Scaling, Fractals and Wavelets by Patrice Abry, Paolo Goncalves, Jacques Levy Vehel
By Patrice Abry, Paolo Goncalves, Jacques Levy Vehel
Scaling is a mathematical transformation that enlarges or diminishes gadgets. The approach is utilized in numerous parts, together with finance and photograph processing. This publication is geared up round the notions of scaling phenomena and scale invariance. a number of the stochastic versions accepted to explain scaling ? self-similarity, long-range dependence and multi-fractals ? are brought. those versions are in comparison and on the topic of each other. subsequent, fractional integration, a mathematical device heavily relating to the suggestion of scale invariance, is mentioned, and stochastic methods with prescribed scaling homes (self-similar procedures, in the neighborhood self-similar procedures, fractionally filtered methods, iterated functionality platforms) are outlined. a few purposes the place the scaling paradigm proved fruitful are specific: picture processing, monetary and inventory industry fluctuations, geophysics, scale relativity, and fractal time-space.
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Extra resources for Scaling, Fractals and Wavelets
Example text
4. It was then gradually realized that fg contains information sometimes more relevant than fh , particularly in signal processing applications. For more details on this topic, see [LEV 98b], where the denominations “Hausdorff spectrum”, “large deviation spectrum” and “Legendre spectrum” were introduced. The next section tackles the problem of calculating the multifractal spectra. 4. Practical calculation of spectra Let us begin with the Hausdorff spectrum. It is clear that the calculation of the exponents in each point, and then of all the associated dimensions of the Hausdorff spectrum, is an extremely difficult task.
When this is the case, we say that the (strong) multifractal formalism holds. Define: 2n VX (Ink )q Sn (q) = k=1 and set: τn (q) = − 1 log2 Sn (q) n τ (q) = lim inf τn (q) n→∞ To understand the link between fh and τ heuristically, let us evaluate τn by grouping together the terms that have the same order of magnitude in the definition of Sn . In that view, fix a “Hölder exponent” α and consider those intervals for which, when n is sufficiently large, VX (Ink ) ∼ |Ink |α . Assume that this approximation is true uniformly in k.
C and τ˜c are increasing and concave functions; – τ c (0) = τ˜c (0) = −Δ(supp(X)); – if X is a probability measure, then τ c (1) = τ˜c (1) = 0; – τ c (q) = lim inf n→∞ log H q (Rηn )/ log ηn , where (ηn ) is a sequence tending to zero such that log ηn / log ηn+1 → 1 when n → ∞. The same is true for τ˜c . The last property is important in numerical applications: it means that τ c and τ˜c may be estimated by using discrete sequences of the type ηn = 2−n . Kernel method A second method to estimate fg , which does not assume that the weak formalism is true (and thus in particular allows us to obtain non-concave spectra), is based on the following.