Scale Invariance and Wavelets

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As a result, FBM presents the advantage (or the disadvantage) of being globally self-similar on the entire frequency axis, the only parameter H controlling, according to the requirements, one or other of the three regimes cited before: self-similarity, long memory and local regularity In terms of modeling, FBM appears as a particularly interesting starting point (as can be the case for white Gaussian noise in stationary contexts) This simplicity (FBM is the only Gaussian process with stationary and self-similar increments and it is entirely determined by the single parameter H) is not of course without counterparts when it is comes to applications, ie as soon as it becomes necessary to consider real data From this theme, numerous variations can be considered, which are not only mentioned here but are also studied in detail in the other chapters of this volume In all cases, it is a matter of replacing the single exponent H by a collection of exponents 226 Beyond the paradigm of scale invariance To begin with, we can consider modifying relation (213) by allowing the exponent to depend on time: E |X(t + ) X(t)|2 C(t)| |2h(t) , 0

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When 0 < h(t) < 1 is a suf ciently regular deterministic function, we describe the process X as multifractional or, when it is Gaussian, as locally self-similar, ie, that locally around t, X(t) possesses similarities with a FBM of parameter H = h(t) (for more details, see 6) The local regularity is no longer a uniform or global quantity along the sample path but, on the contrary, it varies in time, according to h(t), which therefore makes it possible to model time variations of the roughness When h(t) is itself a strong irregular function, possibly a random process, in the sense that, with t xed, h(t) depends on the observed realization of X, the process X is said to be multifractal The variability uctuations are no longer described by h(t), but by a multifractal spectrum D(h) which characterizes the Hausdorff dimension of the set of points t where h(t) = h (see 1 and 3) One of the major consequences of multifractality in the processes is the fact that quantities usually called partition functions, behave according to power laws in the small scale limit: 1 n

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For processes with stationary increments, the time averages (1/n) k=1 |X ( ) (t+ k )|q can be regarded as estimations of the averages of the set E |X ( ) (t)|q Relation (215) thus recalls equation (22), which is a consequence of self-similarity However, a fundamental difference exists: the exponents (q) do not possess a priori any

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Scaling, Fractals and Wavelets

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reason to present a linear behavior qH In other words, the description of scaling laws in data cannot be carried out with a single exponent but requires a whole collection of them Measuring exponents (q) represents a possibility, through a Legendre transform, of estimating the multifractal spectrum However, a detailed discussion of the multifractal processes is beyond the scope of this chapter; to this end, see 1 and 3 Multifractal processes provide a rich and natural extension of the self-similar model insofar as a single exponent is replaced by a set; nevertheless, they are essentially related to the existence of power law behaviors In the analysis of experimental data, such behaviors might not be observed In order to illustrate these situations, the in nitely divisible cascades model exploits an additional degree of freedom: we relax the constraint of a proper power law behavior for the moments, and replace it with a simple behavior that has separable variables q (order of the moment) and (scale analysis) The equations below explain this behavior: self-similar multifractal inf divisib casc E |X ( ) (t)|q = cq | |qH = cq exp(qH log ); E |X ( ) (t)|q = cq | | (q) = cq exp (q) log ; E |X ( ) (t)|q = cq exp H(q)n( ) (216) (217) (218)

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In this scenario, the function n( ) is no longer xed a priori to be log , as much as the function H(q) is no longer a priori linear according to qH The concept of an in nitely divisible cascade was initially introduced by Castaing in the context of turbulence [CAS 90, CAS 96] The complete de nition of this notion is beyond the scope of this chapter and can be found in [VEI 00] It is nonetheless important to indicate that a quantity, called the propagator of the cascade, plays an important role here: it links the probability densities of process increments with two different scales and The in nite divisibility formally translates the notion of the absence of any preferred time scale and demands this propagator be constituted of an elementary function G0 , convoluted with itself a number of times dependent only on the scales and , and therefore with the following functional form: G , (log ) = [G0 (log )] (n( ) n(

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A possible interpretation of this relation is to read the function G0 as the elementary step, ie the building block of the cascade, whereas the quantity n( ) n( ) measures how many times this elementary step must be carried out to evolve from scale to scale The derivative of n with respect to log thus describes, in a sense, the size of the cascade The term of in nitely divisible cascade is ascribed to situations where the function n possesses the speci c form n( ) = log ; otherwise, we only refer to a scaling law behavior The in nitely divisible scale invariant cascades correspond to multiscaling or multifractality when the scaling law

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