Multifractality Signatures in Quasars Time Series. I. 3C 273
A. Bewketu Belete, J. P. Bravo, B. L. Canto Martins, I. C. Leão, J. M. De Araujo, J. R. De Medeiros
MMNRAS , 1–12 (2017) Preprint 21 May 2018 Compiled using MNRAS L A TEX style file v3.0
Multifractality Signatures in Quasars Time Series. I. 3C273
A. Bewketu Belete, (cid:63) J. P. Bravo, , B. L. Canto Martins, , I. C. Le˜ao, J. M. De Araujo, J. R. De Medeiros Departamento de F´ısica Te´orica e Experimental, Universidade Federal do Rio Grande do Norte, Natal, RN 59078-970, Brazil Instituto Federal de Educa¸c˜ao, Ciˆencia e Tecnologia do Rio Grande do Norte, Natal, RN 59015-000, Brazil Observatoire de Gen`eve, Universit´e de Gen`eve, Chemin des Maillettes 51, Sauverny, CH-1290, Switzerland
Accepted 2018 May 15. Received 2018 May 15; in original form 2017 December 21
ABSTRACT
The presence of multifractality in a time series shows different correlations for dif-ferent time scales as well as intermittent behaviour that cannot be captured by asingle scaling exponent. The identification of a multifractal nature allows for a char-acterization of the dynamics and of the intermittency of the fluctuations in non-linearand complex systems. In this study, we search for a possible multifractal structure(multifractality signature) of the flux variability in the quasar 3C 273 time series forall electromagnetic wavebands at different observation points, and the origins for theobserved multifractality. This study is intended to highlight how the scaling behavesacross the different bands of the selected candidate which can be used as an addi-tional new technique to group quasars based on the fractal signature observed in theirtime series and determine whether quasars are non-linear physical systems or not.The Multifractal Detrended Moving Average algorithm (MFDMA) has been used tostudy the scaling in non-linear, complex and dynamic systems. To achieve this goal,we applied the backward ( θ = ) MFDMA method for one-dimensional signals. Weobserve weak multifractal (close to monofractal) behaviour in some of the time seriesof our candidate except in the mm, UV and X-ray bands. The non-linear temporalcorrelation is the main source of the observed multifractality in the time series whereasthe heaviness of the distribution contributes less. Key words: methods: statistical – galaxies: active – (galaxies:) quasar: individual:3C 273
Active galactic nucleus (AGN) is a very small fraction of agalaxy whose luminosity outshines the entire galaxy - indi-cating that the excess luminosity is not produced by stars.The luminosity observed almost in all the electromagnetic(EM) spectrum further ensures that the energy productionmechanism in AGNs is completely different from the onein stars. The exciting modern era of AGN studies beginswith the identification of the first high-luminosity AGNs(Schmidt 1963). Even though the history of AGN researchbegun many years before, the science still remains with alot of open questions, such as understanding the energy pro-duction mechanisms, determining the size of the broad lineregion, and many others. As might be expected AGNs are (cid:63)
E-mail: asnakew@fisica.ufrn.br among the most energetic and complex objects in the uni-verse. Variability is one of the main observational featuresof AGNs noticed at the whole electromagnetic wavebands.Courvoisier et al. (1990) affirm that flux variation is a well-known property of AGNs. Quasars are a very luminous typeof AGNs which are roughly classified as radio loud and ra-dio quiet. The variability observed in quasars light-curvesat short and long time scale, from less than one hour upto several decades, is one of the most important observedcharacteristics of quasars. The physical features of quasarsdepend on several properties such as the mass of the centralblack hole, the rate of gas accretion onto the black hole, theorientation of the accretion disk, the degree of obscurationof the nucleus by dust, and the presence or absence of jets.The quasar 3C 273 is a non-extreme, but a very brighttype of AGN (McHardy et al. 2007). The quasar 3C 273has been extensively studied on diverse time scales across © a r X i v : . [ a s t r o - ph . GA ] M a y A. Bewketu et al. the entire EM spectrum, with measurements made in sin-gle bands and often as well as in multi-bands (Fan et al.2014, 2009; Abdo et al. 2010; Pacciani et al. 2009; Dai et al.2009; Soldi et al. 2008; McHardy et al. 2007; Chernyakovaet al. 2007; Jester et al. 2006; Savolainen et al. 2006; At-tridge et al. 2005; Kataoka et al. 2002; Greve et al. 2002;Sambruna et al. 2001; Collmar et al. 2000; Mantovani et al.2000). Blazars show rapid flux variability across the com-plete EM spectrum (Kalita et al. 2015). Fluctuations rang-ing from a few tens of minutes (and sometimes even morerapid) to less than a day is often known as intra-day vari-ability (IDV) (Wagner & Witzel 1995). It has been reportedby Courvoisier et al. (1990) that the flux variation across thespectra of 3C 273 shows no simple correlation. It was alsonoticed that, at higher frequencies, the strength of the fluxvariations in 3C 273 are higher than in the lower frequencies(e.g. Fan et al. 2009). Fluctuation in the flux of astronomicalobjects is an important and widespread phenomenon whichprovides relevant information on the dynamics of the systemthat drives the fluctuation. The IDV in blazars is the leastwell-understood type of variations but it can provide an im-portant tool for learning about structures on small spatialscales and it also provides us with a better understandingof the different radiation mechanisms that are important inthe emitting regions (e.g. Wagner & Witzel 1995). Any char-acteristic timescale of variability can effectively provide uswith information about the physical structure of the centralregion and complex phenomena such as hot spots on accre-tion discs (e.g. Mangalam & Wiita 1993). It has been knownthat quasars are non-linear physical systems whose tempo-ral evolution is described by non-linear dynamical equationsand their light-curves cannot be analysed only by means ofthe classical method (e.g., PS, SF, and covariance analyses)which are suitable only for dealing the signals of a linearsystem (Vio et al. 1991). The presence of non-linearity in atime series leaves open the possibility that the time series isproduced by a chaotic system. However, as indicated by Vioet al. (1992), non-linearity is a necessary, but not sufficient,condition for a light curve to be produced by a chaotic sys-tem. There are a number of real-world signals from complexsystems that exhibit self-similarity and non-linear power-lawbehaviour that depends on higher-order moments and scale,which is the signature of a fractal signature in the system(Feder 1988).To study the variability seen in quasars light-curves, as-tronomers have been working on the time-frequency analysisusing different approaches to unveil the physics behind theobserved physical characteristics. These days, fractals arewidely used in the modelling and interpretation of many dif-ferent natural phenomena (e.g. Pietronero & Siebesma 1986;Mandelbrot 1983), including astrophysical phenomena (e.g.Heck & Perdang 1991), due to their capability of compact-ing the information on the scaling and clustering behaviourof the system-fractal nature. Fractal and multifractal be-haviour is common in natural and social sciences (Mandel-brot 1983). Fractals can be classified into two categories:monofractals and multifractals. If this scaling behaviour ischaracterized by a single scaling exponent, or equivalently isa linear function