A_Effect of Trends on Detrended Fluctuation Analysis

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Effect of trends on detrended uctuation analysis

Kun Hu, 1 Plamen Ch. Ivanov, 1,2 Zhi Chen, 1 Pedro Carpena, 3 and H. Eugene Stanley 11Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215

2 Harvard Medical School, Beth Israel Deaconess Medical Center, Boston, Massachusetts 022153 Departamento de F sica Aplicada II, Universidad de Ma laga E-29071, Ma laga, Spain

Received 8 March 2001; published 26 June 2001

Detrended uctuation analysis DFA is a scaling analysis method used to estimate long-range power-lawcorrelation exponents in noisy signals. Many noisy signals in real systems display trends, so that the scalingresults obtained from the DFA method become difcult to analyze. We systematically study the effects of threetypes of trends linear, periodic, and power-law trends, and offer examples where these trends are likely tooccur in real data. We compare the difference between the scaling results for articially generated correlatednoise and correlated noise with a trend, and study how trends lead to the appearance of crossovers in thescaling behavior. We nd that crossovers result from the competition between the scaling of the noise and theapparent scaling of the trend. We study how the characteristics of these crossovers depend on i the slopeof the linear trend; ii the amplitude and period of the periodic trend; iii the amplitude and power of thepower-law trend, and iv the length as well as the correlation properties of the noise. Surprisingly, we nd thatthe crossovers in the scaling of noisy signals with trends also follow scaling lawsi.e., long-range power-lawdependence of the position of the crossover on the parameters of the trends. We show that the DFA result of noise with a trend can be exactly determined by the superposition of the separate results of the DFA on thenoise and on the trend, assuming that the noise and the trend are not correlated. If this superposition rule is notfollowed, this is an indication that the noise and the superposed trend are not independent, so that removing thetrend could lead to changes in the correlation properties of the noise. In addition, we show how to use DFAappropriately to minimize the effects of trends, how to recognize if a crossover indicates indeed a transitionfrom one type to a different type of underlying correlation, or if the crossover is due to a trend without anytransition in the dynamical properties of the noise.

DOI: 10.1103/PhysRevE.64.011114 PACS number s : 05.40. a

I. INTRODUCTION

Many physical and biological systems exhibit complexbehavior characterized by long-range power-law correla-

tions. Traditional approaches such as the power-spectrumand correlation analysis are not suited to accurately quantifylong-range correlations in nonstationary signalse.g., sig-nals exhibiting uctuations along polynomial trends. De-trended uctuation analysis DFA 1 4 is a scaling analy-sis method providing a simple quantitative parameterthescaling exponent to represent the correlation propertiesof a signal. The advantages of DFA over many methods arethat it permits the detection of long-range correlations em-bedded in seemingly nonstationary time series, and alsoavoids the spurious detection of apparent long-range correla-tions that are an artifact of nonstationarity. In the past fewyears, more than 100 publications have utilized the DFA asthe method of correlation analysis, and have uncovered long-range power-law correlations in many research elds such ascardiac dynamics 523 , bioinformatics 1,2,2434,68 ,economics 3547 , meteorology 4850 , material science51 , ethology 52 , etc. Furthermore, the DFA method may

help identify different states of the same system according toits different scaling behaviors, e.g., the scaling exponent for heart interbeat intervals is different for healthy and sick individuals 14,16,17,53 .

The correct interpretation of the scaling results obtainedby the DFA method is crucial for understanding the intrinsicdynamics of the systems under study. In fact, for all systems

where the DFA method was applied, there are many issuesthat remain unexplained. One of the common challenges isthat the correlation exponent is not always a constant inde-pendent of scale and crossovers often existi.e., a change

of the scaling exponent for different range of scales5,16,35 . A crossover usually can arise from a change in thecorrelation properties of the signal at different time or spacescales, or can often arise from trends in the data. In this paperwe systematically study how different types of trends affectthe apparent scaling behavior of long-range correlated sig-nals. The existence of trends in times series generated byphysical or biological systems is so common that it is almostunavoidable. For example, the number of particles emittedby a radiation source in a unit time has a trend of decreasingbecause the source becomes weaker 54,55 ; the density of air due to gravity has a trend at a different altitude; the airtemperature in different geographic locations, rainfall andthe water ow of rivers have a periodic trend due to seasonalchanges 49,50,5659 ; the occurrence rate of earthquakes incertain areas has a trend in different time periods 60 . Animmediate problem facing researchers applying a scalinganalysis to a time series is whether trends in data arise fromexternal conditions, having little to do with the intrinsic dy-namics of the system generating noisy uctuating data. Inthis case, a possible approach is to rst recognize and lterout the trends before we attempt to quantify correlations inthe noise. Alternatively, trends may arise from the intrinsicdynamics of the system rather than being an epiphenomenonof external conditions, and thus they may be correlated with

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the noisy uctuations generated by the system. In this case,careful consideration should be given if trends should beltered out when estimating correlations in the noise, sincesuch intrinsic trends may be related to the local propertiesof the noisy uctuations.

Here we study the origin and the properties of crossoversin the scaling behavior of noisy signals, by applying the DFAmethod rst on correlated noise and then on noise withtrends, and comparing the difference in the scaling results.To this end, we generate an articial time seriesanticorrelated, white, and correlated noise with standard de-viation equal to oneusing the modied Fourier lteringmethod introduced by Makse et al. 63 . We consider thecase when the trend is independent of the local properties of the noise external trend . We nd that the scaling behaviorof noise with a trend is a superposition of the scaling of thenoise and the apparent scaling of the trend, and we deriveanalytical relations based on the DFA, which we call thesuperposition rule. We show how this superposition rulecan be used to determine if the trends are independent of thenoisy uctuation in real data, and if ltering these trends out

will not affect the scaling properties of the data.The outline of this paper is as follows. In Sec. II wereview the algorithm of the DFA method, and in Appendix Awe compare the performance of the DFA with the classicalscaling analysisHursts analysis ( R / S analysis andshow that the DFA is a superior method to quantify the scal-ing behavior of noisy signals. In Sec. III we consider theeffect of a linear trend and we present an analytic derivationof the apparent scaling behavior of a linear trend in Appen-dix C. In Sec. IV we study a periodic trend, and in Sec. V westudy the effect of a power-law trend. We systematicallystudy all resulting crossovers, their conditions of existence,and their typical characteristics associated with the differenttypes of trends. In addition, we also show how to use DFAappropriately to minimize or even eliminate the effects of those trends in cases that trends are not choices of the study,that is, trends do not reect the dynamics of the system butare caused by some irrelevant background. Finally, Sec.VI contains a summary.

II. DFA

To illustrate the DFA method, we consider a noisy timeseries, u ( i ) ( i 1, . . . , N ma x ). We integrate the time seriesu ( i ),

y j i 1

j

u i u , 1

where

u1

N ma x j 1

N ma x

u i , 2

and is divided into boxes of equal size n. In each box, we tthe integrated time series by using a polynomial function, y fi t ( i ), which is called the local trend. For order- l DFA

DFA-1 if l 1, DFA-2 if l 2, etc. , the l -order polynomial

function should be applied for the tting. We detrend theintegrated time series y ( i ) by subtracting the local trend y fi t ( i ) in each box, and we calculate the detrended uctua-tion function

Y i y i y fi t i . 3

For a given box size n, we calculate the root mean square

rms uctuation

F n1

N ma x i 1

N ma x

Y i 2 . 4

The above computation is repeated for box sizes n differentscales to provide a relationship between F (n ) and n. Apower-law relation between F (n ) and the box size n indi-cates the presence of scaling: F (n ) n . The parameter ,called the scaling exponent or correlation exponent, repre-sents the correlation properties of the signal: if 0.5, thereis no correlation and the signal is an uncorrelated signalwhite noise ; if 0.5, the signal is anticorrelated; if

0.5, there are positive correlations in the signal.

III. NOISE WITH LINEAR TRENDS

First we consider the simplest case: correlated noise witha linear trend. A linear trend

u i ALi 5

is characterized by only one variable the slope of thetrend AL . For convenience, we denote the rms uctuationfunction for noise without trends by F (n ), linear trends byF L(n ), and noise with a linear trend by F L(n ).

