fractal characterization of complexity in temporal ...€¦ · fractal characterization of...

INSTITUTE OF PHYSICS PUBLISHING PHYSIOLOGICAL MEASUREMENT

Physiol. Meas. 23 (2002) R1–R38 PII: S0967-3334(02)18520-0

TOPICAL REVIEW

Fractal characterization of complexity in temporalphysiological signals

A Eke, P Herman, L Kocsis and L R Kozak

Institute of Human Physiology and Clinical Experimental Research, Semmelweis University,Faculty of Medicine, PO Box 448, 1446 Budapest, Hungary

E-mail: [email protected]

Received 27 July 2001Published 28 January 2002Online at stacks.iop.org/PM/23/R1

AbstractThis review first gives an overview on the concept of fractal geometry withdefinitions and explanations of the most fundamental properties of fractalstructures and processes like self-similarity, power law scaling relationship,scale invariance, scaling range and fractal dimensions. Having laid down thegrounds of the basics in terminology and mathematical formalism, the authorssystematically introduce the concept and methods of monofractal time seriesanalysis. They argue that fractal time series analysis cannot be done in aconscious, reliable manner without having a model capable of capturing theessential features of physiological signals with regard to their fractal analysis.They advocate the use of a simple, yet adequate, dichotomous model offractional Gaussian noise (fGn) and fractional Brownian motion (fBm). Theydemonstrate the importance of incorporating a step of signal classificationaccording to the fGn/fBm model prior to fractal analysis by showing thatmissing out on signal class can result in completely meaningless fractalestimates. Limitation and precision of various fractal tools are thoroughlydescribed and discussed using results of numerical experiments on idealmonofractal signals. Steps of a reliable fractal analysis are explained. Finally,the main applications of fractal time series analysis in biomedical research arereviewed and critically evaluated.

Keywords: time series analysis, complexity, fractal, self-similarity,correlations, laser Doppler-flux, heart rate variability.

1. Introduction

Living organisms are complex both in their structures and functions. Parameters ofhuman physiological functions such as arterial blood pressure (Blaber et al 1996), breathing

0967-3334/02/010001+38$30.00 © 2002 IOP Publishing Ltd Printed in the UK R1

http://stacks.iop.org/pm/23/R1

R2 A Eke et al

(Dirksen et al 1998) and heart rate (Huikuri et al 2000), etc, are not stable but do fluctuate intime (Glass 2001). The actual pattern of these fluctuations results from an interaction betweendisturbances from the external or internal environments and actions on the part of controlmechanisms attempting to maintain the equilibrium state of the system. The actual valueof a parameter is measured by some form of a receptor whose signal is then compared bya control centre to an internal reference value, the set-point. The difference in the two, theerror signal, determines the direction and magnitude of change that brings the actual valueof the controlled parameter near the set-point. The homeostatic view of physiological controlplaces emphasis on the constancy of the controlled parameter in spite of its evident fluctuationsaround the set-point. Temporal fluctuations, however, can also result from intrinsic sources,such as the activity of organism or ageing both affecting the set-point to varying extent. Hencethe homeodynamic concept of physiological control seems more realistic (Goodwin 1997).Understanding the actual mechanisms involved is usually attempted in two ways: (i) thereductionistic approach identifies the elements of the system and attempts to determine thenature of their interactions; (ii) the holistic approach looks at detailed records of the variationsof the controlled parameter(s) and seeks a consistent pattern indicative of the presence of acontrol scheme. These approaches are not mutually exclusive; they are indeed often usedtogether. Both use mathematical (e.g. statistical) methods and lately mathematical models forrendering the findings conclusive or shaping and strengthening the hypotheses.

Fluctuations in physiological systems are nonperiodic. Stochastic, chaotic and noisychaotic models can mathematically treat these patterns. The stochastic (random) modelsassume that the fluctuations result from a large number of weak influences. The chaoticmodels conceptualize that strong nonlinear interactions between a few factors shape thefluctuations. A combination of these two into the noisy chaotic model is possible. Among thestochastic approaches, the fractal models give the best description of reality. In this reviewwe concentrate on the fractal analysis of time series capturing the nonperiodic fluctuations ofphysiological systems.

The classic theory of homeostasis focused on the set-point as determined by the meanof the physiological signal, the fluctuations around the mean were thus discarded as ‘noise’.Research of the last decades revealed that the homeodynamic and holistic concepts, suchas fractal and nonlinear analyses could be very useful in understanding the complexity ofphysiological functions. They revealed that (i) physiological processes can operate far fromequilibrium; (ii) their fluctuations exhibit a long-range correlation that extends across manytime scales; (iii) and underlying dynamics can be highly nonlinear ‘driven by’ deterministicchaos. In our view the fractal and chaotic approaches are not mutually exclusive, because theypresent two ways of looking at physiological complexity (Bassingthwaighte et al 1994). Thefractal approach is aimed at demonstrating the presence of scale-invariant self-similar features(correlation, memory) in a detailed record of temporal variations of a physiological parameter,while the very same record can also be analysed according to the concept of deterministicchaos attempting to find the minimal set of often simple differential equations capable ofproducing the erratic, random dynamics of time series on deterministic grounds.

2. Concept of fractal geometry

Fractal geometry is rooted in the works of late 19th and early 20th century mathematicianswho found their fancy in generating complex geometrical structures from simple objects likea line, a triangle, a square, or a cube (the initiator) by applying a simple rule of transformation(the generator) in an infinite number of iterative steps. The complex structure that resultedfrom this iterative process proved equally rich in detail at every scale of observation, and

Fractal time series analysis R3

1D von-Koch curve

Initiator Generator Fractal

2D Sierpinski triangle

3D Menger sponge

Figure 1. Ideal, mathematical fractals. These structures can be generated by repetitively applyinga single rule of generation (generator) to the object on subsequent scales starting out with the oneon the largest (initiator). Note that this procedure in a mere three generations result in a complex,self-similar structure (fractal).

when their pieces were compared to larger pieces or to those of the whole, they proved similarto each other (see von Koch curve, Sierpinski gasket, Menger sponge (figure 1), Cantor set,Peano and Hilbert curve (Peitgen et al 1992)).

These peculiar-looking geometrical structures lay dormant until Benoit Mandelbrotrealized that they represent a new breed of geometry suitable to describe the essence inthe complex shapes and forms of nature (Mandelbrot 1982). Indeed, traditional Euclideangeometry with its primitive forms cannot describe the elaborated forms of natural objects. AsMandelbrot (1982) put it: ‘Clouds are not spheres, mountains are not cones, coastlines arenot circles, and bark is not smooth, nor does lightning travel in a straight line.’ Euclideangeometry can handle complex structures only by breaking them down into a large numberof Euclidean objects assembled according to an equally large set of corresponding spatialcoordinates. The complex structure is thus converted to an equally complex set of numbersunsuitable to grab the essence of a design or to characterize its complexity. In nature, similarlyto the iterative process of the von Koch curve, complex forms and structures, such as a tree,begin to take shape as a simple initiator, the first sprout that is, and evolve by reapplying thecoded rule of the generator by branching dichotomously over several spatial scales. A holisticgeometrical description of a tree is thus possible by defining the starting condition (initiator),the rule of morphogenesis (generator) and the number of scales to which this rule should beapplied (scaling interval).

3. Properties of fractal structures and processes

Unlike Euclidean geometry that applies axioms and rules to describe an object of integerdimensions (1, 2 and 3), the complex geometrical objects mentioned above can be characterizedby recursive algorithms that extend the use of dimension to the noninteger range (Herman et al2001). Hence, Mandelbrot named these complex structures fractals emphasizing their

R4 A Eke et al

Exact fractal Statistical fractal

Pial arterial networkMandelbrot tree

Figure 2. Two basic forms of fractals. Using the example of spatial fractals, the two basic formsof fractals are demonstrated. An exact fractal is assembled from pieces that are an exact replicaof the whole, unlike the case of statistical fractals where exact self-similar elements cannot berecognized. These structures, like the skeletonized arborization of a pial arterial tree running alongthe brain cortex of the cat, are fractals, too, but their self-similarity is manifested in the power lawscaling of the parameters characterizing their structures at different scales of observation.

fragmented character, and the geometry that describes them as fractal geometry using theLatin word ‘fractus’ (broken, fragmented). Fractals cannot be defined by axioms but as a setof properties instead, whose presence indicates that the observed structure is indeed fractal(Falconer 1990).

3.1. Self-similarity

Pieces of a fractal object when enlarged are similar to larger pieces or to that of the whole.If these pieces are an identical rescaled replica of the other, the fractal is exact (figure 2).When the similarity is present only in between statistical populations of observations of agiven feature made at different scales, the fractal is statistical. Mathematical fractals such asthe Cantor set or the von Koch curve are exact; most natural fractal objects are statistical.

Self-similarity needs to be distinguished from self-affinity. Self-similar objects areisotropic; i.e. the scaling is identical in all directions, therefore when self-similarity is tobe demonstrated the pieces should be enlarged uniformly in all directions. Self-affine objectsare also fractals, but scaling is anisotropic, i.e. in one direction the proportions betweenthe enlarged pieces are different from those in the other. This distinction is, however, oftensmeared and for the purpose of being more expressive, self-similarity is used when self-affinityis meant (Beran 1994). Formally, physiological time series are self-affine temporal structures,because the units of their amplitude is not time (figure 3) (Eke et al 2000).

3.2. Power law scaling relationship

When a quantitative property, q, is measured in quantities of s (or on scale s, or with aprecision s), its value depends on s according to the following scaling relationship:

q = f (s). (1)

When the object is not fractal, the estimates of q using progressively smaller units of measure,s, converge to a single value. (Consider a square of 1 × 1 m, where q is its diagonal. Estimatesof q in this case converge to the value of

√2 m.) For fractals, q does not converge but, instead

exhibits a power law scaling relationship with s, whereby with decreasing s it increases withoutany limit

q = psε (2)


R=20

640

652

508

395

327

0

R=2-3

800

Blo

od C

ell P

erfu

sion

, a.

u.

296

247

R=2-5

200

Time, seconds

Figure 3. Self-affinity of temporal statistical fractals. Fractals exist in space and time. Here, bloodcell perfusion time series monitored by laser-Doppler flowmetry (LDF) from the brain cortex of ananesthetized rat is shown (Eke et al 2000). The first 640 elements of the 217 elements of the LDFtime series are shown that were 3.2 s long in real time. Note the spontaneous, seemingly random(uncorrelated) fluctuation of this parameter. Scale-independent, fractal temporal patterns in theseblood cell perfusion fluctuations can be revealed. Compare the segments of this perfusion timeseries, displayed at different resolutions given by R shown on the right. If any enlarged segmentof the series is observed at scale s = 1/R and its amplitude is rescaled by RH, where H is theHurst coefficient, the enlarged segment is seen to look like the original. The impression is thatthe segments have a similar appearance irrespective of the timescale at which the signal is beingobserved. The degree of randomness resulting from the range of excursions around the signalmean blending different frequencies into a random pattern, indeed seems similar. Because scalingfor this structure is anisotropic in that in one direction (time) the proportions between the enlargedpieces is different than in the other (amplitude of perfusion), this structure is not self-similar butself-affine. (For this particular time series H = 0.23.)

where p is a factor of proportionality (prefactor) and ε is a negative number, the scalingexponent. The value of ε can be easily determined as the slope of the linear regression fit tothe data pairs on the plot of log q versus log s:

log q = logp + ε log s. (3)

Data points for exact fractals are lined up along the regression slope, whereas those ofstatistical fractals scatter around it (figure 4) since the two sides of equation (3) are equal onlyin distribution.

