Integrated approach to the assessment of long range correlation in time series data

PHYSICAL REVIEW E VOLUME 61, NUMBER 5 MAY 2000 Itegrated approach to the assessmet of log rage correlatio i time series data Govida Ragaraja* Departmet of Mathematics ad Cetre for Theoretical Studies, Idia Istitute of Sciece, Bagalore 560 012, Idia Migzhou Dig Ceter for Complex Systems ad Brai Scieces, Florida Atlatic Uiversity, Boca Rato, Florida 33431 Received 22 September 1999; revised mauscript received 21 Jauary 2000 To assess whether a give time series ca be modeled by a stochastic process possessig log rage correlatio, oe usually applies oe of two types of aalysis methods: the spectral method ad the radom walk aalysis. The first objective of this work is to show that each oe of these methods used aloe ca be susceptible to producig false results. We thus advocate a itegrated approach which requires the use of both methods i a cosistet fashio. We provide the theoretical foudatio of this approach ad illustrate the mai ideas usig examples. The secod objective relates to the observatio of log rage aticorrelatio Hurst expoet H1/2) i real world time series data. The very peculiar ature of such processes is emphasized i light of the striget coditio uder which such processes ca occur. Usig examples, we discuss the possible factors that could cotribute to the false claim of log rage aticorrelatios, ad demostrate the particular importace of the itegrated approach i this case. PACS umbers: 05.40.Fb, 05.45.Tp I. INTRODUCTION Radom processes with log rage power law correlatio have bee observed i a variety of fields icludig ecoomics, geoscieces, physics, ad biology 1 9. There are roughly two types of tools used i assessig the presece of such correlatios i time series data: spectral domai methods represeted by power spectrum aalysis 1,9, ad radom walk methods i the time domai represeted by the rescaled rage aalysis 1,9,10. Ofte these two types of tools are applied sigly to a give data set. I Sec. II of this paper, we poit out the pitfalls of this approach through a series of examples, ad advocate a itegrated approach which requires a cosistet applicatio of both types of methods. Log rage correlatios are characterized by a quatity called the Hurst expoet H. Whe H1/2 the process is said to have positive log rage correlatio or persistece, while H1/2 meas the process has log rage aticorrelatio or atipersistece. Whe H1/2 we say that the process has short rage correlatio. I Sec. III of this paper we preset the coditio for H1/2, ad discuss how this striget coditio ca be corrupted i real world data. We the proceed to demostrate the sigificace of the itegrated approach i this case, ad aalyze the reaso uderlyig the may reported examples of H1/2. II. AN INTEGRATED APPROACH TO THE ASSESSMENT OF LONG RANGE CORRELATION IN TIME SERIES First, we derive the relatioships betwee the power law expoets obtaied from aalyzig the time series usig three differet tools spectral, autocorrelatio, ad rescaled rage aalyses. The we demostrate through examples that the use of spectral method or the rescaled rage method aloe ca produce erroeous results. The autocorrelatio method is ot cosidered, sice it is ofte difficult to use i practice. A. Theoretical cosideratios Cosider a statioary stochastic process i discrete time, k, with k 0 ad 2 k 2. Here deotes esemble average. If the autocorrelatio fuctio C() k k scales with the lag as C for large, where 01, the k is called a log rage correlated or log memory process 9. The reaso for the N latter term is that C() decays so slowly that 1 C() diverges as N. The case of 1 will be treated i Sec. III of this paper. The correlatio structure of k ca be coveietly measured by the power spectrum, which is defied as 11 S f C02 Ccos2 f. 1 If C() obeys the scalig relatio i Eq. 1 the 1 2 I this sectio, we argue that a itegrated approach is required to assess log rage correlatios i times series. S f 2 cos2 f. 1 3 *Also associated with the Jawaharlal Nehru Ceter for Advaced Scietific Research, Bagalore, Idia. Electroic address: ragaraj@math.iisc.eret.i Electroic address: dig@walt.ccs.fau.edu I this case, we show below that for small f where 1. S f f 4 1063-651X/2000/615/499111/$15.00 PRE 61 4991 2000 The America Physical Society