of the moments, the process is monofrac-tal. A multifractal system is a generalization of a fractalsystem in which a single exponent is not enough to describeits dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed. The multi-fractal analysis consists of determining whether some typeof power-law scaling exists for various statistical momentsat different scales. The main feature of multifractals is thatthe fractal dimension is not the same on all scales. In thecase of a one-dimensional signal, the fractal dimension canvary from unity to a dot. Thus, a fundamental character-istic of the multifractal structure is that the scaling prop-erties may be different in different segments of the systemrequiring more than one scaling exponent to be completelydescribed. The multifractal spectrum effectively shows thedistribution of scaling exponents for a signal and provides ameasure of how much the local regularity of a signal variesin time. The presence of multifractality in a physical sys-tem indicates the non-linearity, inhomogeneity, complexityof the system dynamics (Kantelhardt et al. 2002), the pres-ence of intermittency (Vio et al. 1991), and provides a wayto describe signals from a complex, non-linear and dynamicsystems. A multifractal analysis performed by Longo et al.(1996) clearly indicated non-linear intermittent behaviourin the long term (1910 - 1991) B-band light curve of NGC4151.Several techniques have been proposed in the literatureto study the scaling and self-similarity or fractality prop-erties in the time series, such as Autocorrelation Function(ACF), detrended fluctuation analysis (DFA), the multifrac-tal detrended fluctuation analysis (MFDFA), rescaled rangestatistical (R/S) analysis that provide type of self-affinity forstationary time series (Bashan et al. 2008), the periodogramregression (GPH) method, the (m, k)-Zipf method, and thedetrended moving average (DMA) analysis (Shao et al. 2012;Carbone et al. 2004). The simplest type of multifractal anal-ysis is based upon the standard partition function multi-fractal formalism, which has been developed for the multi-fractal characterization of normalized, stationary measure-ments (Peitgen et al. 2004; Bacry et al. 2001; Barab´asi &Vicsek 1991; Feder 1988). However, this standard formal-ism does not give correct results for non-stationary timeseries that are affected by trends or that cannot be nor-malized. Therefore, an improved multifractal formalism wasdeveloped known as wavelet transform modulus maxima(WTMM) method (Muzy et al. 1991), which is based onthe wavelet analysis and involves tracing the maxima linesin the continuous wavelet transform over all scales. Thismethod has been widely used in diverse fields of solar ac-tivities to the earth science (e.g. Geology, DNA sequences,neuron spiking, heart rate dynamics, economic time seriesand also weather related and earthquake signals) (Ida et al.2005; Vyushin et al. 2004; Varotsos et al. 2003, 2002; Kan-telhardt et al. 2001; Telesca et al. 2001; Arneodo et al. 1995;Molchanov & Hayakawa 1995). Readers are referred to Oua-habi & Femmam (2011) to better understand WTMM basedmultifractality analysis. By generalizing this standard DFAmethod, Kantelhardt et al. (2002) has been introduced theMFDFA, which allows the global detection of multifractalbehaviour. By applying this MFDFA method to the sunspotnumber time series, Sadegh Movahed et al. (2006) and Huet al. (2009) have found that the presence of multifractal-ity/complexity in the sunspot number fluctuations is almostdue to long-range correlation.In this work, we apply the extended version of DMA,called multifractal detrended moving algorithm (MFDMA)
MNRAS , 1–12 (2017) ultifractal Behaviour of the 3C 273 Light-curves for one-dimensional time series. This technique has been ap-plied by de Freitas et al. (2017) in the search for multifractal-ity traces in the magnetic activity of stars. To our knowledge,this is the first time that this technique is applied to quasartime series analysis. We hope that it will give insights intothe different degrees of non-linearity of the system across itselectromagnetic spectra.This paper is structured as follows. In Section 2, we dis-cuss the data, method and procedures used, and in Section3, we present our results and discuss the multifractal natureof the quasar 3C 273. Summary and conclusions are givenin Section 4. For the present study, we have chosen some of the 3C273 light-curves provided by Integral Science Data Centre(ISDC) linked to the Astronomical Observatory of the Uni-versity of Geneva (Soldi et al. 2008; T¨urler et al. 1999).Specifically, we have selected the time series covering the fol-lowing wavebands: radio at 8 GHz, millimetre (mm) at 1.3mm, infrared (IR) at 2.2 µ m, optical at 5479 ˚A, UV at 1525˚A, and X-ray at 5 keV. The referred data are displayed inFig. 1, where the behaviour of Flux (in Jansky) versus Time(in days) is given for all the wavebands in consideration. Thereason why we chose this candidate is that it is suitable formultifractal analysis due to the well-established flux fluctua-tion in its light-curves, in all observed spectrum regions. Forthe light-curves considered, we only have selected good datapoints (points with flag ≥ ), and disregarded those withnegative flag which are considered as useless, uncertain, anddubious. We have performed no special treatment in our se-lection of the light-curves but we have considered stability(continuity) in the distribution of data points as the firstcriteria and the number of data points as the second crite-ria for all the time series considered, except for the 1.3 mmwhich we selected deliberately to determine whether a gapin data points affects our multifractality analysis or not.Though the emissions of AGNs in general, and of 3C273 in particular, are known to be complex due to the pres-ence of many emission components of different nature, thereare different physical mechanisms known to produce radia-tion at different energy spectra in such astronomical objects.In 3C 273, synchrotron flaring emission from a relativisticjet (non-thermal radiations) dominate the radio to millime-ter energy output and known to contribute in the infrared- optical domains (T¨urler et al. 2000; Robson et al. 1993;Courvoisier et al. 1988), in addition, thermal emission fromdust grains by a UV source is also in part responsible forthe production of infrared continuum (T¨urler et al. 2006;Robson et al. 1993). Matter accreted in clumps is a possiblemechanism for the production of UV radiation rather thanthrough an accretion disc (Courvoisier & T¨urler 2005). Theinteraction of these clumps generates optically thick shocksproducing the UV emission, whereas optically thin shocks Figure 1.
3C 273 light-curves. From top to bottom: radio, mm,IR, optical, UV and X-ray observations at 8 GHz, 1.3 mm, 2.2 µ m (K band), 5479 ˚A (V band), 1525 ˚A, 5 keV, respectively.Reprint from the original source at the Integral Science Data Cen-tre (ISDC database).MNRAS , 1–12 (2017) A. Bewketu et al. closer to the black hole give origin to the X-rays. Comp-tonisation of thermal plasma, for example, in an accretiondisc, would produce X-ray emission similar to that observedin Seyfert galaxies (Grandi & Palumbo 2004). It has beenindicated that the soft-excess could be due to thermal Comp-tonisation of cool-disc photons in a warm corona (Page et al.2004). Inverse Compton processes of a thermal plasma in thedisc, or in a corona, and of a non-thermal plasma associatedto the jet are believed to generate the X-ray to gamma-rayemission (Grandi & Palumbo 2004; Kataoka et al. 2002).