3.3. Scale-invariance

The ratio of two estimates of q measured at two different scales, s1 and s2, q2/q1 depends onlyon the ratio of scales (relative scale), s2/s1, and not directly on the absolute scale, s1 or s2

q2/q1 = psε2/psε1 = (s2/s1)

ε. (4)

For exact fractals s2/s1 is a discrete variable. For instance, in case of the von Koch curve seenin figure 1, s2/s1 can only assume values of a geometrical series such as 1/3, 1/9, . . . etc. Forstatistical fractals, like those of nature, s2/s1 may change in a continuous fashion still leavingthe validity of equation (4) unaffected. This equation is commonly referred to as the (absolute)

R6 A Eke et al

0 1 2 30

1

2

3

4

5

6Mandelbrot tree: +

Pial fractal: .

log

q

log s

Figure 4. Power law scaling relationship for exact and statistical fractals. The scaling exponent, ε,relating to a quantitative property of the object, q, observed at a scale, s, can be easily determinedbased on equation (3) on a log–log scale of q versus s. Data points for exact fractals (Mandelbrottree of figure 2) are lined up along the regression slope, whereas those of a statistical fractal (pialarterial network of figure 2) are scattered along it, since for these fractals equation (3) equals onlyin distributions.

scale-invariant property of fractals. What is behind this scale-invariance is the self-similarityof fractals: quantitative properties of the pieces because the uniform geometrical structuredepends only on the ratio of the scales and not on the absolute scale. With regard to the vonKoch curve in figure 1, if property q is considered its length and is measured with a 1/3 shorterstick, due to the emergence of newer infoldings of the structure it will be found q2/q1 = 4/3times longer, which yields a scale-invariant scaling exponent

4

3=

(1

3

)εε = log 4/3

log 1/3≈ −0.2619. (5)

Equation (4) holds only for exact fractals. For statistical fractals the two sides of equation (4)are equal only in distribution (=d must be used instead of =)

q2/q1 =d psε2/ps

ε1 =d (s2/s1)

ε. (6)

3.4. Scaling range

For natural fractals scale-invariance holds only for a restricted range of the absolute scale(Avnir et al 1998) and these fractals are often referred to as prefractals (Bassingthwaighte et al1994). The upper limit of validity,or the upper cut-off point of equation (7),smax, for prefractalsfalls into the range of the size of the structure itself, likewise the lower cut-off point, smin, fallsinto the dimensions of the smallest structural elements. The scaling range, SR, is given indecades

SR = log10(smax/smin). (7)

3.5. Euclidean, topological and fractal dimensions

The Euclidean or embedding dimension, E, is an integer indicating that the consideredgeometrical object is found on a line (E = 1), on a plane (E = 2), or in three space


.P

Topologicaldimension of PX=1, Y=-1

Euclideandimension of PX=1, Y=-1, Z=0.1

X

Z

Y

Figure 5. Euclidean and topological dimension. On the example of point P on a warped surfacethe principle of these two dimensions are demonstrated.

(E = 3). E gives the number of coordinates needed to determine the position of a pointof the object in space, as it follows from the concept of dimensions in Euclidean geometry.The topological dimension, DT, indicates the numbers needed to determine the position of apoint on the actual geometrical structure. Lines, surfaces and three-dimensional solids haveDT of 1, 2, and 3. Note, however, that because DT � E, for the warped surface shown infigure 5 DT = 2 and E = 3. The fractal dimensions, D, although having multiple definitions,have one thing in common: their value is usually a noninteger, fractional number, hence thisdimension is referred to as fractal. For this very reason Mandelbrot named these structuresfractals. D is related to the scaling exponent of a given attribute of the object in a simple,straightforward manner and characterizes one specific feature of a fractal. Mandelbrot definedfractals as structures with DT � D � E.

Self-similarity dimension (Dss). Dss tells how many structural units of the observed object(i.e. line segments of the von Koch curve), N, are seen at a given resolution, R = 1/s

N = RDss which yields Dss = logN/ logR. (8)

Dss can only be determined for an exact fractal, which strictly obeys a single iterative rule ofgeneration, like the von Koch curve. For this structure Dss = 1 − ε, where ε here is the scalingexponent of the length of the curve (Bassingthwaighte et al 1994), thus using equation (5) Dss

yields 1.2619. As its fractal (fractional) dimension suggests, this structure indeed occupiesmore space than a line (E = 1) but less than a 2D surface (E = 2).

Capacity dimension (Dcap) and its simplest implementation, the box-counting dimension(Dbox). The concept of capacity dimension was developed to estimate Dss for real, nonexact,statistical fractals. Dcap is a generalization of Dss and is calculated as follows. It uses ‘balls’whose dimension equals E of the space in which the object is embedded. For E = 1 the ballis a line segment of length 2r, for E = 2 it is a circle with radius r, and for E = 3 it is a spherewith radius r. The object is to be covered by balls so that its every point is enclosed withinat least one ball. It is important to note that overlapping balls are allowed. The minimumnumber of balls of size r needed to cover the object, N(r), is found, then r is decreased andN(r) is found again. Dcap tells how the number of balls needed to cover the object changes asthe size of the ball is decreased (Bassingthwaighte et al 1994).

N = RDcap where R = 1/r (9)

and the radius of the ball covering the whole object is 1. Equation (9) gives a precisemathematical definition of Dcap, but it does not allow for an effective calculation of it. Dbox isa practical implementation of the algorithm of Dcap. It uses a nonoverlapping grid of boxes

R8 A Eke et al

instead of overlapping balls, which greatly simplifies the procedure and makes the calculationof D applicable to traced 2D or 3D objects feasible. The essence of this method is to coverthe image or structure by this grid of nonoverlapping boxes of various edge lengths, L = 1/R,and to determine the number of boxes covering any part of the object N. N is determined forprogressively smaller box sizes. Dbox is given as the slope of a linear regression fit to datapairs on a log–log plot of N as a function of R

N = RDbox which yields Dbox = logN/ logR (10)

and the edge length of the box containing the whole object is 1. The fractal dimension fora line segment, square and cube equals the Euclidean and the topological dimension and are1, 2, and 3, respectively. For 2D and 3D fractals, D falls in between these landmark valuesand gives a good characterization of the space-filling properties of the structure. The moreD differs from DT and is closer to E, the more the structure invades the Euclidean space. Atthe beginning, Mandelbrot emphasized the fractional value of D. He also noted, however, thatfractals could be found with integer D like the Peano curve whose D = 2 (Mandelbrot 1982,Peitgen et al 1992). Self-similarity is therefore a more important attribute of fractals than theirnoninteger, fractal dimensions.

Definitions of the fractal character of physiological signals are very diverse in the literature.One finds terms like ‘self-similar signal (or process)’, ‘1/f noise’, ‘long-memory process’,‘power-law process’, ‘fractional Gaussian noise (fGn)’, ‘fractional Brownian motion (fBm)’,and ‘fractional ARIMA process (fARIMA)’. Each of these terms has its own definition and infact they are interrelated in specific ways that will be explored in the following.

4. Concept and methods of fractal time series analysis

The aim of fractal time series analysis is to identify the presence of one or more of thefollowing fundamental features: self-similarity, power law scaling relationship and scale-invariance. Fractal analysis of time series cannot be done without an a priori notion of amodel incorporating the most essential features of the underlying physiological process(es).The analytical tool should be selected according to the proposed model. Fractal methods arediverse, but their approaches have one thing in common in that they employ equation (2) infitting their proposed model to data pairs of log feature versus log scale for finding the scalingexponent, ε, from the regression slope. The working hypothesis is that behind the complex—seemingly random—fluctuations of the signal one may find a time-invariant mechanism thatone would wish to describe in the smallest possible number of parameters.

A time series is defined as a discrete series, xi, for i = 1, . . . , N, and is a sampledrepresentation of a temporal process or signal, x(t) sampled at equal intervals in time,�t. Thesampling frequency, fs = 1/�t. The length of the series is often referred to as N, that shouldclearly be discerned from the actual length (duration) of the series in real time, T = N �t.

A time series is therefore a digital signal, thus its analysis is often called digital signalprocessing (Weitkunat 1991). Methods of time series analysis can be used in cases whenx varies not in time but space, in which case the notion of time series is extended to allordered sequences of observed data. Temporal (or for that matter, spatial) sampling requiresthe constancy of the sampling interval, in which case the resultant series is equidistant.Application criteria of the analytical methods are often relaxed, hence they seem to work onnonequidistant datasets, too. Caution is warranted, though, since nonequidistant datasets dorequire special analytical treatment.

The term ‘fractal time series’ in the literature in general refers to a single fractal ormonofractal temporal signal. Multifractal time series are heterogeneous, self-similar only


in local ranges of the structure, their fractal measure does vary in time; hence they can becharacterized by a set of local fractal measures (Bassingthwaighte et al 1994, Ivanov et al1999). This review is about monofractal time series analysis only, for its methods arewidely used in biomedical research in contrast with those for multifractal analyses which aremaking their first, though by no means promising, steps towards a wider scope of application(Ivanov et al 1999).

4.1. Time domain methods

If data are collected as a function of time, the series containing these data is said to be inthe time domain. Analyses done on signals without any prior transformation are referred toas time domain methods (Weitkunat 1991), aimed at identifying the statistical dependenceof the elements of the time series in contrast to descriptive statistical approaches providingsuch measures as the mean, standard deviation etc that treat the elements of the sampleas independent events. In this sense, these two approaches are complementary. In fact,some fractal time domain methods like dispersional analysis, scaled windowed variance, ordetrended fluctuation analysis etc do take descriptive statistical measures of the time seriesalthough in a scale-dependent manner that allows revealing of the scale-invariant properties.

A fundamental property of a signal that the physiologist must be aware of is its stationarity.We will see below that it bears importance in fractal analysis of time series too, becausestationary and nonstationary signals require different methods in their analyses. For astationary signal the probability distribution of signal segments is independent of the (temporal)position of the segment and segment length,which translates into constant descriptive statisticalmeasures such as mean, variance, correlation structure etc over time (figure 6).

According to a simple, yet realistic dichotomous model, signals are seen as realizationsof one of two temporal processes: fractional Brownian motion (fBm), and fractional Gaussiannoise (fGn) (Eke et al 2000). The fBm-signal is nonstationary (figure 6) with stationaryincrements. Physiological signals are generally nonstationary. An fBm signal, x, is self-similar in that its sampled segment xi,n of length n is equal in distribution to a longer segmentxi,sn of length sn when the latter is rescaled (multiplied by s−H). This means that every statisticalmeasure, mn, of an fBm time series of length n is proportional to nH

xi,n =d s−Hxi,sn (11)

mn =d pnH which yields logmn =d logp +H logn (12)

where H is the Hurst-coefficient and p is a suitable prefactor. H ranges between 0 and 1. Happroaching 1 describes a signal of smooth appearance, while H close to 0 is one with roughstructuring (‘hairy’ appearance). Increments yi = xi − xi−1 of a nonstationary fBm signal yielda stationary fGn signal and vice versa, cumulative summation of an fGn signal results in anfBm signal (figure 7). By definition they have the same H as their fractal measure. Note thatmost methods listed below that have been developed to analyse statistical fractal processes,thereby applicable in physiology, share the philosophy of equation (12) in that in their ownways all attempt to capture the power law scaling in the various statistical measures of theevaluated time series.

Note, that the quantitative properties (in equation (2): q) of a process as captured in itsstatistical measures (in equation (12): m), the scale (in equation (2): s) at which the process isinvestigated as determined by the width of the observation window (in equation (12): n) relatein the same power law manner for temporal and spatial fractals alike. Equation (2) gives theoverall formalism of the fractal power law relationship, while equation (12) adapts it to thespecific case of a statistical temporal fractal process.

R10 A Eke et al

-20

-10

0

10

20Stationary (fGn)

Am

plitu

de

Nonstationary (fBm)

0 1000 2000 3000 40000

20

40

60

80

Time

Var

ianc

e

0 1000 2000 3000 4000Time

Figure 6. Stationary and nonstationary time series. The two pure monofractal time series (upperpanels) were generated by the method of Davies and Harte (DHM) (1987) according to thedichotomous model of fractional Gaussian noise (fGn) and fractional Brownian motion (fBm)(Eke et al 2000). DHM produces an fGn signal, whose cumulatively summed series yields an fBmsignal. These two signals differ in their variance (lower panels) in that the fGn signal is stationary,hence its variance is constant, unlike that of the nonstationary fBm signal whose variance increaseswith time. This difference explains why the analysis of these signals in the time domain requirespecial methods capable of accounting for the long-term fluctuations and increasing variance inthe fBm signal.

Hurst’s rescaled range analysis (R/S). The first method for assessing H was invented byHurst (1951) when confronting the question of how high the Aswan dam had to be built so thatit would contain the greatly varying levels of the Nile within a given window of observation,n (figure 8). The logic he followed was governed by the three criteria of an ideal reservoir:(1) the outflow is uniform, (2) the level is the same at the beginning and at the end of theobservation window, (3) the reservoir never overflows (Bassingthwaighte et al 1994). Helooked at restrospective records of water levels, xi, which are proportional to the velocity ofwater inflow to the dam. Hurst assumed a uniform outflow, which can be calculated as themean of the varying inflow. The time series of the increase in water volume in the container isgiven by the summed difference of inflow and outflow yj = ∑j

i=1 (xi − xi)�t . The range ofyj , R = ymax − ymin, determines how high the dam should be built. In addition, Hurst dividedthe range by the standard deviation of inflow fluctuations, S, and found much to the surprise ofcontemporary statisticians that R/Sn showed a power law scaling relationship with the lengthof observation n

(R/S)n = pnH which yields log(R/S)n =d logp +H logn (13)

where p is a prefactor. This method is now known as Hurst’s rescaled range (R/S) analysis. Hiswork led Mandelbrot and Wallis (1969) to discover the widespread occurrence of self-similarfluctuations in natural phenomena. The R/S analysis is a rather complex method, which isapplicable to fGn signals or to differenced fBm signals.


fBmH

0.1

0.5

0.9

fGn

Figure 7. Typical patterns of fractional Gaussian noise (fGn) and fractional Brownian motion(fBm) signals. Three signals are shown in each signal category of the fGn/fBm model with a Hurstcoefficient, H = 0.1, H = 0.5, and H = 0.9. The case of H = 0.5 is the special case of uncorrelated(random) white noise and random walk (Eke et al 2000). The appearance of the fBm signal withH = 0.1 is rather ‘hairy’ due to the strong anticorrelation in between the successive differences(increments) of this series, while due to the noisy appearance of the fGn signals this cannot readilybe seen for the fGn signal with H = 0.1. The case of H = 0.9 shows the correlated structuring offractal motion and noise signals.

