WAVELETS FOR THE ANALYSIS, ESTIMATION, AND SYNTHESIS OF SCALING DATA

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Fig. 2.2 Logscale diagrams. Left: An example of the yj against j plot and regression line for a LRD process with strong short-range dependence. The vertical bars at each octave give 95% con dence intervals for the yj . The series is simulated FARIMA (0; d; 2) with d 0:25 and second-order moving average operator C B 1 2B B2, implying a; cf 0:50; 6:38 . Alignment is observed over scales j1 ; j2 4; 10 , and a weighted regression over this range ^ allows an accurate estimation despite the strong short-range dependence: a 0:55 0:07, ^ ^ cf 6:0 with 4:5 < cf < 7:8. The scaling can be identi ed as LRD as the value is in the ^ correct range, a P 0; 1 , and the alignment region includes the largest scales in the data. ^ Right: Alignment is observed over the full range of scales with a 2:57, corresponding to ^ H 0:79, consistent with the self-similarity of the simulated FBM (H 0:8) series analyzed.

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the identi cation of the kind of scaling is made by interpreting the estimated value in the context of the observed range. These different aspects of the aims and use of the logscale diagram are expanded upon next. 2.3.1.2 The Detection of Scaling A priori it is not known over which scales, if any, a scale-invariant property may exist. By the detection of scaling in the logscale diagram we mean the identi cation of region(s) of alignment and the determination of their lower and upper cutoff octaves, j1 and j2, respectively, which are taken to correspond to scaling regimes. In a sense this is an insoluble problem, as scaling often occurs asymptotically or has an asymptotic de nition, with no clear way to de ne how a scaling range begins or ends. Nonetheless experience shows that good estimates are possible. Note the semantic difference between the term scaling region or range, a theoretical concept that refers to where scaling is truly present (an unknown in real data), and alignment region or range, an estimation concept corresponding to what is actually observed in the logscale diagram for a given set of data. The rst essential point here is that the concept of alignment is relative to the con dence intervals for the yj , and not to a close alignment of the yj themselves. Indeed, an undue alignment of the actual estimates yj indicates strong correlations between them, a highly undesirable feature typical of time domain log log based methods such as variance time plots. As mentioned earlier the mj , and hence the yj , are weakly dependent, resulting in a natural and desirable variation around

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2.3 WAVELETS AND SCALING: ESTIMATION

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the calculated regression line as seen, for example, in Fig. 2.2. Using weighted regression incorporates the varying con dence intervals into the estimation phase; however, the selection of the range of scales de ning the alignment region is prior to this, and great care is required to avoid poor decisions. We now discuss the selection, in practice, of the cutoff scales j1 and j2 . A preliminary comment is that for the regression to be well de ned at least two scales are required, for a Chi-squared goodness of t test three, and in practice four, are needed before any estimate can be taken seriously: it is simply too easy for three points to align fortuitously if the con dence intervals are not very small. A useful heuristic in the selection of a range is that the regression line should cut, or nearly so, each of the con dence intervals within it. This can help avoid the following two errors: (1) the nondetection of an alignment region due to the apparently wild variation of the yj , when in fact to within the con dence intervals the alignment is good (this typically occurs when the slope is small, such as in the right-hand plot in Fig. 2.4, as the vertical scale on plots is reduced, increasing the apparent size of variations), and (2) the erroneous inclusion of extra scales to the left of an alignment region, since to the eye they appear to accurately continue a linear trend, whereas in fact the small con dence intervals about the yj for small j reveal that they depart signi cantly from it. The above heuristic can be formalized somewhat by a Chi-squared goodness of t test [9], where the critical level of the goodness of t statistic is monitored as a function of the endpoints of the alignment range. At least in the case of the lower scale, this can make a very clear and relatively objective choice of cutoff possible, eliminating the error of type (2) above. An example of this is afforded by the lefthand plot of Fig. 2.2, where the octave j 3, if included, results in a drop of the Chisquared goodness of t of several orders of magnitude! An even subtler example is that of the right-hand plot in Fig. 2.3, where j 2 was excluded from the alignment region for the same reason, whereas in the left-hand plot in the same gure it is clear even to the eye that, given the small size of the con dence interval about octave j 8, it should not be included. Further work is required to develop reliable automated methods of cutoff scale determination. This is especially true for upper cutoff scales, where the dif culties are compounded by a lack of data. On the other hand, at smaller scales the technical assumptions used in the calculation of the con dence intervals (see below) may be less reliable, whereas at large scales the data are highly aggregated and therefore Gaussian approximations are reasonable. 2.3.1.3 The Interpretation of Scaling By the interpretation of scaling we mean the identi cation of the kind of underlying scaling phenomenon LRD, H-ss, and so on generating the observed alignment in the logscale diagram. The task is the meaningful interpretation of the estimated value of a in the context of the range of scales de ning the alignment region, informed where possible by other known or assumed properties of the time series such as stationarity. It is in fact partly a question of model choice, and there may be no unique solution. We now consider, nonexhaustively, a number of important cases.

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