A. DFA-1 on noise with a linear trend

Using the algorithm of Makse et al. 63 , we generate acorrelated noise with a standard deviation one, with a givencorrelation property characterized by a given scaling expo-nent . We apply DFA-1 to quantify the correlation proper-ties of the noise and nd that only in a certain good t regioncan the rms uctuation function F ( n ) be approximated by apower-law function see Appendix A

F n b 0n , 6

where b 0 is a parameter independent of the scale n. We ndthat the good t region depends on the correlation exponent see Appendix A . We also derive analytically the rmsuctuation function for a linear trend only for DFA-1 andnd that see Appendix C

F L n k 0 ALn L, 7

where k 0 is a constant independent of the length of trend N ma x , of the box size n, and of the slope of the trend AL .We obtain L 2.

Next we apply the DFA-1 method to the superposition of a linear trend with correlated noise and we compare the rmsuctuation function F L(n ) with F (n ) see Fig. 1 . We

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observe a crossover in F L ( n ) at scale n n . For nn , the behavior of F L ( n ) is very close to the behavior

of F ( n ), while for n n , the behavior of F L ( n ) is veryclose to the behavior of F L( n ). A similar crossover behavioris also observed in the scaling of the well-studied biasedrandom walk 61,62 . It is known that the crossover in the

biased random walk is due to the competition of the unbiasedrandom walk and the bias see Fig. 5.3 of 62 . We illustratethis observation in Fig. 2, where the detrended uctuationfunctions Eq. 3 of the correlated noise, Y ( i ), and of thenoise with a linear trend, Y L ( i ), are shown. For the box sizen n as shown in Figs. 2 a and 2 b , Y L ( i ) Y ( i ). Forn n as shown in Figs. 2 c and 2 d , Y L ( i ) has a distin-guishable quadratic background signicantly different fromY

( i ). This quadratic background is due to the integration of the linear trend within the DFA procedure and represents thedetrended uctuation function Y L of the linear trend. Theserelations between the detrended uctuation functions Y ( i ) atdifferent time scales n explain the crossover in the scalingbehavior of F

L ( n ): from very close to F

( n ) to very close

to F L( n ) observed in Fig. 1 .The experimental results presented in Figs. 1 and 2 sug-

gest that the rms uctuation function for a signal which is asuperposition of a correlated noise and a linear trend can beexpressed as

F L

n 2 F L n2 F

n 2 . 8

We provide an analytic derivation of this relation in Appen-dix B, where we show that Eq. 8 holds for the superposi-tion of any two independent signalsin this particular case

noise and a linear trend. We call this relation the superpo-sition rule. This rule helps us understand how the compe-tition between the contribution of the noise and the trend tothe rms uctuation function F L ( n ) at different scales nleads to appearance of crossovers 61 .

Next, we ask how the crossover scale n depends on ithe slope of the linear trend A L , ii the scaling exponent of the noise, and iii the length of the signal N ma x . Surpris-ingly, we nd that for noise with any given correlation ex-ponent the crossover scale n itself follows a power-lawscaling relation over several decades: n ( A L)

see Fig.3 . We nd that in this scaling relation, the crossover expo-nent is negative and its value depends on the correlationexponent of the noisethe magnitude of decreaseswhen increases. We present the values of the crossoverexponent for different correlation exponents in Table I.

To understand how the crossover scale depends on thecorrelation exponent of the noise we employ the superpo-sition rule Eq. 8 and estimate n as the intercept betweenF

( n ) and F L( n ). From Eqs. 6 and 7 , we obtain the

following dependence of n on :

n A L

k 0b 0

1/( L )

A L

k 0b 0

1/( 2)

. 9

This analytical calculation for the crossover exponent1/( L ) is in a good agreement with the observed val-

ues of obtained from our simulations see Fig. 3 andTable I .

A L=216

A L=212

A L=28

Correlated noise withlinear trend: F

L(n)

n x

DFA1

100

101

102

103

104

105

n

10 6

10 4

10 2

100

102

104

106

F

( n )

Correlated noise : F(n)

linear trends: F L(n)

2

2

FIG. 1. Crossover behavior of the root-mean-square uctuationfunction F L ( n ) for noise of length N ma x 2

17 and correlation ex-ponent 0.1) with superposed linear trends of slope A L

2 16,2 12,2 8 . For comparison, we show F ( n ) for the noisethick solid line and F L( n ) for the linear trends dot-dashed lineEq. 7 . The results show a crossover at a scale n for F L ( n ).

For n n , the noise dominates and F L ( n ) F ( n ). For nn , the linear trend dominates and F L ( n ) F L( n ). Note that

the crossover scale n increases when the slope A L of the trenddecreases.

0 150 300 6

0

6

Y

L

Correlated noise + linear trend

(b) n < n x

0 500 1000

i

20

0

20

Y

L

(d) n > n x

0 150 300 6

0

6

Y

Correlated noise

(a) n < n x

0 500 1000

i

20

0

20

Y

(c) n > n x

FIG. 2. Comparison of the detrended uctuation function fornoise Y ( i ) and for noise with linear trend Y L ( i ) at differentscales. a and c are Y for noise with 0.1; b and d are Y Lfor the same noise with a linear trend with slope A L 2

12 thecrossover scale n 320, see Fig. 1 . a and b For scales n

n the effect of the trend is not pronounced and Y Y L i.e.,Y

Y L). c and d For scales n n , the linear trend is domi-

nant and Y Y L .

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Finally, since the F L( n ) does not depend on N ma x as weshow in Eq. 7 and in Appendix C, we nd that n does notdepend on N ma x . This is a special case for linear trends anddoes not always hold for higher-order polynomial trends seeAppendix D .

B. DFA-2 on noise with a linear trend

Application of the DFA-2 method to noisy signals withoutany polynomial trends leads to scaling results identical to thescaling obtained from the DFA-1 method, with the exceptionof some vertical shift to lower values for the rms uctuationfunction F ( n ) see Appendix A . However, for signalswhich are a superposition of correlated noise and a lineartrend, in contrast to the DFA-1 results presented in Fig. 1,F

L ( n ) obtained from DFA exhibits no crossovers, and isexactly equal to the rms uctuation function F ( n ) obtained

from DFA-2 for correlated noise without trend see Fig. 4 .These results indicate that a linear trend has no effect on thescaling obtained from DFA-2. The reason for this is that bydesign the DFA-2 method lters out linear trends, i.e.,Y L( i ) 0 Eq. 3 and thus F L ( n ) F ( n ) due to the su-perposition rule Eq. 8 . For the same reason, polynomialtrends of order lower than l superposed on correlated noisewill have no effect on the scaling properties of the noisewhen DFA- l is applied. Therefore, our results conrm thatthe DFA method is a reliable tool to accurately quantifycorrelations in noisy signals embedded in polynomial trends.Moreover, the reported scaling and crossover features of F ( n ) can be used to determine the order of polynomialtrends present in the data.

IV. NOISE WITH SINUSOIDAL TREND

In this section we study the effect of sinusoidal trends onthe scaling properties of noisy signals. For a signal which isa superposition of correlated noise and sinusoidal trend, wend that based on the superposition rule Appendix B theDFA rms uctuation function can be expressed as

F S

n 2 F

n 2 F S n2 , 10

where F S ( n ) is the rms uctuation function of noise with asinusoidal trend, and F S( n ) is for the sinusoidal trend. Firstwe consider the application of DFA-1 to a sinusoidal trend.Next we study the scaling behavior and the features of cross-overs in F S( n ) for the superposition of a correlated noiseand a sinusoidal trend employing the superposition rule Eq.10 . At the end of this section we discuss the results ob-

tained from higher-order DFA.

10 6

10 5

10 4

10 3

10 2

10 1

AL

101

102

103

n x

=0.1

=0.3

=0.5

=0.7

=0.9

DFA1

FIG. 3. The crossover n of F L( n ) for noise with a lineartrend. We determine the crossover scale n based on the difference

between log F noise and log F L noise with a linear trend . Thescale for which 0.05 is the estimated crossover scale n . Forany given correlation exponent of the noise, the crossover scale

n exhibits a long-range power-law behavior n ( A L)

, where thecrossover exponent is a function of see Eq. 9 and Table I .

