as a natural renormalization tool that reveals, in a well-de ned manner, the possible occurrence of long-range dependence in an observed process. By construction, aggregation amounts to averaging the analyzed process over adjacent blocks of larger and larger support. This is of course reminiscent of the way the Haar MRA, whose scaling function is simply the indicator function of [0, 1], is built. More precisely, we have the exact equivalence [6] X 2 k 2 j=2 hX ; f Haar i; j;k
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from which we can deduce that Var a Haar j; k % 2ja ; X j 3 I:
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Revisiting the aggregation procedure from the Haar perspective allows for two levels of generalization [6]. First, we can relax the choice of the Haar system and move toward any MRA. Second, we can replace approximation coef cients by details, since we also have, for any admissible wavelet c, Var dX j; k % 2ja ; J 3 I:
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2.3.5.2 Allan Variance Although well rooted in wavelet theory, this idea of using details rather than approximations can itself be viewed as a generalization of a method put forward by Allan in 1966 [13]. Speci cally, whereas Eq. (2.31) shows the theoretical link between variance and the scaling behavior associated with longrange dependence, Allan showed that the estimation of a is greatly improved when the standard variance estimate 1=N n X T n 2 is replaced by the ``Allan T variance,'' de ned as 1=N n X n 1 X T n 2 . In wavelet words, this just amounts when dyadic intervals of the form T 2j are considered to computing Haar details [32]. As explained previously, the reason for the improved ef ciency of such an approach is to be found in the ability of wavelets to almost decorrelate LRD processes. Moreover, revisiting the Allan variance in the light of the
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WAVELETS FOR THE ANALYSIS, ESTIMATION, AND SYNTHESIS OF SCALING DATA
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Haar system suggests that more versatile generalizations could be considered based on wavelets with more vanishing moments. This indeed not only guarantees smaller residual correlations but also an increased robustness of the analysis to polynomial trends. 2.3.5.3 Fano Factor Apart from aggregation and the Allan variance, wavelets also offer a way of generalizing standard approaches in other areas, for example, in the case of point processes [4]. Let us consider, for instance, processes of the form P t