4992 GOVINDAN RANGARAJAN AND MINGZHOU DING PRE 61 The proof of this relatioship betwee ad draws o well kow results i trigoometric series theory 12. Cosider the Taylor expasio of the fuctio (1y) 1, 1y 1 A y, 0 where by defiitio we have A 0 1 ad, for 1, 5 ad 1 s1 scs s1 For 01, this meas R 2 2 s 1 1 s 1 2. 11 12 This meas 1 A 12! 1. y 11y 1 1. Replacig, yre i2 f, ad 0r1 i the above equatio leads to 1 r e i2f 11re i2 f 1 1. Lettig r 1 ad f 0, ad takig the real part, we obtai 1 cos 2f12 f 1 cos1/2. Substitutig this ito Eq. 3 yields S f 212 f 1 cos1/2 f 1. Comparig the above with Eq. 4, we obtai 1. Aother way to assess the correlatio structure of k is to covert the statioary process to a radom walk by usig partial sums, R 1 1, R 2 1 2,..., R 1 2,..., where R is the positio of the walker at time. The mea rage of the radom walk trajectory as a fuctio of time bears specific relatios with the scalig relatio Eq. 1. For the ease of aalytical evaluatio we cosider the mea square displacemet as a measure of the rage of the radom walk, which is defied as R 2 i1 1 2 i 2 scs s1 1 1 2 2 s1 Cs2 scs. s1 Let C(s) obey the scalig law i Eq. 1. The sums i the above equatio are estimated as 1 s1 Cs s1 s 1 s 1 6 7 8 9 10 for large. Covetioally, the mea square displacemet is characterized by the Hurst expoet H as where R 2 2H, H2/21/2. 13 14 Thus we obtai a set of cosistet relatios betwee the scalig expoets,, ad H. B. Examples As show i Sec. II A, the scalig expoets obtaied from spectral aalysis ad from radom walk aalysis must be cosistet through Eq. 14. We first provide three examples of two simulated correlated processes ad oe experimetal log rage correlated process demostratig this cosistecy. The we proceed to show that the spectral method or the radom walk method used aloe ca be susceptible to artifacts i the data ad produce erroeous results. The combiatio of the two methods ca ofte detect such artifacts through icosistecies with Eq. 14. The radom walk aalysis tool that we will use i this paper is the rescaled rage aalysis 1,5. A brief descriptio of this method follows. For a give data set i, cosider s the sum L(,s) i1 i, where L(,s) ca be regarded as the positio of a radom walk after s steps. Defie the tred-corrected rage R(,s) of the radom walk as R,smaxL,ppL,s/s,1ps mil,ppl,s/s,1ps. 15 Let S 2 s (,s) deote the variace of the data set i i1.if the data has log rage correlatio, the average rescaled statistic Q(s)R(,s)/S(,s) where deotes the average over ) scales with s as a power law for large s, Qss H, 16 where H is the Hurst expoet itroduced earlier. This power law maifests itself as a straight lie i the log-log plot of Q(s) versus s. Spectral aalysis was doe usig fast Fourier trasform, ad the Bartlett widow was employed 13. 1. Geuie log rage Correlated processes To geerate a process whose spectral desity scales with the frequecy f as a power law f, we start with a realizatio of a discrete zero mea white Gaussia oise process k, k0,1,...,n1, with variace 2. Usig Fourier trasform, we obtai

PRE 61 INTEGRATED APPROACH TO THE ASSESSMENT OF... 4993 FIG. 1. a Log-log plot of the rescaled rage statistic Q(s) agaist the widow size s for a true log rage correlated process with 0.6, variace 0.25, ad a log data set of 8192 poits. b Spectral desity of the same data. N1 k 0 expi2k/n, k0,1,...,n1. 17 Next we multiply k by the factor f /2 (k/n) /2 to obtai the scaled quatity k. Both k ad Nk are multiplied by the same factor sice we wat a real-valued time series. Fially we perform a iverse Fourier trasform to obtai N1 x 1 N k0 k exp2k/n, 0,1,...,N1. 18 FIG. 2. a Log-log plot of the rescaled rage statistic Q(s) agaist the widow size s for a true log rage correlated process with 0.6, variace 0.25, ad a short data set of 256 poits. b Spectral desity of the same data. The discrete process x, by costructio, has a mea power spectrum that scales as f with the frequecy. The variability aroud the mea spectrum is provided by the white oise process. For our first example we geerated a log rage correlated process usig the above costructio with 0.6 ad variace 0.25. The first data set has 8192 poits. Whe we apply rescaled rage aalysis to this data, we obtai a Hurst expoet H0.74 see Fig. 1a. The power spectrum exhibits a power law behavior by costructio with 0.58 see Fig. 1b. Note that these values are cosistet with Eq. 14 14. Next we trucated the above time series data to obtai a short data set with oly 256 poits. The results from rescaled rage ad power spectrum aalyses