TABLE I. The crossover exponent from the power-law rela-tion between the crossover scale n and the slope of the linear trend A L , n ( A L)

, for different values of the correlation exponents of the noise Fig. 3 . The values of obtained from our simulationsare in good agreement with the analytical prediction 1/(2 )Eq. 9 . Note that 1/(2 ) are not always exactly equal to

because F ( n ) in simulations is not a perfect simple power-lawfunction and the way we determine numerically n is just approxi-mated.

1/(2 )

0.1 -0.54 -0.530.3 -0.58 -0.590.5 -0.65 -0.670.7 -0.74 -0.770.9 -0.89 -0.91

100

101

102

103

104

n

10 1

100

101

102

103

F ( n )

= 0.1

= 0.3

= 0.5

= 0.7

= 0.9

Noise

Noise with linear trend (A L=212

):

DFA2

optimal fitting range

FIG. 4. Comparison of the rms uctuation function F ( n ) fornoise with different types of correlations lines and F L ( n ) for thesame noise with a linear trend of slope A L 2

12 symbols forDFA-2. F L ( n ) F ( n ) because the integrated linear trend can beperfectly ltered out in DFA-2, thus Y L( i ) 0 from Eq. 3 . Wenote that to estimate accurately the correlation exponents, one hasto choose an optimal range of scales n , where F ( n ) is tted. Fordetails see Appendix A.


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A. DFA-1 on sinusoidal trend

Given a sinusoidal trend u ( i ) ASsin(2 i / T ) ( i1 , . . . , N ma x ), where AS is the amplitude of the signal and T

is the period, we nd that the rms uctuation function F S(n )does not depend on the length of the signal N ma x , and hasthe same shape for different amplitudes and different periodsFig. 5 . We nd a crossover at scale corresponding to the

period of the sinusoidal trend

n 2 T , 11

and it does not depend on the amplitude AS . We call thiscrossover n 2 for convenience, as we will see later. For nn 2 , the rms uctuation F S( n ) exhibits an apparent scal-

ing with the same exponent as F L( n ) for the linear trend seeEq. 7 ,

F S n k 1 AST

n S, 12

where k 1 is a constant independent of the length N ma x , of theperiod T , of the amplitude AS of the sinusoidal signal, and of the box size n. As for the linear trend Eq. 7 , we obtain S 2 because at small scales box size n ) the sinusoidal

function is dominated by a linear term. For n n 2 , due tothe periodic property of the sinusoidal trend, F S(n ) is a con-stant independent of the scale n,

F S n1

2 2 AST . 13

The period T and the amplitude AS also affects the verticalshift of F S(n ) in both regions. We note that in Eqs. 12 and13 , F S(n ) is proportional to the amplitude AS , a behavior

which is also observed for the linear trend Eq. 7 .

B. DFA-1 on noise with sinusoidal trend

In this section we study how the sinusoidal trend affectsthe scaling behavior of noise with different types of correla-tions. We apply the DFA-1 method to a signal which is asuperposition of correlated noise with a sinusoidal trend. Weobserve that there are typically three crossovers in the rmsuctuation F S( n ) at characteristic scales denoted by n 1 ,n 2 , and n 3 Fig. 6 . These three crossovers divide F S( n )into four regions, as shown in Fig. 6 a the third crossovercannot be seen in Fig. 6 b because its scale n 3 is greaterthan the length of the signal . We nd that the rst and third

100

101

102

103

104

105

n

10 2

100

102

104

106

F S

( n )

A S=64, T=211

A S=64, T=212

A S=32, T=211

A S=32, T=212

2

n2 x

DFA1

FIG. 5. Root-mean-square uctuation function F S( n ) for sinu-soidal functions of length N ma x 2

17 with different amplitude ASand period T . All curves exhibit a crossover at n 2 T /2, with aslope S 2 for n n 2 and a at region for n n 2 . There aresome spurious singularities at n j(T /2) ( j is a positive integer

shown by the spikes.

FIG. 6. Crossover behavior of the root-mean-square uctuationfunction F S(n ) circles for correlated noise of length N ma x

2 17) with a superposed sinusoidal function characterized by pe-

riod T 128 and amplitude AS 2. The rms uctuation functionF (n ) for noise thick line and F S(n ) for the sinusoidal trend thinline are shown for comparison. a F S(n ) for correlated noisewith 0.9. b F S(n ) for anticorrelated noise with 0.9. Thereare three crossovers in F S(n ), at scales n 1 , n 2 , and n 3 thethird crossover cannot be seen in b because it occurs at scalelarger than the length of the signal . For n n 1 and n n 3 thenoise dominates and F S( n ) F (n ) while for n 1 n n 3 thesinusoidal trend dominates and F S(n ) F S( n ). The crossovers atn 1 and n 3 are due to the competition between the correlatednoise and the sinusoidal trend see Fig. 7 , while the crossover atn 2 relates only to the period T of the sinusoidal Eq. 11 .


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crossovers at scales n 1 and n 3 , respectively see Fig. 6 ,result from the competition between the effects on F S ( n ) of the sinusoidal signal and the correlated noise. For n n 1region I and n n 3 region IV , we nd that the noise has

the dominating effect F ( n ) F S( n ) , so the behavior of F

S ( n ) is very close to the behavior of F ( n ) Eq. 10 . Forn

1n n

2region II and n

2n n

3region III the

sinusoidal trend dominates F S( n ) F ( n ) , thus the behav-ior of F S( n ) is close to F S( n ) see Figs. 6 and 7 .

To better understand why there are different regions in thebehavior of F S( n ), we consider the detrended uctuationfunction Eq. 3 and Appendix B of the correlated noiseY

( i ), and of the noise with sinusoidal trend Y S . In Fig. 7we compare Y ( i ) and Y S ( i ) for anticorrelated and corre-lated noise in the four different regions. For very small scalesn n 1 , the effect of the sinusoidal trend is not pronounced,Y

S( i ) Y ( i ), indicating that in this scale region the signalcan be considered as noise uctuating around a constant

trend which is ltered out by the DFA-1 procedure Figs.7 a and 7 b . Note that the behavior of Y S Fig. 7 b isidentical to the behavior of Y L Fig. 2 b , since both asinusoidal with a large period T and a linear trend with smallslope A L can be well approximated by a constant trend forn n 1 . For small scales n 1 n n 2 region II , we ndthat there is a dominant quadratic background for Y S ( i )Fig. 7 d . This quadratic background is due to the integra-

tion procedure in DFA-1, and is represented by the detrendeductuation function of the sinusoidal trend Y S( i ). It is similarto the quadratic background observed for linear trend Y L ( i )Fig. 2 d i.e., for n 1 n n 2 the sinusoidal trend be-

haves as a linear trend and Y S( i ) Y L( i ). Thus in region IIthe linear trend effect of the sinusoidal is dominant, Y S

Y , which leads to F S ( n ) F S( n ). This explains also

why F S ( n ) for n n 2 Fig. 6 exhibits crossover behaviorsimilar to the one of F L ( n ) observed for noise with a lineartrend. For n 2 n n 3 region III the sinusoidal behavioris strongly pronounced Fig. 7 f , Y S( i ) Y ( i ) , andY

S ( i ) Y S( i ) changes periodically with period equal to the

period of the sinusoidal trendT

. SinceY

S (i) is boundedbetween a minimum and a maximum value, F S ( n ) cannot

increase and exhibits a at region Fig. 6 . At very largescales, n n 3 , the noise effect is again dominant Y S( i )remains bounded, while Y grows when increasing the scalewhich leads to F S ( n ) F ( n ) and a scaling behavior thatcorresponds to the scaling of the correlated noise.

First we consider n 1 . Surprisingly, we nd that fornoise with any given correlation exponent the crossoverscale n 1 exhibits long-range power-law dependence of theperiod T , n 1 T

T1, and the amplitude A S , n 1 ( A S) A1 of

the sinusoidal trend see Figs. 8 a and 8 b . We nd thatthe crossover exponents T1 and A1 have the same mag-nitude but different sign

T1is positive while

A1is nega-

tive. We also nd that the magnitudes of T1 and A1 in-crease for larger values of the correlation exponents of thenoise. We present the values of T1 and A1 for the differentcorrelation exponents in Table II. To understand thepower-law relations between n 1 and T , between n 1 and A S , and also how the crossover scale n 1 depends on thecorrelation exponent , we employ the superposition ruleEq. 10 and estimate n 1 analytically as the rst intercept

of F ( n ) and F S( n ). From Eqs. 12 and 6 , we obtain thefollowing dependence of n 1 on T , A S and :

n 1

b 0k 1

T

A S

1/(2 )

14

From this analytical calculation we obtain the following re-lation between the two crossover exponents T1 and A1 andthe correlation exponent : T1 A1 1/(2 ), which isin a good agreement with the observed values of T1 , A1obtained from simulations see Figs. 8 a and 8 b and TableII .

