A Brief Overview of Multifractal Time Series...Ivanov.) Another interesting property of the wavelet transform is that the coefficients at these maxima---which are a small fraction

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A Brief Overview of Multifractal Time Series

Part 1: Fractal behavior in time seriesPart 2: Using wavelets to detect singular behaviorPart 3: The fractal dimension of the singular behaviorPart 4: The singularity spectra of multifractal signalsPart 5: What one learns from the singularity spectra of multifractal signalsPart 6: Multifractality of healthy human heart rateBibliograpy

Part 1: Fractal behavior in time series

The functions f(t) typically studied in mathematical analysis are continuous and have continuous derivatives.Hence, they can be approximated in the vicinity of some time ti by a so-called Taylor series or power series

(eqn. 1)For small regions around ti, just a few terms of the expansion (eqn. 1) are necessary to approximate the functionf(t). In contrast, most time series f(t) found in "real-life" applications appear quite noisy (Fig. 1). Therefore, atalmost every point in time, they cannot be approximated either by Taylor series (or by Fourier series) of just a fewterms. Moreover, many experimental or empirical time series have fractal features--i.e., for some times ti, theseries f(t) displays singular behavior. By this, we mean that at those times ti, the signal has components withnon-integer powers of time which appear as step-like or cusp-like features, the so-called singularities, in the signal (see Figs. 1b,c).

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Figure 1: (a) A common example of numerically-generated "noise" with long-range power-law correlations. Thissignal has a power-law distributed power spectrum which increases as (frequency)0.6. (b) Cardiac interbeat intervals (in arbitrary units) for a healthy subject under ambulatory conditions. Note that the interbeat intervals areplotted against beat number. It is known that heart rate variability has long-range correlations characterized by apower spectrum that decreases as (frequency)-1.0. (c) Another example of numerically-generated noise also with long-range correlations but of a different type. In this case, the power spectrum decreases as (frequency)-1.4. Note how the high-frequency features of the signal decrease from (a) to (c). Note also: plotted in green are two instancesof cusp-singularities (c1) and plotted in blue (b1) is one example of a step-singularity. Panel (b) also illustratesanother complicating factor in many instances, singularities are not isolated, but may instead appear very close toone another, making their characterization rather more complex.

Formally, one can write (2, 4):

(eqn.2) where t is inside a small vicinity of ti, and hi is a non-integer number quantifying the local singularity of f(t) at t = ti.

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Part 1: Fractal behavior in time seriesPart 2: Using wavelets to detect singular behaviorPart 3: The fractal dimension of the singular behaviorPart 4: The singularity spectra of multifractal signalsPart 5: What one learns from the singularity spectra of multifractal signalsPart 6: Multifractality of healthy human heart rateBibliography

Part 2: Using wavelets to detect singular behaviorIn a signal with fractal features, an immediate question one faces is "how to quantify the fractal properties of such asignal?'' The first problem is to find the set of locations of the singularities {ti}, and to estimate the value of h for each ti.

Figure 2: Wavelet function at different scales and positions. We consider as the wavelet function the first derivative ofthe Gaussian and plot it (a) centered at position b = 0.25 and at scale a = 0.015, (b) centered at position b = 0.5 and at scale 2a, and (c) centered at position b = 0.75 and at scale 4a. Note that the wavelet takes values significantly differentfrom zero just for small ranges.

In contrast to the Fourier transform, which assumes that the signal is stationary at the time scales of interest, thewavelet transform instead determines the frequency content of a signal as a function of time (1). In the Fouriertransform, one determines the coefficients that best approximate a function f(t) as a sum of sines and cosines. Similarly,in the wavelet transform, one approximates a function f(t) as a sum of properly weighted basis functions. The basis in thewavelet transform are functions that, like the sines or cosines, can be considered at different frequency but, unlike thesines or cosines, are localized in time and hence have to be translated along the signal. An example of a wavelet basis isthe set of functions (see Fig. 2)

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(eqn. 3)where G'(t,a,b) is the first derivative of the Gaussian function, a is an inverse frequency and b is the time location. Onedetermines the coefficients of the wavelet transform by convolving f(t) with G'(t,a,b).

Besides being naturally suited to handle nonstationary signals, the wavelet transform easily removes polynomialcontributions that would otherwise mask singular (fractal) behavior. To illustrate this fact, consider a signal f(t) that one can expand for t close to ti as a series of the form of Eqn. (2). In a fractal analysis, one wants to measure hi, but for small

values of t - ti, the "trends'' (t - ti) k with k < hi will dominate the sum. Hence, one ideally wants to remove all terms (t - ti) k for which k < hi. By convolving f(t) with an appropriate wavelet function, one can put to zero all coefficients that wouldarise from such polynomial contributions. For instance, the derivative of order k of the Gaussian convolves to zero all polynomial terms up to order k - 1.