are show i Figs. 2a ad 2b, respectively. Agai these two results are cosistet with oe aother. This illustrates the fact that whe you have a process with geuie log rage correlatio, eve a short data set is ofte sufficiet to reveal this property. As a secod example, we cosider a differet type of log rage correlated process a fractioal auto regressive itegrated movig average ARIMA (0,d,0) process 9 with 0d0.5. It ca be show that the autocorrelatio fuctio C() for this process scales with lag as C() 2d1. Thus from Eq. 14, 12d ad Hd0.5. The fractioal ARIMA(0,d,0) process for d0.25 (H0.75) was geerated 15 usig its kow autocovariace fuctio 9. The results from rescaled rage aalysis ad spectral aalysis of this data, show i Figs. 3a ad 3b are mutually cosistet with oe aother cf. Eq. 14. The fial example aalyzes the data from a figer tappig experimet ivolvig the huma sesorimotor coordiatio 7. I this experimet, subjects cyclically tapped their idex figer agaist a computer key i sychroy with a periodic

4994 GOVINDAN RANGARAJAN AND MINGZHOU DING PRE 61 FIG. 3. a Log-log plot of the rescaled rage statistic Q(s) agaist the widow size s for a fractioal ARIMA(0,d,0) process with d0.25. b Spectral desity of the same data. series of auditory beeps, delivered through a headphoe. The data collected were the sychroizatio or tappig errors defied as the time betwee the computer recorded respose time ad the metroome oset see Ref. 7 for further details. Here we aalyze the sychroizatio error time series from this experimet usig rescaled rage ad spectral aalyses. The results, exhibited i Figs. 4a ad 4b, are agai mutually cosistet demostratig that the error time series has log rage correlatios 7. 2. Failures of the rescaled rage aalysis I this sectio, we cosider various situatios where rescaled rage aalysis used aloe ca give wrog results. As our first example, we cosider the superpositio of a expoetial tred over a white oise process. See, Ref. 16 for more examples i this area. Specifically, we geerated the followig discrete process x k exp0.01k k, k1,2,..., 19 where k is a white oise process with zero mea ad variace 0.16. A total of 8192 poits were geerated. This example ca be realized i situatios where the process uder ivestigatio has a expoetially decayig trasiet, ad oe does ot discard the iitial portio of the data cotaiig this trasiet while recordig it. FIG. 4. a Log-log plot of the rescaled rage statistic Q(s) agaist the widow size s for the sychroizatio error time series data from the figer tappig experimet. b Spectral desity of the same data. Whe the above process is subject to rescaled rage aalysis, we obtai a Hurst expoet equal to 0.75 see Fig. 5a, idicatig log rage correlatio where there is oe. O the other had, the power spectrum is flat see Fig. 5b, ad does ot show ay power law behavior. This icosistecy betwee the spectral method ad rescaled rage method serves as a warig sig poitig to the eed for further more careful examiatio of the data. As our secod example, we cosider the followig autoregreesive AR process of order oe AR1 11: x k x k1 k, k1,2,..., 20 where k is a zero mea white oise process with variace 0.25, ad the coefficiet is close to 1 0.9 i our case. The autocorrelatio fuctio of the x process decays expoetially: C(k) k. This meas there is o log rage correlatio i the x process. However, as show i Fig. 6a, the rescaled rage aalysis of the above process with 1024 poits idicates the presece of log rage correlatio by producig H0.76. The power spectrum Fig. 6b, othe other had, exhibits a flatteig tred at low frequecies ad cotradicts the result from the rescaled rage aalysis. Eve if oe misses this flatteig tred ad fits a straight lie to the remaiig portio of the spectrum o a log-log scale, we