Autocorrelation analysis (AC). The aim is to determine to what extent the value ofone given event, xi, of the time series depends on its past values k lag apart, xi−k. The k-lagautocorrelation coefficient, ρk, defines how strongly the momentary value of the signal dependson the one k lag before. The k-lagged autocorrelation may be positive, zero, or negative withinboundaries of 1 and –1 (figure 9, note that for fGn signals the lower bound is −0.5!). Apositive correlation indicates that the trend of deviation of xi and xi−k relative to the meanof the signal is in the same direction, while with negative values of ρk the trend is opposite(anticorrelation). Estimates of autocorrelation for lags k = 0, 1, . . . , can be calculated as

ρk =1

N − k − 1

∑Ni=k+1 (xi − µ) (xi−k − µ)

1N − 1

∑Nt=1 (xi − µ)2

where µ = 1

N

N∑i=1

xi. (14)

The autocorrelation function of an fGn-signal with 0 � H � 1; k = 0, 1, . . . , is

ρk = 12 (|k + 1|2H − 2|k|2H + |k − 1|2H) (15)

where H is the Hurst-coefficient of the fBm signal whose increments yield the fGn signal inquestion. Equation (15) can be derived from equation (12) (Beran 1994). The closer H isto 1 the slower ρk decays, in other words the longer is the memory of the process (figure 9).On this ground, fractal signals are often called long-memory processes. Note, however,that in the mathematical sense long-memory correlation (or long-range dependence) isfor 0.5 < H < 1 (Beran 1994), hence H = 0.5 when there is no correlation in the fGnsignal (a typical random white noise), and when H < 0.5 the fGn signal shows anticorrelation(Eke et al 2000). The correlation for the nearest neighbours were determined by van Beeket al (1989) as

ρ1 = 22H−1 − 1. (16)

R12 A Eke et al

Water level (fGn signal)

Cumulative difference of inflow and outflow (fBm signal)

xi

yi R

S

n

Figure 8. Basic concept of Hurst’s rescaled range method. This is a time domain method. Itcalculates a descriptive statistical measure, the standard deviation, S, of a noise signal, xi, withinwindows of observation marked by dashed lines, and the range of excursions, R, from the cumulativeseries of xi, a motion signal, yi. It uses the scaling relationship between the rescaled range (R/S)and the scales of observation (window length, n), for determining the Hurst coefficient, 0<H< 1.

Bassingthwaighte and Beyer (1991) worked out a method to determine H by fitting thetheoretical autocorrelation function (equation (15) to the first few estimates of ρk obtainedby equation (14). This method is termed autocorrelation analysis and can be applied to fGnsignals or to differenced fBm signals.

Scaled windowed variance analyses (SWV ). Fluctuations of a parameter over time canbe characterized by calculating the variance, or according to equation (17) its square root, thestandard deviation

σ =√√√√ 1

N − 1

N∑i=1

(xi − µ)2. (17)

For fBm processes of length N when divided into nonoverlapping windows of size n(figure 10)—as equation (12) predicts—the standard deviation within the window, σn,depends on the window size n in a power law fashion:

σn =d pnH . (18)

This method, originally introduced by Mandelbrot (1985) was further developed by Peng et al(1994). By taking the logarithm of equation (18) the following practical formula is obtained:

log σn =d logp +H logn. (19)


0 5 10 -0.5

0

0.5

1

k

ρ k

H=0.99

H=0.9

H=0.6

H=0.4

H=0.2

H=0.01

for fGn-signals(stationary)

Figure 9. Autocorrelation functions of exact fractal noise time series with different Hurstcoefficients. The autocorrelation coefficient, ρk, given by equation (15) is shown as a function ofthe lag, k, of the elements in the series. The correlation is maximum (ρk = 1) at lag 0, a trivialcase. The correlation decays slower than an exponential function of time (lag) for cases H > 0.5,a phenomenon which is termed long-memory or long-range dependence. At the level of H = 0.5(uncorrelated white noise) ρk = 0. Anticorrelated fractal noises have negative nearest neighbourcorrelations (−0.5 < ρk < 0) (van Beek et al 1989).

H can be found as the slope of the linear regression fit to the data pairs on the plot of log σ n

versus log n. In practice, σ n’s calculated for each segment of length n of the time series areaveraged for the signal at each window size. Note, that n cannot vary from 1 to N, becauseusing too short (N pieces of length n = 1) or too few (1 piece of length n = N ) segmentswould distort the result. The standard method applies no trend correction. Trends in the signalseen within a given window can be corrected either by subtracting a linearly estimated trend(line-detrended version) or the values of a line bridging the first and last values of the signal(bridge-detrended version, bdSWV) (Cannon et al 1997). This method can only be applied tofBm signals or cumulatively summed fGn signals.

Dispersional analysis (Disp). This statistical approach—originally using relativedispersion of spatial data—was introduced by Bassingthwaighte (1988). Later it was extendedto the temporal domain (Bassingthwaighte and Raymond 1995). It is similar to SWV with thedifference being that σ is the standard deviation of the windows’ means (figure 10)

σn =d pnH−1 (20)

log σn =d logp + (H − 1) logn. (21)

This method can only be applied to fGn processes or to differenced fBm signals.Detrended fluctuation analysis (DFA). This method, introduced by Peng et al (1994), was

developed to improve on root mean square analysis of highly nonstationary data, by removingnonstationary trends from long-range-correlated time series. First the signal is summed andthe mean is subtracted

yj =j∑i=1

xi − µ. (22)

R14 A Eke et al

fBm

SWV

diffe

renc

ing

sum

min

g

fGn

Disp

....

....

µi µi+1

σµ

µi+n

σi σi+1

µσ

σi+n

Figure 10. Basic concept of dispersional (Disp) and scaled windowed variance (SWV) methods.These are time domain methods. The signal for Disp was generated by the DHM method andcumulatively summed for SWV according to the relations of the fGn/fBm model shown on theleft. Disp and SWV are similar in that they calculate the same descriptive statistical measuresas a function of scale for the purpose of deriving the transscale parameter, H. For Disp the localmeasure is the mean (µ), the partition-based measure is the standard deviation of local means (σµ),for SWV the local measure is the standard deviation (σ ), the partition-based measure is the meanof local standard deviations (µσ).

Then the local trend yj,n is estimated in nonoverlapping boxes of equal length n, using a leastsquare fit on the data. For a given box-size n the fluctuation is determined as a variance uponthe local trend, i.e. the root mean square (RMS) of differences from it,

Fn =√√√√ 1

N

N∑j=1

(yj − yj,n)2 (23)

in which the DFA method is closely related to the line-detrended SWV method. Recall thatSWV computes standard deviation relative to the mean, after subtracting the local trend, whileDFA directly calculates the standard deviation-like measure, RMS variance, upon the local


trend. For self-similar fBm processes of length N with nonoverlapping windows of size n thefluctuation, F, depends on the window size n in a power law fashion

Fn =d pnα (24)

and

logFn = logp + α logn. (25)

If xi is an fGn signal, then yj will be an fBm signal. Fn then is equivalent to mn of equation (12)yielding Fn =d pnH; therefore in this case α = H. If xi is an fBm signal then yj will be a summedfBm signal. Then Fn =d pn

H+1, where α = H + 1 (Peng et al 1994).

4.2. Frequency domain methods

The time domain is not the only domain in which sampled data could be represented. Out ofthe many domains resulting from transformations of time series data, one of the most usefulis the frequency domain (Weitkunat 1991).

Power spectral density analysis (PSD, periodogram method, Fourier analysis). A timeseries can be represented as a sum of cosine wave components of different frequencies

xi =N/2∑n=0

An cos[ωnti + ϕn] =N/2∑n=0

An cos

[2πn

Ni + ϕn

](26)

where An is the amplitude and ϕn is the phase of the cosine-component with ωn angular (i.e.fn = ωn/2π cyclic) frequency. The An(fn), φn(fn) and A2

n( fn) functions are termed asamplitude, phase and power spectrum of the signal, respectively. These spectra can bedetermined by an effective computational technique, the fast Fourier transform (FFT). Thepower spectrum (periodogram, power spectral density or PSD) of a fractal process is a powerlaw relationship

A2n =d pω

−βn which yields logA2

n =d logp − β log fn (27)

where β is termed spectral index. Signals with this form of power spectrum are referred to as1/f noise. The power law relationship expresses the idea that as one doubles the frequency thepower changes by the same fraction (2−β) regardless of the chosen frequency, i.e. the ratio isindependent of where one is on the frequency scale. The signal has to be preprocessed beforeapplying the FFT, which means subtracting the mean, multiplying with a parabolic window(windowing) and bridge detrending (endmatching). After calculating the power spectrumusing a FFT, the high-frequency part of the spectrum should be excluded before fitting theregression line (Fougere 1985, Eke et al 2000). 1/f noises with –1 < β < 1 and 1 < β < 3are almost identical with fGn and fBm signals, respectively (Beran 1994). Disregarding thealmost has led to some misunderstanding and misinterpretation in fractal time series analysis(see below).

Coarse graining spectral analysis (CGSA) separates the fractal and the harmoniccomponents in the signal and can thus estimate the spectral index, β, without the interferenceof the latter (Yamamoto and Hughson 1991). The power of the fractal components can alsobe given as a per cent fraction. Its concept is based on the self-affine structuring of the fractalprocesses as defined by equations (11) and (12). When an n length segment of an N lengthsignal is rescaled in intensity by (n/N )H, the original signal and its rescaled (coarse-grained)segment remain equal in distribution in all of their statistical measures. The cross-powerspectrum of the two is obtained by calculating the power from the spectral estimates of theoriginal and the coarse-grained series. If the original signal is a fractal process, the cross-power

R16 A Eke et al

spectrum equals that of the original. In the case of a simple harmonic signal, the cross-powerspectrum approaches 0. Based on these relations, a mixed signal yields a cross-power spectrumof the broad band 1/f, fractal type. Hence, a ratio of the summed cross-power and that ofthe original gives the relative power of the fractal fraction in the signal. This original versionof the method had to be improved since a pure harmonic signal with frequencies scaled by(n/N) was seen by the original method as pure fractal. This problem later was eliminated bytesting for the phase angle of the coarse-grained spectrum (Yamamoto and Hughson 1993).The phase relationship in the fractal time series is random, while in the harmonic oscillationsit is fixed, thereby allowing a discrimination of fractal and harmonic components. Followingtheir separation, β is calculated from the fractal power slope, the slope of the log–log fractalpower spectrum as shown earlier.

4.3. Time-frequency (time-scale) domain analysis

Dennis Gabor (1946) adapted the Fourier transform to analyse only small segments of thesignal at a time—a technique called windowing the signal—and produced the short-timefourier transform (STFT). STFT maps a signal into a two-dimensional function of time andfrequency. It provides simultaneous information about the location in time and the particularfrequency at which a signal event occurs. However, this information is of limited precisiondetermined by the size of the window, which is the same for all frequencies. Many signalsrequire a more flexible approach.

Fractal wavelet analysis represents a windowing technique with variable-sized regionsthus allowing an equally precise characterization of low- and high-frequency dynamics in thesignal. A wavelet is a waveform of limited duration with an average value of zero. Fourieranalysis breaks up a signal into cosine wave components of various frequencies. The waveletanalysis breaks up a signal into shifted and stretched versions of the original wavelet. Inother words, instead of a time-frequency domain it rather uses a time-scale domain, whichis extremely useful in the fractal analysis. The Hurst coefficient can be estimated in threeways: by the wavelet transform modulus maxima method (WTMM) (Arneodo et al 1995),the wavelet packet analysis (Jones et al 1996), and the averaged wavelet coefficient (AWC)method (Simonsen and Hansen 1998), the latter being the most effective and the easiest toimplement (figure 11). The shifted (translated) and stretched (dilated, scaled) versions of theoriginal wavelet function ψ(t) can be defined as

ψa;b(t) = ψ

(t − ba

)(28)

where the scale (dilation) parameter is a > 0, and the translation parameter −∞ < b < ∞.The continuous wavelet transform (CWT) of a given a function x(t), is defined as

W [x](a, b) = 1√a

∫ ∞

−∞ψ∗a;b(t)x(t) dt . (29)

Here ψa,b∗(t) denotes the complex conjugate of ψa,b(t). From equation (11), one can easily

derive how the self-affinity of an fBm signal x(t) determines its CWT coefficients:

W [x](sa, sb) =d s12 +HW [x](a, b). (30)

The AWC method is based on equation (30) (Simonsen and Hansen 1998). This method canbe applied to fBm signals or to cumulatively summed fGn signals.