Next we consider n 2 . Our analysis of the rms uctua-tion function F S( n ) for the sinusoidal signal in Fig. 5 sug-gests that the crossover scale F S( n ) does not depend on the-

0 200 400 600 50

50

Y

(e)

n2x


7/19

amplitude AS of the sinusoidal. The behavior of the rms uc-tuation function F S( n ) for noise with a superposed sinu-soidal trend in Figs. 6 a and 6 b indicates that n 2 does notdepend on the correlation exponent of the noise, since forboth correlated ( 0.9) and anticorrelated ( 0) noise ( T

and AS are xed , the crossover scale n 2 remains un-changed. We nd that n 2 depends only on the period T of the sinusoidal trend and exhibits a long-range power-law be-havior n 2 T

T2 with a crossover exponent T2 1 Fig.8 c which is in agreement with the prediction of Eq. 11 .

102

103

104

T

101

102

103

n 1 x

=0.1

=0.3

=0.5

=0.7

=0.9

(a) Noise + sin. trend (A S=5.0)

1

DFA1

10 1

100

101

102

AS

101

102

103

n 1 x

=0.1 =0.3 =0.5 =0.7 =0.9

(b) Noise + sin. trend (T=211

)

A1

DFA1

102

103

104

T

102

103

104

n 2 x

(c) Noise + sin. trend

1.0

DFA1

101

102

T

102

103

104

n 3 x

=0.4

=0.5

=0.6

=0.7

=0.8

=0.9

(d) Noise + sin. trend (A S=2)

T3

DFA1

100

101

AS

102

103

104

n 3 x

=0.4

=0.5

=0.6

=0.7

=0.8

=0.9

(e) Noise + sin. trend (T=16)

A3

DFA1

FIG. 8. Dependence of the three crossovers in F S( n ) for noise with a sinusoidal trend Fig. 6 on the period T and amplitude AS of thesinusoidal trend. a Power-law relation between the rst crossover scale n 1 and the period T for xed amplitude AS and varying correlationexponent : n 1 T T1 , where T1 is a positive crossover exponent see Table II and Eq. 14 . b Power-law relation between the rstcrossover n 1 and the amplitude of the sinusoidal trend AS for xed period T and varying correlation exponent : n 1 AS

A1 where A1 isa negative crossover exponent Table II and Eq. 14 . c The second crossover scale n 2 depends only on the period T : n 2 T

T2, where T2 1. d Power-law relation between the third crossover n 3 and T for xed amplitude AS and varying trend: n 3 T

T3. e Power-lawrelation between the third crossover n 3 and AS for xed T and varying : n 3 ( AS)

A3. We nd that A3 T3 Table III and Eq. 15 .


011114-7


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For the third crossover scale n 3 , as for n 1 we nd apower-law dependence on the period T , n 3 T

T 3 , and onthe amplitude AS , n 3 ( AS)

A3 , of the sinusoidal trend seeFigs. 8 d and 8 e . However, in contrast to the n 1 case,we nd that the crossover exponents Tp3 and A3 are equaland positive with decreasing values for increasing correlation

exponents . In Table III we present the values of these twoexponents for different correlation exponent . To under-stand how the scale n 3 depends on T , AS , and the correla-tion exponent simultaneously, we again employ the super-position rule Eq. 10 and estimate n 3 as the secondintercept n 3

th of F (n ) and F S( n ). From Eqs. 13 and 6 ,we obtain the following dependence:

n 31

2 2 b 0 AST

1/

. 15

From this analytical calculation we obtain T3 A3 1/ which is in good agreement with the values of T3 and A3observed from simulations Table III . Finally, our simula-tions show that all three crossover scales n 1 , n 2 , and n 3do not depend on the length of the signal N ma x , since F (n )and F S( n ) do not depend on N ma x as shown in Eqs. 6 , 10 ,12 , and 13 .

C. Higher-order DFA on pure sinusoidal trend

In Sec. IV B we discussed how sinusoidal trends affectthe scaling behavior of correlated noise when the DFA-1

method is applied. Since DFA-1 removes only constanttrends in data, it is natural to ask how the observed scalingresults will change when we apply DFA of order l designedto remove polynomial trends of order lower than l . In thissection we rst consider the rms uctuation F S for a sinu-soidal signal and then we study the scaling and crossoverproperties of F S for correlated noise with a superposedsinusoidal signal when higher-order DFA is used.

We nd that the rms uctuation function F S does notdepend on the length of the signal N ma x , and preserves asimilar shape when a different order- l DFA method is usedFig. 9 . In particular, F S exhibits a crossover at a scale n 2

proportional to the period T of the sinusoidal: n 2 T T2

with T2 1. The crossover scale shifts to larger values forhigher order l Figs. 5 and 9 . For the scale n n 2 F Sexhibits an apparent scaling: F S n

S with an effective ex-ponent S l 1. For DFA-1, we have l 1 and recover S 2 as shown in Eq. 12 . For n n 2 , F S( n ) is a con-stant independent of the scale n and of the order l of the DFAmethod in agreement with Eq. 13 .

Next, we consider F S(n ) when DFA- l with a higher or-der l is used. We nd that for all orders l , F S(n ) does notdepend on the length of the signal N ma x and exhibits threecrossovers at small, intermediate, and large scales; similarbehavior is reported for DFA-1 in Fig. 6. Since both thecrossover at small scales n 1 and the crossover at large scale

n 3 result from the competition between the scaling of the correlated noise and the effect of the sinusoidal trendFigs. 6 and 7 , by using the superposition rule Eq. 10 we

can estimate n 1 and n 3 as the intercepts of F (n ) andF S(n ) for the general case of DFA- l .

For n 1 we nd the following dependence on the periodT , amplitude AS , the correlation exponent of the noise,and the order l of the DFA- l method:

n 1 T / AS1/( l 1 ) . 16

For DFA-1, we have l 1 and we recover Eq. 14 . In addi-

TABLE II. The crossover exponents T1 and A1 characterizingthe power-law dependence of n 1 on the period T and amplitude ASobtained from simulations: n 1 T

T1 and n 1 ( AS) A1 for differ-

ent values of the correlation exponent of noise Figs. 8 a and8 b . The values of T1 and A1 are in good agreement with theanalytical predictions T1 A1 1/(2 ) Eq. 14 .

T1 - A1 1/(2 )

0.1 0.55 0.54 0.530.3 0.58 0.59 0.590.5 0.66 0.66 0.670.7 0.74 0.75 0.770.9 0.87 0.90 0.91

TABLE III. The crossover exponents T3 and A3 for the power-law relations: n 3 T

T3 and n 3 ( AS) A3 for different values of

the correlation exponent of noise Figs. 8 c and 8 d . The values

of p3 and a 3 obtained from simulations are in good agreementwith the analytical predictions T3 A3 1/ Eq. 15 .

T3 A3 1/

0.4 2.29 2.38 2.500.5 1.92 1.95 2.000.6 1.69 1.71 1.670.7 1.39 1.43 1.430.8 1.26 1.27 1.250.9 1.06 1.10 1.11

102

103

104

n

10 2

10 1

100

101

102

103

F S

( n )

DFA1

DFA2

DFA3

2

34

FIG. 9. Comparison of the results of different order DFA on asinusoidal trend. The sinusoidal trend is given by the function64sin(2 i /211 ) and the length of the signal is N ma x 2

17. The spu-rious singularities spikes arise from the discrete data we use forthe sinusoidal function.


011114-8


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tion, n 1 is shifted to larger scales when higher-order DFA-l is applied, due to the fact that the value of F S( n ) decreaseswhen l increases ( S l 1, see Fig. 9 .