PRE 61 INTEGRATED APPROACH TO THE ASSESSMENT OF... 4995 FIG. 5. a Log-log plot of the rescaled rage statistic Q(s) agaist the widow size s for the superpositio of a expoetial tred over a white oise process. b Spectral desity of the same data. obtai a value for equal to 1. Here the cosistecy relatio H(1)/2 is ot satisfied, thus idicatig the absece of log rage correlatios. As our third example, we agai cosider a AR1 process cf. Eq. 20, but this time with the coefficiet close to 1 (0.9 i our case. I this example, the applicatio of the rescaled rage aalysis gives a Hurst expoet H0.33 Fig. 7a. Naively, this Hurst value ca be iterpreted as a idicator of log rage atipersistece 1. As before, the power spectrum cotradicts this result see Fig. 7b, ad shows a flatteig tred i the low frequecies. This observatio is further stregtheed by aalyzig a log data set 100 000 poits usig rescaled rage aalysis. The results see Fig. 7c show that H approaches a value of 0.5 as the data set becomes loger. I Sec. III of this paper we will discuss i more detail processes with the Hurst expoet H 1/2. We would like to make oe poit regardig the applicatio of the surrogate data aalysis 17, which is ofte used i combiatio with may aalysis methods to stregthe their results by demostratig that a completely radom process could ot have exhibited the observed results. We show below that this is ot fool proof whe used i cojuctio with rescaled rage aalysis. We have already see i the secod example above that rescaled rage aalysis idicates FIG. 6. a Log-log plot of the rescaled rage statistic Q(s) agaist the widow size s for a AR1 process with 0.9. b Spectral desity of the same data. the presece of log rage correlatio i the AR1 process with 0.9) where there is oe. We ow apply surrogate data aalysis by shufflig the data radomly ad reapplyig rescaled rage aalysis to the shuffled data sets. Figure 8 shows compariso betwee the Hurst expoet obtaied from the ushuffled origial data with the average value of Hurst expoets obtaied from five realizatios of radomly shuffled data. We see that the shuffled data gives a average value of H aroud 0.5 as compared to 0.76 for the origial data. The two results are well separated. Therefore, the applicatio of surrogate data aalysis would idicate that the result obtaied by rescaled rage aalysis of the origial data is correct, idicatig the presece of log rage correlatio. This is obviously a false coclusio. This example demostrates that surrogate data aalysis caot be used idiscrimiately for this type of problems. 3. Failure of the power spectrum aalysis Here we give a example where the use of power spectrum aalysis with iappropriate parameters ca lead to wrog results. If we ivestigate the geuie log rage process itroduced above usig the power spectrum aalysis with a Parze widow ad M20 that is, with a lot of smoothig 11, the we obtai the spectrum give i Fig. 9. This power spectrum shows a flat portio at low frequecies,

4996 GOVINDAN RANGARAJAN AND MINGZHOU DING PRE 61 FIG. 8. Compariso betwee the Hurst expoet obtaied from the ushuffled origial data of the AR1 process with 0.9) ad the average value of Hurst coefficiets obtaied from five realizatios of the above data, radomly shuffled. the choice of iput parameters. I this case the icosistecies betwee the two aalysis methods will prompt more careful examiatios of the methods employed. 4. Failures of the combied use of rescaled rage ad power spectrum aalysis All the above examples illustrate the fact that oe should ot rely o a sigle tool to aalyze time series data. A itegrated approach requirig the cosistet use of several available tools is more desirable. But eve a itegrated approach is ot foolproof as we show below. Cosider a process that is the superpositio of AR1 processes. I particular, we cosider a variable that is the sum of the followig five idepedet processes: x k x k1 k, k1,2,..., 21 where the coefficiets for the idividual processes are give by 0.99, 0.9, 0.4, 0.2, ad 0.1, ad the variaces are give by 0.05, 0.1, 0.3, 0.4, ad 0.5, respectively. I this case, for a relatively short data set of 256 poits, both the rescaled rage ad power spectrum aalyses idicate the FIG. 7. a Log-log plot of the rescaled rage statistic Q(s) agaist the widow size s for a AR1 process with 0.9, variace 0.25, ad 1024 data poits. b Spectral desity of the same data. c Log-log plot of the rescaled rage statistic Q(s) agaist the widow size s for the same process but with 100 000 poits. idicatig wrogly the absece of log rage correlatios. This is ot a problem with power spectrum aalysis per se, but is a example of usig it with iappropriate parameters. We do ot ru ito such problems with rescaled rage aalysis, sice it does ot have such free parameters that ca be wrogly tued. The above example is ot as artificial as it seems, sice caed power spectrum aalysis routies are ofte used i data aalysis without proper thought goig ito FIG. 9. Spectral desity of the geuie log rage correlated process cosidered i Fig. 1 usig a Parze widow with M20.

