WAVELET AND SCALING: THEORY in Visual Studio .NET

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2.2 WAVELET AND SCALING: THEORY
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dence. FGN is close to an ``ideal'' model because its spectral density is-close to n1 2H  n a for a large range of frequencies n in the interval [0, 1], and because its 2 correlation function, r k 1 f k 1 2H 2k 2H jk 1j2H g; 2 2:9
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is invariant under aggregation (see Section 2.3.5.1). We now recall the properties of the wavelet coef cients of H-sssi processes (such as FBM) and LRD processes (such as FGN) and show that they can be gathered into a uni ed framework. We subsequently show that other stochastic processes exhibiting scaling behavior also t into this framework, opening up the prospect of a single approach covering diverse forms of scaling.
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Wavelet Transform of Scaling Processes
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2.2.3.1 Discrete Wavelet Transform of Stochastic Processes Whereas the wavelet theory was rst established for deterministic nite-energy processes, it has clearly been demonstrated in the literature that the wavelet transform can be applied to stochastic processes; for example, see Cambanis and Houdre [20] and Masry [49]. More speci cally, for the second-order random processes of interest here, it is well known that the wavelet transform is a second-order random eld, on the condition that the scaling function f0 (and hence the wavelet c0 ) satisfy certain mild conditions [20, 49] related to the covariance structure of the analyzed process. We will assume hereafter that the scaling functions and wavelets decay at least exponentially fast in the time domain, so that the second-order statistics of the wavelet transform exist for all of the random processes we discuss here. 2.2.3.2 Wavelet Transform (WT) of H-ss and H-sssi Processes Let X be an H-ss process. Its wavelet coef cients dX j; k exactly reproduce the self-similarity through the following central scaling property; see Delbeke [25] and Delbeke and Abry [26] or Pesquet-Popescu [57]:  P0 SS: For the DWT, dX j; k hX ; cj;k i, so that dX j; 0 ; dX j; 1 ; . . . ; dX j; Nj 1 2j H 1=2 dX 0; 0 ; dX 0; 1 ; . . . ; dX 0; Nj 1 : For the CWT, TX a; t hX ; ca;t i, and hence TX ca; ct1 ; . . . ; TX ca; ctn cH 1=2 TX a; t1 ; . . . ; TX a; tn ;
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WAVELETS FOR THE ANALYSIS, ESTIMATION, AND SYNTHESIS OF SCALING DATA
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These equations mimic the self-similarity of the process. Let us emphasize that this, nontrivially, results from the fact that the analyzing wavelet basis is designed from the dilation operator and is therefore, by nature, scale invariant (F1). For second-order processes, a direct consequence of Eq. (2.10) is EdX j; k 2 2j 2H 1 EdX 0; k 2 : 2:11
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Moreover, if we add the requirement that X has stationary increments (i.e., X is H-sssi), ingredients F1 and F2 combine, resulting in:  P1 SS: The wavelet coef cients with xed scale index fdX j; k ; k P Zg form a stationary process. This follows from the stationary increments property of the analyzed processes [20, 25, 49]. This property is not trivial, given that self-similar processes are nonstationary processes, and is a consequence of N !1 (F2). In this case, Eq. (2.11) reduces to the fundamental result: EdX j; k 2 2j 2H 1 C H; c0 s2 ; Vk; 2:12
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with C H; c0 jtj2H c0 u c0 u t du dt and s2 EX 1 2 .  P2 SS: Using the speci c covariance structure of an H-sssi process X t , namely, EX t X s s2 2H fjtj jsj2H jt sj2H g; 2 2:13
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it can be shown [32, 73] that the correlations between wavelet coef cients located at different positions is extremely small as soon as N ! H 1 and their 2 decay can be controlled by increasing N : EdX j; k dX jH ; k H % j2j k 2j k H j2H 2N ; j2j k 2j k H j 3 I:
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These two results have been obtained and illustrated originally in the case of the FBM [31 34] (see also Tew k and Kim [73]) and have been stated in more general contexts [20, 25, 26, 49]. 2.2.3.3 WT of LRD Processes Let X be a second order stationary process, its wavelet coef cients dX j; k satisfy the following:  P0 LRD: EdX j; k GX n 2j jC0 2j n j2 dn
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