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(417)
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Multifractal Scaling: General Theory and Approach by Wavelets
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This exponent measures the variability of Y and is related to the dimension of the paths of Y Deducting constant, resp linear terms in formulae (416) and (417) reminds us of the use of wavelets with one, resp two vanishing moments 43 Multifractal analysis Multifractal analysis has been discovered and developed in [MAN 74, FRI 85, KAH 76, GRA 83, HEN 83, HAL 86, CUT 86, CAW 92, BRO 92, BAC 93, MAN 90b, HOL 92, FAL 94, OLS 94, ARB 96, JAF 97, PES 97, RIE 95a, MAN 02, BAC 03, BAR 04, BAR 02, CHA 05, JAF 99] to give only a short list of some relevant work done in this area The main insight consisted of the fact that local scaling exponents on fractals as measured by h(t), (t) or w(t) are not uniform or continuous as a function of t, in general In other words, h(t), (t) and w(t) typically change in an erratic way as a function of t, thus imprinting a rich structure on the object of interest This structure can be captured either in geometric terms, making use of the concept of dimensions, or in statistical terms based on sample moments A useful connection between these two descriptions emerges from the multifractal formalism As we will see, as far as the multifractal formalism is concerned there is no restriction in choosing a singularity exponent which seems t for describing scaling behavior of interest To express this fact we consider in this section the arbitrary scaling exponents s(t) := lim inf snn k
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s(t) := lim sup snn , k
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(418)
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where sn (k = 0, , 2n 1, n N) is any sequence of random variables To keep k a connection with what was said before, think of sn as representing a coarse scaling k n exponent of Y over the dyadic interval Ik 431 Dimension based spectra A geometric description of the erratic behavior of a multifractal s scaling exponents can be achieved using a quanti cation of the prevalence of particular exponents in terms of fractal dimensions as follows: We consider the sets Ka which are de ned pathwise in terms of limiting behavior of snn as n , as k Ea := {t : s(t) = a}, E a := {t : s(t) = a}, Ka := {t : s(t) = a} (419)
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These sets Ka are typically fractal , meaning loosely that they have a complicated geometric structure and more precisely that their dimensions are non-integer A compact description of the singularity structure of Y is therefore in terms of the following so-called Hausdorff spectrum d(a) := dim(Ka ), where dim(E) denotes the Hausdorff dimension of the set E [TRI 82] (420)
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The sets Ea (a ) and also E a form a multifractal decomposition of the support of Y We will loosely address Y as a multifractal if this decomposition is rich, ie if the sets Ea (a ) are highly interwoven or even dense in the support of Y However, the study of singular measures (deterministic and random) has often been restricted to the simpler sets Ka and their spectrum d(a) [KAH 76, CAW 92, FAL 94, ARB 96, OLS 94, RIE 98, RIE 95a, RIE 95b, BAR 97] With the theory developed here (Lemma 42) it becomes clear that most of these results extend to provide formulae for dim(Ea ) and dim(E a ) as well This aspect of multifractal analysis has been of much interest to the mathematical community 432 Grain based spectra An alternative description of the prevalence of singularity exponents, statistical in nature due to the counting involved, is f (a) := lim lim sup
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