PRE 61 INTEGRATED APPROACH TO THE ASSESSMENT OF... 4997 cosistet positive results from both these aalyses at least idicates the presece of multiple time scales i the data set. III. CONDITION FOR HË1Õ2 AND ITS IMPLICATIONS FOR REAL WORLD DATA ANALYSIS As metioed earlier, rescaled rage aalysis 1,5 is ofte used i determiig the presece of log rage correlatio i data sets. Results of rescaled rage aalyses are typically quatified usig the Hurst expoet H (0H1). I priciple, aalysis of a data set ca lead to ay value of H betwee 0 ad 1. I this sectio, we will argue that processes with H1/2 are rather special, i that they must satisfy the coditio that the sum of the autocorrelatio fuctio be zero. May physical processes are kow to meet this coditio 4. However, this coditio ca be easily corrupted i real world data where oise urelated to the physical process eters the measuremet. We demostrate the importace of the itegrated approach i the proper diagosis of processes with H1/2. We also discuss a commo way i which a misjudgmet of log rage aticorrelatio ca occur. I this regard we idetify the three cotributig factors: 1 the variable beig recorded is ot fudametal see below, 2 the data set is short, ad 3 oly a radom walk type of aalysis is employed. FIG. 10. a Log-log plot of the rescaled rage statistic Q(s) agaist the widow size s for a superpositio of five AR1 process. b Spectral desity of the same data. c Q(s) versus s for a much loger data set from the same process. presece of log rage correlatio i the data see Figs. 10a ad 10b, respectively by yieldig H0.96 ad 0.82. Clearly, the results are cosistet with H(1)/2. But, for a much loger data set of 50 000 poits we see the true Hurst expoet of H0.5 i Fig. 10c. This example shows that eve the superpositio of a few AR1 processes ca mimic a log rage correlated process for short data sets. Theoretically it is kow that the superpositio of a ifiite umber of AR1 processes ca, i some cases, give rise to a log rage correlated process 18. Eve though a itegrated approach usig both rescaled rage ad power spectrum aalyses ca give spurious results, A. Coditio for HË1Õ2 Refer to Eq. 9. Suppose that C(s)s. Whe 0 1, we showed that both the secod ad third terms i the above equatio diverge ad scale with as 2. Therefore, R 2 scales with as 2 for large, ad this leads to H1/2 cf. Eq. 14. For 2 that is, for ay C(s) that decays faster tha C(s)s 2 ], both sums i the above equatio coverge, ad we geerally obtai H1/2. Thus the oly remaiig rage of is 12. For such the sum s1 C(s) is fiite. Note that, strictly speakig, this type of process ca o loger be termed a log memory process based o the defiitio i Sec. II A. But, sice it has the potetial of givig H1/2, we will still use the term log rage correlatio. Therefore the first two terms scale as 1. The sum 1 s1 sc(s) i the third term is evaluated to be 1 s1 scs 2, 22 where (2)1. This meas that the rate of divergece of the third term is slower tha the first two terms. Therefore, i the large limit R 2, we would still observe H1/2. The oly situatio whe this will ot happe occurs whe the followig equatio is precisely satisfied: 2 2 s1 s Cs s Cs0. 23

4998 GOVINDAN RANGARAJAN AND MINGZHOU DING PRE 61 FIG. 11. a Log-log plot of the rescaled rage statistic Q(s) agaist the widow size s for a true log rage correlated process with 0.5, variace 0.25, ad a log data set 8192 poits. b Spectral desity of the same data. I this case the first two terms i R 2 drop out givig R 2 2. Therefore, we obtai H(2)/2, which is smaller tha 1/2. It is clear that for H1/2 to occur the process must meet a very striget coditio Eq. 23. It has bee show that may physical systems satisfy this coditio 4. But, whe such a physical system is subject to measuremet, oise is a ievitable factor. For the oisy measuremet it is likely