For the third crossover observed in F S ( n ) at large scalen 3 we nd for all orders l of the DFA- l the followingscaling relation:

n 3 TA S1/ . 17

Since the scaling function F ( n ) for correlated noise shiftsvertically to lower values when higher-order DFA- l is usedsee the discussion in Appendix A and Sec. V B , n 3 ex-

hibits a slight shift to larger scales.For the crossover n 2 in F S ( n ) at F S( n ) at intermedi-

ate scales, we nd n 2 T . This relation is independent of the order l of the DFA and is identical to the relation foundfor F S( n ) Eq. 11 . n 2 also exhibits a shift to larger scaleswhen higher-order DFA is used see Fig. 9 .

The features reported here of the crossovers in F S ( n )can be used to identify low-frequency sinusoidal trends innoisy data and to recognize their effects on the scaling prop-erties of the data. This information may be useful when

quantifying correlation properties in data by means of a scal-ing analysis.

V. NOISE WITH POWER-LAW TRENDS

In this section we study the effect of power-law trends onthe scaling properties of noisy signals. We consider the caseof correlated noise with a superposed power-law trend u ( i )

A Pi , when A P is a positive constant, i 1 , . . . , N ma x , and N ma x is the length of the signal. We nd that when theDFA-1 method is used, the rms uctuation function F P ( n )exhibits a crossover between two scaling regions Fig. 10 .

This behavior results from the fact that at different scales n ,either the correlated noise or the power-law trend is domi-nant, and can be predicted by employing the superpositionrule

F P

n 2 F

n 2 F P n2 , 18

whereF

(n

) andF

P(n

) are the rms uctuation function of noise and the power-law trend, respectively, and F P ( n ) isthe rms uctuation function for the superposition of the noiseand the power-law trend. Since the behavior of F ( n ) isknown Eq. 6 and Appendix A , we can understand thefeatures of F P ( n ) if we know how F P( n ) depends on thecharacteristics of the power-law trend. We note that the scal-ing behavior of F P ( n ) displayed in Fig. 10 a is to someextent similar to the behavior of the rms uctuation functionF

L ( n ) for correlated noise with a linear trend Fig. 1 e.g.,the noise is dominant at small scales n , while the trend isdominant at large scales. However, the behavior F P( n ) ismore complex than that of F L( n ) for the linear trend, sincethe effective exponent for F P( n ) can depend on thepower of the power-law trend. In particular, for negativevalues of , F P( n ) can become dominated at small scalesFig. 10 b while F ( n ) dominates at large scalesa situ-

ation completely opposite of noise with a linear trend Fig. 1or with a power-law trend with positive values for the power

. Moreover, F P( n ) can exhibit crossover behavior at smallscales Fig. 10 b for negative which is not observed forpositive . In addition, F P( n ) depends on the order l of theDFA method and the length N ma x of the signal. We discussthe scaling features of the power-law trends in the followingthree sections, V A V C.

101

103

n

10 2

100

102

104

F ( n )

Noise+ positive powerlaw trendPositive powerlaw trend: = 0.4Correlated noise: =0.9

=1.9 DFA1

(a) Positive

=0.9

nx 101

102

103

104

n

10 2

10 1

100

101

102

103

F ( n )

Noise+negative powerlaw trendNegative powerlaw trend: = 0.7Correlated noise: =1.5

DFA1

(b) Negative

=0.8

=1.5

nx

FIG. 10. Crossover behavior of the rms uctuation function F P ( n ) circles for correlated noise of length N ma x 217) with a super-

posed power-law trend u ( i ) A Pi . The rms uctuation function F ( n ) for noise solid line and the rms uctuation function F P( n ) dashedline are also shown for comparison. The DFA-1 method is used. a F P ( n ) for noise with correlation exponent 0.9 and the power-lawtrend with amplitude A P 1000/( N ma x )

0.4 and positive power 0.4. b F P ( n ) for Brownian noise integrated white noise, 1.5) and

the power-law trend with amplitude A P 0.01/( N ma x )0.7

and negative power 0.7. Note that although in both cases there is a similarcrossover behavior for F P ( n ), the results in a and b represent completely opposite situations: while in a the power-law trend withpositive power dominates the scaling of F P ( n ) at large scales, in b the power-law trend with negative power dominates the scalingat small scales. The arrow in b indicates a weak crossover in F P( n ) dashed lines at small scales for negative power .


011114-9


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A. Dependence of F P n on the power

First we study how the rms uctuation function F P( n ) fora power-law trend u ( i ) A Pi depends on the power . Wend that

F P n A Pn

, 19

where is the effective exponent for the power-law trend.For positive we observe no crossovers in F P( n ) Fig.10 a . However, for negative there is a crossover in F P( n )at small scales n Fig. 10 b , and we nd that this crossoverbecomes even more pronounced with decreasing or in-creasing the order l of the DFA method, and is also shifted tolarger scales Fig. 11 a .

Next, we study how the effective exponent for F P( n )depends on the value of the power for the power-law trend.

We examine the scaling of F P( n ) and estimate for 44. In the cases when F P( n ) exhibits a crossover, in

order to obtain we t the range of larger scales to theright of the crossover. We nd that for any order l of theDFA- l method there are three regions with different relationsbetween and Fig. 11 b . They are as follows:

i l 1 for l 0.5 region I .ii 1.5 for 1.5 l 0.5 region II .iii 0 for 1.5 region III .

Note that for integer values of the power (0,1, . . . , m 1), i.e., polynomial trends of order m 1, the

DFA- l method of order l m 1 ( l is also an integer leadsto F P( n ) 0, since DFA- l is designed to remove polynomialtrends. Thus for integer values of the power there is noscaling and the effective exponent is not dened if aDFA- l method of order l is used Fig. 11 . However, it is

100

101

102

103

104

105

n

10 2

100

102

104

F P

( n )

= 0.6

= 1.6 = 2.6

= 3.6DFA3

(a)

4 2 0 2 4

0

1

2

3

4

DFA1

DFA2

DFA3

(b)

100

101

102

103

104

105

n

10 12

10 10

10 8

10 6

10 4

10 2

F P

( n )

=1.001

=1.0001

=1.00001

=1.000001

2.5

DFA2

(c)

FIG. 11. Scaling behavior of the rms uctuation function F P( n ) for power-law trends, u ( i ) i , where i 1 , . . . , N ma x and N ma x 217 is

the length of the signal. a For 0, F P( n ) exhibits crossover at small scales which is more pronounced with increasing the order l of DFA- l and decreasing the value of . Such crossover is not observed for 0 when F P( n ) n

for all scales n see Fig. 10 a . bDependence of the effective exponent on the power for different order l 1,2,3 of the DFA method. Three regions are observed,depending on the order l of the DFA: region I ( l 0.5), where l 1; region II ( 1.5 l 0.5), where 1.5; region III( 1.5), where 0. We note that for integer values of the power 0,1, . . . , l 1, where l is the order of DFA we used, there is noscaling for F P( n ) and is not dened, as indicated by the arrows. c Asymptotic behavior near integer values of . F P( n ) is plotted for

1 when DFA-2 is used. Even for 1 10 6 , we observe at large scales n a region with an effective exponent 2.5. This region isshifted to innitely large scales when 1.


011114-10


11/19

of interest to examine the asymptotic behavior of the scalingof F P( n ) when the value of the power is close to an inte-ger. In particular, we consider how the scaling of F P(n ) ob-tained from the DFA-2 method changes when 1 Fig.11 c . Surprisingly, we nd that even though the values of F P(n ) are very small at large scales, there is a scaling forF P(n ) with a smooth convergence of the effective exponent 2.5 when 1, according to the dependence

1.5 established for region II Fig. 11 b . At smaller scalesthere is a at region which is due to the fact that the uctua-tion function Y ( i ) Eq. 3 is smaller than the precision of the numerical simulation.

B. Dependence of F P n on the order l of DFA

Another factor that affects the rms uctuation function of the power-law trend F P( n ) is the order l of the DFA methodused. We rst take into account the following.

1 For integer values of the power , the power-law trendu ( i ) APi is a polynomial trend which can be perfectlyltered out by the DFA method of order l , and as dis-

cussed in Secs. III B and V A see Figs. 11 b and 11 c ,there is no scaling for F P(n ). Therefore, in this section weconsider only noninteger values of .

2 For a given value of the power , the effective expo-nent can take different values depending on the order l of the DFA method we use see Fig. 11 e.g., for xed l

0.5, l 1. Therefore, in this section we consider onlythe case when l 0.5 regions II and III .