Sca

le,

a

Time

Averaged wavelet coefficient method

Tim

e se

ries

Am

plitu

de

Htrue = 0.3fBm-signal

0

0

1 3 51.5

4.5

3

1

128

a=48

a=96

130

10

70

96

48

1000

1000

Wav

elet

anal

ysis

Wav

elet

coe

ffici

ent m

atrix

Log

AW

C

AW

C: W

[x](

a,b)

aver

aged

by

bat

a

W[x

](a,

b)

Time, b

Log a

H=slope - 0.5Hest =0.31

Figure 11. Steps of analysis by the averaged wavelet coefficient method (AWC). This is a time-frequency domain analysis applicable to fractal motion signals (upper panel). The signal wasgenerated at H = 0.3 by the DHM method and cumulatively summed (fBm). The length of thesignal is 1000 in this example. First a wavelet series is defined by equation (28) (‘Mexican hat’)with dilation parameter, a, and while being moved along the timescale in steps given by itstranslation parameter, b, at each step a wavelet coefficient is obtained between the wavelet andthe given segment of the series. The wavelet coefficient matrix (middle) is scaled by W [x](a, b)shown as an intensity scale on the right. Its values are averaged at every step of scale, a, yieldingthe AWC shown in the intensity code on the right. The Hurst coefficient is estimated from theslope of log AWC versus log a. In the given example the true and estimated H’s are very similar,which is indicative of the precision of this method.

R18 A Eke et al

4.4. Linear system analysis of fractal time series (fARIMA)

This approach builds on the memory in the analysed system. Any given value xi of the timeseries depends on the preceding values with lags k = 1, . . . , r as a linear combination ofthe values within this range. The simplest variant of the ARIMA (autoregressive integratedmoving average) models is the AR(r) model (autoregressive model of order r). It can beformulated as (Beran 1994)

xi =r∑k=1

αkxi−k + εi (31)

where αk are the autoregressive coefficients. The value of xi—in addition to previous valuesof the series—is also modified by a random noise εi, which represents a noisy perturbationacting on the system. The value of αk can be estimated by

α = (X′X)−1X′Y

α =

α1

...

αr

X =

xr · · · x1

.... . .

...

xN−1 · · · xN−r

Y =

xr+1

...

xN

. (32)

In the fAR (fractional AR) model for αk the following equation holds:

αk = − −-(k − d)-(k + 1)-(−d) . (33)

For large lag k we have

αk =d pk−d−1 where d = H − 1

2and p = −1

-(−d) (34)

from where the Hurst coefficient, H, can be obtained. This model can be applied to an fGnsignal or to a differenced fBm signal.

4.5. Methods of signal classification

The fGn/fBm dichotomous model is simple, yet effective in providing a framework forunderstanding the basic characteristics of random statistical processes with self-similar scalingproperties. The signals in physiology can be regarded as analogous to one of these two casesof discretely sampled fractal processes. From the above survey of fractal tools it follows thatsome of them can either be applied to stationary fractional Gaussian-type or the nonstationaryfractional Brownian-type signals, at least in particular cases after cumulative summing ofthe former or differencing of the latter (Eke et al 2000). As shown above, signals must beclassified according to this model prior to analysis, a circumstance not clearly recognizedearlier (Eke et al 2000).

Classification by the power spectral analysis (its improved variant, lowPSDw,e). Signalclassification should begin with the application of the PSD method, since it can be applied tofractal noises and motions alike and provides a basic parameter of a time series, its spectralindex, β. The purpose is to determine the signal class based on β. This can best be foundif the high-frequency spectral estimates are excluded from fitting for the spectral slope, −β(Eke et al 2000). If the estimate of β, β < 1 then the signal is fGn, if β > 1 then it is fBmwith some range of uncertainty in between.


Table 1. Conversion of various fractal measuresa.

α β HfBm HfGn D

D 3 − α 5−β2 2 − HfBm – –

HfBm α − 1 β−12 – – 2 − D

HfGn αβ+1

2 – – –

β 2α − 1 – 2HfBm + 1 2HfGn − 1 5 − 2D

α – β+12 HfBm + 1 HfGn 3 − D

ρ1 – – – 22HfGn−1 − 1 –

a α: a parameter of DFA method, β: spectral index of the PSD method, HfBm: the Hurst coefficientof an fBm signal, HfGn: the Hurst coefficient of an fGn signal, D: fractal dimension (for fBmsignals, only!), ρ1: nearest neighbour correlation coefficient.

Classification by the detrended fluctuation analysis (DFA) (Peng et al 1994) is possiblebased on the value of α. If α < 1 then the signal is fGn, if α > 1 then it is fBm with somerange of uncertainty in between.

Classification by the signal summation conversion method (SSC). This method was recentlydeveloped by Eke et al (2000) in order to refine the analysis near the fGn/fBm boundary.First it converts an fGn to an fBm signal or an fBm to a summed fBm signal by calculatingits cumulative sum. Then it applies bdSWV to calculate from the cumulant series the Hurstcoefficient H

′. When 0 < H

′ � 1 the signal is an fGn with H = H′. Alternatively, when

H′> 1 SSC finds the signal an fBm with H = H

′ − 1.

5. Converting various fractal measures (spectral index, Hurst coefficient, fractaldimension, etc)

Although fractal methods differ in their operational domain and the way they producequantitative measures of self-similarity, there are still common grounds in the ‘fractaldomain’. Examples: the Hurst coefficient (exponent, H), and the DFA-exponent (α)assume and characterize power law scaling in the time domain, and the spectral index(β) does the same in the frequency domain: they are still (almost) equivalent to eachother. The wavelet fractal analysis operates in the time-scale domain and fARIMAis a linear system analysis. They in fact assume time domain power law scaling.The box-counting method generally used to calculate the fractal dimension of spatialobjects can also be applied to a self-similar temporal signal (fBm) treating it as apicture, a two dimensional object (Peitgen et al 1992). Irrespective of these differences,fractal measures do relate to each other in a simple manner as shown in table 1.A concise overview of most of these fractal measures and a pictorial demonstration of thecorresponding signal character can be found in the paper of Eke et al (2000, figure 2). The mostuniversal parameter is the spectral index, β, that covers the full range of the fGn/fBm modelfrom the roughest fractal noise signals (β = −1) to the smoothest fractal motions (β = 3).Hence,β can be regarded as the base for converting fractal estimates, like from the H of SWV orDisp methods, to theα of the DFA. This way, a reliable comparison of reported fractal estimatesobtained by different methods can be carried out. Note that D can be calculated for a self-similar

R20 A Eke et al

low high

ff

f f

|A|2 N=3 N=3

N=6N=6

fN fN

fNfN

Sampling frequency, fs

Frequencycomponents

Aliasing

weakstrong

Signallength, N

few

many

short

long

|A|2 |A|2

|A|2

Figure 12. Influence of the principal aspects of frequency domain analysis on the precision ofestimating the true spectrum of the signal. Signal length and sampling frequency, representedfrequency components of the spectrum, and an artifactual contribution to power known as aliasingare related to the measured (dashed line) and true spectrum (continuous line) as a function ofabsolute frequency, f . The Nyquist frequency, fN = fs/2, is the upper bound of the representedfrequencies. Overestimation of power due to aliasing can be minimized if fs is chosen to be a highvalue. Representation of different frequency components is the most complete if N is larger.

signal of the fBm type only (see equation (11)). Also note that H is scaled between 0 and 1irrespective of the class of the signal. Due to the dichotomy of the fGn/fBm model underlyingthe analysis, it is strongly recommended to report values of H along with the class of the signalas HfGn and HfBm (Eke et al 2000). This practice will guarantee that values of H obtained bya class-specific method for the wrong (incompatible) class do not get published.

6. Limitations of fractal time series analysis

For some methods the limitations are better understood, than for others. Herewith, the casesof the PSD and AC methods are presented for they are fundamental in defining the two mostimportant fractal properties of time series, power law scaling relationship of spectral poweras a function of frequency (assessed by the PSD method by equation (27)) and the basicallypower law correlation structure (assessed by the AC method based on equation (15)).

The standard version of the power spectral density method is weak in estimating the truespectral index of exact fractal time series (Eke et al 2000). One known problem is aliasing,an artifactual contribution to the power of spectral estimates aliased from beyond fs/2 (theNyquist frequency, fN) into ranges below it in a fashion mirroring the shape of the spectrum(figure 12). Selecting fs high enough to minimize the power at the Nyquist frequency couldtreat this problem. Beran (1994) pointed out that a fractal signal with an exact correlationstructure (generated by the method of Davies and Harte (1987)) deviates from a pure 1/f β

power spectrum at high frequencies. This observation was confirmed by Eke et al (2000)in numerical experiments, which demonstrated that the high end of the frequency spectrum( fs/8 < f < fs/2) needs to be excluded when fitting for the power slope to yield a reliable


Autocorrelation Function

Power Spectral Density Function

k H0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91

-0.5

0.5

1

0

2 3 4 5 6 7 8 9

log(

|A|2

)

log(f) log(f)fN1/4fN

H=0.2

H=0.8

ρk ρ1

Fractal motion (fBm)DHMSSM

DHMSSM

theoretical. DHM+ SSM

Fractal noise (fGn)DHMSSM

H=0.8H=0.2

fN1/4fN

Figure 13. Characterization of the power law fractal signal generated by the spectral synthesismethod (SSM) and the exact fractal signal produced by the DHM method. These signals canbe described in their autocorrelation (top) and power spectral densities (bottom). Autocorrelationfunctions of DHM signals exactly follow the prediction of equation (15), while those of SSM signalstend to 0 at every lag and H (top left with H values shown top right). In other words, the nearestneighbour correlation coefficient, ρ1, of DHM signals follows exactly the theoretically predicatedvalues (equation (16)), while correlated SSM signals underestimate and anticorrelated SSM signalsoverestimate it. The SSM signals have ideal 1/f β power slopes while the DHM signals deviatein the high-frequency range, more so when the signal is anticorrelated. (The upward deviation forDHM motion signals can be due to the summing operation needed to generate these signals fromthe raw fGn signal produced by the DHM method.)

measure of power law scaling when exact self-similar signals generated by the Davies andHarte method based on equation (15) were used.

The power spectrum and the autocorrelation function are interrelated in that the Fouriertransform of the latter yields the power spectrum of the signal. Note, that high frequenciesof the PSD function where dynamics are the fastest, are represented in the short lagged rangeof the AC function (figure 13). The long-lagged region of the AC function (at large k’s)corresponds to the low-frequency range of the PSD function. The autocorrelation analysisdoes not use all the lags for estimating H, for longer lags proved statistically unreliable dueto their values being close to 0. Usually the first five lags are used. Most of the signals inphysiology are, however, dominated by low frequencies, hence the AC method can present

R22 A Eke et al

problems in their analysis for it fits the high-frequency range of these signals where spectralestimates may well deviate from the ideal spectral slope (figure 13). This limitation of the ACmethod in the analysis of physiological signals cannot be compensated—although would belogical—by using its long lagged values for fitting H because of their being in the unreliablerange as mentioned above.

7. Evaluating the methods’ performance

None of the methods we have looked at so far were ideal; they cannot be, for their performanceeven on ideal fractal signals will depend on the algorithmic implementation of testing for thepresence of statistical self-affinity defined by equation (12) and the way the method and signalin question will ‘interact’. Numerical testing on pure monofractal signals is helpful in providinga ranked list of methods according to their precision in estimating the fractal properties orsignal class and in identifying cases when limitation of the method, or incompatibility ofmethod and signal, can be demonstrated.

7.1. Generating ideal monofractal signals

Ideal fractal signals of known H can be synthesized based on the two properties of fractal timeseries: power law scaling in the time domain (equation (12)) and in the frequency domain(equation (27)). Based on power law scaling properties in the frequency domain defined byequation (27) and using the Fourier transform algorithm the spectral synthesis method (SSM)produces 1/f noises, which are almost exact fGn- or fBm-type signals, depending on thevalue of β used. The method of Davies and Harte (1987) (DHM) produces an exact fGn signalusing its special correlation structure (equation (15)), which is a consequence of the timedomain power law scaling of the related fBm signal (equation (12)). Cumulative summing ofthe fGn signal generates the corresponding fBm signal. Both of these signals are fractals.

7.2. Testing fractal tools on ideal monofractal signals

The fractal analytical methods can be tested on synthesized fractal signals of known H(Bassingthwaighte and Raymond 1994, 1995, Caccia et al 1997, Cannon et al 1997, Eke et al2000). For each set of series of known H and particular signal length N the variance and biasare calculated. The mean squared error (MSE = var + bias2) gives a combined characterizationof bias and variance in the estimates of H, H . Figure 14 gives an overview of the performancefor a long enough signal length of 217 at which length all of the tested methods should performat their best. For other signal lengths table 2 gives the MSE values obtained for DHM signals.Methods with the best overall performance can be selected from a rank-ordered list by MSE.