that the equality i Eq. 23 o loger holds. The implicatio is that i the log ru oe may observe H1/2, ad therefore ot be able to correctly idetify the uderlyig process. I what follows we show that the itegrated approach advocated i Sec. II is agai a essetial tool i revealig strog clues as to the true ature of the physical process. Moreover, we will show that the itegrated approach is also idispesable i guardig agaist misjudgmet of H1/2 for systems where this is ot true. B. Itegrated approach to oisy HË1Õ2 data For a geuie H1/2 process, Eq. 14 still holds, albeit will be a egative umber. We geerated a f process artificially usig the procedure i Sec. II B 1. A value of 0.5 was used. Whe the data are subject to rescaled rage aalysis, we obtai a value of H0.28 see Fig. 11a. The spectral aalysis gives a power law curve with 0.5 as expected Fig. 11b. We ote the results of the rescaled rage ad spectral aalyses are mutually cosistet i this case, sice the applicatio of Eq 14 gives a H value of 0.25. Now we cosider the effect of additive oise o the same data set. Whe oise is added, Eq. 23 is o loger strictly satisfied. Hece we would expect the Hurst expoet of the process to asymptotically approach H0.5 for log data sets. This is bore out by our umerical simulatios. We start with the true log rage correlated process described above (H0.25), ad add Gaussia white oise to the data with variace 0.01. This simulates the effect of oisy measuremet i experimets. Figure 12a shows the results of rescaled rage aalysis applied to the above process with 262, 144 poits. We see that H asymptotically teds to 0.5. As the variace of the added oise is icreased, the H0.5 value is reached eve faster. However, all is ot lost. The itegrated approach allows us to approximately recover the true process hidde by the oise. We start by trucatig the data set which prevets the asymptotic limit for H beig reached. Figs. 12b ad 12c display the results of applyig the itegrated approach to the trucated data set with 8192 poits all other parameter values remai the same as i Fig. 12a. We obtai a H value of 0.34 from rescaled rage aalysis, ad the spectral aalysis is cosistet with this value. This cosistecy tells us that the data represet a true log rage correlated process the H value obtaied is higher tha that for the true process because of the added oise. We commet that this cosistecy is i marked cotrast to what was observed for the AR1 process with 0.9 see Sec. II B. There, eve though the H value was 0.33 for small data sets ad rose to H0.5 for large data sets, the results of spectral aalysis were completely icosistet with this. This led us to coclude that there was o true log rage correlated process i that example. The above examples agai demostrate that the itegrated approach is very useful i revealig the true ature of a process represeted by time series data. C. Possible causes for false idetificatio of HË1Õ2 I the literature oe ofte comes across reports where aalysis of real world data, usig the method of radom walk aloe, yield H1/2. The examples i Sec. II B show the shortcomig of usig just oe type of aalysis method. Upo further examiatio we realize that there is a commo thread i these reports that has to do with the fact that the data aalyzed do ot come from a fudametal process which we discuss below. 1. Notio of a fudametal process ad data differecig Cosider a statioary process k. By defiitio a statioary process is ot diffusive. I other words, 2 k is a costat. Cosider the partial sum R k1 k.ifr 2 icreases with, that is, R is a diffusive process, we say that k is a fudametal process. If the time series data comig from a fudametal process are subject to radom walk type of aalysis such as the rescaled rage aalysis, it ca be trusted to correctly assess the correspodig Hurst expoet.

