Wavelet versus detrended fluctuation analysis of multifractal structures

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 monofractal 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 the 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 must 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 $f\left(\alpha \right)$, but MFDFA suggests that this multifractality is poorer than in the case of WTMM.

Figure 1

Example of a monofractal function: the devil’s staircase obtained by integrating the Cantor measure. (a) Analyzed signal (top) and its wavelet transform $T_{\psi }$ (bottom) are presented together with (b) the singularity spectrum $f\left(\alpha \right)$ (main) and the scaling exponent $\tau \left(q\right)$ (inset). The wavelet used in this calculation was ${\psi }^{\left(3\right)}$.

Figure 2

(Color online) Classical Brownian motion $H=0.5$: (a) Fluctuation function $F_q\left(s\right)$ (open symbols) and rescaled partition function $\left[sZ\right(q,s∕41){]}^{1∕q}$ (filled symbols) for different values of Rényi parameter: $q=-5$ (squares), $q=-2$ (triangles up), $q=2$ (circles), and $q=5$ (triangles down). Note that the plots for different $q$’s were vertically shifted in order to improve readability. Calculations were carried out on time series of length $N=130000$ by applying $P^{\left(2\right)}$ and ${\psi }^{\left(3\right)}$. Functional dependence of mean scaling exponent $\overline{\tau \left(q\right)}$ was derived from $F_q\left(s\right)$ in MFDFA (b) and from $Z(q,s^{\prime })$ in WTMM (c) and compared with its theoretical form (solid line).

Figure 3

(Color online) Classical Brownian motion $H=0.5$: Mean singularity spectra $\overline{f\left(\alpha \right)}$ (open circles) obtained by using MFDFA ($P^{\left(2\right)}$, left-hand column) and WTMM (${\psi }^{\left(3\right)}$, right-hand column) procedures for different range of Rényi parameter $q$: $-3\le q\le 3$ (top), $-5\le q\le 5$ (middle), and $-10\le q\le 10$ (bottom). Time series of length $N=130000$ were used; error bars indicate standard deviation of data points calculated from 10 independent realizations of the process.

Figure 4

Classical Brownian motion $H=0.5$: Fraction $R^{\left(q\right)}$ of the total number of signal segments (MFDFA) and of the total number of the maxima lines $l∊L\left(s^{\prime }\right)$ (WTMM) effectively used in calculation of $F_q\left(n\right)$ for MFDFA ($P^{\left(2\right)}$, left-hand column) and of $Z(q,s^{\prime })$ for WTMM (${\psi }^{\left(3\right)}$, right-hand column). Plots for different negative values of $q$ are shown: $q=-3$ (top), $q=-5$ (middle), and $q=-10$ (bottom), with three different independent process realizations of length $N=130000$ (denoted by distinct symbols and lines). Note that for both methods only such ranges of $n$ and $s^{\prime }$ that were associated with the central stable parts of $R^{\left(q\right)}$ were considered in calculation of $\tau \left(q\right)$.

Figure 5

(Color online) Classical Brownian motion $H=0.5$: $\overline{f\left(\alpha \right)}$ (open circles) for different time series lengths: $N=15,000$ (top), $N=65000$ (middle), and $N=130000$ (bottom). Performance of MFDFA ($P^{\left(2\right)}$, left column) and WTMM (${\psi }^{\left(3\right)}$, right column) is compared. Error bars indicate standard deviation of data points calculated from 10 independent realizations of the process.

Figure 6

(Color online) Classical Brownian motion $H=0.5$: $\overline{f\left(\alpha \right)}$ (open circles) obtained with MFDFA for different polynomials: from $P^{\left(1\right)}$ (top left) to $P^{\left(4\right)}$ (bottom right). Time series of length $N=130000$ were used; error bars indicate standard deviation of data points calculated from 10 independent realizations of the process.

Figure 7

(Color online) Classical Brownian motion $H=0.5$: $\overline{f\left(\alpha \right)}$ (open circles) obtained with WTMM for different wavelets: from ${\psi }^{\left(1\right)}$ (top left) to ${\psi }^{\left(4\right)}$ (bottom right). Time series of length $N=130000$ were used; error bars indicate standard deviation of data points calculated from 10 independent realizations of the process.