In this work, we use the Multifractal Detrended Moving Av-erage algorithm, an extended version of DMA algorithm,which consists of a multifractal characterization of non-stationary time series (Gu & Zhou 2010). Our choice ofMFDMA is because of the fact that it details the signal ona wide Hurst exponent spectrum multifractality in the timeseries (Gu & Zhou 2010). We apply the backward ( θ = )MFDMA algorithm as suggested by Gu & Zhou (2010) forbetter computation accuracy of the multifractal scaling ex-ponent τ ( q ) . In multifractality study the most crucial param-eters to describe the structural properties of a time series x(t) are (i) a q th -order fluctuation function F q ( n ) , (ii) themultifractal scaling exponent τ ( q ) , and (iii) the multifractalspectrum f ( α ). These parameters are obtained according tothe procedure introduced by Gu & Zhou (2010) as givenbelow:1. First, the time series is reconstructed as a sequence ofcumulative sums given by: y ( t ) = t (cid:213) i = x ( t ) , t = , , , ..., N , (1)where N is the length of the signal.2. Second, we calculate the moving average function ˜ y ( t ) ofEq. 1 in a moving window using the relation (Gu & Zhou2010; Halsey et al. 1986): ˜ y ( t ) = n (cid:100)( n − )( n − θ )(cid:101) (cid:213) k = −(cid:98)( n − ) θ (cid:99) y ( t − k ) , (2)where n is the window size, (cid:100) (.) (cid:101) is the smallest integernot smaller than argument (.), (cid:98) (.) (cid:99) is the largestinteger not larger than argument (.). The referencepoint of the moving average window can be changed bythe introduction of a parameter θ with ≤ θ ≤ . Inthe present work, θ is adopted as zero, referring to thebackward moving average. The moving average function ˜ y ( t ) is calculated by averaging the n -past value in eachsliding window of length n and the reference point of theaveraging process is the past point of the window. Hence,the moving average function considers (cid:100)( n − )( n − θ )(cid:101) data points in the past and (cid:98)( n − ) θ (cid:99) points in the future: ˜ y ( t ) contains half-past and half-future information ineach window. The index k in ˜ y ( t ) is accordingly setwithin the segment n . 3. The trend (the change in the mean of the signal seriesover time) is removed from the reconstructed time series y ( t ) using the function ˜ y ( t ) and the residual sequence (cid:15) ( t ) is obtained from: (cid:15) ( i ) = y ( i ) − ˜ y ( i ) , (3)where n − (cid:98)( n − ) θ (cid:99) ≤ i ≤ N − (cid:98)( n − ) θ (cid:99) . The residual timeseries (cid:15) ( t ) is subdivided into N n disjoint segments withthe same size n given by N n = (cid:98) Nn − (cid:99) . In this sense, theresidual sequence (cid:15) ( t ) for each segment is denoted by (cid:15) v ,where (cid:15) v ( i ) = (cid:15) ( l + ) for ≤ i ≤ n and l = ( v − ) n .4. Then we calculate the root-mean-square function as fol-lows: F v ( n ) = (cid:40) n n (cid:213) i = (cid:15) v ( i ) (cid:41) / . (4)5. The q th -order overall generating (fluctuation function) F q ( n ) is determined using the following relation: F q ( n ) = (cid:40) N n N n (cid:213) v = F v ( n ) (cid:41) / q (5)for all q (cid:44) . When q = , according to L’Hˆospital’s rule,we have: F q ( n ) = exp (cid:40) N n N n (cid:213) v = ln [ F v ( n )] (cid:41) . (6)6. By varying the values of segment size n , we determinethe power-law relation between the function F q ( n ) and thesize scale n . The scaling relation between the detrendedfluctuation function F q ( n ) and the size scale n can be cal-culated as: F q ( n ) = n h ( q ) , (7)where the exponent h ( q ) is known as generalized Hurstexponent. This relation allows to estimate the scalingexponent of h ( q ) and thus of the Hurst exponent of theseries y ( t ). In general, it tells us how the fluctuationsof the profile (the cumulative sum) in a given timewindow of size n , increase with n . For q = it reducesto the ordinary Hurst exponent ( H ) (Feder 1988). H (cid:44) . indicates the presence of long-range correlationin the system. An exponent . < H < correspondsto a positive correlation or persistent in the systemand < H < . corresponds to the process whichis long-range dependent with negative correlations oranti-persistent. (Carbone et al. 2004).7. Using the calculated h ( q ) , we determine the multifractalscaling exponent τ ( q ) as follows: τ ( q ) = qh ( q ) − D f , (8)where D f is the fractal dimension of the geometricsupport of the multifractal measure (Kantelhardtet al. 2002). For our case, D f = , since we are apply-ing the MFDMA for one-dimensional time series analysis.8. At last, we determine the singularity strength function(the Holder exponent) α ( q ) which is related to τ ( q ) via MNRAS000
In this work, we use the Multifractal Detrended Moving Av-erage algorithm, an extended version of DMA algorithm,which consists of a multifractal characterization of non-stationary time series (Gu & Zhou 2010). Our choice ofMFDMA is because of the fact that it details the signal ona wide Hurst exponent spectrum multifractality in the timeseries (Gu & Zhou 2010). We apply the backward ( θ = )MFDMA algorithm as suggested by Gu & Zhou (2010) forbetter computation accuracy of the multifractal scaling ex-ponent τ ( q ) . In multifractality study the most crucial param-eters to describe the structural properties of a time series x(t) are (i) a q th -order fluctuation function F q ( n ) , (ii) themultifractal scaling exponent τ ( q ) , and (iii) the multifractalspectrum f ( α ). These parameters are obtained according tothe procedure introduced by Gu & Zhou (2010) as givenbelow:1. First, the time series is reconstructed as a sequence ofcumulative sums given by: y ( t ) = t (cid:213) i = x ( t ) , t = , , , ..., N , (1)where N is the length of the signal.2. Second, we calculate the moving average function ˜ y ( t ) ofEq. 1 in a moving window using the relation (Gu & Zhou2010; Halsey et al. 1986): ˜ y ( t ) = n (cid:100)( n − )( n − θ )(cid:101) (cid:213) k = −(cid:98)( n − ) θ (cid:99) y ( t − k ) , (2)where n is the window size, (cid:100) (.) (cid:101) is the smallest integernot smaller than argument (.), (cid:98) (.) (cid:99) is the largestinteger not larger than argument (.). The referencepoint of the moving average window can be changed bythe introduction of a parameter θ with ≤ θ ≤ . Inthe present work, θ is adopted as zero, referring to thebackward moving average. The moving average function ˜ y ( t ) is calculated by averaging the n -past value in eachsliding window of length n and the reference point of theaveraging process is the past point of the window. Hence,the moving average function considers (cid:100)( n − )( n − θ )(cid:101) data points in the past and (cid:98)( n − ) θ (cid:99) points in the future: ˜ y ( t ) contains half-past and half-future information ineach window. The index k in ˜ y ( t ) is accordingly setwithin the segment n . 