Multifractal detrended fluctuation analysis of analog random multiplicative processes

Chaos, Solitons and Fractals 41 (2009) 2806–2811

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos

Multifractal detrended fluctuation analysis of analog random multiplicative processes L.B.M. Silva, M.V.D. Vermelho, M.L. Lyra *, G.M. Viswanathan Instituto de Física, Universidade Federal de Alagoas, Maceió - AL, 57072-970, Brazil

a r t i c l e

i n f o

Article history: Accepted 17 October 2008

a b s t r a c t We investigate non-Gaussian statistical properties of stationary stochastic signals generated by an analog circuit that simulates a random multiplicative process with weak additive noise. The random noises are originated by thermal shot noise and avalanche processes, while the multiplicative process is generated by a fully analog circuit. The resulting signal describes stochastic time series of current interest in several areas such as turbulence, finance, biology and environment, which exhibit power-law distributions. Specifically, we study the correlation properties of the signal by employing a detrended fluctuation analysis and explore its multifractal nature. The singularity spectrum is obtained and analyzed as a function of the control circuit parameter that tunes the asymptotic power-law form of the probability distribution function. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction The emergence of power-law distributions in equilibrium thermodynamic systems is a result of scale-free cooperative behavior that takes place at finely tuned conditions leading to the vicinity of a second-order transition. On the other hand, power-laws are quite ubiquitous in non-equilibrium systems. They appear in turbulence phenomena, in the size distribution of earthquakes and avalanches, in econometric time series, in biological and environmental records, among many others. Their wide spread occurrence are consistent with the idea of an underlying self-organized process which drives these non-equilibrium dynamical systems to a critical point without the need of a fine tuning [1,2]. In some cases, the distribution exhibits multi-scaling, with distinct and independent exponents governing the several moments of the distribution, a feature usually named as multifractality [3,4]. Multifractal distributions have been identified in a very broad range of scenarios and length scales including turbulence and chaos [5,6], high-energy physics [7], metal–insulator transition [8,9], quasi-crystals [10,11] and traffic flows [12], among others. Simple mathematical models which effectively reproduce the main features of the distribution of collective stochastic quantities without reference to the microscopic dynamics are of unquestionable value in clarifying the general mechanism behind the emergence of scale-free distributions. One of the simplest mechanisms leading to power-law distributions is the random multiplicative process (RMP), in which the stochastic variable is driven by a multiplicative noise in addition to some weak additive noise [13–16]. The statistical properties resulting from such process have been widely investigated and used as a basic model to study the fluctuations in economic activities [17–20], population dynamics [21], on–off intermittency [22], conformation of polymers [23], non-linear coupled oscillators [24] and air pollutant concentration [25], among others. A relevant feature of the RMP is that the power-law exponent of the distribution function can be tuned by varying the basic statistical characteristics of the multiplicative noise. The intensity of the additive noise influences only the crossover to the asymptotic power-law regime. Also, the resulting stochastic series exhibits multi-scaling and its moments scale as a power of the additive noise strength [26].

* Corresponding author. E-mail address: [email protected] (M.L. Lyra). 0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.10.027

L.B.M. Silva et al. / Chaos, Solitons and Fractals 41 (2009) 2806–2811

2807

Analog simulations of the RMP have been recently developed [27–29]. Analog studies represent an important complementary tool to investigate stochastic systems due to the relative simplicity and high speed which allow large volumes of the parameter’s space to be surveyed quickly without the accumulation of truncation errors present in digital simulations. In particular, it has been successfully used to investigate chaotic and stochastic phenomena in non-linear systems [30–32] A circuit tailored to generate a random multiplicative noise was introduced by Sato [27] which was able to reproduce the power-law distribution of the output signal and to exploit its auto-correlation function. More recently, aiming to ensure all components to operate within their specific regimes, an alternative configuration for the electrical circuit was proposed and used to study stochastic resonance in a sub-threshold system driven by power-law noise [28]. The present work will be devoted to investigate the multi-scaling properties of the generated non-Gaussian output signal of the electrical circuit proposed in [28]. After a brief discussion about the role of correlations, we will employ an extension of the detrended fluctuation analysis [33,34] that has been tailored to identify multi-scaling properties [35–38]. With this technique, we will obtain the singularity spectrum of the signal. We obtain the degree of deviation from the mono-fractal character can be finely tuned by controlling a circuit parameter and consequently the power-law decay of the signal distribution function. 2. Analog circuit to generate a random multiplicative noise The power-law dependence of the non-Gaussian temporal series analyzed in this paper were originally obtained from a Gaussian white noise source due to an electronic circuit. More detailed description of the circuit characteristics can be found elsewhere [28]. The block diagram indicating the white noise generator and the multiplicative process is depicted in Fig. 1. In brief, the initial white noise is obtained by way of the thermal noise, shot noise, and other contributions from a reverse biased PN junction of transistor (BF494) producing avalanche noise. This resulting signal is doubly amplified in cascade via operational amplifiers (LF356) for the purpose of driving following stage of the multiplication function. The choice of this configuration was made bearing in mind to amplify the largest bandwidth signal. The non Gaussian signals obey the RMP described by the Langevin equation