Since higher-order DFA- l provides a better t for thedata, the uctuation function Y ( i ) Eq. 3 decreases withincreasing order l . This leads to a vertical shift to smallervalues of the rms uctuation function F (n ) Eq. 4 . Such avertical shift is observed for the rms uctuation function

F (n ) for correlated noise see Appendix A , as well as forthe rms uctuation function of power-law trend F P( n ). Herewe ask how this vertical shift in F (n ) and F P(n ) dependson the order l of the DFA method, and if this shift has dif-ferent properties for F (n ) compared to F P(n ). This infor-mation can help identify power-law trends in noisy data, andcan be used to differentiate crossovers separating scaling re-gions with different types of correlations and crossovers thatare due to effects of power-law trends.

We consider correlated noise with a superposed power-law trend, where the crossover in F P ( n ) at large scales nresults from the dominant effect of the power-law trendF P (n ) F P(n ) Eq. 18 and Fig. 10 a . We choose thepower 0.5, a range where for all orders l of the DFAmethod the effective exponent of F P(n ) remains thesame, i.e., 1.5 region II in Fig. 11 b . For a super-position of an anticorrelated noise and power-law trend with

0.4, we observe a crossover in the scaling behavior of F P (n ), from a scaling region characterized by the correla-tion exponent 0.1 of the noise, where F P (n ) F (n ),to a region characterized by an effective exponent 1.9,where F P (n ) F P(n ), for all orders l 1,2,3 of the DFA- lmethod Fig. 12 a . We also nd that the crossover of F P (n ) shifts to larger scales when the order l of DFA- lincreases, and that there is a vertical shift of F P (n ) to lower

values. This vertical shift in F P (n ) at large scales, whereF P (n ) F P(n ), appears to be different in magnitude whendifferent order l of the DFA- l method is used Fig. 12 a .We also observe a less pronounced vertical shift at smallscales where F P (n ) F (n ).

Next, we ask how these vertical shifts depend on the orderl of DFA- l . We dene the vertical shift as the y interceptof F P(n ): F P(n 1). We nd that the vertical shift inF P(n ) for the power-law trend follows a power law:

l ( ) . We tested this relation for orders up to l 10, and wend that it holds for different values of the power of thepower-law trend Fig. 12 b . Using Eq. 19 we can writeF P(n )/ F P( n 1) n , i.e., F P(n ) F P(n 1). Since F P(n

1) l ( ) Fig. 12 b , we nd that

F P n l ( ) . 20

We also nd that the exponent is negative and is a decreas-ing function of the power Fig. 12 c . Because the effec-tive exponent which characterizes F P(n ) depends on the

power see Fig. 11 b , we can express the exponent as afunction of as we show in Fig. 12 d . This representationcan help us compare the behavior of the vertical shift inF P(n ) with the shift in F (n ). For correlated noise with adifferent correlation exponent , we observe a similarpower-law relation between the vertical shift in F ( n ) andthe order l of DFA- l : l ( ) , where is also a negativeexponent that decreases with . In Fig. 12 d we compare ( ) for F P(n ) with ( ) for F (n ), and nd that for any , ( ) ( ). This difference between the verticalshift for correlated noise and for a power-law trend can beutilized to recognize effects of power-law trends on the scal-ing properties of data.

C. Dependence of F P n on the signal length N max

Here we study how the rms uctuation function F P( n )depends on the length N ma x of the power-law signal u ( i )

APi ( i 1 , . . . , N ma x ). We nd that there is a vertical shiftin F P(n ) with increasing N ma x Fig. 13 a . We observe thatwhen doubling the length N ma x of the signal the vertical shiftin F P( n ), which we dene as F P

2 N ma x / F P N ma x , remains the

same, independent of the value of N ma x . This suggests apower-law dependence of F P( n ) on the length of the signal:

F P n N ma x , 21

where is an effective scaling exponent.Next, we ask if the vertical shift depends on the power

of the power-law trend. When doubling the length N ma x of the signal, we nd that for l 0.5, where l is the order of the DFA method, the vertical shift is a constant independentof Fig. 13 b . Since the value of the vertical shift whendoubling the length N ma x is 2 from Eq. 21 , the results inFig. 13 b show that is independent of when l 0.5,and that log2 0.15, i.e. The effective exponent

0.5.


011114-11


12/19

For l 0.5, when doubling the length N ma x of the sig-

nal, we nd that the vertical shift 2

exhibits the followingdependence on : log 102 log 102

l , and thus the effec-tive exponent depends on l . For positive in-teger values of ( l ), we nd that 0, and there is noshift in F P( n ), suggesting that F P(n ) does not depend on thelength N ma x of the signal, when DFA of order l is used Fig.13 . Finally, we note that depending on the effective expo-nent , i.e., on the order l of the DFA method and the valueof the power , the vertical shift in the rms uctuation func-tion F P(n ) for the power-law trend can be positive ( l ),negative ( l ), or zero ( l ).

D. Combined effect on F P n of , l , and N max

We have seen that by taking into account the effects of the power Eq. 19 , the order l of DFA- l Eq. 20 , andthe effect of the length of the signal N ma x Eq. 21 , wereach the following expression for the rms uctuation func-tion F P(n ) for a power-law trend u ( i ) APi :

F P n APn l ( ) N ma x

( ) . 22

For correlated noise, the rms uctuation function F (n ) de-

101

102

103

104

n

10 1

100

101

102

103

F

P

( n )

DFA1DFA2DFA3

n x

=0.1

=1.9

(a) Noise with powerlaw trend

1 10

Order l of the DFA method

10 18

10 14

10 10

10 6

10 2

10

=0.6=0.2=0.2=0.6=1.2=1.6

()

(b) Dependence of vertical shift on l

2 3 4 5 6 7 8 9

1 0 1 2

7

5

3

1

( )

(c)

0 1 2 3

7

5

3

1

1

Powerlaw trend: vs. Correlated noise: vs.

(d)

FIG. 12. Effect of higher-order DFA- l on the rms uctuation function F P ( n ) for correlated noise with a superposed power-law trend.a F P ( n ) for anticorrelated noise with the correlation exponent 0.1 and a power-law u ( i ) APi , where AP 25/( N ma x )

0.4, N ma x2 17, and 0.4. Results for different order l 1,2,3 of the DFA method show i a clear crossover from a region at small scales where the

noise dominates F P ( n ) F (n ) to a region at larger scales where the power-law trend dominates F P ( n ) F P(n ), and ii a vertical shiftin F P with increasing l . b Dependence of the vertical shift in the rms uctuation function F P(n ) for a power-law trend on the order

l of DFA- l for different values of : l ( ) . We dene the vertical shift as the y intercept of F P( n ): F P( n 1). Note, that weconsider only noninteger values for and that we consider the region l 0.5. Thus, for all values of the minimal order l that can beused in the DFA method is l 0.5, e.g., for 1.6 the minimal order of the DFA that can be used is l 3 for details see Fig. 11 b .c Dependence of on the power error bars indicate the regression error for the ts of ( l ) in b . d Comparison of ( ) for F P( n )

and ( ) for F ( n ). Faster decay of ( ) indicates larger vertical shifts for F P(n ) compared to F (n ) with increasing order l of theDFA- l .


011114-12


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pends on the box size n Eq. 6 and on the order l of DFA-l Sec. V B and Fig. 12 a , d , and does not depend on thelength of the signal N ma x . Thus we have the following ex-pression for F ( n ):

F

n n l ( ) . 23

To estimate the crossover scale n observed in the appar-

ent scaling of F P ( n ) for a correlated noise superposed witha power-law trend Figs. 10 a , 10 b , and 12 a , we employthe superposition rule Eq. 18 . From Eqs. 22 and 23 ,we obtain n as the intercept between F P( n ) and F ( n ),

n A l ( ) ( ) N ma x 1/( ) . 24

To test the validity of this result, we consider the case of correlated noise with a linear trend. For the case of a lineartrend ( 1) when DFA-1 ( l 1) is applied, we have

2 see Appendix C and Sec. V A, Fig. 11 b . Since in thiscase l 1 l 0.5 we have l 0 see Sec.V C,Fig. 13 b , and from Eq. 24 we recover Eq. 9 .