Figure 8

(Color online) Antipersistent fractional Brownian motion $H=0.3$: $\overline{f\left(\alpha \right)}$ (open circles) for different polynomials $P^{\left(l\right)}$ of MFDFA (left-hand column) and for different Gaussian derivatives ${\psi }^{\left(m\right)}$ of WTMM (right-hand column). Time series of length $N=130000$ were used; error bars indicate standard deviation of data points calculated from 10 independent realizations of the process.

Figure 9

(Color online) Persistent fractional Brownian motion $H=0.75$: $\overline{f\left(\alpha \right)}$ (open circles) for different polynomials $P^{\left(l\right)}$ of MFDFA (left-hand column) and for different Gaussian derivatives ${\psi }^{\left(m\right)}$ of WTMM (right-hand column). Time series of length $N=130000$ were used; error bars indicate standard deviation of data points calculated from 10 independent realizations of the process.

Figure 10

(Color online) Fractional Brownian motion: (a) MFDFA with $P^{\left(2\right)}$ and (b) WTMM with ${\psi }^{\left(3\right)}$: Mean singularity spectra (open circles) for different time series lengths: $N=15000$ (top) and $N=65000$ (bottom) for the fractional Brownian motion in its antipersistent ($H=0.3$, left) and persistent ($H=0.75$, right) variants. Error bars indicate standard deviation of data points calculated from 10 independent realizations of each process. Plots should be compared with the respective panels of Figs. 8, 9, where signals of $N=130000$ were used.

Figure 11

(Color online) Lévy process ${\alpha }_L=1.5$: (a) Fluctuation function $F_q\left(s\right)$ (open symbols) and rescaled partition function $\left[sZ\right(q,s∕41){]}^{1∕q}$ (filled symbols) for different values of Rényi parameter $q=-5$ (squares), $q=-2$ (triangles up), $q=2$ (circles), and $q=5$ (triangles down). Calculations were carried out on time series of length $N=250000$ by applying $P^{\left(2\right)}$ and ${\psi }^{\left(3\right)}$. Functional dependence of mean scaling exponent $\overline{\tau \left(q\right)}$ was derived from $F_q\left(s\right)$ in MFDFA (b) and from $Z(q,s^{\prime })$ in WTMM (c) and compared with its theoretical form (solid line).

Figure 12

(Color online) Lévy process ${\alpha }_L=1.5$: $\overline{f\left(\alpha \right)}$ (open circles) obtained with MFDFA procedure (left-hand column) for different polynomial orders: from $P^{\left(1\right)}$ (top) to $P^{\left(4\right)}$ (bottom) and with WTMM procedure (right-hand column) for different wavelets: from ${\psi }^{\left(1\right)}$ (top) to ${\psi }^{\left(4\right)}$ (bottom) compared with the corresponding exact theoretical values of $f\left(\alpha \right)$ (filled squares). Time series of length $N=250000$ were used; error bars indicate standard deviation of data points calculated from 10 independent realizations of the process.

Figure 13

(Color online) Lévy process ${\alpha }_L=1.5$: $\overline{f\left(\alpha \right)}$ (open circles) for different time series lengths: $N=100000$ (top) and $N=250000$ (bottom) together with the corresponding theoretical values of $f\left(\alpha \right)$ (filled squares). Performance of MFDFA ($P^{\left(2\right)}$, left-hand column) and WTMM (${\psi }^{\left(3\right)}$, right-hand column) is compared. Error bars indicate standard deviation of data points calculated from 10 independent realizations of the process.

Figure 14

(Color online) Deterministic binomial cascade $a=0.65$: (a) Examples of fluctuation function $F_q\left(s\right)$ and rescaled partition function $\left[sZ\right(q,s∕41){]}^{1∕q}$ for different values of Rényi parameter: $q=-5$ (squares), $q=-2$ (triangles up), $q=2$ (circles), and $q=5$ (triangles down). (b) and (c) Scaling exponents $\tau \left(q\right)$ for MFDFA $\left(P^{\left(2\right)}\right)$ and for WTMM $\left({\psi }^{\left(3\right)}\right)$ are presented together with their theoretical behavior (solid line). Cascade was generated in $k=17$ stages and its length was equal to $N=131072$ points.

