Multifractal surrogate-data generation algorithm that preserves pointwise Hölder regularity structure, with initial applications to turbulence

An algorithm is described that can generate random variants of a time series while preserving the probability distribution of original values and the pointwise Hölder regularity. Thus, it preserves the multifractal properties of the data. Our algorithm is similar in principle to well-known algorithms based on the preservation of the Fourier amplitude spectrum and original values of a time series. However, it is underpinned by a dual-tree complex wavelet transform rather than a Fourier transform. Our method, which we term the iterated amplitude adjusted wavelet transform can be used to generate bootstrapped versions of multifractal data, and because it preserves the pointwise Hölder regularity but not the local Hölder regularity, it can be used to test hypotheses concerning the presence of oscillating singularities in a time series, an important feature of turbulence and econophysics data. Because the locations of the data values are randomized with respect to the multifractal structure, hypotheses about their mutual coupling can be tested, which is important for the velocity-intermittency structure of turbulence and self-regulating processes.

Figure 1

A multifractional Brownian motion $m\left(t\right)$ is shown as a thick black line at the origin and is generated from ${\alpha }_p\left(t\right)=0.33+0.2sin\left(6\pi t\right)$. Two example IAAFT surrogates are shown in black and displaced vertically above $m\left(t\right)$, while two example Paluš surrogates are in black and displaced below $m\left(t\right)$. Surrogates generated by our algorithm are shown in gray and are immediately below $m\left(t\right)$.

Figure 2

(a) A multifractal signal $B$ generated using the Benzi et al. [43] algorithm, shown as a black line. (b) Close-up of a portion of $B$. Surrogate series generated using the IAAWT algorithm are shown in gray and displaced above the black line, and those from the Paluš [21] algorithm are in gray and displaced below in (a) and (b). (c) Structure function exponents ${\xi }_n$ as a function of moment number $n$, with error bars indicating $\pm 2$ standard deviations. The dark lines with circles are for 20 different realizations of the process with constant $P\left(\eta \right)$. The squares give the structure function scaling for the black lines in (a) and (b), the triangles are the results for 25 IAAWT surrogates of this data series, and diamonds are the results for 25 Paluš surrogates. Values are displaced horizontally from the integer value of $n$ for clarity.

Figure 3

An analysis of the third order moments of the velocity differences (increments) for a turbulence time series of the longitudinal velocity component $u_x$. The left-hand column (data as circles) shows the third moments of the absolute differences, while the right-hand column (data as squares) shows the third moments of the differences. The $z$ value in each plot is the distance of the data from the mean of the surrogates as a function of the standard deviation of the surrogates (the $z$ score). The values for the increment separation $r$ range from the Taylor scale, $\sim 0.02L$, to the integral scale, $L$, and are given on the left. The box plots indicate the values for the surrogates: the median is shown by a thick vertical line, the edges of the box are the lower and upper quartiles, and the whiskers extend up to $1.5$ times the interquartile range, with outliers shown by crosses.

Figure 4

(a) A fractional Brownian motion with an embedded chirp is shown in black. (b) An example IAAFT surrogate is shown in gray. (c) A multifractional Brownian motion with an embedded chirp is shown in black. (d) An IAAFT surrogate for (c) is shown in gray. The panels show a smaller segment of the full data set in each case, which has a support of $0\le t\le 1.2$.

Figure 5

(a) A multifractional Brownian motion with an embedded chirp is shown in black. (b)–(d) Three example IAAWT surrogates are shown in gray. The preservation of the pointwise Hölder exponents and the location of the oscillating components is clear. The panels show a smaller segment of the full data set in each case, which has a support of $0\le t\le 1.2$.

Figure 6

Multifractal spectra $D\left({\alpha }_p\right)$ estimated using the large deviation method (black) and compared to surrogate data (gray) for (a) and (b) a fractional Brownian motion with an embedded chirp and (c) and (d) a multifractional Brownian motion with an embedded chirp. (a) and (c) show IAAFT surrogate data, while IAAWT surrogates are used in (b) and (d). The multimodal structure of the spectrum indicates the presence of oscillating singularities in the original data in all cases, but this disappears in the IAAFT surrogates (although it is retained by the IAAWT surrogates).

Figure 7

Time series of $z$-score estimates based on the the original data and the mean and standard deviation for surrogates for ${\alpha }_p$ (black) and ${\alpha }_L$ (gray). The results are (a) for IAAFT surrogates and (b) for IAAWT surrogates. An almost sure threshold [62] is indicated by the dashed lines.

Figure 8

Box plots of the results for the velocity-intermittency quadrant analysis at various dimensionless distances downwind of the fence. Values for the actual data are shown as open triangles. The boxes themselves are based on the values for 39 IAAWT surrogates, with outliers indicated by a cross, the box consisting of the median (center line) and first and third quartiles (bottom and top edges of the box, respectively), and the whiskers extending up to 1.5 times the interquartile deviation.

Figure 9

An analysis of velocity intermittency from various experiments. The various lines correspond to data from a turbulent jet experiment [73] (solid black line with open squares), wake data at 8.5 m ${\mathrm{s}}^{-1}$ (gray dotted lines) and 24.3 m ${\mathrm{s}}^{-1}$ (gray solid lines) [32], and data below 150 wall units (solid black lines) and above 700 wall units (dotted black lines) at 6 m ${\mathrm{s}}^{-1}$ (open diamonds) and 8 m ${\mathrm{s}}^{-1}$ (solid circles) for a zero pressure gradient boundary layer [33]. Data for flow over bed forms [71] are shown as a solid black line. This figure is modified from C. J. Keylock et al. [Geophys. Res. Lett. 40, 1351 (2013)]. Copyright the American Geophysical Union (AGU) and reproduced with the permission of AGU.

Figure 10

A summary of the results in Fig. 8 where the distance between the data and the mean of the surrogates, normalized by the standard deviation of the surrogates (hence a $z$ score $z_{(u,\alpha )})$, is reported, with an average taken over all four quadrants.