dv ¼ kðtÞv ðtÞ þ gðtÞ; dt

ð1Þ

where v(t) is a stochastic variable, k(t) a multiplicative noise, and g(t) an additive noise. Here, both k(t) and g(t) will be taken as uncorrelated (white) and Gaussian, with average and variance given by hk(t)i = k and h(k(t)  k)(k(t0 )  k)i = 2Dkd(t  t0 ), and hg(t)i = 0 and hg(t)g(t0 )i = 2Dgd(t  t0 ). It has been demonstrated, both analytically [26] and numerically [16,28], that the resulting signal v(t) displays a non-Gaussian probability distribution function with a slowly decaying tail given by P(v) / va, with a = 1  k/Dk. To implement an analog circuit equivalent to the above differential equation, the power law noise source was built in two main blocks, namely, a variable white noise source and a multiplicative stage. The last is responsible to obtain the required negative and positive feedback according to the multiplicative noise theory and consist of an analog multiplier (analog devices – AD734) and an operational amplifier in the Miller integrator configuration. Following [28], the Langevin equivalent equation as a result of the multiplicative noise of the circuit is given by

s

  dv R7 þ v 0 ðR6 ; tÞ v ðtÞ þ gðtÞ; ¼ dt R8

ð2Þ

where s = C3R7 is the integration time scaling of the circuit, and v0(R6, t) is the amplified semiconductor white noise. Although the additive term is not explicitly added in the circuit, it derives from either thermal noises of the operational amplifiers or

Fig. 1. Block diagram of the analog circuit employed to generate a non-Gaussian noise with an asymptotic power-law distribution. The intrinsic thermal noise, combined with the noise from external forces, acts as an additive noise.

2808

L.B.M. Silva et al. / Chaos, Solitons and Fractals 41 (2009) 2806–2811

from external electromagnetic noises. The data acquisition system employed in the characterization of this noise source consists of 16-bits analog-to-digital card (ADC) – NI6036E (200kS/s) – and a personal computer (PC). Appropriated software was also used for data processing. A sampling frequency of 125 kHz was used during the experiments. Typical time series of v(t), which obey the above Langevin equation, are shown in Fig. 2. The series depicted correspond to values of the control resistance R6 which were chosen to illustrate the distinct statistical characteristics that can be explored. For small values of the control resistance, the series is quite well described by a white Gaussian noise. For large resistances, the voltage v(t) exhibits random spikes. The frequency of the spikes introduces a power law tail in the PDF as shown in the right panel of Fig. 2. Since small correlations exist in the series thus generated, we shuffle the data before analysis, to guarantee independent and identically distributed variables. It is important to stress here that nonlinear effects saturate the response of the active components of the circuit. This naturally warrants the condition of null probability flux needed to achieve stationarity, which was imposed by considering reflecting boundaries in the analytical calculations of [26]. 3. Statistical data analysis: multifractality Power laws can arise in time series either in the probability density distribution or in correlation functions or both. Distributions can belong to the family of Gaussian (i.e. Normal) distributions or to ‘‘anomalous” ones such as the Lévy or powerlaw distributions, among others. The correlation properties can be classified in terms of short-range correlations and longrange correlations. While short-term correlations decay exponentially or faster and thus cannot alter the long-term behavior, long-range power-law correlations can lead to significant effects due to their scale invariant properties, leading typically to non-Markovian behavior. If a time series does not have long-range power law correlation it is refered as ‘‘white”. On the other hand, if long-range power laws exist, it is termed as ‘‘colored”. To quantify the high-order moments of time series, one usually computes their multifractal spectra. The multifractal spectrum of a time series contains information about n-point correlations [4] and thus provides more information compared to two-point correlation functions. We apply the multifractal detrended fluctuation analysis (MF-DFA) method to study the multifractal spectrum of the generated series [35]. Like the wavelet transform modulus maximus (WTMM) method [39], the MF-DFA method avoids the difficulties of managing diverging negative moments. We briefly summarize the MF-DFA method P as follows: First, we integrate the time series v(k) to generate a profile yðtÞ  tk¼1 v ðkÞ, which we can visualize as a one dimensional random walk. We then divide the series y(t) into Ns segments of size s. We use non-overlapping segments, so the product s Ns gives the total length of the complete time series. For each segment m, we apply linear regression to calculate a best fitting ‘‘trend” ym(i), where i = 1, 2, . . . , s. For each segment m we can thus define a mean square fluctuation