3. The trend (the change in the mean of the signal seriesover time) is removed from the reconstructed time series y ( t ) using the function ˜ y ( t ) and the residual sequence (cid:15) ( t ) is obtained from: (cid:15) ( i ) = y ( i ) − ˜ y ( i ) , (3)where n − (cid:98)( n − ) θ (cid:99) ≤ i ≤ N − (cid:98)( n − ) θ (cid:99) . The residual timeseries (cid:15) ( t ) is subdivided into N n disjoint segments withthe same size n given by N n = (cid:98) Nn − (cid:99) . In this sense, theresidual sequence (cid:15) ( t ) for each segment is denoted by (cid:15) v ,where (cid:15) v ( i ) = (cid:15) ( l + ) for ≤ i ≤ n and l = ( v − ) n .4. Then we calculate the root-mean-square function as fol-lows: F v ( n ) = (cid:40) n n (cid:213) i = (cid:15) v ( i ) (cid:41) / . (4)5. The q th -order overall generating (fluctuation function) F q ( n ) is determined using the following relation: F q ( n ) = (cid:40) N n N n (cid:213) v = F v ( n ) (cid:41) / q (5)for all q (cid:44) . When q = , according to L’Hˆospital’s rule,we have: F q ( n ) = exp (cid:40) N n N n (cid:213) v = ln [ F v ( n )] (cid:41) . (6)6. By varying the values of segment size n , we determinethe power-law relation between the function F q ( n ) and thesize scale n . The scaling relation between the detrendedfluctuation function F q ( n ) and the size scale n can be cal-culated as: F q ( n ) = n h ( q ) , (7)where the exponent h ( q ) is known as generalized Hurstexponent. This relation allows to estimate the scalingexponent of h ( q ) and thus of the Hurst exponent of theseries y ( t ). In general, it tells us how the fluctuationsof the profile (the cumulative sum) in a given timewindow of size n , increase with n . For q = it reducesto the ordinary Hurst exponent ( H ) (Feder 1988). H (cid:44) . indicates the presence of long-range correlationin the system. An exponent . < H < correspondsto a positive correlation or persistent in the systemand < H < . corresponds to the process whichis long-range dependent with negative correlations oranti-persistent. (Carbone et al. 2004).7. Using the calculated h ( q ) , we determine the multifractalscaling exponent τ ( q ) as follows: τ ( q ) = qh ( q ) − D f , (8)where D f is the fractal dimension of the geometricsupport of the multifractal measure (Kantelhardtet al. 2002). For our case, D f = , since we are apply-ing the MFDMA for one-dimensional time series analysis.8. At last, we determine the singularity strength function(the Holder exponent) α ( q ) which is related to τ ( q ) via MNRAS000 , 1–12 (2017) ultifractal Behaviour of the 3C 273 Light-curves the Legendre transform (Halsey et al. 1986), as shownbelow, and the multifractality spectrum f ( α ) as follows: α ( q ) = d τ ( q )/ dq , (9)and f ( α ) = q α − τ ( q ) . (10)The singularity strength, α , is extracted from the slopeof the scaling exponent curve, i.e., its value is dependent onthe nature of the scaling exponent curve (on the relationshipbetween τ ( q ) and q ) which in turn depends on the local fluc-tuations or the q dependency of the local Hurst exponent h ( q ). The ∆ α (the range of singularity strength α ) is themeasure of the range of multifractal singularity strength. Ifthere is strong non-linearity in the scaling exponent curve,i.e., different slope for the negative and positive q values,then we can have wider ∆ α , which reflects strong multifrac-tality behaviour in the time series considered. In general,the wide range corresponds to strong multifractality, andcontrarily the narrow range corresponds to weak multifrac-tality or monofractal. The multifractal spectrum providesdetailed information about the relative importance of sev-eral types of fractal exponents present in the signal (Telesca& Lapenna 2006), and also the detection of multifractal-ity behaviour indicates the presence of intermittency in thedata under consideration (McHardy et al. 2007). A physicalsystem is said to be intermittent if it has a wide multifrac-tal spectrum or concentrates into a small-scale features withlarge magnitude of fluctuations enclosed by extended areasof less strong fluctuations (Monin & Yaglom 2007; Frisch1995; Moffatt 1994).In our multifractality analysis, we have characterizedthe multifractal spectrum based on the behaviour of (i) thewidth of α (given by ∆ α = α max − α min ) and (ii) the symme-try in the shape of α defined as A = ( α max − α )/( α − α min ) ,where α is the value of α when f ( α ) assumes its maximumvalue. Concerning to the width of α , Shimizu et al. (2002)and Ashkenazy et al. (2003) proposed that the width of amultifractal spectrum is the measure of the degree of mul-tifractality. Broader the spectrum (larger the value of ∆ α ),richer the multifractality (Telesca et al. 2004). Smaller valuesof ∆ α (i.e., ∆ α gets close to zero) indicates the monofractallimit whereas larger values indicates the strength of the mul-tifractal behaviour in the signal (Gu & Zhou 2010). For thesymmetry in the shape of α , the asymmetry presents threeshapes: asymmetry to the right-skewed ( A > ), left-skewed( < A < ), or symmetric ( A = ). It has been reportedby Ihlen (2012) that the symmetric spectrum is originatedfrom the levelling of the q th -order generalized Hurst expo-nent for both positive and negative q values. The levelling of q th -order Hurst exponent reflects the q th -order fluctuationis insensitive to the magnitude of local fluctuation. Whenthe multifractal structure is sensitive to the small-scale fluc-tuation with large magnitudes, the spectrum will be foundwith right truncation; whereas, the multifractal spectrumwill be found with left-side truncation when the time se-ries has a multifractal structure that is sensitive to the localfluctuations with small magnitudes. Thus, it may be notedthat the width and shape of the multifractal spectrum areable to classify a small and large magnitude (intermittency)fluctuations in the considered time series. It has been under-stood that there are two possible sources of multifractality in a time series data (Kantelhardt et al. 2002), namely (i)multifractality due to long-range time correlations of thesmall and large fluctuations and (ii) multifractality due toa fat-tailed probability distribution function of the values inthe series. The first kind of multifractality can be removedby random shuffling of the given series. Shuffling a time se-ries destroys the long-range temporal correlation for smalland large fluctuations. The corresponding shuffled series willexhibit monofractal scaling, i.e., leaves the original time se-ries the same but without memory, α = . . Obviously, theprobability distribution will not be changed by random shuf-fling and hence the multifractality of the second kind willremain intact. For the latter case, the non-Gaussian effectscan be weakened by creating phase-randomized surrogates.In this context, the procedure preserves the amplitudes ofthe Fourier transform and the linear properties of the orig-inal series but randomizes the Fourier phases while elimi-nating nonlinear effects (Norouzzadeh et al. 2007). On theother hand, the surrogate (phase randomization) analysis isan empirical technique of testing non-linearity for a time se-ries. First, we interpolate our data to get evenly sampleddata and generate surrogate series using Fourier transformmethod (phase randomization). We have applied the sameprocedure for the shuffled and surrogate time series as theoriginal one. These procedures are used to study the degreeof complexity of time series to distinguish different sourcesof multifractality in the time series. The basic idea of thesurrogate data method is to first specify some kind of linearstochastic process that mimics linear properties of the origi-nal data. Thus, if the shuffled signal only presents long-rangecorrelations, we should find that h shu f f ( q ) = . . However,if the source of multifractality is due to heavy-tailed dis-tributions obtained by the surrogate method, the values of h surr ( q ) will be independent of q . If a given series containsboth kinds of multifractality, the corresponding shuffled se-ries will exhibit weaker multifractality than the original one(Sadegh Movahed et al. 2006). If h ( q ) is not enough to iden-tify the source of multifractality, an alternative method toassess the behaviour of multifractality is to compare the non-linearity in the multifractal scaling exponent τ ( q ) for theoriginal, shuffled, and surrogate (phase-randomized) data.Differences between these two scaling exponents and theoriginal exponent reveal the presence of long-range correla-tions and/or heavy-tailed distributions. These comparisonscan be shown by the scaling exponent τ ( q ) versus q plots,which present the following relations (Gu & Zhou 2010): τ ( q ) − τ shu f f ( q ) = q [ h ( q ) − h shu f f ( q )] = qh corr ( q ) (11)and τ ( q ) − τ surr ( q ) = q [ h ( q ) − h surr ( q )] = qh tail ( q ) . (12)The linear behaviour of τ ( q ) (and the narrow width ofthe singularity spectrum, α ( q ) ) indicates the presence of amonofractal, while non-linear behaviour (and a wide singu-larity spectrum, α ( q ) ) indicates multifractality. In this work, we perform the backward MFDMA for one-dimensional time series and analyse the non-linear and mul-tifractal properties of the quasar 3C 273 time series cover-ing all bands at different observation points. The scenario