Which of the two signal generating methods, SSM or DHM, is more suited for numericaltesting? Before answering this question a fundamental issue about the strategy of numericaltesting, not fully understood earlier, needs to be clearly seen. When methods of analysisand synthesis are based on the same algorithm, as in the case of the PSD method andSSM signal or the AC method and DHM signal, the result is almost perfect, provided thesignal length is adequate: estimates of true H are with no bias and very small variance(Schepers et al 1992, Eke et al 2000) (figure 14, PSD/SSM/fBm and AC/DHM/fGn panels,respectively). The inference of these methods’ excellent performance under these specificcircumstances should by no means be that these methods are ideal in a universal sense for theanalysis of physiological signals, because physiological processes are produced by complexinteractions in their underlying mechanisms, which do not necessarily result in ideal signal


0

0.5

1PSD

Est

imat

ed H

0.0029

lowPSDw,e

0.0010

DFA

0.0001

AC

0.0000

Disp

0.0004

bdSWV

Nonapplicable

0

0.5

1

Est

imat

ed H

0.0712 0.0011 0.0017

Nonapplicable

Nonapplicable

0.0000

0

0.5

1

Est

imat

ed H

0.0101 0.0002 0.0036 0.0339 0.0074

Nonapplicable

0 0.5 10

0.5

1

Est

imat

ed H

0.0000

0 0.5 1

0.0002

0 0.5 1

0.0002

0 0.5 1

Nonapplicable

0 0.5 1

Nonapplicable

0 0.5 1

0.0012

SS

MD

HM

True H

fGn

fBm

fGn

fBm

Figure 14. Result of numerical experiments on pure monofractal signals to characterize theprecision of different fractal tools. The test signals were generated according to the dichotomousfGn/fBm model by the DHM and the spectral synthesis method (SSM). DHM yields an fGnsignal, while SSM produces an fBm signal. The corresponding signals in the fGn/fBm modelwere created by differencing (fBm to fGn) or cumulative summing (fGn to fBm). A signal lengthof 217 was chosen to compare methods to ensure that the signal length has a minimal impact onprecision (Eke et al 2000). A hundred signals were generated for each level of H from 0.1 to 0.9 insteps of 0.1. Vertical error bars (±SD) are shown indicating the variance of the method. Deviationfrom the line of identity between true H and estimated H (dotted line) characterizes the bias inthe estimates of H. A combined measure of these two is given in the mean squared error (MSE)normalized by the step in H displayed in the lower right corner of each panel at a precision of fourdecimal places.

structuring of the SSM or DHM type. PSD would be ideal for their analysis only if theywere pure realizations of the SSM algorithm as predicted by equation (12). Similarly, ACwould yield highly reliable H for a physiological signal resembling the correlation structureof the DHM model extending into its high-frequency range. Since structuring of physiologicalsignals may be due to self-similarity in the time or frequency domain, both methods arerecommended to be used in evaluating the performance of fractal tools for the purpose ofgaining a deeper understanding of their interactions with the prospective physiological signals(figure 14).

Variants of the standard method (Fougere 1985, Eke et al 2000) yield different precisionsin their fractal estimates (figure 14, columns 1 and 2). Employing signal preprocessing steps(such as parabolic windowing, endmatching) in a proper sequence and excluding the highfrequencies from estimating the power slope as shown by Eke et al (2000) yields a variant of thePSD method that gives the best results in fractal analysis of DHM signals (figure 14, column 2,

R24 A Eke et al

Table 2. Precision of Disp, DFA and lowPSDw,e methods as a function of signal lengtha.

fGn fBm

n Disp DFA lowPSDw,e bdSWV DFA lowPSDw,e

28 0.00422 0.00920 0.03998 0.00461 0.04164 0.0986429 0.00274 0.00457 0.01902 0.00249 0.02321 0.04397210 0.00180 0.00240 0.01039 0.00134 0.01457 0.01954211 0.00126 0.00131 0.00678 0.00084 0.01015 0.01003212 0.00092 0.00076 0.00515 0.00047 0.00807 0.00609213 0.00067 0.00045 0.00422 0.00033 0.00696 0.00454214 0.00050 0.00031 0.00374 0.00017 0.00631 0.00378215 0.00039 0.00024 0.00350 0.00010 0.00599 0.00358216 0.00031 0.00020 0.00338 0.00006 0.00587 0.00350217 0.00024 0.00018 0.00330 0.00004 0.00582 0.00347218 0.00020 0.00017 0.00325 0.00002 0.00577 0.00347

a Mean squared errors are shown characterizing bias and variance in the fractal estimates of these methods.

panels 1 and 2) and is designated as the lowPSDw,e method. Although bdSWV, Disp and DFAproved more robust and precise than the PSD method, the latter is still recommended to beused as the basic method in fractal analysis for visualizing the power spectrum, demonstratingthe presence of a proper scaling range and as a guide to deeper analysis (Eke et al 2000). Wehave seen that spectral methods with the exception of the variant identified by Eke et al (2000)are sensitive to high-frequency disturbances, while DFA, Disp and bdSWV are not, becausethese methods bin the high frequencies. Each point in DFA, Disp and bdSWV can be taken tobe (roughly) equivalent to the average of a group of spectral estimates. Hence, DFA, Disp andbdSWV use fewer points in fitting for H than PSD for β. Discarding the largest and smallestbins further improves the precision of Disp and bdSWV (Cannon et al 1997).

As seen in figure 15, even the best variant of the PSD method and two robust andprecise methods, Disp and bdSWV, do show greatly varying performance. A precisionof �H = 0.05 can be expected in 100% of the cases for bdSWV if N > 213. However,this holds only for a restricted range of conditions (N > 215 and H < 0.7) when Dispis applied to fGn signals generated by the DHM algorithm. In this test lowPSDw,e is theweakest with a range of 100% performance for N > 214 and H > 0.2 but with muchless precision for H’s < 0.2. The larger the H is, the longer the signal length should befor obtaining the same degree of precision in H as summarized in table 3 for cases ofsmall H (H < 0.15) and large H (H > 0.8) signals and signals in between these extremesanalysed by four reliable methods

(lowPSDw,e,Disp, bdSWV and DFA

)at a precision of

95%. Most reports of fractal analysis of various signals obtained from physiologicalmeasurements use a signal length of much less than that demonstrated by Bassingthwaighteet al as reliable (Bassingthwaighte and Raymond, 1995). Acceptable ranges are shown infigure 15 and table 3. It is still very common that results of analyses done on signals of afew hundred data points in length are reported. Estimating H in these cases is a meaninglessexercise, especially if no class of the signal is reported that would at least aid the readerin finding out more on the precision of the estimate using results of evaluation studies(Bassingthwaighte and Raymond, 1994, 1995, Caccia et al 1997, Cannon et al 1997, Eke et al2000). DFA, Disp and SWV are directly related methods (Raymond and Bassingthwaighte1998). This also explains these methods’ comparably robust performance (figures 14 and 15)(Eke et al 2000) and reliable use in the dichotomous fGn/fBm model to evaluate the stationaryand nonstationary fractal physiological processes (Eke et al 2000). Among the three variants


0.99

0.8

0.6

0.4

0.2

0.01

0.99

0.8

0.6

0.4

0.2

0.01

100

%

80

60

40

20

0

100

80

60

40

20

0

Hur

st c

oeffi

cien

t, H

fGn

DH

M

fBm

Dispersional analysis

bdSWV

Length, n

0.99

0.8

0.6

0.4

0.2

0.01

0.99

0.8

0.6

0.4

0.2

0.01

100

80

60

40

20

0

28 210 212 214 216 218

100

80

60

40

20

028 210 212 214 216 218

Hur

st c

oeffi

cien

t, H

fGn

SS

M

fBm

lowPSDw,e

lowPSDw,e

lowPSDw,e

lowPSDw,e

Dispersional analysis

bdSWV

Figure 15. Evaluation of the precision of an improved variant of the PSD method and two robustmethods, Disp and bdSWV as a function of H and signal length. Signal generation was the sameas described for figure 12 except for the following two aspects. Signal length varied from 28 to218. True H was increased from 0.01 to 0.99 in steps of 0.01. The intensity-coded parameter isthe per cent of estimates falling into the range of Htrue ± 0.05. Light areas designate the range oflengths and H’s where the tested methods’ performance is good.

R26 A Eke et al

Table 3. Signal lengths at which Disp, lowPSDw,e, DFA and bdSWV method have a precision of95%.

H < 0.15 0.15 < H < 0.8 H > 0.8

fGn Disp 213 214 218

lowPSDw,e – 213 214

DFA 210 212 213

fBm bdSWV 212 211 212

lowPSDw,e – 213 215

DFA 215 216

of the SWV method (standard, line detrended and bridge detrended), the bridge detrendedmethod (bdSWV) by removing the ‘bridge’, the line between the first and last points withina window, results in the smallest bias and variance in its estimates of H (figure 14, panelbdSWV/DHM/fBm, table 2) (Eke et al 2000).

Both versions of the CGSA analysis in order to undertake coarse graining have tocalculate H by an independent method. Hurst’s rescaled range method was originallychosen, which in numerical testing proved quite weak (Bassingthwaighte and Raymond1994). Precision of the CGSA method may be further decreased by the fact that itemploys the standard periodogram method with a limited range of applicability (β � 2)(Fougere 1985) (figure 14, panel PSD/DHM/fBm). Newer, more precise variants of theFourier method applicable in the full range of the fGn/fBm model (−1 � β � 3), such aslowPSDw,e eliminating the high-frequency range from the spectral estimates and using signalpreprocessing steps like windowing and endmathching—in this order (Eke et al 2000)—shouldbe considered in future improvements. Given the fact that the CGSA method is widely usedin the analysis of heart rate variability, the impact of these circumstances needs to be lookedat more closely.

Classification of fractal signals always leaves a range of uncertainty where fractal noisesignals cannot be separated. For a comparison of the SSC, lowPSDw,e and DFA methods seefigure 16 where the misclassification rate is shown as a function of signal length 28 � N �218. In these tests SSM and DHM signals were generated with either ideal frequency or timedomain power law scaling, respectively. For all lengths tested, SSC proved to be a very robustmethod for signal classification irrespective of the model used to generate the signal.

8. How to apply fractal methods to physiological signals?

Application of the complex arsenal of fractal tools to the analysis of physiological temporalprocesses is inherently complex. The difficulty is due to a number of reasons, among which wefind the mathematical aspects of the approach. Indeed, before application, all attempts shouldbe made to decipher the algorithms into a code meaningful to the physiologist, a practice wehave tried to follow in this review. The application itself is not a one-step procedure, likecalculating a fractal measure (H, β or D, etc.) by a method chosen based on mere availabilityfor a signal of a length determined by some arbitrary circumstance. It must have been clearlyrealized by now, that this would be a high-risk operation and should be abandoned irrespectiveof how common this practice still may be. A flowchart given in figure 17 gives a summary ofthe steps needed to be taken, from deciding on signal length and sampling rate when producinga time series to choosing the proper method for analysis (rounded boxes in figure 17). Another


Length, 2n8 13 18

0

20

40

60

80

100

Length, 2n8 13 18

SS

M

Est

imat

ed β

Length, 2n8 13 18

−1

0

1

2

3

SSC

%

%

0

20

40

60

80

100DFA

DH

M

Est

imat

ed β

lowPSDw,e

−1

0

1

2

3

Figure 16. Precision of signal classification. An improved variant of the PSD method (lowPSDw,e)

(Eke et al 2000), the detrended fluctuation analysis (DFA) (Peng et al 1994), and the signalsummation conversion method (Eke et al 2000) were compared on various signal lengths 28 �N � 218 in 11 steps. Fractal estimates by these methods were converted into β. A total of 19 800monofractal DHM signals were generated at each signal length in increments of �β = 0.02 forthe whole range of the fGn/fBm model (−1 � β � 3). The misclassification rate was displayedintensity coded as a per cent in total trials at any β and N. lowPSDw,e and DFA have a relativelywide range of uncertainty on the noise side of the 1/f boundary for the DHM signal. This rangeis much narrower for SSM signals due to the signals and methods sharing the same algorithmicbases (Eke et al 2000). SSC proved a very robust and precise method of signal classification onboth DHM and SSM signals from short to long signals with a narrow range of uncertainty on thenoise side of the 1/f boundary only.

flowchart focusing more on the technical details of the fractal analysis can be found elsewhere(Eke et al 2000).