F 2 ðm; sÞ 

s 1X jyððm  1Þs þ iÞ  ym ðiÞj2 ; s i¼1

which corresponds to a measure of the second moment for segment m. We then define a qth-order fluctuation function

F q ðsÞ 

Ns 1 X F 2 ðm; sÞq=2 : Ns m¼1

Positive values of q weight large fluctuations while negative moments weight small fluctuations. The fact that F2(m, s) always remains positive avoids the divergence of the negative moments. The scaling of the fluctuation for moment q follows

F q ðsÞ  sqHðqÞ ;

ð3Þ 2

4

10

v(t)

v(t)

2 0 -2 0

10

0

P(v)

-4 0.3 0.2 0.1 0 -0.1 -0.2

-2

10

0.8

1.6

2.4

t (in Seconds)

3.2

4.0

-2

10

-1

10

0

10

v (in Volts)

Fig. 2. (Typical time series generated by the analog circuit for two distinct values of the control resistance, namely R6 = 1690 X (top left) and R6 = 12.7 X (bottom left). Data correspond to 105 records of the voltage during a total period of 4 s. Right panel: probability distribution functions of these series. The solid line fitting the data from R6 = 12.7 X (squares) correspond to a Gaussian distribution, while that fitting the data from R6 = 1690 X (circles) correspond to P(v) / (1 + (v/s)2)a/2, with a = 2.8.

L.B.M. Silva et al. / Chaos, Solitons and Fractals 41 (2009) 2806–2811

2809

where H(q) represents a generalized Hurst exponent, with the traditional Hurst exponent given by H = H(2). Monofractal time series have a unique Hurst exponent H(q) = H, while for multifractal series the value of H(q) depends on q. In Fig. 3(a), we report the H(q) curves obtained for a set of noise time-series generated by the presently developed analog circuit. Notice the deviation from the mono-fractal behavior, in qualitative agreement with the predicted scaling for uncorrelated power-law distributed series [35]. We can relate H(q) to the exponents s(q) conventionally used to quantify the scaling of the partition function

Z q ðsÞ  ssðqÞ in the standard textbook multifractal formalism [35] via

sðqÞ ¼ qHðqÞ  1:

ð4Þ

We can obtain this relation by noting that for any positive, normalized and stationary time series x(k), we can define a ‘‘box probability”

a

0.54 0.53

H(q)

0.52 0.51 0.50 0.49 0.48

-4

-2

0

2

4

2

4

q

b

τ-q/2

-1.0

-1.1

-1.2 -4

-2

0

q

c

1.0

f(h)

0.9

0.8

0.7 0.4

0.45

0.5

0.55

0.6

h Fig. 3. (a) Generalized Hurst exponents H(q) estimated using MF-DFA, for time series generated using distinct control resistances R6 = 816 X (circles), R6 = 694 X (squares), R6 = 555 X (diamonds) and R6 = 423 X (triangles) Notice the progression from multifractal to more monofractal behavior as the control resistance decreases. (b) Partition function scaling exponents s(q) plotted as s(q)  q/2 for easier visualization. (c) Multifractal singularity spectrum f(h) as a function of the Hölder exponent or singularity strength h. The spectrum of singularities becomes narrower as the time series distribution function approaches the Gaussian behavior. Notice that h centers around 1/2, as expected due to the lack of correlations (H = h = 1/2).