VI. CONCLUSION AND SUMMARY

In this paper we show that the DFA method performsbetter than the standard R / S analysis to quantify the scalingbehavior of noisy signals for a wide range of correlations,and we estimate the range of scales where the performanceof the DFA method is optimal. We consider different typesof trends superposed on correlated noise, and we study howthese trends affect the scaling behavior of the noise. Wedemonstrate that there is a competition between a trend and anoise, and that this competition can lead to crossovers in the

scaling. We investigate the features of these crossovers, theirdependence on the properties of the noise, and the super-posed trend. Surprisingly, we nd that crossovers which area result of trends can exhibit power-law dependences on theparameters of the trends. We show that these crossover phe-nomena can be explained by the superposition of the separateresults of the DFA method on the noise and on the trend,assuming that the noise and the trend are not correlated, andthat the scaling properties of the noise and the apparent scal-ing behavior of the trend are known. Our work may providesome help to differentiate between different types of cross-overs, e.g., crossovers that separate scaling regions with dif-ferent correlation properties may differ from crossovers thatare an artifact of trends. The results we present here could beuseful for identifying the presence of trends and to accuratelyinterpret correlation properties of noisy data. Related work on trends 64 and other forms of nonstationarity 65 will bepublished separately.

ACKNOWLEDGMENTS

We thank NIH/National Center for Research ResourcesGrant No. P41RR13622 , NSF, and the Spanish Govern-ment Grant No. BIO99-0651-CO2-01 for support, and alsoA. L. Goldberger, C.-K. Peng, and Y. Ashkenazy for helpfuldiscussions.

APPENDIX A: NOISE

The standard signals we generate in our study are uncor-related, correlated, and anticorrelated noise. First we musthave a clear idea of the scaling behaviors of these standardsignals before we use them to study the effects from other

101

102

103

104

105

n

10 6

10 4

10 2

100

102

104

F P

( n )

Nmax =217

Nmax =219

Nmax =221

DFA1

(a) Powerlaw trend: =0.4

=1.9

2 0 2 4

0.35

0.25

0.15

0.05

0.05

0.15

0.25

l o g

1 0 [ F

2 N m a x

/ F N m a x

]

DFA1DFA2DFA3

log 102

(b) Vertical shift due to length doubling

FIG. 13. Dependence of the rms uctuation function F P( n ) for a power-law trend u ( i ) A Pi , where i 1 , . . . , N ma x , on the length of thetrend N ma x . a A vertical shift is observed in F P( n ) for different values of N ma x N 1 ma x and N 2 ma x . The gure shows that the vertical shift,dened as F P

N 1 max ( n )/ F P N 2 max ( n ), does not depend on N ma x but only on the ratio N 1 ma x / N 2 ma x , suggesting that F P( n ) ( N ma x )

. bDependence of the vertical shift on the power . For l 0.5 ( l is the order of DFA , we nd a at constant region characterized withan effective exponent 0.5 and negative vertical shift. For l 0.5, we nd an exponential dependence of the vertical shift on . Inthis region, l , and the vertical shift can be negative if l ) or positive if l ). The slope of log 10 F P

2 N max ( n ) / F P N max ( n ) vs

is log 102 due to doubling the length of the signal N ma x . This slope changes to log 10m when N ma x is increased m times while remainsindependent of N ma x . For l there is no vertical shift, as marked with . Arrows indicate integer values of l , for which values theDFA- l method lters out completely the power-law trend and F P 0.


011114-13


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aspects. We generate noises by using a modied Fourier l-tering method 63 . This method can efciently generatenoise u ( i ) ( i 1,2,3,..., N max ), with the desired power-lawcorrelation function that asymptotically behaves as

j ii t u ( j) 2 t 2 . By default, a generated noise has stan-

dard deviation 1. Then we can test DFA and R / S byapplying it on generated noises since we know the expectedscaling exponent .

Before doing that, we want to briey review the algorithmof R / S analysis. For a signal u ( i ) ( i 1 , . . . , N max ), it is di-vided into boxes of equal size n. In each box, the cumulativedeparture X i for the k th box, i kn 1 , . . . , kn n ) is cal-culated

X i j kn 1

i

u j u , A1

where u n 1 i kn 1(k 1) n u ( i ) , and the rescaled range R / S is

dened by

R / S S 1 maxkn 1 i (k 1) n

X i minkn 1 i (k 1) n

X i , A2

where S n 1 j 1n u ( j) u 2 is the standard deviation

in each box. The average of rescaled range in all the boxes of equal size n, is obtained and denoted by R / S . Repeat theabove computation over different box size n to provide arelationship between R / S and n. According to Hursts ex-perimental study 66 , a power-law relation between R / Sand the box size n indicates the presence of scaling: R / S

n .Figure 14 shows the results of R / S , DFA-1, and DFA-2

on the same generated noises. Loosely speaking, we can see

that F (n ) for DFA and R / S for R / S analysis show apower-law relation with n as expected: F ( n ) n and R / S

n . In addition, there is no signicant difference betweenthe results of different order DFA except for some verticalshift of the curves and the little bend-down for small box sizen. The bend-down for a very small box of F (n ) from higher-order DFA is because there are more variables to t thosefew points.

Ideally, when analyzing a standard noise, F (n ) DFAand R / S ( R / S analysis will be power-law functions with agiven power: , no matter which region of F (n ) and R / S is

100

101

102

103

104

105

n

100

101

102

103

104

R / S

=0.1

=0.3

=0.5

=0.7

=0.9

1

2 3

2

(a) R/S analysis

100

101

102

103

104

n

10 1

100

101

102

103

F ( n )

=0.1

=0.3

=0.5

=0.7

=0.9

1

2 3

(b) DFA1

2

100

101

102

103

104

n

10 1

100

101

102

103

F ( n )

= 0.1

= 0.3

= 0.5

= 0.7

= 0.9

(c) DFA2

FIG. 14. Scaling behavior of noise with the scaling exponent . The length of noise N max 217. a Rescaled range analysis ( R / S) . b

Order-1 detrended uctuation analysis DFA-1 . c Order-2 detrended uctuation analysis DFA-2 . We do the linear tting for the R / S

analysis and the DFA-1 in three regions as shown and get

1 ,

2 , and

3 for estimated

, which are listed in Tables IV and V. We nd thatthe estimation of is different in the different regions.


011114-14


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chosen. However, a careful study shows that the scaling ex-ponent depends on scale n . The estimated is different forthe different regions of F ( n ) and R / S as illustrated by Figs.14 a and 14 b and by Tables IV and V. It is very importantto know the best tting region of the DFA and R / S analysis

in the study of real signals. Otherwise, an inaccurate valuefor will be obtained if an inappropriate region is selected.

In order to nd the best region, we rst determine thedependence of the locally estimated , loc , on the scale n .First, generate a standard noise with given scaling exponent

TABLE V. Estimation of the correlation exponent for corre-lated noise from DFA-1 in the three regions as shown in Fig. 14 b . is the input value of the scaling exponent, 1 is the estimationfrom region 1 (4 n 32), 2 from region 2 (32 n 3162), and 3 from region 3 (3126 n 2

17).

1 2 3

0.1 0.28 0.15 0.080.3 0.40 0.31 0.220.5 0.55 0.50 0.350.7 0.72 0.69 0.550.9 0.91 0.91 0.69

10 0 10 1 10 2 10 3 10 4n

0.1

0.3

0.5

0.7

0.9

1.1

l o c

=0.1 =0.3 =0.5 =0.7 =0.9integrated =0.1

n min

(a) R/S N max =214

c o r r e

l a t e d

a n t i

u n c o r r e

l a t e d

10 0 10 2 10 4n

0.1

0.3

0.5

0.7

0.9

1.1

l o c

=0.1 =0.3 =0.5 =0.7 =0.9integrated =0.1

n min

(b) R/S N max =220

c o r r e

l a t e d

a n t i

u n c o r r e

l a t e d

100

101

102

103

104

n

0.1

0.3

0.5

0.7

0.9

1.1

l o c

=0.1 =0.3 =0.5 =0.7 =0.9integrated =0.1

(c) DFA1 N max =214

n min

c o r r e

l a t e

d

u n c o r r e

l a t e d

a n t

i

10 1 10 1 10 3 10 5

n

0.1

0.3

0.5

0.7

0.9

1.1

l o c

=0.1 =0.3 =0.5 =0.7 =0.9integrated =0.1

(d) DFA1 N max =220

n min

c o r r e

l a t e d

a n t

i

u n c o r r e

l a t e d

FIG. 15. The estimated from the local t a R / S analysis, the length of signal N max 214. b R / S analysis, N max 2

20. c DFA-1, N max 2

14. d DFA-1, N max 220. loc come from the average of 50 simulations. If a technique is working, then the data for the scaling

exponent should be a weakly uctuating horizontal line centered about loc . Note that such a horizontal behavior does not hold for allthe scales. Generally, such an expected behavior begins from some scale n min , holds for a range, and ends at a larger scale n max . For DFA-1,n min is quite small 0.5. For the R / S analysis, n min is small only when 0.7.