Figure 3 shows the wavelet decomposition of a heart rate signal for a healthy subject. The self-similar "arch''-likestructures in the figure indicate maxima of the modulus of the wavelet transform. They indicate the time locations (at eachscale) of the singularities in the signal. The figure helps illustrate two points. First, the singularities are not present forall times. Second, the location of the singularities, as a function of scale and time, have a fractal structure.


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Figure 3: Wavelet decomposition of heart rate signal for healthy subject. Top: Interbeat interval, in second, plottedagainst beat number for beats 1600 to 3400. Bottom: Modulus of the wavelet decomposition of the signal. The colorcode is displayed in the color bar at the bottom of the figure. Dark regions correspond to the maxima discussed in the text.The horizontal axis indicates beat number and the vertical axis indicates the "scale" of the wavelet (from 2 to 80 beats). Ahorizontal cut through the figure reveals the fluctuations on a specific time-scale. The analyzing wavelet used in thisdecomposition is the second derivative of the Gaussian. Note the fractal structure of maxima (and minima). Continuousmaxima line from small scales to large scales determine the time of the singularity at different scales. Merging maximalines indicate that at scales larger than the merging scale the singularities are no longer isolated. (Courtesy of Plamen Ch.Ivanov.)

Another interesting property of the wavelet transform is that the coefficients at these maxima---which are a small fractionof the total number of coefficients---are enough to encode the information contained in the signal (3). Moreover, as onefollows a maxima line from the lowest scale to higher and higher scales, one is following the same singularity. This factallows for the calculation of hi by a power law fit to the coefficients of the wavelet transform along the maxima line (5).

We ask the question "what comes out of our analysis of the signal?" The first possibility is that we find a single value hi = H for all singularities ti, the signal is then said to be monofractal (6, 7). The second, more complex, possibility is thatwe find several distinct values for h, the signal is then said to be multifractal (8, 9).

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Part 3: The fractal dimension of the singular behavior

The next problem is to quantify the "frequency" in the signal of a particular value h of the singularity exponents hi.Let us first assume that our signal is monofractal. Different possibilities can be considered. For example, the set oftimes with singular behavior {ti} may be a finite fraction of the time series and homogeneously distributed over thesignal. But {ti} may also be an asymptotically infinitesimal fraction of the entire signal and have a veryheterogeneous structure. That is, the set {ti} may be a fractal itself. In either case, it is useful to quantify theproperties of the sets of singularities in the signal by calculating their fractal dimensions (8).

Consider the signal in Fig. 4. This type of signal is usually called a Devil's staircase because it takes constant values except at a subset of points where it changes discontinuously (2, 4). At those points, the function f(t) has singularities. Moreover, all singularities are of the same type --i.e., the signal is monofractal.

Figure 4: A monofractal Devil's staircase. Top: Four iteration steps in the building of a Cantor set. The set isgenerated by removing from the middle of a segment a region with half the length of the segment. This rulegenerates a "dust'' of points with equal mass and a fractal distribution in time. The distribution is fractal because

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there are holes of all sizes between the dust. The fractal dimension of this dust is D = 1/2 (see text for details). Bottom: One can generate a Devil's staircase type of signal by integrating the fractal dust generated according tothe previous rule. Such a signal is shown in this panel. Note the discontinuities in the signal (2 k for the kth iteration). These discontinuities are the times where singularities occur. The singularities in this case are all of thesame type. Hence one has a single value of h in the signal. The corresponding fractal dimension is also D = 1/2. Because the signal of Fig. 4 is deterministic, we can easily identify the position of the singularities. Theirpositions are shown in the top panel of Fig. 4. One can see that the singularity points arise from the iteration of aCantor set rule. The signal in the bottom panel of Fig. 4 arises from integrating the "dust'' generated by theCantor rule (8).

One can easily calculate the fractal dimension of the Cantor set of singularities by using box counting methods. Thefractal dimension is, as usual, given by the relation

(4)where n(r) is the number of boxes of radius r needed to cover the fractal dust. For the deterministic fractal shown,after k iterations, one has r = (1/4) k , and n(r) = 2 k , yielding

(5)In the multifractal formalism, one says that the signal of Fig. 4 has a single type of singularity hi = 1/2 and that thesupport of that singularity has fractal dimension D(h=1/2) = 1/2. The curve D(h) is called the singularity spectrum of the time series, which for this case is zero everywhere except at a single point h = 1/2.

The signals in Figs. 1a,c are also monofractal. They are usually called fractional Brownian motion. For the signalin Fig. 1a we have h = -0.8 while for the signal in Fig. 1c we have h = 0.2. But in contrast with the devil's staircase of Fig. 4, for which singularities appear only for a very small and heterogeneous set of times, singularities appearuniformly throughout the signals in Figs. 1a,c. Hence, the fractal dimension of the set of singularities is one, thedimension of a line.

