We perform a comparative study of applicability of the Multifractal Detrended Fluctuation Analysis (MFDFA) and the Wavelet Transform Modulus Maxima (WTMM) method in proper detecting of mono-and multifractal character of data. We quantify the performance of both methods by using different sorts of artificial signals generated according to a few well-known exactly soluble mathematical models: monofractal fractional Brownian motion, bifractal Lévy flights, and different sorts of multifractal binomial cascades. Our results show that in majority of situations in which one does not know a priori the fractal properties of a process, choosing MFDFA should be recommended. In particular, WTMM gives biased outcomes for the fractional Brownian motion with different values of Hurst exponent, indicating spurious multifractality. In some cases WTMM can also give different results if one applies different wavelets. We do not exclude using WTMM in real data analysis, but it occurs that while one may apply MFDFA in a more automatic fashion, WTMM has to be applied with care. In the second part of our work, we perform an analogous analysis on empirical data coming from the American and from the German stock market. For this data both methods detect rich multifractality in terms of broad f (α), but MFDFA suggests that this multifractality is poorer than in the case of WTMM.
Drozdz and Speth Reply: The Comment by Biswas andAzam [1] is an attempt to interpret our results [2] using a formalism based on the periodic-orbit expansion for the spectral density [3] which puts the related considerations into the context of semiclassical quantization. This approach seems, however, more adequate to explain the spectral rigidity measured in terms of the A3 statistics than to explain the nearest-neighbor spacing (NNS) distribution. Certain predictions of the Comment concerning NNS find no strict confirmation in our further calculations performed in order to clarify this point. In particular, for rectangular billiards the global character of deviations from universalities for a sequence of say 400 states turns out to depend on how such a sequence is centered and develops in a different way for different dimensions N.More precisely, as shown in our original paper, in the low-energy domain (states between 50 and 450) the repulsion parameter r defined in terms of the generalized Wigner distribution always takes positive values and clearly increases with increasing TV. The calculations show that at higher energies the deviations still remain but the difference for different TV essentially dissolves, and r fluctuates approximately within the interval (-0.12, 0.12). It is interesting to notice the appearance of negative values of r, although the positive ones are more frequent. However, the region of this type of fluctuation shifts to the higher energies when TV increases. For TV =5 it begins at around n = 1000 (n denotes here the label of the lowest state in the sequence), for TV = 3 at A*«400, while for TV = 2 it reaches the lowest energies. Using longer sequences of states leads to analogous behavior but, of course, the amplitude of fluctuations becomes smaller. A similar tendency is observed for the other Hamiltonians as well; for instance, for the five-dimensional case considered in Fig. 3(b) [21, r=0.45 for 400 states at low energies. Shifting up such a sequence of states causes a systematic decrease of r, and at n ~ 1000, r=0.19. Then the fluctuations begin. This suggests that the number of low-lying states to be discarded for statistical analysis should be correlated with the dimensionality of the problem. This is an important practical point which demands further, more careful and extensive study.As the sequence of states is moved up the spectrum be-
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