PRE 61 INTEGRATED APPROACH TO THE ASSESSMENT OF... 4999 FIG. 12. a Log-log plot of the rescaled rage statistic Q(s) agaist the widow size s for a true log rage correlated process corrupted by added white oise with 0.5, variace 0.25, ad 262 144 poits. The variace of added white oise is 0.01. b Same as above but with 8192 poits. c Spectral desity of the same process with 8192 poits. A differeced process refers to a process k which is obtaied by k k1 k. Clearly, k is ot a fudametal process, sice its partial sums give k which is ot diffusive. This meas if we iput data from a differeced process ito the rescaled rage type of radom walk aalysis, we will ot be able to assess the correlatio properties of the origial process, ad possibly eve be fooled by the appearace of the rescaled rage plot see below. Differeced data ca arise i practice i a umber of ways. First, differecig is a commoly applied techique FIG. 13. a Log-log plot of the rescaled rage statistic Q(s) agaist the widow size s for a differeced Gaussia white oise process with 1024 data poits. b Same as above but with 40 000 data poits. c Spectral desity of the data i a. for tred removal 11. Secod, the measured physical variable is a derivative of aother fudametal variable. We believe that the use of differeced data, i combiatio with a rescaled rage type of aalysis, uderlies some of the reported cases of H1/2. Below we demostrate this poit by examples. 2. Gaussia white oise Cosider a Gaussia white oise process k. The partial sums of this process yield a diffusive Browia motio with a Hurst expoet equal to 1/2. Suppose what is beig mea-

5000 GOVINDAN RANGARAJAN AND MINGZHOU DING PRE 61 FIG. 15. a Log-log plot of the rescaled rage statistic Q(s) agaist the widow size s for the iterrespose iterval time series data from the figer tappig experimet. b Spectral desity of the same data. FIG. 14. a Log-log plot of the rescaled rage statistic Q(s) agaist the widow size s for the velocity variable ẋ of a Lagevi process with 5 ad 250 data poits. b Same as above but with 60 000 data poits. c Log-log plot of the rescaled rage statistic Q(s) agaist the widow size s for the positio variable x of a Lagevi process with 5 ad 250 data poits. sured is ot k but the differeced variable k k1 k. To see the effect of this, a Gaussia white oise process k with zero mea ad variace 0.25 was first geerated. If the rescaled rage aalysis is performed o k, the result is show i Fig. 13a. The data legth is 1024 poits. If we force a liear fit to the ed part of the log-log plot, we observe a Hurst expoet equal to about 0.12. However, this expoet is ot a reflectio of the process but is caused by the fiite size of the data set. If we aalyze a much loger data set 40 000 poits, we observe that the slope of 0.12 that we had see earlier is oly a trasiet effect see Fig. 13b. It ca be show that the true slope goes to zero as the time lag s icreases. The H value of 0.12 obtaied for the differeced data set of 1024 poits ca also be easily rejected by the itegrated approach. Subjectig the same data set to the spectral aalysis yields 1.95 Fig. 13c. This value is totally icosistet with the 0.76 predicted by Eq. 14 based o H0.12. This icosistecy should be used as a clue to further examie the ature of the data set. 3. Lagevi equatio I this sectio, we cosider a more physical example the Lagevi equatio. The Lagevi process that we studied is ẋxt, 24 where (t) is a white oise process with zero mea ad variace 0.25 ad 5. The above stochastic differetial equatio was itegrated usig a efficiet algorithm 19. Suppose that the variable beig measured is the velocity ẋ, ad successive values of ẋ by the umerical itegratio scheme costitute our data set. The rescaled rage aalysis applied to a short data set of 250 poits yields a Hurst expoet equal to 0.18 see Fig. 14a. O the other had, if the

PRE 61 INTEGRATED APPROACH TO THE ASSESSMENT OF... 5001 size of the data set is icreased to 60 000 poits, the value of H becomes early zero for large s values Fig. 14b. Theoretically, it ca be show that x, after a iitial trasiet, is a statioary fudametal process. To demostrate this we apply rescaled rage aalysis to the values of x. I this case, eve for short data sets, we obtai a value of H close to 0.5 see Fig. 14c, as predicted by theory. The result see i Fig. 14a is therefore a cosequece of ẋ beig a overdiffereced variable. 