MNRAS , 1–12 (2017)
A. Bewketu et al. emerging for each time series is discussed separately as fol-lows. In Fig. 2, we present the multi-scaling behaviour of thefluctuation functions F q ( n ) corresponding to the time seriesby the log-log plots of the fluctuation function F q ( n ) againstthe time scale (segment) n for the original, shuffled and sur-rogate time series. The q th -order weights the influence ofsegments with large and small fluctuations. For negative q values, F q ( n ) is influenced by segments with small fluctua-tion whereas for positive q ’s, F q ( n ) is influenced by segmentswith large fluctuation. The midpoint q = is neutral to theinfluence of segments with small and large fluctuations. Theslope of the regression line of q th -order fluctuation functionand time segment gives the well-known generalized Hurstexponent h ( q ) . The slopes h ( q ) of the straight lines obtainedby a least-square fitting method are given in Table 1. Formonofractal data sets, the fluctuation function has a con-stant slope at all q values (Kantelhardt et al. 2002). In Fig. 2 (top panels), we observed a power-law relationshipbetween the fluctuation function F q ( n ) and the segment n .We remark that the q dependency of h ( q ) reveals the pres-ence of multifractality in the time series. The slope of thebest fit line decreases by a small amount from the nega-tive to the positive moments indicating the q dependencyof the slope h ( q ) which in turn implies the presence of weakmultifractality signature in the time series. The differencein the slope h ( q ) between the original, shuffled, and surro-gate data shows that the three data have different multi-fractality strengths. As we have discussed in Sect. 2, since h shu f f ( q ) (cid:44) . and h surr ( q ) is q -dependent, it is not possi-ble, at this stage, to determine whether the multifractal sig-nature (though it is weak) is due to the temporal correlationor the fat-tailed probability distribution in the time seriesconsidered; and therefore, we referred to the relations givenby Eqs. 11 and 12. As shown in Fig. 3 (top panels) for allvalues of q , the three time series for the radio waveband ex-hibit almost a straight form indicating that the multifractalbehaviour in this time series is weak (close to monofractal)which is in support of our h shu f f ( q ) based analysis.For the time series under consideration, we have calcu-lated the width ∆ α value and obtained ∆ α = . / . / . for the original, shuffled, and surrogate data, respec-tively as shown in Table 1. These results of ∆ α , with re-spect to zero, further strengthened our analysis based on theslopes h ( q ) and scaling exponent τ ( q ) that the multifractal-ity strength in this time series is weak (close to monofractal)and the temporal correlation is the possible source of the de-tected weak multifractality. Following same analysis techniques and procedures used forthe radio observation, we have detected multifractality be-haviour in this time series. The difference between the q th -order fluctuation function for positive and negative q val-ues or/and at the smallest segment sizes compared to thelarge segment sizes reveal the multiscaling properties ofthe time series considered Ihlen (2012). The curves of theoriginal, shuffled and surrogate time series decrease with q which show that the slopes h ( q ) of the regression lines are q -dependent indicating the presence of a multifractal signa-ture in the time series under consideration (Sadegh Movahedet al. 2006).The q dependency of h ( q ) leads to a non-linear τ ( q ) dependence on q (Sadegh Movahed et al. 2006). The non-linearity indicates that there is a non-linear interaction be-tween the different scale events and multifractal nature ofthe system (Kantelhardt et al. 2001; Moffatt 1994). The q dependency of τ ( q ) shown in Fig. 3 (2nd panels) confirmsmultifractality in the time series. Also notice that in theparts of the curves corresponding to small variance regimes q < , in Fig. 2 (2nd panels), the shuffled series seem de-viated more (i.e., close to linearity) from the original datawith respect to the surrogate data, revealing that the shuf-fling process affects the original time series. The differencein the non-linearity between the shuffled and surrogate datawith respect to the original time series tells us about the pos-sible source of the observed multifractal behaviour. In thiscase, the temporal correlation is the main source of the ob-served multifractal signature than the fat-tail distribution.The presence of crossover in τ ( q ) versus q plot (Fig. 3), i.e.,different slopes for q < and q > , shows that the multi-fractality in the time series is strong.The calculated width ( ∆ α ) values for the original timeseries and the corresponding shuffled, and surrogate dataare given by ∆ α = . / . / . , respectively (Table1). Therefore, multifractal parameters h ( q ) , τ ( q ) , and themultifractal spectrum for the shuffled data clearly demon-strate that the observed multifractality originates more fromthe temporal correlation in the time series rather than thefat-tailed probability density distribution. To control the ef-fect due to some gaps in the data points of the time se-ries considered, we interpolate the data and produced evenlydistributed points. Applying same procedures for the datapoints evenly distributed (interpolated data) we obtainedsimilar results. The wider ∆ α with regard to ∆ α = richerand stronger the multifractality is. The left-side truncatedspectrum reflects the existence of a multifractal structuresensitive to the local fluctuations with small magnitudes (in-termittency). Here also, there is a q dependence of h ( q ) though it changesslowly as seen in the radio observation (Fig. 2, 3rd panels).The slope of the surrogate data is also changing with q show-ing the presence of temporal correlation in the time series.To support our h ( q ) based analysis, we can see in the scalingexponent plot τ ( q ) in Fig. 3 (3rd panels), that all the datahave weak non-linear behaviour both for positive and nega-tive q values, indicating the existence of weak multifractalityin the time series.As we have discussed in the methodology part, the rel-evance of calculating ∆ α for the shuffled and surrogate datain addition to the original one is to identify the possiblesource(s) for the multifractal signature (if any) in a giventime series. In this time series, the slowly changing localHurst exponent (the local slope), the scaling exponent curvewhich shows weak non-linearity in q , and, of course, the cal-culated width value for the original time series, which is nar-row though not extremely narrow, indicate the presence of MNRAS000