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Part 4: The singularity spectra of multifractal signalsOur analysis becomes more complex if instead of a single type of singularity, the signal of interest has multipletypes of singularities. As an example, consider the signal in Fig. 5 which is also a Devil's staircase (i.e., Fig. 4)because of its many singularities. But in contrast to the signal of Fig. 4, the types of singularities varyconsiderably. The reason for this variation is made clear by the top panel in Fig. 5. The type of fluctuations in localincrements vary considerably even for the fourth iteration.

Figure 5: A multifractal Devil's staircase. Top: Four iteration steps in the building of a multiplicative binomialcascade. The set is generated by partitioning the mass of the segment into two parts of equal length but un-equaldensities. For the case shown, the left half of the segment receives 1/4th of the mass while the right half receives3/4th of the mass. Bottom: One can generate a Devil's staircase type of signal by integrating the set generatedaccording to the previous rule. Such a signal is shown in this panel. Note the presence of numerous cusp-likefeatures in the signal. These cusps indicate the times where singularities occur. Because of the local variations inthe mass distribution of the binomial cascade of the Top panel, the singularities in this case are of several differenttypes.

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To quantify the variation in the local singularities of the signal of Fig. 5, we calculate the value of h at every singularity. Figure 6 shows the signal again and also, by a color coding, the value of h. Clearly hi can take many different values. Moreover, by focusing on a single color, i.e., a single value of h, one can uncover the fractal structure of the corresponding set of singularities.

Figure 6: Singularity decomposition of the multiplicative binomial process of Fig. 5. (a) Devil staircase after 9iterations. (b) Position and value of the different singularities for the signal in (a). (c) Color coding of (b). The dark blue background indicates absence of singularities. The color spectrum goes from dark blue to green to yellowand to reddish brown. Blue indicates small values of h while reddish brown indicates large values of h. Note that nosingularities appear at the edges because we do not enforce periodic boundary conditions on the signal and hencecannot perform calculations close to the edges. (d) Decomposition of the singularities into different setscorresponding to different values of h. The top panel displays singularities with values of h approximately twostandard deviations smaller than the mean h = 0.6. The middle panel displays singularities with the average h = 1.1.Finally, the bottom panel displays singularities with values of h approximately two standard deviations larger thanthe mean h = 1.6. (Note: The color panels in (d) have bars of a single color, unfortunately color and resolutionconflicts may give rise to bars of different colors.)

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Part 4: The singularity spectra of multifractal signalsOur analysis becomes more complex if instead of a single type of singularity, the signal of interest has multipletypes of singularities. As an example, consider the signal in Fig. 5 which is also a Devil's staircase (i.e., Fig. 4)because of its many singularities. But in contrast to the signal of Fig. 4, the types of singularities varyconsiderably. The reason for this variation is made clear by the top panel in Fig. 5. The type of fluctuations in localincrements vary considerably even for the fourth iteration.

Figure 5: A multifractal Devil's staircase. Top: Four iteration steps in the building of a multiplicative binomialcascade. The set is generated by partitioning the mass of the segment into two parts of equal length but un-equaldensities. For the case shown, the left half of the segment receives 1/4th of the mass while the right half receives3/4th of the mass. Bottom: One can generate a Devil's staircase type of signal by integrating the set generatedaccording to the previous rule. Such a signal is shown in this panel. Note the presence of numerous cusp-likefeatures in the signal. These cusps indicate the times where singularities occur. Because of the local variations inthe mass distribution of the binomial cascade of the Top panel, the singularities in this case are of several differenttypes.


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To quantify the variation in the local singularities of the signal of Fig. 5, we calculate the value of h at every singularity. Figure 6 shows the signal again and also, by a color coding, the value of h. Clearly hi can take many different values. Moreover, by focusing on a single color, i.e., a single value of h, one can uncover the fractal structure of the corresponding set of singularities.

Figure 6: Singularity decomposition of the multiplicative binomial process of Fig. 5. (a) Devil staircase after 9iterations. (b) Position and value of the different singularities for the signal in (a). (c) Color coding of (b). The dark blue background indicates absence of singularities. The color spectrum goes from dark blue to green to yellowand to reddish brown. Blue indicates small values of h while reddish brown indicates large values of h. Note that nosingularities appear at the edges because we do not enforce periodic boundary conditions on the signal and hencecannot perform calculations close to the edges. (d) Decomposition of the singularities into different setscorresponding to different values of h. The top panel displays singularities with values of h approximately twostandard deviations smaller than the mean h = 0.6. The middle panel displays singularities with the average h = 1.1.Finally, the bottom panel displays singularities with values of h approximately two standard deviations larger thanthe mean h = 1.6. (Note: The color panels in (d) have bars of a single color, unfortunately color and resolutionconflicts may give rise to bars of different colors.)