Figure 15

(Color online) Deterministic binomial cascade: Singularity spectra (open circles) for different mass allocations between subintervals: $a=0.55$ (top), $a=0.65$ (middle), and $a=0.75$ (bottom). Left-hand column: results from MFDFA with different polynomials: $P^{\left(1\right)}$ (squares), $P^{\left(2\right)}$ (triangles up), $P^{\left(3\right)}$ (diamonds), and $P^{\left(4\right)}$ (triangles down). Right-hand column: results from WTMM with different wavelets: ${\psi }^{\left(1\right)}$ (circles), ${\psi }^{\left(2\right)}$ (triangles up), ${\psi }^{\left(3\right)}$ (diamonds), and ${\psi }^{\left(4\right)}$ (triangles down). Theoretical $f\left(\alpha \right)$ spectra are denoted by solid lines. Cascade was generated in $k=17$ stages and its length was equal to $N=131072$ points.

Figure 16

(Color online) Deterministic binomial cascade with $a=0.55$ (top), $a=0.65$ (middle), and $a=0.75$ (bottom): Singularity spectra (open circles) for different cascade stages: $k=14$ ($N=16384$, circles), $k=16$ ($N=65536$, squares), and $k=17$ ($N=131072$, triangles) together with the corresponding theoretical spectrum (solid line). Left-hand column: results from MFDFA with $P^{\left(2\right)}$. Right-hand column: results from WTMM; for each value of parameter $a$ a wavelet with the best performance (see Fig. 15) is used: ${\psi }^{\left(1\right)}$ (top), ${\psi }^{\left(3\right)}$ (middle), and ${\psi }^{\left(2\right)}$ (bottom).

Figure 17

(Color online) Log-Poisson binomial cascade with $\gamma =1.4$: Mean singularity spectra $\overline{f\left(\alpha \right)}$ (open circles) obtained with MFDFA (left-hand column) for different polynomials: from $P^{\left(1\right)}$ (top) to $P^{\left(4\right)}$ (bottom) and with WTMM (right-hand column) for different wavelets: from ${\psi }^{\left(1\right)}$ (top) to ${\psi }^{\left(4\right)}$ (bottom), compared with the corresponding theoretical spectrum (solid lines). Cascade was generated in $k=17$ stages $(N=131072)$. Error bars indicate standard deviation of data points calculated from 10 independent realizations of the process.

Figure 18

(Color online) Log-Poisson binomial cascade with $\gamma =1.4$: $\overline{f\left(\alpha \right)}$ (open circles) for different cascade stages: $k=14$ ($N=16384$, top), and $k=16$ ($N=65536$, bottom) together with theoretical spectrum (solid lines). Performance of MFDFA ($P^{\left(2\right)}$, left-hand column) and WTMM (${\psi }^{\left(3\right)}$, right-hand column) is compared. Error bars indicate standard deviation of data points calculated from 10 independent realizations of the process. Plots should be compared with the corresponding panels of Fig. 17, where the results for signals of $k=17$ $(N=131072)$ are shown.

Figure 19

(Color online) Log-gamma binomial cascade with $\gamma =1$, $\beta =\ln 2$: $\overline{f\left(\alpha \right)}$ (open circles) obtained with MFDFA (left-hand column) for different polynomials: from $P^{\left(1\right)}$ (top) to $P^{\left(4\right)}$ (bottom) and with WTMM (right-hand column) for different wavelets: from ${\psi }^{\left(1\right)}$ (top) to ${\psi }^{\left(4\right)}$ (bottom), compared with theoretical spectra (solid lines). Cascade was generated in $k=17$ stages $(N=131072)$. Error bars indicate standard deviation of data points calculated from 10 independent realizations of the process.