A. Bewketu et al. emerging for each time series is discussed separately as fol-lows. In Fig. 2, we present the multi-scaling behaviour of thefluctuation functions F q ( n ) corresponding to the time seriesby the log-log plots of the fluctuation function F q ( n ) againstthe time scale (segment) n for the original, shuffled and sur-rogate time series. The q th -order weights the influence ofsegments with large and small fluctuations. For negative q values, F q ( n ) is influenced by segments with small fluctua-tion whereas for positive q ’s, F q ( n ) is influenced by segmentswith large fluctuation. The midpoint q = is neutral to theinfluence of segments with small and large fluctuations. Theslope of the regression line of q th -order fluctuation functionand time segment gives the well-known generalized Hurstexponent h ( q ) . The slopes h ( q ) of the straight lines obtainedby a least-square fitting method are given in Table 1. Formonofractal data sets, the fluctuation function has a con-stant slope at all q values (Kantelhardt et al. 2002). In Fig. 2 (top panels), we observed a power-law relationshipbetween the fluctuation function F q ( n ) and the segment n .We remark that the q dependency of h ( q ) reveals the pres-ence of multifractality in the time series. The slope of thebest fit line decreases by a small amount from the nega-tive to the positive moments indicating the q dependencyof the slope h ( q ) which in turn implies the presence of weakmultifractality signature in the time series. The differencein the slope h ( q ) between the original, shuffled, and surro-gate data shows that the three data have different multi-fractality strengths. As we have discussed in Sect. 2, since h shu f f ( q ) (cid:44) . and h surr ( q ) is q -dependent, it is not possi-ble, at this stage, to determine whether the multifractal sig-nature (though it is weak) is due to the temporal correlationor the fat-tailed probability distribution in the time seriesconsidered; and therefore, we referred to the relations givenby Eqs. 11 and 12. As shown in Fig. 3 (top panels) for allvalues of q , the three time series for the radio waveband ex-hibit almost a straight form indicating that the multifractalbehaviour in this time series is weak (close to monofractal)which is in support of our h shu f f ( q ) based analysis.For the time series under consideration, we have calcu-lated the width ∆ α value and obtained ∆ α = . / . / . for the original, shuffled, and surrogate data, respec-tively as shown in Table 1. These results of ∆ α , with re-spect to zero, further strengthened our analysis based on theslopes h ( q ) and scaling exponent τ ( q ) that the multifractal-ity strength in this time series is weak (close to monofractal)and the temporal correlation is the possible source of the de-tected weak multifractality. Following same analysis techniques and procedures used forthe radio observation, we have detected multifractality be-haviour in this time series. The difference between the q th -order fluctuation function for positive and negative q val-ues or/and at the smallest segment sizes compared to thelarge segment sizes reveal the multiscaling properties ofthe time series considered Ihlen (2012). The curves of theoriginal, shuffled and surrogate time series decrease with q which show that the slopes h ( q ) of the regression lines are q -dependent indicating the presence of a multifractal signa-ture in the time series under consideration (Sadegh Movahedet al. 2006).The q dependency of h ( q ) leads to a non-linear τ ( q ) dependence on q (Sadegh Movahed et al. 2006). The non-linearity indicates that there is a non-linear interaction be-tween the different scale events and multifractal nature ofthe system (Kantelhardt et al. 2001; Moffatt 1994). The q dependency of τ ( q ) shown in Fig. 3 (2nd panels) confirmsmultifractality in the time series. Also notice that in theparts of the curves corresponding to small variance regimes q < , in Fig. 2 (2nd panels), the shuffled series seem de-viated more (i.e., close to linearity) from the original datawith respect to the surrogate data, revealing that the shuf-fling process affects the original time series. The differencein the non-linearity between the shuffled and surrogate datawith respect to the original time series tells us about the pos-sible source of the observed multifractal behaviour. In thiscase, the temporal correlation is the main source of the ob-served multifractal signature than the fat-tail distribution.The presence of crossover in τ ( q ) versus q plot (Fig. 3), i.e.,different slopes for q < and q > , shows that the multi-fractality in the time series is strong.The calculated width ( ∆ α ) values for the original timeseries and the corresponding shuffled, and surrogate dataare given by ∆ α = . / . / . , respectively (Table1). Therefore, multifractal parameters h ( q ) , τ ( q ) , and themultifractal spectrum for the shuffled data clearly demon-strate that the observed multifractality originates more fromthe temporal correlation in the time series rather than thefat-tailed probability density distribution. To control the ef-fect due to some gaps in the data points of the time se-ries considered, we interpolate the data and produced evenlydistributed points. Applying same procedures for the datapoints evenly distributed (interpolated data) we obtainedsimilar results. The wider ∆ α with regard to ∆ α = richerand stronger the multifractality is. The left-side truncatedspectrum reflects the existence of a multifractal structuresensitive to the local fluctuations with small magnitudes (in-termittency). Here also, there is a q dependence of h ( q ) though it changesslowly as seen in the radio observation (Fig. 2, 3rd panels).The slope of the surrogate data is also changing with q show-ing the presence of temporal correlation in the time series.To support our h ( q ) based analysis, we can see in the scalingexponent plot τ ( q ) in Fig. 3 (3rd panels), that all the datahave weak non-linear behaviour both for positive and nega-tive q values, indicating the existence of weak multifractalityin the time series.As we have discussed in the methodology part, the rel-evance of calculating ∆ α for the shuffled and surrogate datain addition to the original one is to identify the possiblesource(s) for the multifractal signature (if any) in a giventime series. In this time series, the slowly changing localHurst exponent (the local slope), the scaling exponent curvewhich shows weak non-linearity in q , and, of course, the cal-culated width value for the original time series, which is nar-row though not extremely narrow, indicate the presence of MNRAS000 , 1–12 (2017) ultifractal Behaviour of the 3C 273 Light-curves weak multifractal signature in this time series. Though themultifractality signature observed in this time series is weak,it is scientific to determine the possible source(s) of this weakmultifractal behaviour. For that end, we have applied thesame procedure and calculated the width, ∆ α , for the shuf-fled and surrogate data in addition to the original one as wehave done for all wave bands to detect the origin(s) of theobserved multifractality behaviour in each time series. Thewidth values for the original, shuffled, and surrogate dataare calculated as ∆ α = . / . / . , respectively(Table 1), which further shows