8.1. Choose adequate signal length and sampling frequency

Signal length and sampling frequency from the standpoint of fractal analysis are interrelatedparameters, which fundamentally determine the outcome of the analysis (see figure 12). Notall frequency components are fully represented in the power spectrum of a series, xi, of atemporal signal, x(t) sampled at fs . The upper limit is the Nyquist frequency fN = fs/2. fsshould be chosen high enough to minimize the artifactual influence of aliasing. When selectingfs , the sampling theorem of Claude Shannon should be observed (Taylor 1994) prescribingfs > 2fmax, wherefmax is the Nyquist rate representing the upper bound of the highest possiblefrequencies in the signal. If fs is chosen as a magnitude larger than 2fmax, this condition issurely met. The higher the fs chosen, the lower the power at fN is found beyond which aliasinginto the spectrum occurs. Increasing fs as a measure to treat aliasing cannot continue beyondsome upper limit for exceeding it would increase the chance that high-frequency estimates inthe power spectrum would not reflect physiology. This would not be much of a problem if theupper 75% of the spectral estimates were to be discarded as recommended (Eke et al 2000)

R28 A Eke et al

Create time series

Isolate case of fractal

H

Apply descriptive statistics

Scaling range ≥ 2 decades

H

Use

ClassificationSSC

fGn signal

Class cannot befound

fBm signal

DetermineDisp

DeterminebdSWV

class-independentmethods of

analysis

Determine

DFA

HlowPSD w,e

PSD

Figure 17. Flowchart of the strategy for fractal time series analysis. Steps from creating the timeseries from the temporal record of a physiological signal to calculating the Hurst coefficient areshown. Note the central importance of signal classification allowing the selection of one of thetwo robust methods of high precision for obtaining H. A more detailed flowchart can be foundelsewhere (Eke et al 2000).

and if these physiologically irrelevant estimates would fall into the discarded range. They can,however, extend into ranges below the upper 75% if signal length, N, is chosen much too lowrelative to fs the latter already having been chosen high enough to minimize aliasing. N shouldbe large if we want to capture low-frequency dynamics in our analysis. Since most signals inphysiology are known to be dominated by low-frequency fluctuations, this interaction betweenfs and N needs to be fully understood. Their values should be chosen to meet the optimumbetween the need to capture relevant dynamics, to minimize aliasing, to sufficiently extend thefrequency range into the slow dynamics known to be physiologically relevant and to containthe artifactual or physiologically irrelevant high-frequency components within the upper 75%of the spectral range.

Exclusion of the upper 75% of the spectral estimates from the spectral analysis makeslowPSDw,e less sensitive to high-frequency disturbance as compared to the standard version.Similarly, the robustness and precision of bdSWV, Disp and DFA at least in part stems fromtheir weighting estimates in larger size bins (equivalent to lower frequencies) over those inshort bins. By the same token, Disp’s weaker performance as compared to bdSWV in theirrespective signal class is most likely due to the comparable impact of high and low frequencieson the spectral slope tilting up or down from the horizontal position of the white noise, whilethe low frequencies always dominate the nonstationary fBm signal resulting in a slope tiltedin one direction, only.


8.2. Characterize measurement noise

No measurement is without noise. In addition to the customary techniques to characterizenoise in the signal, like signal-to-noise ratio, the PSD method is recommended to be used. Theaim is to demonstrate that the signal contains only insignificant power due to instrument noiseand signal contribution of some origin irrelevant to the study as compared to the fluctuationsdue to the sampled (relevant) physiological process within the range of frequencies of thefractal analysis. Instrument noise can be characterized in the power spectral density plotsof the signal sampled from the instrument’s signal output when it is disconnected from thephysiological signal source. The second aim can be achieved by optimizing the scaling rangein such a way that it excludes signal components unrelated to the observed physiologicalprocess.

Bassingthwaighte and Raymond (1995) found that adding random noise with SD equalto the SD of the fractal signal with H > 0.7 (strongly correlated fGn signal) caused less thana 10% downward error in the estimate of H. Cannon et al (1997) found the same for the fBmsignals with added random motion. This effect can and should also be tested numericallyin the experimental studies by finding SDnoise/SDsignal. In our studies analysing the fractalcorrelation in the spontaneous fluctuations of regional blood cell perfusion in the rat braincortex (Eke et al 2000) we found SDnoise/SDsignal = 7.6% (unpublished data). When thepure motion signal of β calculated from the spectrum of instrument noise signal was addedat an SD ratio of 7.6% to a pure motion signal of H fBm = 0.22 (H found for the blood cellperfusion time series of the fBm type), the downward error in the estimate of H was found to beonly 2.2%.

8.3. Demonstrate the presence of fractal scaling range

The PSD method is a key tool in demonstrating that power law scaling as predicted by equation(27) is present in the time domain in the signal throughout a continuous range of 2 decades offrequencies (Cannon et al 1997). Note that assessment of the scaling range is influenced byour choice of fs and N (figure 12). In the case of a shorter than 2 decades of scaling range,fs and N can be modified in an attempt to find the right combination that would identify thescaling range beyond SR = 2 decades. It is also important to determine the physiologicalmeaning and significance of the upper and lower cut-off point (fmin and fmax) of the scalingrange.

8.4. Determine signal class

Identifying the class of the signal according to the dichotomous fGn/fBm model is usefulnot only in signal analysis, but in interpreting the underlying physiology, too (Eke et al2000). Typically the mechanism of signal generation in the studied physiological process isunknown, hence it is best to use robust methods, like SSC, for signal classification. The rangeof uncertainty for SSC is very small irrespective of whether the signal is generated by the SSMor the DHM model (figure 16). Even in cases, when the improved version of PSD is used andthe signal character is known to be SSM, the misclassification rate of lowPSDw,e shows a verystrong dependence on signal length and it still falls behind that of SSC. Proper classificationof the signal is a prerequisite to using the robust time domain methods (Disp and bdSWV)with improved precision. Failure to find the right class or, what is common in the literature,not using classification as a prelude to analysis, results in gross error in the estimates of H(figure 18).

R30 A Eke et al

differenced dataraw data

Time series

fGnfBm

Signal Classification and Analysis by lowPSDw,e

class-compatibleraw class-incompatibleSignal Analysis by Disp Method

680

580

4800 650 0 650

60

0

-60Blo

od c

ell p

erfu

sion

Time, seconds Time, seconds

1012

103

10-60 100 0 100

Pow

er

Frequency, Hz Frequency, Hz

4

-1

-60 10 0 10

log

SD

(n)

log (n) log (n)

class-incompatibleraw class-compatibleSignal Analysis by bdSWV Method

3.2

2.5

1.82.5 7 2.5 7

log

SD

(n)

log (n) log (n)

Figure 18. Exemplary analysis of blood cell perfusion time series obtained by LDF-flowmetry fromthe rat cerebral microcirculation with special emphasis on the importance of signal classification.The raw signal (first row) was classified as fBm by lowPSDw,e, based on having an estimatedspectral index, β > 1 with all spectral estimates above 25 Hz (marked by dashed lines) excludedwhen fitting for β (second raw). Therefore the best estimate of H can be obtained as the mean ofH’s by lowPSDw,e and bdSWV. The conclusion is that the raw signal is a fractal motion with H =0.22. Disp cannot be applied to fBm or bdSWV to fGn signals, while lowPSDw,e can be applied toboth signal classes. Note that Disp does not give an estimate of H on a motion-type physiologicalsignal (bottom panel) (Eke et al 2000).


8.5. Choose the appropriate method for analysis

What we can learn from the results of numerical experiments evaluating the performance ofthe fractal tools on the two classes of the fGn/fBm model and using the DHM and SSMsignals (figures 14 and 15 and tables 2 and 3), is that there is no single recipe but instead thereare choices we should be aware of. Signal classification by SSC is recommended as the firststep, for the class found will identify our potential tools. If we have good reason to believeor better yet, we could prove that our real signal is very close in character to the ideal onegenerated by the SSM or DHM models, the lowPSDw,e or the AC methods are likely to give veryprecise estimates of H, respectively. If this is not the case, it is better to use the average ofestimates by robust methods, like Disp on noise and bdSWV on motion, and that of lowPSDw,e

(Eke et al 2000). If the signal class cannot be determined, we should settle for the less preciselowPSDw,e and DFA methods which have the advantage of being applicable to both signalcategories, keeping in mind that their estimates are biased on anticorrelated DHM/noise andmotion, and on anticorrelated DHM/motion and SSM/noise signals, respectively. When DFAis compared to bdSWV, the latter is evidently more robust. Consequently, we should also takeinto account the actual value of H in further refining our analysis and select methods with thesmallest bias and variance in the particular range of H.

9. Applications of fractal time series analysis in physiological research with emphasison pitfalls of analysis

Application of fractal approaches in biomedical research leads to new insight into the complexpatterns of signals emerging from physiological systems. These studies have demonstrated that(i) behind the seemingly random fluctuations of a number of physiological parameters a specialorder, the structure of fractal geometry can be found; (ii) a single measure of this complexstructure (one of the fractal measures) can describe the complexity of fluctuations; and (iii) thatthis parameter does change in response to disturbances of the system such as activity, ageingand disease. These steps were the first, and in cases reported before that the details of fractaltheory, and the prerequisites of its application to real, physiological systems in particular wereto be outlined, and possible sources of error identified. This latter point will be emphasizedin the brief survey below on the application of fractal time series in physiological research.

In the last 15 years more than a thousand studies have been published using thefractal approach to analyse fluctuations in physiological signals, aimed at extracting usefulinformation from the seemingly random excursions of the observed parameter aroundthe set-point of the controlled system. Concise studies can also be found (West 1990,Bassingthwaighte et al 1994, Nonnenmacher et al 1994) that give an excellent overview onthe fractals discovered in living structures and processes. Following a brief overview of theareas where fractal time series analysis has set foot in biomedicine, we focus on two areas inparticular: applications relating to cerebral blood flow fluctuations—including studies of ourown—and heart rate variations, the latter being responsible for over half of the fractal timeseries publications to date.

Posture of our body is maintained by a complex processing of the continuous influx ofsignals (from the proprioceptors located in muscles and joints, receptors in the vestibularsystem and those in the retina of the eye) producing a continuous efflux of neural impulsesfrom several centres in the central nervous system via the motor fibres innervating the skeletalmuscles. It has been demonstrated that the random wobbling around a stable position has afractal character (Riley et al 1997, Peterka 2000) and that the rhythm in normal and abnormalgait is also a fractal process (Hausdorff et al 1999, Hausdorff et al 2000.).

R32 A Eke et al

Rhythm of normal breathing pattern is controlled by a complex interaction of chemicaland neural parameters. The partial pressure of carbon dioxide gas in the blood is monitoredby central chemoreceptors and by the extent of inflation of the lungs by mechanoreceptors.Neuronal circuits receiving additional inputs from internal rhythm generators as well as highercentres in the central nervous system evaluate these signals. The resulting neural outflow fromthis control system generates a normal breathing pattern that fluctuates over time in a fractal,self-similar manner (Szeto et al 1992, Dirksen et al 1998, Frey et al 1998).

When the layout of DNA sequences is converted into a one-dimensional series, DFAanalysis—typically used in time series analysis—can be applied in its fractal characterization.Interestingly, these ‘DNA-landscapes’ proved fractal, suggesting that the biological code mayhave aspects using long-range fractal correlation patterns along the gene (Peng et al 1994,Arneodo et al 1998, Yu et al 2000). Such extended coding seems a likely possibility, sincecoding in triplet sequences was demonstrated to be more important than in sequences of thebases (Arneodo et al 1998).

The functional integrity of our brain depends on an uninterrupted supply of blood flowthrough its regions and microregions. The ultrasonic transcranial Doppler (TCD) and the laser-Doppler (LDF-flowmetry) methods have proved especially useful in the continuous monitoringof blood flow (indeed one of its components) in large arteries supplying the brain or throughregions of the brain cortex.

Fractal analysis has been applied quite early on to continuous records of blood cell velocityin large vessels supplying the brain as monitored by the TCD-method (Rossitti and Stephensen1994, Blaber et al 1996, Zhang et al 2000). These velocity time series, though proved to befractals, might require reevaluation for two reasons. First, none of the signals analysed waslonger than 512 points, which make the results suspect of unacceptable error in their fractalestimates. Second, when published fractal estimates are expressed in β, they do seem to bein serious conflict (∼0.52 (Rossitti and Stephensen 1994), ∼1.66 (Blaber et al 1996), ∼0.34(Zhang et al 2000)) because they fall at a considerable distance on the two sides of the 1/fboundary of the fGn/fBm model. The only possible explanation for this extreme scatter of β isthat in these studies the signal class was not determined, not even when a class-specific methodof analysis (i.e. Disp) was applied. As shown earlier, this leads to grave error in estimating H(figure 18). Typically instead of H ∼ 0.3 (β∼ 1.6) identifying an fBm signal with anticorrelatedincrements, H ∼ 0.8 (β ∼ 0.6) is found, mistakenly indicating a correlated fGn signal.