4. Figer tappig data Geerally, we suggest that whe a H1/2 is obtaied from a radom walk type of aalysis, oe should also perform a spectral aalysis o the same data set. If the result is icosistet with Eq. 14, the oe should coclude that this data set is ot from a fudametal process ad partial sums of this data set should istead be cosidered for a aalysis of the correlatio properties. As a example, we agai cosider the figer tappig experimet 7 described i Sec. II B. But ow we aalyze the iterrespose itervals IRI s istead of the sychroizatio errors. The IRI s ca be obtaied from the sychroizatio error data by differecig it 7 ad is itself a importat physiological variable. Rescaled rage aalysis of this IRI time series data appears to give a value of H0.25 see Fig. 15a. But this is a artifact of the fiite data size. This ca be see by performig a spectral aalysis o the same IRI data see Fig. 15b. We see that this gives results icosistet with Eq. 14. Thus the H value obtaied i Fig. 15a is a cosequece of the IRI beig a overdiffereced variable, combied with fiite data size ad oly oe type of method. IV. SUMMARY Suppose that the autocorrelatio fuctio C(s) for a statioary process scales with s as C(s)s. Depedig o the values of ad the behavior of s s C(s) we have the followig classificatio for the process: 1 if 01, we have 1/2H1, ad the process is said to have log rage persistet correlatio or log memory 9. 2 If 12 ad s s C(s)0, we have 0H1/2 ad the process is said to have log rage atipersistet correlatio or aticorrelatio. 3 If 1 ad s s C(s)0 we have H1/2, ad the process is said to have short rage correlatio. It is worth otig i this classificatio processes with 12 ca be classified as either havig log rage aticorrelatio or short rage correlatio depedig whether s s C(s) is zero. We preserve the log rage aticorrelatio termiology, i keepig with the traditioal amig of such processes. The mai goal of this work has bee to demostrate the importace of the itegrated approach, combiig both spectral ad radom walk aalyses, to the assessmet of correlatio behavior i time series data. We showed that the cosistet use of both spectral ad radom walk aalyses is ot oly essetial i revealig the true ature of a give process, it ca also prevet the false coclusio of log rage correlatio resultig from artifacts or wrog measuremet variables combied with just oe type of aalysis method. ACKNOWLEDGMENTS This work was supported by US ONR Grat No. N00014-99-1-0062. G.R. thaks the Ceter for Complex Systems ad Brai Scieces, Florida Atlatic Uiversity, where this work was performed, for hospitality. 1 B.B. Madelbrot ad J.R. Wallis, Water Resour. Res. 4, 909 1968; B.B. Madelbrot ad J.W. Va Ness, SIAM Rev. 10, 422 1968. 2 C.W.J. Grager, Ecoometrica 34, 150 1966. 3 M. Cassadro ad G. Joa-Lasio, Adv. Phys. 27, 913 1978. 4 P.M. Richards, Phys. Rev. B 16, 1393 1977. 5 H.E. Hurst, Tras. Am. Soc. Civ. Eg. 116, 770 1951; H.E. Hurst, R.P. Black, ad Y.M. Simaiki, Log-term Storage: A Experimetal Study Costable, Lodo, 1965. 6 G. Mathero ad G. De Marsily, Water Resour. Res. 16, 901 1980. 7 Y. Che, M. Dig, ad J.A. Scott Kelso, Phys. Rev. Lett. 79, 4501 1997. See the refereces cited i this paper for may other examples of log rage correlated processes. 8 G. Ragaraja ad D.A. Sat, Geophys. Res. Lett. 24, 1239 1997. 9 J. Bera, Statistics for Log-memory Processes Chapma & Hall, New York, 1994. 10 For a radom walk type of aalysis other tha the rescaled rage method, see C.K. Peg, S.V. Buldyrev, M. Simos, H.E. Staley, ad A.L. Goldberger, Phys. Rev. E 49, 1685 1994; J.J. Collis ad C.J. De Luca, Exp. Brai Res. 103, 151 1995; M.S. Taqqu, V. Teverovsky, ad W. Williger, Fractals 3, 785 1995. 11 C. Chatfield, The Aalysis of Time Series: A Itroductio, 4th ed. Chapma & Hall, Lodo, 1989. 12 A. Zygmud, Trigoometric Series Cambridge Uiversity Press, New York, 1979. 13 W.H. Press, S.A. Teukolsky, W.T. Vetterig, ad B.P. Flaery, Numerical Recipes i Fortra, 2d ed. Cambridge Uiversity Press, New York, 1992. 14 We commet that cosistecy here is defied i a rather loose fashio. I the future we will use more rigorous statistical criteria to provide the basis for evaluatig whether the estimates from the two methods are cosistet with Eq. 14. 15 R.B. Davies ad D.S. Harte, Biometrika 66, 153 1987. 16 D.C. Boses ad J.D. Salas, Water Resour. Res. 14, 135 1978; R.N. Bhattacharya, V.K. Gupta, ad E. Waymire, J. Appl. Probab. 20, 649 1983. 17 J. Theiler, S. Eubak, A. Logti, B. Galdrikia, ad J.D. Farmer, Physica D 58, 771992. 18 C.W.J. Grager, J. Ecoometrics 14, 150 1980. 19 N.J. Rao, J.D. Borwakar, ad D. Ramkrisha, SIAM J. Cotrol 12, 124 1974; R. Maella ad V. Palleschi, Phys. Rev. A 40, 3381 1989.