that the multifractality sig-nature in this time series is weak and it is mainly due to thetemporal correlation in the original time series. The shapeof the multifractal spectrum reflects the temporal variationof the local Hurst exponent, and is helpful in classifying asmall and large scale fluctuations (intermittency) in the timeseries. Therefore, in this case, the right-side truncated spec-trum indicates the existence of small-scale fluctuation withlarge magnitudes (Tanna & Pathak 2014) unlike the natureof fluctuations (intermittencies) observed in the mm, UV,and X-ray bands where the spectrum is left-side truncated. Also in this time series, the slope h ( q ) is slightly changingwith q revealing weak multifractal nature in the time se-ries (Fig. 2, 4th panels). Furthermore, the q dependency of h corr ( q ) tells us that the temporal correlation is the possiblesource for the observed weak multifractality.In Fig. 3 (4th panels), the scaling exponent τ ( q ) showsno strong q dependency which confirms weak multifractal-ity behaviour in the time series considered. Exceptionally inthis time series, the contribution due to the fat-tail is vis-ible despite the fact that the temporal correlation is stillthe dominant contributor to the observed weak multifractalsignature.We have calculated the degree of multifractality as ∆ α = . / . / . for the original, shuffled, and sur-rogate, respectively (Table 1). Based on the calculated widthvalue for the original series, we have observed weak multi-fractal (close to monofractal) behaviour in this time series.This supports our analysis based on the q dependency of h ( q ) and the scaling exponent τ ( q ) . Comparing the generalized Hurst exponent h ( q ) for the orig-inal, shuffled, and surrogate data we observe multifractal be-haviour in this time series. In Fig. 2 (5th panels), we presentthe fluctuation function against the scale, and it shows aclear power-law relationship between them. The slopes forthe original, shuffled and surrogate data are q -dependentwhich indicate the presence of multifractality in the timeseries under consideration and as the degree of multifrac-tality for the three data is different. Since the slope of theshuffled data is not equal to 0.5 and of the surrogate datais q -dependent, the information we have is not enough todetermine the origin of the observed multifractal behaviour.In this case, we should make use of relations 11 and 12.As shown in Fig. 3 (5th panels), the original data isaway from linearity which tells us the presence of multifrac-tal behaviour in the time series. The crossover in τ ( q ) in Fig. 3 (5th panels), further confirms the presence of a multifrac-tal signature in the time series. The shuffled series is almostclose to linearity whereas that of the surrogate series is awayfrom linearity with respect to the original data. This differ-ence in the degree of non-linearity between the shuffled andsurrogate data with respect to the original one shows thatthe shuffling process has removed the temporal correlationin the time series; the temporal correlation contributes moreto the detected multifractality behaviour.Both, h ( q ) and τ ( q ) , speak in favour of multifractal be-haviour in the time series considered. The calculated spec-trum width ( ∆ α ) values for the original time series, thecorresponding shuffled and surrogate data are given by ∆ α = . / . / . , respectively (Table 1). This isfurther strengthened the presence of a multifractal signaturein the time series and the fact that the temporal correlationis the dominant source of multifractality in the time series.The symmetry is left-truncated reflecting the existence of amultifractal structure sensitive to the local fluctuations withsmall magnitudes (intermittency). Similar to all our previous analysis, in addition to the cal-culations of q th -order fluctuation functions and the gener-alized Hurst exponent, exploring the scaling exponent andthe multifractal spectrum indicates the presence of a multi-fractal behaviour in this time series.In Fig. 2 (bottom panels), we observe excellent growingsimilarity (power-law relation) between the q th -order fluc-tuation function and the segment size which leads to a de-creasing in h ( q ) indicating a multifractality signature in thetime series. The original time series has the biggest slopesthroughout its way while the shuffled time series has thesmallest one. Therefore, the three time series reveal a multi-fractal behaviour with different strengths. As shown in Fig.2 (bottom panels), the q dependency of h surr ( q ) confirms thepresence of temporal correlation in the time series or showsthat the multifractality is reduced more by the shuffling pro-cedure than by the phase-randomization procedure. Theseshow that the temporal correlation contributes mainly tothe multifractality behaviour of the time series with respectto the fat-tail distribution.The non-linearity in τ ( q ) (Fig. 3, bottom panels) clearlyconfirms the existence of multifractality in the time series.We can see that the original time series has non-linearityshape with respect to the shuffled time series, and the curvefor the surrogate data is away from linearity in compari-son with the shuffled one. This proves that the shufflingtechnique reduces the multifractality strength of the originaltime series. This is in agreement with our previous analysisbased on the generalized Hurst exponent h ( q ) that the tem-poral correlation is the origin of the multifractal behaviourin the time series. As we have already seen, the time se-ries under analysis has multifractal nature. We measure thedegree of multifractality in the time series by the calculusof the spectrum width ∆ α = α max − α min , and obtained ∆ α = . / . / . for the original, shuffled, and sur-rogate data, respectively (Table 1). This supports our analy-sis based on the generalized Hurst exponent and the scalingexponent by demonstrating again that there is multifrac-tal behaviour in the time series considered and the temporal MNRAS , 1–12 (2017)
A. Bewketu et al.
Figure 2.
The logarithm of the fluctuation function, log( F q ( n ) ), versus the log of the time scale (segment), log( n ), for the original,shuffled and surrogate time series considering, from top to bottom, radio, mm, IR, optical, UV and X-ray observations. For all cases: q = − (green), q = − (blue), q = (black), q = (yellow) and q = (red). The solid lines are the best fit lines. The slopes h ( q ) versus q are displayed in the far right side column. MNRAS000
The logarithm of the fluctuation function, log( F q ( n ) ), versus the log of the time scale (segment), log( n ), for the original,shuffled and surrogate time series considering, from top to bottom, radio, mm, IR, optical, UV and X-ray observations. For all cases: q = − (green), q = − (blue), q = (black), q = (yellow) and q = (red). The solid lines are the best fit lines. The slopes h ( q ) versus q are displayed in the far right side column. MNRAS000 , 1–12 (2017) ultifractal Behaviour of the 3C 273 Light-curves Figure 3.
The scaling exponent τ ( q ) (left), the multifractal spectrum f ( α ) (middle) and the deviation in the scaling exponent ∆ τ ( q ) (right) considering, from top to bottom, radio, mm, IR, optical, UV and X-ray observations.MNRAS , 1–12 (2017) A. Bewketu et al. correlation is responsible for the observed multifractal signa-ture than the fat-tailed distribution. The left-side truncatedspectrum reflects the existence of a multifractal structuresensitive to the local fluctuations with small magnitudes (in-termittency).
