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q J = sup H q R ( )
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Set: c (q) = lim inf
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PROPOSITION 110 flc and flc are concave functions; for any , f c ( ) f c ( ) and fg ( )
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c fg ( );
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c c if is a multinomial measure (see section 146), then fg ( ) = fg ( ) = flc ( ) = flc ( ) = fg ( ) c c THEOREM 111 If fg (respectively fg ) is equal to outside a compact interval, then, for any : c flc ( ) = fg ( )
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PROPOSITION 111 c and c are increasing and concave functions; c (0) = c (0) = (supp(X)); if X is a probability measure, then c (1) = c (1) = 0; c (q) = lim inf n log H q (R n )/ log n , where ( n ) is a sequence tending to zero such that log n / log n+1 1 when n The same is true for c The last property is important in numerical applications: it means that c and c may be estimated by using discrete sequences of the type n = 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 Let K denote the rectangular kernel , ie K(x) = 1 for t x [ 1, 1] and K(x) = 0 elsewhere Let K (t) = 1 K( ) Then, by de nition, n+1 K n ( ), where the symbol represents convolution It is not Nn ( ) = 2 hard to check that replacing K by any positive kernel with compact support whose
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integral equals 1 in the de nition of Nn ( ) will not change the value of fg A basic idea is then to use a more regular kernel than the rectangular one to improve the estimation A more elaborate approach is to use ideas from density estimation to try and remove the double limit in the de nition of fg : this is performed by choosing to be a function of n in such a way that appropriate convergence properties are obtained [LEV 96b] We may, for instance, show the following result:
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PROPOSITION 112 Assume that the studied signal is a nite sum of multinomial measures (see section 146) Let:
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fn ( ) = log Nn ( ) log n n log( n )
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Then, if n is a sequence such that limn constant:
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Even without making such a strong assumption on the signal structure, it is still possible, in certain cases, to obtain convergence results, as above, with = (n), using more sophisticated statistical techniques To conclude our presentation of multifractal spectra, let us emphasize that no spectrum is better than the others in all respects All the spectra give similar but different information As was already observed for the local regularity measures, each one has advantages and drawbacks, and the choice has to be made in view of the considered application If we are interested in the geometric aspects, a dimension spectrum should be favored The large deviation spectrum will be used in statistical and most signal processing applications When the number of data is small or if it appears that the estimations are not reliable, we will resort to the Legendre spectrum To be able to compare the different information and also to assert the quality of the estimations, it is important to dispose of general theoretical relations between the spectra It is remarkable that, as seen above, such relations exist under rather weak hypotheses 145 Re nements: analysis of the sequence of capacities, mutual analysis and multisingularity In this section, some re nements of multifractal analysis, useful in applications, will be brie y discussed The rst re nement stems from the following consideration Assume we are interested in the analysis of road traf c and that the signal X(k) at hand is the ow