By Patrice Abry, Paolo Goncalves, Jacques Levy Vehel
Scaling is a mathematical transformation that enlarges or diminishes gadgets. The strategy is utilized in numerous components, together with finance and photograph processing. This booklet is prepared round the notions of scaling phenomena and scale invariance. many of the stochastic types universal to explain scaling ? self-similarity, long-range dependence and multi-fractals ? are brought. those versions are in comparison and relating to each other. subsequent, fractional integration, a mathematical software heavily on the topic of the proposal of scale invariance, is mentioned, and stochastic tactics with prescribed scaling homes (self-similar techniques, in the neighborhood self-similar procedures, fractionally filtered techniques, iterated functionality structures) are outlined. a couple of purposes the place the scaling paradigm proved fruitful are precise: photo processing, monetary and inventory industry fluctuations, geophysics, scale relativity, and fractal time-space.
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Additional resources for Scaling, Fractals and Wavelets
Now let α = 2. For any plane surface S, the value of H 2 (S) is proportional to the area of S. For non-plane surfaces, H 2 provides an appropriate mathematical deﬁnition of area – using a triangulation of S is not acceptable from a theoretical point of view. More generally, when α is an integer, H α is proportional to the α-dimensional volume. However, α can also take non-integer values, which makes it possible to deﬁne the dimension of any set. The use of the term dimension is justiﬁed by the following property: if aE is the image of E by a homothety of ratio a, then H α (aE) = aα H α (E) Measures estimated using boxes If we want to restrict the class of sets from which coverings are taken even more, one option would be to cover E with centered balls or dyadic boxes.
Indeed, f (d) − f (c) is also the oscillation value of f in [c, d]. However, in general, f is not monotonous and it is therefore necessary to carry out a more accurate analysis, as we will see below. As in the case of measures, we may also consider the “symmetric” exponent deﬁned with an upper limit, and also the exponents obtained as lower and upper limits by using particular grids of intervals, like the dyadic intervals. Oscillation considered as a measurement of the local variability of a function possesses many advantages.
For simplicity, let us assume that our signal X is deﬁned on [0, 1], and let u denote an interval in [0, 1]. We ﬁrst choose a descriptor VX (u) of the relevant information of X in u: if X is a measure μ, we will most often take Vμ (u) = μ(u). If X is a function f , then Vf (u) may for instance be 50 Scaling, Fractals and Wavelets the absolute value of the increment of f in u, that is |f (umax ) − f (umin )| (where u = [umin , umax ]). A more precise descriptor is obtained by considering the oscillation instead, and setting Vf (u) = β(f, u).