2810

L.B.M. Silva et al. / Chaos, Solitons and Fractals 41 (2009) 2806–2811

ps ðmÞ 

ms X

xðkÞ

k¼ðm1Þsþ1

for box size s, given the normalization

Z q ðsÞ 

Ns X

PsNs

k¼1 xðkÞ

¼ 1. The partition function is defined as

jps ðmÞjq :

m¼1

The relation given by Eq. 4 can be seen by noting that Zq(s)  NsFq(s) for the case v(t) = x(t). The advantage of methods such as WTMM and MF-DFA arises due to not being limited to stationary, positive, and normalized time series. Fig. 3(b) shows that as the control resistance increases s(q) becomes strongly non-linear, indicating multifractality. Note that uncorrelated white noise has Hurst exponent H(q) = 1/2 and s(q) = q/2  1, so plotting s  q/2 will give a horizontal (i.e., constant) line. We have therefore plotted s(q)  q/2 instead of s(q) for easier visualization. The Legendre transform of Eq. 4 gives the multifractal singularity spectrum f(h):

h ¼ dsðqÞ=dq;

ð5Þ

f ðhÞ ¼ qh  sðqÞ:

ð6Þ

Here, f gives the dimension of the subset of the series characterized by the singularity strength h. The literature also refers to h as the Hölder exponent, which represents a measure of the number of continuous derivatives that the underlying signal possesses. The singularity spectrum of monofractal time series will show a unique Hölder exponent, while a multifractal series will show a range of values for h. Note that sometimes s and f(as) appear defined differently (e.g., what we define as f(as) here would equal f(a) + 1 in the terminology of [40,41]). We use the definitions contained in [35,42]. Fig. 3(c) shows that we obtain nonuniversal multifractal singularity spectra. A couple series appear nearly monofractal, with a narrow f(h). The above results indicate that the degree of multifractality of the generated time series can be continuously tuned by controlling the characteristic of a single component of the analogue circuit. In order to explicitly show this feature, we measured the second derivative of the singularity spectrum in the vicinity of its maximum for distinct values of the circuit control parameter used to tune the exponent of the power-law tail of the distribution function. In Fig. 4, we report our results for the inverse derivative at the maximum as a function of the effective exponent a. We found a roughly exponential relation, with the width of the multifractal spectrum exhibiting a pronounced increase as the distribution function develops longer tails which favors the occurrence of large events. In the inset, we report the measured characteristic function of the circuit relating the control resistance with the effective power-law exponent of the PDF. 4. Summary and conclusions In summary, we designed and developed an analog circuit to simulate a RMP. We investigated the fractal character of the resulting analog non-Gaussian noise whose PDF exhibits power-law tails. The analog circuit was designed to allow the power-law exponent of the noise PDF to be finely tuned through the control of a variable resistance. We employed a multifractal detrended fluctuation analysis of a set of time-series and computed the singularity spectrum of shuffled series. The shuffling process eliminated the presence of short-range correlations and allowed for a better characterization of the scaling laws associated just with the non-Gaussian nature of the noise distribution function. We found that the RMP generates a noise with mul-

-1

10

30 25

α

20 15 10 -2

10

5

m

0

0

500

1000 1500 2000

R6 (Ohms) -3

10

0

5

10

15

20

25

30

α Fig. 4. (a) The inverse of the second derivative at the maximum of the multifractal singularity spectrum versus the effective power-law exponent of the probability distribution function. In the limit of Gaussian noise, the detrended fluctuation analysis results in a mono-fractal statistics with a single scaling exponent. As the noise assumes a non-Gaussian character, the fluctuations become multifractal with the range of scaling exponents continuously growing as the PDF departs further from the Gaussian behavior. This growth is roughly exponential. The inset shows the measured relation between the control resistance and the power-law exponent of the distribution function.