TABLE IV. Estimation of the correlation exponent for corre-lated noise from the R / S analysis in three regions as shown in Fig.14 a . is the input value of the scaling exponent, 1 is the esti-mation from region 1 (4 n 32), 2 from region 2 (32 n

3162), and 3 from region 3 (3126 n 217). The same corre-

lated noise is used in Table V.

1 2 3

0.1 0.44 0.23 0.120.3 0.52 0.37 0.230.5 0.62 0.52 0.470.7 0.72 0.70 0.450.9 0.81 0.87 0.63


011114-15


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; then calculate F (n ) or R / S), and obtain loc(n ) by localtting of F (n ) or R / S). The same random simulation isrepeated 50 times for both the DFA and R / S analyses. Theresultant average loc(n ), respectively, is illustrated in Fig.15 for the DFA-1 and R/S analyses.

If a scaling analysis method is working properly, then theresult loc(n ) from simulation with would be a horizontalline with a slight uctuation centered about loc(n ) . Note

from Fig. 15 that such a horizontal behavior does not holdfor all the scales n but for a certain range from n min to n max .In addition, at small scale, the R / S analysis gives loc if 0.7 and loc if 0.7, which has been pointed outby Mandelbrot 67 , while DFA gives loc if 1.0 and loc if 1.0.

It is clear that the smaller the n min and the larger the n max ,the better the method. We also perceive that the expectedhorizontal behavior stops because the uctuations becomelarger due to the undersampling of F (n ) or R / S when n getscloser to the length of the signal N max . Furthermore, it canbe seen from Fig. 15 that n max

110 N max independent of if

the best-t region exists , which is why one-tenth of the sig-nal length can be considered as the maximum box size whenusing a DFA or R / S analysis.

On the contrary, n min does not depend on the N max since loc( n ) at small n hardly changes as N max varies but it doesdepend on . Thus, we obtain n min quantitatively as shownin Fig. 16. For the R / S analysis, n min is small only when 0.7. When 0.7 and 0.7, n min becomes very largeand close to n max , indicating that the best-t region willvanish and the R / S analysis does not work at all.

Compared to R / S , DFA works better since n min is quitesmall for correlated signals with 0.5. However, for 0.5 n min is still relatively large. We can improve this

situation by rst integrating the correlated noise and thenapplying the DFA to the integrated signal. The resultant ex-ponent for the integrated signal will be 0 1. Wend that n min for the integrated signal becomes much smalleras shown in Fig. 16 shaded area 1). Therefore, for cor-related noise with 0.5, it is best to estimate rst the scal-ing exponent of the integrated signal and then to obtain by 1.

APPENDIX B: SUPERPOSITION LAW FOR THE DFA

For two uncorrelated signals f ( i ) and g ( i ), their root-mean-square rms uctuation functions are F f ( n ) andF g ( n ), respectively. We want to prove that for the signal f ( i ) g ( i ), its rms uctuation function

F f g n F f n2 F g n

2 . B1

Consider three signals in the same box rst. The inte-grated signals for f , g, and f g are y f ( i ), yg ( i ), and y f g( i )and their corresponding trends are y f

fi t , yg fi t , and y f g

fi t ( i1,2, . . . , n , n is the box size . Since y f g ( i ) y f ( i ) yg ( i ) and combines the denition of the detrended uctua-

tion function Eq. 3 , we have that for all boxes

Y f g i Y f i Y g i , B2

where Y f g is the detrended uctuation function for the sig-nal f g , Y f ( i ) is for the signal f , and Y g ( i ) for g. Further-more, according to the denition of the rms uctuation, wecan obtain

F f g n1

N ma x i 1

N ma x

Y f g i2

1 N ma x i 1

N ma x

Y f i Y g i2 , B3

where l is the number of boxes and k means the k th box. If f and g are not correlated, neither are Y f ( i ) and Y g ( i ) and,thus,

i 1

N ma x

Y f i Y g i 0. B4

From Eq. B4 and Eq. B3 we have

F f g n1

N ma x i 1

N ma x

Y f i2 Y g i

2

F f n2 F g n

2 . B5

APPENDIX C: DFA-1 ON LINEAR TREND

Let us suppose a linear time series u ( i ) ALi . The inte-grated signal y L( i ) is

0 0.5 1 1.5

100

102

104

106

n m i n

R/S

DFA1

minimum box size

FIG. 16. The starting point of a good-t region, n min , for theDFA-1 and R / S analyses. The results are obtained from 50 simula-tions, in which the length of noise is N max 2

20. The condition for agood t is loc 0.01. The data for 1.0 shown in theshading area are obtained by applying an analysis on the integra-tions of noises with 1.0. It is clear that the DFA-1 works betterthan the R / S analysis because its n min is always smaller than that of the R / S analysis.


011114-16


17/19

y L i j 1

i

AL j ALi 2 i

2. C1

Let as call N ma x the size of the series and n the size of thebox. The rms uctuation F L(n ) as a function of n and N ma xis

F L n

AL1

N ma x k 1

N ma x / n

i ( k 1) n 1

kni 2 i

2a k b k i

2

,

C2

where a k and b k are the parameters of a least-squares t of the k th box of size n. a k and b k can be determined analyti-cally, thus giving

a k 11

12 n2 1

2 n2k 112 n

12 k

2n 2 , C3

b k 112 n kn

12 . C4

With these values, F L(n ) can be evaluated analytically,

F L n AL1

60 5 n 4 25n 3 25n 2 25n 30 C5

The dominating term inside the square root is 5 n 4 and thenone obtains

F L n5

60ALn

2 , C6

leading directly to an exponent of 2 in the DFA. An impor-tant consequence is that as F (n ) does not depend on N ma x ,

for linear trends with the same slope, the DFA must giveexactly the same results for series of different sizes. This isnot true for other trends, where the exponent is 2, but thefactor multiplying n 2 can depend on N ma x .

APPENDIX D: DFA-1 ON A QUADRATIC TREND

Let us suppose now a series of the type u ( i ) AQi2 . The

integrated time series y ( i ) is

y i AQ j 1

i

j2 AQ2 i 3 3 i 2 i

6. D1

As before, let us call N ma x and n the sizes of the series andbox, respectively. The rms uctuation function F Q(n ) mea-suring the rms uctuation is now dened as

F Q n AQ1

N ma x k 1

N ma x / n

i (k 1) n 1

kn2 i 3 3 i 2 i

6a k b k i

2

, D2

where a k and b k are the parameters of a least-squares t of the k th box of size n. As before, a k and b k can be determinedanalytically, thus giving

a k 1

15 n3 n 3k 2 715 n

3k 1730 n2k 760 n

2 120 n

23 k

3n 3 12 n2k 2 115 kn , D3

b k 3

10 n2 n 2k 2 n 2k kn 25 n

110 . D4

Once a k and b k are known, F (n ) can be evaluated, giving

F Q n AQ1

126021 n 4 5 n 3 5 n 2 5 n 6 32n 2 6 n 81 210 N ma x 140 N ma x

2 . D5

As N ma x n , the dominant term inside the square root is given by 140 N ma x2 21n 4 AQ2940 n

4 N ma x2 , and then one has

approximately

F Q n AQ1

1260 2940 n 4 N ma x2 AQ

190 15 N ma x n

2 D6

leading directly to an exponent 2 in the DFA analysis. An interesting consequence derived from Eq. D6 is that F Q( n )

depends on the length of the signal N ma x , and the DFA line logF Q(n ) vs log n for the quadratic series u ( i ) AQi2

of different N ma x does not overlap as is the case for linear trends .

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A_Effect of Trends on Detrended Fluctuation Analysis