Figure 20

(Color online) Log-gamma binomial cascade with $\gamma =1$, $\beta =\ln 2$: $\overline{f\left(\alpha \right)}$ (open circles) for different cascade stages: $k=14$ ($N=16384$, top) and $k=16$ ($N=65536$, bottom) together with theoretical $f\left(\alpha \right)$ (solid lines). Performance of MFDFA ($P^{\left(2\right)}$, left-hand column) and WTMM (${\psi }^{\left(3\right)}$, right-hand column) is compared. Error bars indicate standard deviation of data points calculated from 10 independent realizations of the process. Plots should be compared with the corresponding panels of Fig. 19, where the results for signals of $k=17$ $(N=131072)$ are shown.

Figure 21

(Color online) Log-normal binomial cascade with $\lambda =1.1$: $\overline{f\left(\alpha \right)}$ (open circles) obtained with MFDFA (left-hand column) for different polynomials: from $P^{\left(1\right)}$ (top) to $P^{\left(4\right)}$ (bottom) and with WTMM (right-hand column) for different wavelets: from ${\psi }^{\left(1\right)}$ (top) to ${\psi }^{\left(4\right)}$ (bottom) compared with the theoretical spectra (solid lines). Cascade was generated in $k=17$ stages $(N=131072)$. Error bars indicate standard deviation of data points calculated from 10 independent realizations of the process.

Figure 22

(Color online) Log-normal binomial cascade with $\lambda =1.1$: $\overline{f\left(\alpha \right)}$ (open circles) for different cascade stages: $k=14$ ($N=16384$, top), and $k=16$ ($N=65536$, bottom) together with theoretical $f\left(\alpha \right)$ (solid lines). Performance of MFDFA ($P^{\left(2\right)}$, left-hand column) and WTMM (${\psi }^{\left(3\right)}$, right-hand column) is compared. Error bars indicate standard deviation of data points calculated from 10 independent realizations of the process. Plots should be compared with the corresponding panels of Fig. 19, where the results for signals of $k=17$ $(N=131072)$ are shown.

Figure 23

(Color online) (a) Transaction-to-transaction logarithmic price increments of exemplary stock traded on Deutsche Börse: Volkswagen (VOW): Examples of fluctuation function $F_q\left(s\right)$ and rescaled partition function $\left[sZ\right(q,s∕41){]}^{1∕q}$ for different values of Rényi parameter: $q=-5$ (squares), $q=-2$ (triangles up), $q=2$ (circles), and $q=5$ (triangles down). (b) and (c) Mean scaling exponent $\overline{\tau \left(q\right)}$ for MFDFA $\left(P^{\left(2\right)}\right)$ and for WTMM $\left({\psi }^{\left(3\right)}\right)$ calculated from stocks corresponding to 30 companies of DAX (empty circles). Straight lines in bottom panels denote a monofractal, linearly uncorrelated reference signal.

Figure 24

(Color online) Mean $f\left(\alpha \right)$ spectra obtained by MFDFA (a) and by WTMM (b) for 30 stocks corresponding to DAX companies (empty circles) together with their randomly reshuffled counterparts (full squares). Error bars for the original data denote standard deviation of data points calculated from 30 stocks.

Figure 25

(Color online) (a) Transaction-to-transaction logarithmic price increments of exemplary DJI stock: IBM Corp.: Examples of fluctuation function $F_q\left(s\right)$ and rescaled partition function $\left[sZ\right(q,s∕41){]}^{1∕q}$ for different values of Rényi parameter: $q=-5$ (squares), $q=-2$ (triangles up), $q=2$ (circles), and $q=5$ (triangles down). (b) and (c) Mean scaling exponent $\overline{\tau \left(q\right)}$ for MFDFA $\left(P^{\left(2\right)}\right)$ and for WTMM $\left({\psi }^{\left(3\right)}\right)$ calculated from stocks corresponding to 30 DJI companies (empty circles). Straight lines in bottom panels denote a monofractal, linearly uncorrelated reference signal.

Figure 26

(Color online) $\overline{f\left(\alpha \right)}$ obtained by MFDFA (a) and by WTMM (b) for 30 stocks corresponding to DJI companies (empty circles) together with their randomly reshuffled counterparts (full squares). Error bars for the original data denote standard deviation of data points calculated from 30 stocks.