L.B.M. Silva et al. / Chaos, Solitons and Fractals 41 (2009) 2806–2811

2811

tifractal statistics. The range of multifractal exponents becomes wider as the probability distribution deviates further from the Gaussian behavior. We obtained an exponential growth of the multifractal singularity spectrum width for decreasing values of the power-law exponent characterizing the non-Gaussianity of the noise signal,thus evidencing the relevance played by the rare events of large sizes. The presently proposed analog scheme can, therefore, be used to generate non-trivial (non-Gaussian, multifractal) noise signals with pre-determined PDF. The saturation of the active components response due to nonlinear effects naturally keeps the condition of null probability flux satisfied, necessary to achieve the stationarity of the present Langevin process [26]. Due to the fast computation capability of analog approaches, it can be widely used as a technique to investigate dynamical phenomena driven by complex stochastic signals. Further, since the occurrence of large events and fractal scaling are quite common in complex systems exhibiting non-Gaussian behavior ranging from sub-nuclear to cosmological scales [7,43,44], the multifractal detrended fluctuation analysis may be used as an important tool to identify the most relevant parameters leading to complex behavior, as demonstrated here for a random multiplicative process. Acknowledgements We acknowledge partial financial support from CAPES, CNPq and FINEP (Brazilian research agencies) as well as from FAPEAL (Alagoas state research agency). References [1] Bak P, Tang C, Wiesenfeld K. Self-organized criticality – an explanation of 1/f noise. Phys Rev Lett 1987;59:381–4. [2] Bak P, Tang C, Wiesenfeld K. Self-organized criticality. Phys Rev A 1988;38:364–74. [3] Halsey TC, Jensen MH, Kadanoff LP, Procaccia I, Shraiman BI. Fractal measures and their singularities – the characterization of strange sets. Phys Rev A 1986;33:1141–51. [4] Barabasi AL, Vicsek T. Multifractality of self-affine fractals. Phys Rev A 1991;44:2730–3. [5] Benzi R, Paladin G, Parisi G, Vulpiani A. On the multifractal nature of fully-developed turbulence and chaotic systems. J Phys A 1984;17:3521–31. [6] da Silva CR, da Cruz HR, Lyra ML. Low-dimensional non-linear dynamical systems and generalized entropy. Braz J Phys 1999;29:144–52. [7] Iovane G, Chinnici M, Tortoriello FS. Multifractals and El Naschie E-infinity Cantorian space–time. Chaos Solitons & Fractals 2008;35:645–58. [8] Schreiber M, Grussbach H. Multifractal wave-functions at the Anderson transition. Phys Rev Lett 1991;67:607–10. [9] Lima RPA, Lyra ML, Cressoni JC. Multifractality of one electron eigenstates in 1D disordered long range models. Physica A 2001;295:154–7. [10] Albuquerque EL, Cottam MG. Theory of elementary excitations in quasiperiodic structures. Phys Rep 2003;376:225–337. [11] de Oliveira IN, Lyra ML, Albuquerque EL. Specific heat anomalies of non-interacting fermions with multifractal energy spectra. Physica A 2004:343. 424–432. [12] Li XW, Shang PJ. Multifractal classification of road traffic flows. Chaos Solitons & Fractals 2007;31:1089–94. [13] Levy M, Solomon S. Power laws are logarithmic Boltzmann laws. Int J Mod Phys C 1996;7:595–601. [14] Sornette D, Cont R. Convergent multiplicative processes repelled from zero: power laws and truncated power laws. J Phys I 1997;7:431–44. [15] Takayasu H, Sato A-H, Takayasu M. Stable infinite variance fluctuations in randomly amplified Langevin systems. Phys Rev Lett 1997;79:966–9. [16] Sato AH. Explanation of power law behavior of autoregressive conditional duration processes based on the random multiplicative process. Phys Rev E 2004;69:047101–4. [17] Solomon S, Richmond P. Power laws of wealth, market order volumes and market returns. Physica A 2001;299:188–97. [18] Anteneodo C, Riera R. Additive-multiplicative