We have used LDF-flowmetry to obtain sufficiently long (655 s, 217 data points) timeseries of blood cell perfusion from the brain cortex of anesthetized rats (Eke et al 1997, 2000).The time constant of the LDF-instrument was set to its minimum in order to capture the fastestpossible dynamics in the signal. fs was chosen at 200 Hz to observe the sampling theorem andavoid aliasing, setting the upper bound of the fastest dynamics to 20 Hz when our lowPSDw,e

is used. All of the perfusion time series we have analysed so far proved fractal motion (fBm)signals, hence we chose the bdSWV method to analyse these signals. Their fractal estimatesscatter in a narrow range around H ∼ 0.23 (β ∼ 1.46).

Since its first description (Hon and Lee 1965), heart rate variability (HRV) has attractedmuch attention in the cardiovascular literature due to the vital functions of the heart, andalso because the discovered irregular but apparently normal pattern challenged the notion ofreliability associated with the regularly beating normal heart.

Descriptive statistics and frequency decomposition (spectral) techniques have been usedextensively in the analysis of different aspects of HRV and were standardized in 1996(European Society of Cardiology and North American Society of Pacing and Electrophysiology1996). The advance of fractal and chaotic approaches in HRV research has been quite rapid inthe last decades; however, the standards for selecting the proper methods and conditions for


measurement and analysis have not yet been established. A task force of the European Societyof Cardiology and the North American Society of Pacing and Electrophysiology (Task Force)(European Society of Cardiology and North American Society of Pacing and Electrophysiology1996) gave guidelines for the use of spectral analytical methods in system identification, thatwas most certainly regarded as a reference for setting the scale in fractal HRV studies also.The task force recommended two scales to be used: a shorter, approximately 5-min period,and a substantially longer 24-h period. The report specified the length as 512–1024 and 218

data elements, respectively.The most often used fractal tools in HRV analyses are PSD, DFA and CGSA. Signal

classification is not required for these methods; it is embedded in PSD and CGSA, and DFAcan handle fGn and fBm signals, too. Nevertheless, investigators should realize that reportingon the class of the signal provides valuable information on the signal character and can facilitateexpanding the studies into the direction of using other robust methods of high precision (Dispand bdSWV) for the purpose of confirming the fractal measures found, especially if differentialchanges are small.

Comparison of results from different investigations is difficult, because (i) fractal methodsperform increasingly better as signal length increases, with N = 8192 being the point oftransition (figure 15, DFA on fBm/DHM signals); (ii) comparison of fractal measures foundon different scales is questionable because the evaluated spectra represent different scalesof frequencies rendering the scaling range vastly different in short and long record fractalanalyses (figure 12); and (iii) role of the motion effects as described by Fortrat et al (1999)and Serrador et al (1999) can further complicate the issue.

Control values of the spectral exponent (β) vary greatly from 0.8 to 1.9 depending onsignal length and methods of analysis (table 4). Note that these values are scattered around thefGn/fBm boundary (table 4). The uncertainty in the estimates of the spectral index can alsostem from the greatly different scaling ranges involved in these studies. The longer the recordthe more low-frequency components get represented (figure 12). It has been shown that high-frequency components of HRV are influenced by the vagal tone and the ratio of low to highfrequencies changes due to sympathetic control or with the sympatho-vagal balance (EuropeanSociety of Cardiology and North American Society of Pacing and Electrophysiology 1996),while the ultra-low-frequency components are related to motion effects (Serrador et al 1999).To ensure that the high frequencies are properly represented in the spectra of short-term records,the task force recommends that the length of the record should be ten times the wavelength ofthe lower frequency bound of the investigated component (European Society of Cardiologyand North American Society of Pacing and Electrophysiology 1996). This rule applies only toshort-term records when the standard PSD method is used for frequency decomposition andthe analysis is focused on separating peaks. Records of this length, however, are not suitablefor fractal analysis by PSD or for that matter any other fractal method (Bassingthwaighte andRaymond, 1995). Results of numerical testing of different fractal tools (Bassingthwaighte andRaymond, 1995, Caccia et al 1997, Cannon et al 1997, Eke et al 2000) can serve as the basisfor estimating the length of the signal that would be adequate for capturing a specific aspectof HRV within a given scaling range.

The clinical benefits resulting from fractal HRV analysis can be best evaluated fromsurveying the data in the literature from studies on HRV in ageing (Pikkujamsa et al 1999,Sakata et al 1999), disease states (Peng et al 1995b, Bigger et al 1996, Lombardi et al 1996,Butler et al 1997, Makikallio et al 1997, 1998, 1999, Huikuri et al 2000), and survival ratefollowing heart attack (Huikuri et al 1998, 2000, Makikallio et al 1999). The long-rangecorrelation in HRV increases with age irrespective of the method and the scale used (fromβ∼ 1of adolescent to β ∼ 1.3 in the elderly) (Pikkujamsa et al 1999, Sakata et al 1999). Ischemic

R34 A Eke et al

Table 4. Fractal estimates of human heart rate variability in control conditions.a

Row Method Length Subjects Reference Group Mean β �β

1 PSD 218 274 Bigger et al (1996) 1.060 ± 0.120Hayano et al

2 PSD 218 30 (1997) 0.940 ± 0.0203 Mean of 1, 2 1.000 ± 0.020

Aoyagi et al4 CGSA 214 5 (2000) 1.180 ± 0.050 4.50%

Huikuri et al5 PSD 24 h 24 (1999) 1.260 ± 0.150

Pikkujamsa et al6 PSD 24 h 29 (1999) 1.120 ± 0.1907 CGSA 24 h 23 Sakata et al (1999) 0.800 ± 0.1308 Mean of 4–7 1.060 ± 0.236 1.50%9 CGSA 1024 8 Fortrat et al (1998) 1.010 ± 0.12010 CGSA 1024 9 Fortrat et al (1999) 1.120 ± 0.07011 Mean of 9,10 1.065 ± 0.078 1.63%

Yamamoto et al 1.900 22.50%12 CGSA 10 min 6 (1992)

Hoshikawa and13 CGSA 512 8 Yamamoto (1997) 1.290 ± 0.370

Togo and14 CGSA 512 9 Yamamoto (2000) 1.150 ± 0.33015 Mean of 12–14 1.220 ± 0.099 5.50%16 CGSA 400 6 Lucy et al (2000) 1.020 ± 0.20017 CGSA 400 7 Lucy et al (2000) 1.000 ± 0.23018 Mean of 16,17 1.010 ± 0.014 0.25%

Blaber et al19 CGSA 256 11 (1996) 1.450 ± 0.10020 CGSA 256 20 Butler et al (1997) 1.150 ± 0.060

Hughson et al21 CGSA 256 14 (1995) 1.160 ± 0.090

Hughson et al22 CGSA 256 14 (1995) 1.090 ± 0.080

Hughson et al23 CGSA 256 14 (1995) 1.130 ± 0.09024 CGSA 256 9 Satoh et al (1999) 0.890 ± 0.06025 Mean of 19–24 1.145 ± 0.180 3.62%

a β is the mean estimate of the spectral indices, �β is the differential change in β expressed as per cent of the fullrange of the fGn/fBm model (−1 < β < 3) relative to the value of β in line 3 where the estimate is the most precisedue to the longest signal length.

and cardiomyopathic conditions (Peng et al 1995b, Bigger et al 1996, Lombardi et al 1996,Butler et al 1997, Makikallio et al 1997, 1998, 1999, Huikuri et al 2000) also increase the long-range correlations in HRV as compared to control populations. The changes in β are, however,very moderate (3.5 ± 3.6%) and values of β are within the transitional range between fractalnoises and motions. Given these circumstances, and the performance of the PSD method, astested on DHM signals, these estimates should be considered weak unless the HRV signalsmimic those of the SSM model (figure 15). In this case the PSD method should be regarded asvery precise, hence in spite of the small changes found inβ the conclusion on an increased long-term correlation could be much firmer; a possibility we will investigate in depth in the future.The heart is known to function under a strong autonomic influence. Some of these influenceshave earlier been identified in HRV studies and these have recently been confirmed by studies


with this influence being removed, for example, following heart transplantation (Hughsonet al 1995, Bigger et al 1996) (from β ∼ 1 to β ∼ 1.5–2) or autonomic failure (from β ∼ 1.4to β ∼ 1.6) (Blaber et al 1996). The correlation seem to increase (differential change in β =9.1 ± 9.55%) both in the spectra of short (Hughson et al 1995, Blaber et al 1996) and long(Bigger et al 1996) HRV signals. We should remember that short signal length decreases thereliability of fractal estimates, which is particularly problematic since autonomic influencehas been demonstrated to effect the high frequency range that is dominant in the spectra ofshort signals. The only long-range study (Bigger et al 1996) demonstrates the importanceof autonomic control in providing long-term adaptability of the cardiac rhythm. In survivalstudies the DFA and the PSD methods yield adverse results (Huikuri et al 2000). PSD suggeststhat increased correlation decreases survival rate, while DFA suggests the opposite. However,the differences obtained using DFA are very small, in one case insignificant.

When the current state of our understanding of the prerequisites for a reliable fractal timeseries analysis outlined in this review is applied to the results of fractal HRV studies, we mayconclude that they seem influenced by methodological factors due to short signal lengths andvastly different scaling ranges. Further discrimination and refinement of the fractal estimatesof human HRV would be possible if signal classification was part of the analysis followedby the application of class-specific methods offering robust and more precise estimates ofH. Further work is needed to characterize the performance of the methods widely used byresearchers in the HRV field on ideal fractal signals produced by different models and alsoon real physiological signals. These studies could then help in confirming the notion ofa proposed shift in HRV dynamics to a more correlated state in diseased states or ageing(Peng et al 1995a).

10. Future trends and closing remarks

10.1. Need to identify faulty data in literature

The potential of monofractal analysis is far from being fully exploited, since it was only veryrecently that attention has been drawn to the need for a thorough testing of the monofractaltools (Bassingthwaighte and Raymond, 1994, 1995, Caccia et al 1997, Cannon et al 1997,Eke et al 2000), and a much deeper understanding of the nature of the physiological signalsand their interactions with the fractal methods used for their analysis. The central importanceof signal classification in reliable fractal time series analysis has only been recently identified(Eke et al 2000). Most of the published data in the literature, has resulted from earlier studiesunaware of these issues and hence their revision for potentially being in error seems warranted.A likely possibility is that where the published fractal measure indicates the presence of astrong correlation in the signal for which the dispersional method gave H> 0.8, while in realitythe signal analysed by the PSD or signal summation conversion method would have proven anfBm for which the proper choice, the bridge trended scaled windowed variance method wouldhave yielded an H ≈ 0.3, would result in a completely different conclusion being drawn on thecorrelation structure of the signal: i.e. a strong anticorrelation of the increments (figure 18)(Rossitti and Stephensen 1994).

10.2. Stepping beyond monofractal analysis

Monofractal analysis can give a much deeper insight into the internal make-up ofphysiological signals than traditional approaches like descriptive statistics or a simplefrequency decomposition approach by Fourier analysis. Monofractal analysis has alreadyshown that the self-similar, fractal, 1/f -type patterns seems ubiquitous in nature including

R36 A Eke et al

many biological phenomena. Some considerations imply that the universality of 1/f -typepatterns in nature could be explained theoretically (Schlesinger 1987): 1/f noises have asimilar role among stochastic processes (i.e. dependent data series), like normal distributionshave among other distributions of independent datasets. Other authors assume stochasticfeedback mechanisms as being behind physiological 1/f noises (Ivanov et al 1998). The firststeps have already been taken to explore these signals further by showing whether their self-similarity is indeed isotropic, a fundamental property of monofractals. One should anticipatethat physiological processes could, in principle, produce even more complex, anisotropic,multifractal fluctuations of which the monofractal analysis can only give a measure of thelargest of their fractal dimensions. One of the first physiological applications of multifractaltools (Ivanov et al 1999) has demonstrated that this is a real possibility as it reveals multifractaldynamics in human HRV time series.

Acknowledgments

The computer program ‘FracTool’ and ‘HRVtool’ running on the Macintosh, PC, and Unixplatforms for estimating the Hurst coefficient of physiological time series is available on theWeb at http://www.elet2.sote.hu/eke/FRACTPHYS/. These programs can be freelydistributed as long as the ‘readme’ file is included and the use of the programs is properlyacknowledged. This study was supported by OTKA grants I/3 2040, T 016953, T 034122,NIH Grants TW00442 and RR1243.

References

Aoyagi N, Ohashi K, Tomono S and Yamamoto Y 2000 Temporal contribution of body movement to very long-termheart rate variability in humans Am. J. Physiol. (Heart Circ. Physiol.) 278 H1035–41

Arneodo A, Bacry E and Muzy J 1995 The thermodynamics of fractals revisited with wavelets Physica A 213 232–75Arneodo A, d’Aubenton-Carafa Y, Audit B, Bacry E, Muzy J and Thermes C 1998 Nucleotide composition effects

on the long-range correlations in human genes Eur. Phys. J. B 1 259–63Avnir D, Biham O, Lidar D and Malcai O 1998 Is the geometry of nature fractal? Science 279 39–40Bassingthwaighte J B 1988 Physiological heterogeneity: fractals link determinism and randomness in structures and

functions NIPS 3 5–10Bassingthwaighte J B and Beyer R P 1991 Fractal correlation in heterogeneous systems Physica D 53 71–84Bassingthwaighte J B and Raymond G M 1994 Evaluating rescaled range analysis for time series Ann. Biomed. Eng.

