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ei (l, j) =
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and: e(l, j) = e0 (l, j) + e1 (l, j) (j)
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where is a positive increasing function The quantity e(l, j) is the error function and is introduced to account for the fact that errors made on coarse scales have more impact than that made on ne scales Now, the proposed segmentation algorithm can be formulated (see Algorithm 91) The result of this algorithm is the segmentation of f into consecutive parts, each being properly represented by a WSA function From this point of view, this segmentation approach is a new type of tool Instead of dividing the original signal into homogenous parts according to usual criteria such as local average or fractal dimension, we use a criterion based on multifractal stationarity Indeed each segment has a well-de ned multiplicative structure, with a multifractal spectrum given by Theorem 92 As an application of this new segmentation scheme, we will examine in the next section how it enables to estimate non-concave multifractal spectra
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Fix > 0; node = root node; (this is the initialization) function segmentation (node) Begin (l, j) = number of the node; If we have e(l, j) < , then: {f (m), m I(l, j, J)} is approximated by the weak self-similar function de ned by { n (l, j), n = j + 1, , J 1, i = 0, 1} i Otherwise segmentation (left line of the node); segmentation (right line of the node); End
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Algorithm 91 Segmentation algorithm
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NOTE 94 Our algorithm suffers from a weakness: segmentation can only occur at dyadic points, the consequence being an important loss if the real segments are not lined up with dyadic points This dif culty frequently arises when using dyadic wavelets and it can be solved using standard techniques such as non-decimated wavelet transforms 99 Estimation of the multifractal spectrum Representation by means of weak self-similar functions offers a semi-parametric approach to the estimation of the spectrum d( ) At rst, f is segmented into homogenous parts by using the algorithm described in the previous section Each subpart Pi , with i = 1, , p, is represented by a single WSA function Fi whose spectrum di can be calculated using Theorem 92 Because the number of segments is nite, the dimension d( ) associated with the H lder exponent for any signal is thus a maximum of di ( ), with i = 1, , p Therefore, the semi-parametric estimation of d( ) is: d( ) = max di ( )
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Obviously, each di is concave, although, in general, this is not the case for d coincides The concatenation example of two IFS shows that the estimated spectrum d exactly with the theoretical spectrum and exhibits two modes, as is characteristic for phase transition It is important to note that, for a given > 0, it is easy to construct two functions f1 and f2 such that ||f1 f2 ||L2 < or ||f1 f2 || < , yet with very different spectra Therefore, we cannot draw the conclusion that, in general, the original signal spectrum is close to that of the approximating WSA function However, based on criteria which allow us to con rm that the physical properties of the original signal and
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the approximating signal are close, then this conclusion can make sense For example, in the case of speech signals, studied later, an obvious criterion is that of auditory comparison As far as Internet traf c applications are concerned, the chosen criterion will be the comparison between our estimation of the spectrum and that proposed by other approaches Indeed, since approaches are qualitatively different and more or less yield equivalent spectra (as we will see later), this gives credit to the quality of the proposed estimation 910 Experiments The rst example consists of the representation of the word welcome pronounced by a male speaker The signal contains 215 samples, we assume that j0 = 8 and we use the Daubechies-16 wavelet (as explained before, the choice of wavelet is based on the criteria (C1) and (C2)) In this experiment, as in the following one, we assume that (j) = j 2 By using a threshold = 50, we obtain a representation with seven WSA functions, where 64% of the coef cients are processed (the remaining 36% correspond to the tree levels coarser to j0 or to values of cj larger than 1) Figure 98 shows that the original and the approximating k signals are visually almost identical In a more signi cant manner, we cannot distinguish the two signals from an auditory comparison, as can be checked at: http://www-rocqinriafr/fractales In addition, the segmentation (see the crosses in Figure 98) is phonetically consistent, since it coincides almost perfectly with the sounds: silence, /w/, / l /, silence, /k/, /om/, silence The slight difference between the positions of the segmentation marks and the exact transition points between phonetic units is due to the fact that, in our actual implantation, the segmentation is restricted to the dyadic points
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Figure 98 The word welcome pronounced by a male speaker (in black) with its approximation (superimposed in gray) and the segmentation marks (the crosses)
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In the second example, we present an application for Internet traf c signals We use a signal of 512 traf c samples coming out from Berkeley, measured in bytes by time
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