A wide f ( α ) spectrum is a very good indicative of multifrac-tality signature in the time series considered. As we can seefrom the fluctuation function plot, in Fig. 4a, the one withadded noise of highest variance fluctuates more mainly at thelowest scales which is expected for a multifractal data. Thisis further strengthened by the non-linearity in the scalingexponent function (Fig. 4b) and width of the multifractalspectrum function (Fig. 4c). The scaling function, τ ( q ) , ismore non-linear (different slope for negative and positive q )compared to the other two data sets with noise of lowestvariance as a result, the calculated width value is highestfor the data set with noise of highest variance. Therefore,based on the analysis given here, we roughly can say thatwidening of the f ( α ) curve can also happen due to noisecontamination (Harikrishnan et al. 2009). This is the first ever attempt to analyse the multifractalbehaviour of the quasar 3C 273 time series using the Multi-fractal Detrended Moving Average algorithm. In this study,we investigate the multifractal properties of the flux timeseries of quasar 3C 273 using the backward ( θ = ) MFDMAanalysis for one-dimensional time series analysis. We firstcalculate the fluctuation functions from which we estimatethe generalized Hurst exponents using least square fittingmethod. Then we deduce the scaling exponents and the mul-tifractal spectrum of the time series. Our results indicate thepresence of a weak multifractal (close to monofractal) sig-nature in some of the time series considered except in themm, UV, and X-ray bands where the degree of multifractal-ity is the strongest with respect to other bands. Moreover,in order to detect the origin of the observed multifractality,we perform the shuffling and phase-randomization (surro-gate) techniques on the original time series. By observingthe curves representing the generalized Hurst exponents, thescaling exponents, as well as the multifractal spectrum andcalculating the width ( ∆ α ), we conclude that the non-lineartemporal correlations are the dominant source of the ob-served multifractality signature in the time series while thefat-tail probability distribution contributes less in all of thetime series considered. Moreover, by adding noise with differ-ent variance to the radio band we pointed out that wideningof the multifractal spectrum could also happen due to noisecontamination. Based on the result obtained, we observedthat the quasar 3C 273 is a non-linear, complex and rich dy-namic system. Our analysis shows that some of the 3C 273light-curves have self-similar (intermittent) feature, it is amultifractal time series. The multifractality strength acrossthe EM spectra of the quasar 3C 273 is clearly different, andfrom which we can conclude that the nature of flux fluctua-tions across the different bands of the quasar 3C 273 is notthe same which in turn provides a good implication as the Figure 4.
A: The logarithm of the fluctuation function,log( F q ( n ) ), versus the log of the time scale (segment), log( n ),B: The scaling exponent, τ ( q ) , versus the moment, q , and C: Themultifractal spectrum f ( α ) . physical mechanisms that cause the flux fluctuation in thesystem are not the same for all spectral regions.There is much evidence that light-curves of AGN areboth non-linear and non-stationary. For example, optical(V, R and I band) studies show the variable flux in the3C 273 changes with different amplitudes on differenttimescales revealing the non-linear variability characteristicof blazars (Kalita et al. 2015). It has been shown by Kalitaet al. (2015) that there is high flux variation in all bandsof the quasar 3C 273 in the long term. In our work, wehave shown that the quasar 3C 273 time series have amultifractal feature mainly due to temporal correlationwhich is assumed to be due to flux variation across thedifferent bands of the source. The differences in the degreeof multifractality imply that all bands of the source 3C 273are not only emitting from different regions but also theyare driven by different mechanisms. In the work of Kalita MNRAS000
A: The logarithm of the fluctuation function,log( F q ( n ) ), versus the log of the time scale (segment), log( n ),B: The scaling exponent, τ ( q ) , versus the moment, q , and C: Themultifractal spectrum f ( α ) . physical mechanisms that cause the flux fluctuation in thesystem are not the same for all spectral regions.There is much evidence that light-curves of AGN areboth non-linear and non-stationary. For example, optical(V, R and I band) studies show the variable flux in the3C 273 changes with different amplitudes on differenttimescales revealing the non-linear variability characteristicof blazars (Kalita et al. 2015). It has been shown by Kalitaet al. (2015) that there is high flux variation in all bandsof the quasar 3C 273 in the long term. In our work, wehave shown that the quasar 3C 273 time series have amultifractal feature mainly due to temporal correlationwhich is assumed to be due to flux variation across thedifferent bands of the source. The differences in the degreeof multifractality imply that all bands of the source 3C 273are not only emitting from different regions but also theyare driven by different mechanisms. In the work of Kalita MNRAS000 , 1–12 (2017) ultifractal Behaviour of the 3C 273 Light-curves Table 1.
The best fit parameters; the slopes h ( q ) , and the widths( ∆ α ). Radio q original shuffled surrogate-10 1.0547 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ∆ α q original shuffled surrogate-10 1.3675 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ∆ α q original shuffled surrogate-10 1.0697 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ∆ α q original shuffled surrogate-10 1.0595 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ∆ α q original shuffled surrogate-10 1.1313 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ∆ α q original shuffled surrogate-10 1.0997 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ∆ α et al. (2015), it has been presented that there is no relationbetween X-ray and optical/UV emission of 3C 273 whichis more or less in agreement with our work considering thedifference in the degree of multifractality observed in thesebands. McHardy et al. (1999) has shown that there is nocorrelation between the X-ray and mm observations of the 3C 273. These all including our results support the presenceof high flux variation in the spectra of the source 3C 273and the variation is not the same in all bands. Multifractalbehaviour indicates the presence of non-linearity in a timeseries which leaves open the possibility that the time seriesis produced by a chaotic system. However, non-linearity isa necessary but not sufficient condition for a light-curveto be produced by a chaotic system (Vio et al. 1992).Therefore, our work, at least, shows that 3C 273 is a non-linear system. We believe that understanding the fractalbehaviour in the light-curves also contributes in terms ofproviding information from a different point of view aboutthe complexity of the dynamics interior to the system underconsideration. Understanding the nature of the scaling inthe 3C 273 flux could tell us fundamental things abouthow the flux varies in each band which in turn providessufficient information in order to answer basic AGN relatedquestions, such as energy production mechanisms, emissionlines problem, and many others. Having clear picture ofhow the scaling in a time series (the fractal nature) behavescan provide fundamental information for those working onmodel construction to fully uncover the AGN science. ACKNOWLEDGEMENTS
This paper includes data collected by the Integral ScienceData Centre (ISDC) linked to the Astronomical Observa-tory of the University of Geneva. Research activities of theAstronomy Observational Board of the Federal Universityof Rio Grande do Norte are supported by continuous grantsfrom the Brazilian agencies CNPq and FAPERN. We alsowarmly thank Prof. Wei-Xing ZHOU, at the East ChinaUniversity of Science and Technology for his continuouscomments and communication. A.B.B. acknowledges finan-cial support from the CAPES Brazilian agency. J.P.B. ac-knowledges a CAPES/PNPD post-doctorate fellowship. Fi-nally, we warmly thank the anonymous Reviewer for pro-viding very helpful comments and suggestions that largelyimproved this work.
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