22 432–44Bassingthwaighte J B and Raymond G M 1995 Evaluation of the dispersional analysis method for fractal time series

Ann. Biomed. Eng. 23 491–505Bassingthwaighte J, Liebovitch L and West B 1994 Fractal Physiology (New York: Oxford University Press)Beran J 1994 Statistics for Long-Memory Processes (New York: Chapman and Hall)Bigger J J, Steinman R, Rolnitzky L, Fleiss J, Albrecht P and Cohen R 1996 Power law behavior of RR-interval

variability in healthy middle-aged persons, patients with recent acute myocardial infarction, and patients withheart transplants Circulation 93 2142–51

Blaber A P, Bondar R L and Freeman R 1996 Coarse graining spectral analysis of HR and BP variability in patientswith autonomic failure Am. J. Physiol. 271 H1555–64

Butler G C, Ando S and Floras J S 1997 Fractal component of variability of heart rate and systolic blood pressure incongestive heart failure Clin. Sci. Colch. 92 543–50

Caccia D C, Percival D, Cannon M J, Raymond G and Bassingthwaighte J B 1997 Analyzing exact fractal time series.Evaluating dispersional analysis and rescaled range methods Physica A 246 609–32

Cannon M J, Percival D B, Caccia D C, Raymond G M and Bassingthwaighte J B 1997 Evaluating scaled windowedvariance methods for estimating the Hurst coefficient of time series Physica A 241 606–26

Davies R B and Harte D S 1987 Test for Hurst effect Biometrika 74 95–101Dirksen A, Holstein-Rathlou N-H, Madsen F, Skovgaard L, Ulrik C, Heckscher T and Kok-Jensen A 1998 Long-range

correlations of serial FEV1 measurements in emphysematous patients and normal subjects J. Appl. Physiol. 85259–65


Eke A, Herman P, Bassingthwaighte J B, Raymond G M, Cannon M, Balla I and Ikrenyi C 1997 Temporal fluctuationsin regional red blood cell flux in the rat brain cortex is a fractal process Adv. Exp. Med. Biol. 428 703–9

Eke A, Herman P, Bassingthwaighte J B, Raymond G M, Percival D B, Cannon M, Balla I and Ikrenyi C 2000Physiological time series: distinguishing fractal noises from motions Pflugers Arch.—Eur. J. Physiol. 439403–15

European Society of Cardiology and North American Society of Pacing and Electrophysiology 1996 Heart ratevariability: standards of measurement, physiological interpretation and clinical use. Task force of the EuropeanSociety of Cardiology and the North American Society of Pacing and Electrophysiology Circulation 931043–65

Falconer K 1990 Fractal geometry: mathematical foundations and applications (New York: Wiley)Fortrat J O, Formet C, Frutoso J and Gharib C 1999 Even slight movements disturb analysis of cardiovascular

dynamics Am. J. Physiol. 277 H261–7Fortrat J-O, Nasr O, Duvareille M and Gharib C 1998 Human cardivascular variability, baroreflex and hormonal

adaptations to a blood donation Clin. Sci. 95 269–75Fougere P F 1985 On the accuracy of spectrum analysis of red noise processes using maximum entropy and

periodogram methods: simulation studies and application to geophysical data J. Geophys. Res. 90 43–66Frey U, Silverman M, Barabasi A and Suki B 1998 Irregularities and power law distributions in the breathing pattern

in preterm and term infants J. Appl. Physiol. 86 789–97Gabor D 1946 Theory of Communication J. Inst. Electr. Eng. 93 429–57Glass L 2001 Synchronization and rhythmic processes in physiology Nature 410 277–84Goodwin B 1997 Temporal organisation and disorganization in organisms Chronobiol. Int. 14 531–6Hausdorff J, Lertratanakul A, Cudkowicz M, Peterson A, Kaliton D and Goldberger A 2000 Dynamic markers of

altered gait rhythm in amyotrophic lateral sclerosis J. Appl. Physiol. 88 2045–53Hausdorff J M, Zemany L, Peng C-K and Goldberger A L 1999 Maturation of gait dynamics: stride-to-stride

variability and its temporal organization in children J. Appl. Physiol. 86 1040–7Hayano J, Yamasaki F, Sakata S, Okada A, Mukai S and Fujinami T 1997 Spectral characteristics of ventricular

response to atrial fibrillation Am. J. Physiol. 273 H2811–6Herman P, Kocsis L and Eke A 2001 Fractal branching pattern in the pial vasculature in the cat J. Cereb. Blood Flow

Metab. 21 741–54Hon E H and Lee S T 1965 Electronic evaluations of the fetal heart rate patterns preceding fetal death: further

observations Am. J. Obstet. Gynecol. 87 817–26Hoshikawa Y and Yamamoto Y 1997 Effects of Stroop color-word conflict test on the autonomic nervous system

responses Am. J. Physiol. 272 H1113–21Hughson R L, Maillet A, Dureau G, Yamamoto Y and Gharib C 1995 Spectral analysis of blood pressure variability

in heart transplant patients Hypertension 25 643–50Huikuri H V, Makikallio T H, Airaksinen K E, Seppanen T, Puukka P, Raiha I J and Sourander L B 1998 Power-law

relationship of heart rate variability as a predictor of mortality in the elderly Circulation 97 2031–6Huikuri H V, Makikallio T H, Peng C K, Goldberger A L, Hintze U and Moller M 2000 Fractal correlation properties

of R-R interval dynamics and mortality in patients with depressed left ventricular function after an acutemyocardial infarction [see comments] Circulation 101 47–53

Huikuri H V, Poutiainen A M, Makikallio T H, Koistinen M J, Airaksinen K E, Mitrani R D, Myerburg R J andCastellanos A 1999 Dynamic behavior and autonomic regulation of ectopic atrial pacemakers Circulation 1001416–22

Hurst H 1951 Long-term storage capacity of reservoirs Trans. Am. Soc. Civ. Eng. 116 770–808Ivanov P, Amaral L, Goldberger A, Havlin S, Rosenblum M, Struzik Z and Stanley H 1999 Multifractality in human

heartbeat dynamics Nature 6 461–5Ivanov P, Amaral L, Goldberger A and Stanley H 1998 Stochastic feedback and the resolution of biological rhytms

Europhys. Lett. 43 363–8Jones C, Lonergan G and Mainwaring D 1996 Wavelet packet computation of the Hurst exponent J. Phys. A: Gen.

Phys. 29 2509–27Lombardi F, Sandrone G, Mortara A, Torzillo D, La Rovere M T, Signorini M G, Cerutti S and Malliani A 1996 Linear

and nonlinear dynamics of heart rate variability after acute myocardial infarction with normal and reduced leftventricular ejection fraction Am. J. Cardiol. 77 1283–8

Lucy S D, Hughson R L, Kowalchuk J M, Paterson D H and Cunningham D A 2000 Body position and cardiacdynamic and chronotropic responses to steady-state isocapnic hypoxaemia in humans Exp. Physiol. 85 227–37

Makikallio T H, Høiber S, Køber L, Torp-Pedersen C, Peng C K, Goldberger A L and Huikuri H V 1999 Fractalanalysis of heart rate dynamics as a predictor of mortality in patients with depressed left ventricular functionafter acute myocardial infarction Am. J. Cardiol. 83 836–9

R38 A Eke et al

Makikallio T H, Ristimae T, Airaksinen K E, Peng C K, Goldberger A L and Huikuri H V 1998 Heart rate dynamicsin patients with stable angina pectoris and utility of fractal and complexity measures Am. J. Cardiol. 8127–31

Makikallio T H, Seppanen T, Airaksinen K E, Koistinen J, Tulppo M P, Peng C K, Goldberger A L and Huikuri H V1997 Dynamic analysis of heart rate may predict subsequent ventricular tachycardia after myocardial infarctionAm. J. Cardiol. 80 779–83

Mandelbrot B 1982 The Fractal Geometry of Nature (New York: W H Freeman)Mandelbrot B 1985 Self-affine fractals and fractal dimension Phys. Scripta 32 257–60Mandelbrot B B and Wallis J R 1969 Some long-run properties of geophysiological records Water Resources Research

5 321–40Nonnenmacher T, Losa G and Weibel E 1994 Fractals in Biology and in Medicine (Basel: Birkhauser)Peitgen H-O, Jurgens H and Saupe D 1992 Chaos and Fractals. New Frontiers of Science (New York: Springer)Peng C K, Havlin S, Hausdorff J M, Mietus J E, Stanley H E and Goldberger A L 1995a Fractal mechanisms and

heart rate dynamics. Long-range correlations and their breakdown with disease J. Electrocardiol. 28 59–65Peng C-K, Buldyrev S, Havlin S, Simons M, Stanley H and Goldberger A 1994 Mosaic organization of DNA

nucleotids Phys. Rev. E 49 1685–9Peng C-K, Havlin S, Stanley H and Goldberger A 1995b Quantification of scaling exponents and crossover phenomena

in nonstationary heartbeat time series Chaos 5 82–7Peterka R 2000 Postural control model interpretation of stabilogram diffusion analysis Biol. Cybern. 82 335–43Pikkujamsa S M, Makikallio T H, Sourander L B, Raiha I J, Puukka P, Skytta J, Peng C K, Goldberger A L and Huikuri

H V 1999 Cardiac interbeat interval dynamics from childhood to senescence: comparison of conventional andnew measures based on fractals and chaos theory Circulation 100 393–9

Raymond G M and Bassingthwaighte J B 1998 Deriving dispersional and scaled windowed variance analyses usingthe correlation function of discrete fractional Gaussian noise Physica A 265 85–96

Riley M A, Wong S, Mitra S and Turvey M T 1997 Common effects of touch and vision on postural parameters Exp.Brain Res. 117 165–70

Rossitti S and Stephensen H 1994 Temporal heterogeneity of the blood flow velocity at the middle cerebral artery inthe normal human characterized by fractal analysis Acta. Phys. Scan. 151 191–8

Sakata S, Hayano J, Mukai S, Okada A and Fujinami T 1999 Aging and spectral characteristics of the nonharmoniccomponent of 24-h heart rate variability Am. J. Physiol. 276 R1724–31

Satoh K, Koh J and Kosaka Y 1999 Effects of nitroglycerin on fractal features of short-term heart rate and bloddpressure variability J. Anesth. 13 71–6

Schepers H E, van Beek J H G M and Bassingthwaighte J B 1992 Four methods to estimate the fractal dimensionfrom self-affine signals IEEE Eng. Med. Biol. 11 57–71

Schlesinger M F 1987 Fractal time and 1/f noise in complex systems Ann. N.Y. Acad. Sci. 504 214–28Serrador J M, Finlayson H C and Hughson R L 1999 Physical activity is a major contributor to the ultra low frequency

components of heart rate variability Heart (Br. Card. Soc. Online) 82 e9Simonsen I and Hansen A 1998 Determination of the Hurst Exponent by use of wavelet transforms Phys. Rev. E 58

79–87Szeto H H, Cheng P Y, Decena J A, Cheng Y, Wu D L and Dwyer G 1992 Fractal properties in fetal breathing

dynamics Am. J. Physiol. 263 R141–7Taylor F 1994 Principles of Signals and Systems (McGraw-Hill Series in Electrical and Computer Engineering.

Communications and Signal Processing) (New York: McGraw Hill)Togo F and Yamamoto Y 2000 Decreased fractal component of human heart rate variability during non-REM sleep

Am. J. Physiol. (Heart Circ. Physiol.) 280 H17–21van Beek J, Roger S and Bassigthwaigthe J 1989 Regional myocardial flow heterogeneity explained with fractal

networks Am. J. Physiol. 257 1670–80Weitkunat R e 1991 Digital Biosignal Processing (Amsterdam: Elsevier)West B 1990 Fractal Physiology and Chaos in Medicine (Singapore: World Scientific)Yamamoto Y and Hughson R L 1991 Coarse-gaining spectral analysis: new method for studying heart rate variability

J. Appl. Physiol. 71 1143–50Yamamoto Y and Hughson R L 1993 Extracting fractal components from time series Physica D 68 250–64Yamamoto Y, Hughson R L and Nakamura Y 1992 Autonomic nervous system responses to exercise in relation to

ventilatory threshold Chest 101 206s–10sYu Z, Anh V and Wang B 2000 Correlation property of length sequences based on global structure of the complete

genome Phys. Rev. E 63 1–8Zhang R, Zuckerman J and Levine B 2000 Spontaneous fluctuations in cerebral blood flow: insights from extended-

duration recordings in human Am. J. Physiol. (Heart Circ. Physiol.) 278 H1848–55

fractal characterization of complexity in temporal ...€¦ · fractal characterization of...

Documents