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Part 5: What one learns from the singularity spectra of multifractalsignals

The singularity spectrum D(h) quantifies the degree of nonlinearity in the processes generating the output f(t)in a very compact way (see Fig. 7). For a linear fractal process the output of a system will have the same fractalproperties (i.e., the same type of singularities) regardless of initial conditions or of driving forces. In contrast,nonlinear fractal processes will generate outputs with different fractal properties that depend on the inputconditions or the history of the system. That is, the output of the system over extended periods of time will displaydifferent types of singularities.

Figure 7: Singularity spectra of the two signals considered in Figs. 4 and 5. Note the broad range of values of hwith non-zero fractal dimensions for the multiplicative binomial process and contrast it to the "pulse"-likespectrum for the Cantor signal.

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A classical example from physics is the Navier-Stokes equation for fluid dynamics (10). In the turbulent regime,this nonlinear equation generates a multifractal output with a characteristic singularity spectrum D(h) similar, for some types of turbulence, to D(h) for the binomial multiplicative process.

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Part 6: Multifractality of healthy human heart rate

Multifractality has been uncovered in a number of fundamental physical and chemical processes (9). Recently, itwas also reported that heart rate fluctuations of healthy individuals are multifractal (11). This finding posed newchallenges to our understanding of heart rate regulation as most modeling of heart rate fluctuations over long timescales had concerned itself only with monofractal properties (12). For example, it appears that a majorlife-threatening condition, congestive heart failure, leads to a loss of multifractality (Fig. 8).

Figure 8: Singularity spectra of the of heart rate signals for healthy and diseased hearts. Note the broad range ofvalues of h with non-zero fractal dimensions for the healthy heart beat. This is indicative of multifractal dynamics.These dynamics are quite different from the ones responsible for the binomial multiplicative process as the twosingularity spectra are quite different. For diseased patients, suffering from a life-threatening condition namedcongestive heart failure, we find a narrow range of values of h with non-zero fractal dimension. Even though thisrange is not really pulse-like, it is still likely that the dynamics are monofractal. The reason is that for finitesignals there will always be some small error in the estimation of the value of h at a singularity. This error will lead to a small widening of the singularity spectrum (3).

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More importantly, neither monofractal nor multifractal behaviors are accounted for by current understanding ofphysiological regulation based on homeostasis. Hence it would be beneficial, perhaps, to uncover howmultifractality in the healthy heart dynamics arises. Two distinct possibilities can be considered. The first is thatthe observed multifractality is primarily a consequence of the response of neuroautonomic control mechanisms to activity-related fractal stimuli. If this were the case, then in the absence of such correlated inputsthe heartbeat dynamics would not generate such a heterogeneous multifractal output. The second is that theneuroautonomic control mechanisms---in the presence of even weak external noise---endogenously generatemultifractal dynamics.

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Next: Bibliography

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Bibliography: 1. Daubechies, I., Ten Lectures on Wavelets (S.I.A.M., Philadelphia, 1992).

2. Feder, J., Fractals (Plenum Press, 1988).

3. Muzy, J. F., Bacry, E. & Arneodo, A. (1994) The multifractal formalism revisited with wavelets. Int. J. Bifurc. Chaos.4, 245-302.

4. Vicsek, T., Fractal Growth Phenomena, 2nd ed. (World Scientific, Singapore, 1993).

5. Struzik. Z. R. (2000) Determining local singularity strengths and their spectra with the wavelet transform. Fractals 8, 163-179.

6. Hurst, H. E. (1951) Long-term storage capacity of reservoirs. Trans. Am. Soc. Civ. Eng. 116, 770-808.

7. Bunde, A. & Havlin, S., Fractals and Disordered Systems, 2nd ed. (Springer-Verlag, Berlin, 1996).

8.Mandelbrot, B. B., The Fractal Geometry of Nature (W. H. Freeman, 1983).

9. Stanley, H.E. & Meakin, P. (1988) Multifractal phenomena in physics and chemistry. Nature, 335, 405-409.

10. Meneveau, C. & Sreenivasan, K.R. (1987) Simple multifractal cascade model for fully developed turbulence. Phys. Rev. Lett. 59, 1424-1427.

11. Ivanov, P. Ch., Amaral, L. A. N., Goldberger, A. L., Havlin, S., Rosenblum, M. B., Struzik, Z. & Stanley, H. E.(1999) Multifractality in healthy heartbeat dynamics. Nature 399, 461-465.

12. Amaral, L.A.N., Goldberger, A.L., Ivanov, P.Ch. & Stanley H.E. (1998) Scale-independent measures and pathologiccardiac dynamics. Phys. Rev. Lett. 81, 2388-2391.

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A Brief Overview of Multifractal Time Series...Ivanov.) Another interesting property of the wavelet transform is that the coefficients at these maxima---which are a small fraction