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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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The detrended cross-correlation coefficient ρDCCA has recently been proposed to quantify the strength of cross-correlations on different temporal scales in bivariate, non-stationary time series. It is based on the detrended cross-correlation and detrended fluctuation analyses (DCCA and DFA, respectively) and can be viewed as an analogue of the Pearson coefficient in the case of the fluctuation analysis. The coefficient ρDCCA works well in many practical situations but by construction its applicability is limited to detection of whether two signals are generally cross-correlated, without possibility to obtain information on the amplitude of fluctuations that are responsible for those cross-correlations. In order to introduce some related flexibility, here we propose an extension of ρDCCA that exploits the multifractal versions of DFA and DCCA: MFDFA and MFCCA, respectively. The resulting new coefficient ρq not only is able to quantify the strength of correlations, but also it allows one to identify the range of detrended fluctuation amplitudes that are correlated in two signals under study. We show how the coefficient ρq works in practical situations by applying it to stochastic time series representing processes with long memory: autoregressive and multiplicative ones. Such processes are often used to model signals recorded from complex systems and complex physical phenomena like turbulence, so we are convinced that this new measure can successfully be applied in time series analysis. In particular, we present an example of such application to highly complex empirical data from financial markets. The present formulation can straightforwardly be extended to multivariate data in terms of the q-dependent counterpart of the correlation matrices and then to the network representation.

We propose an algorithm, multifractal cross-correlation analysis (MFCCA), which constitutes a consistent extension of the detrended cross-correlation analysis and is able to properly identify and quantify subtle characteristics of multifractal cross-correlations between two time series. Our motivation for introducing this algorithm is that the already existing methods, like multifractal extension, have at best serious limitations for most of the signals describing complex natural processes and often indicate multifractal cross-correlations when there are none. The principal component of the present extension is proper incorporation of the sign of fluctuations to their generalized moments. Furthermore, we present a broad analysis of the model fractal stochastic processes as well as of the real-world signals and show that MFCCA is a robust and selective tool at the same time and therefore allows for a reliable quantification of the cross-correlative structure of analyzed processes. In particular, it allows one to identify the boundaries of the multifractal scaling and to analyze a relation between the generalized Hurst exponent and the multifractal scaling parameter λ(q). This relation provides information about the character of potential multifractality in cross-correlations and thus enables a deeper insight into dynamics of the analyzed processes than allowed by any other related method available so far. By using examples of time series from the stock market, we show that financial fluctuations typically cross-correlate multifractally only for relatively large fluctuations, whereas small fluctuations remain mutually independent even at maximum of such cross-correlations. Finally, we indicate possible utility of MFCCA to study effects of the time-lagged cross-correlations.

Based on the Multifractal Detrended Fluctuation Analysis (MFDFA) and on the Wavelet Transform Modulus Maxima (WTMM) methods we investigate the origin of multifractality in the time series. Series fluctuating according to a qGaussian distribution, both uncorrelated and correlated in time, are used. For the uncorrelated series at the border (q = 5/3) between the Gaussian and the Levy basins of attraction asymptotically we find a phase-like transition between monofractal and bifractal characteristics. This indicates that these may solely be the specific nonlinear temporal correlations that organize the series into a genuine multifractal hierarchy. For analyzing various features of multifractality due to such correlations, we use the model series generated from the binomial cascade as well as empirical series. Then, within the temporal ranges of well developed power-law correlations we find a fast convergence in all multifractal measures. Besides of its practical significance this fact may reflect another manifestation of a conjectured q-generalized Central Limit Theorem.

We adopt the concept of the correlation matrix to study correlations among sequences of time-extended events occurring repeatedly at consecutive time intervals. As an application we analyze the magnetoencephalography recordings obtained from the human auditory cortex in the epoch mode during the delivery of sound stimuli to the left or right ear. We look into statistical properties and the eigenvalue spectrum of the correlation matrix C calculated for signals corresponding to different trials and originating from the same or opposite hemispheres. The spectrum of C largely agrees with the universal properties of the Gaussian orthogonal ensemble of random matrices, with deviations characterized by eigenvectors with high eigenvalues. The properties of these eigenvectors and eigenvalues provide an elegant and powerful way of quantifying the degree of the underlying collectivity during well-defined latency intervals with respect to stimulus onset. We also extend this analysis to study the time-lagged interhemispheric correlations, as a computationally less demanding alternative to other methods such as mutual information.