stochastic models of financial mean-reverting processes. Phys Rev E 2005;72:026106–7. [19] Fulco UL, Lyra ML, Petroni F, Serva M, Viswanathan GM. A stochastic model for multifractal behavior of stock prices. Int J Mod Phys B 2004;18:681–9. [20] Serva M, Fulco UL, Gleria IM, Lyra ML, Petroni F, Viswanathan GM. A Markov model of financial returns. Physica A 2006;363:393–403. [21] Sornette D. Linear stochastic dynamics with nonlinear fractal properties. Physica A 1998;250:295–314. [22] Venkataramani SC, Antonsen TM, Ott E, Sommerer JC. On–off intermittency: power spectrum and fractal properties of time series. Physica D 1996;96:66–99. [23] Deutsch JM. Polymer conformation in random velocity-fields. Phys Rev Lett 1992;69:1536–9. [24] Kuramoto Y, Nakao H. Origin of power-law spatial correlations in distributed oscillators and maps with nonlocal coupling. Phys Rev Lett 1996;76:4352–5. [25] Lee CK. Multifractal characteristics in air pollutant concentration time series. Water Air Soil Pollut 2002;135:389–409. [26] Nakao H. Asymptotic power law of moments in a random multiplicative process with weak additive noise. Phys Rev E 1998;58:1591–600. [27] Sato A, Takayasu H, Sawada Y. Power law fluctuation generator based on analog electrical circuit. Fractals 2000;8:219–25. [28] da Silva MP, Lyra ML, Vermelho MVD. Analog study of the first passage time problem driven by power-law distributed noise. Physica A 2005;348:85–96. [29] Duarte JRR, Vermelho MVD, Lyra ML. Stochastic resonance of a periodically driven neuron under non-Gaussian noise. Physica A 2008;387:1446–54. [30] Miliou AN, Valaristos AP, Stavrinides SG, Kyritsi K, Anagnostopoulos AN. Characterization of a non-autonomous second-order non-linear circuit for secure data transmission. Chaos Solitons & Fractals 2007;33:1248–55. [31] Oya T, Asai T, Amemiya Y. Stochastic resonance in an ensemble of single-electron neuromorphic devices and its application to competitive neural networks. Chaos Solitons & Fractals 2007;32:855–61. [32] Ruiz P, Gutiérrez JM, Güémez J. Experimental mastering of nonlinear dynamics in circuits by sporadic pulses. Chaos Solitons & Fractals 2008;36:635–45. [33] Peng CK, Buldyrev SV, Havlin S, Simons M, Stanley HE, Goldberger AL. Mosaic organization of DNA nucleotides. Phys Rev E 1994;49:1685–9. [34] Hu K, Ivanov PC, Zhi Chen Z, Carpena P, Stanley HE. Effect of trends on detrended fluctuation analysis. Phys Rev E 2001;64:011114–9. [35] Kantelhardt JW, Zschiegner SA, Koscielny-Bunde E, Havlin S, Bunde A, Stanley HE. Multifractal detrended fluctuation analysis of nonstationary time series. Physica A 2002;316:87–114. [36] Oswiecimka P, Kwapien J, Drozdz S. Wavelet versus detrended fluctuation analysis of multifractal structures. Phys Rev E 2006;74:016103–17. [37] Lim G, Kim S, Lee H, Kim K, Lee DI. Multifractal detrended fluctuation analysis of derivative and spot markets. Physica A 2007;386:259–66. [38] Shang PJ, Lu YB, Kamae S. Detecting long-range correlations of traffic time series with multifractal detrended fluctuation analysis. Chaos Solitons & Fractals 2008;36:82–90. [39] Muzy JF, Bacry E, Arneodo A. Wavelets and multifractal formalism for singular signals – application to turbulence data. Phys Rev Lett 1991;67:3515–8. [40] Paladin G, Vulpiani A. Anomalous scaling laws in multifractal objects. Phys Rep 1987;156:147–225. [41] Paladin G, Vulpiani A. Anomalous scaling and generalized Lyapunov exponents of the one-dimensional Anderson model. Phys Rev B 1987;35:2015–20. [42] Matia K, Ashkenazy Y, Stanley HE. Multifractal properties of price fluctuations of stock and commodities. Europhys Lett 2003;61:422–8. [43] Iovane G. Mohamed El Naschie’s (1) Cantorian space–time and its consequences in cosmology. Chaos Solitons & Fractals 2005;25:775–9. [44] Iovane G. Cantorian space–time and Hilbert space: Part I – foundations. Chaos Solitons & Fractals 2006;28:857–78.