Complete Description of All Self-Similar Models Driven by Levy Stable Noise

Complete description of all self-similar models driven by Levy stable noise

Aleksander Weron∗ and Krzysztof Burnecki Hugo Steinhaus Center, Institute of Mathematics, Wroclaw University of Technology, Wyb. Wyspianskiego 27, 50-370 Wroclaw, Poland

Szymon Mercik and Karina Weron Institute of Physics, Wroclaw University of Technology, Wyb. Wyspianskiego 27, 50-370 Wroclaw, Poland (Dated: October 14, 2004) A canonical decomposition of H-self-similar Lévy symmetric α-stable processes is presented. The resulting components completely described by both deterministic kernels and the corresponding stochastic integral with respect to the Lévy symmetric α-stable motion, are shown to be related to the dissipative and conservative parts of the dynamics. This result provides new stochastic analysis tools for study the anomalous diffusion phenomena in the Langevin equation framework. For example, a simple computer test for testing the origins of self-similarity is implemented for four real empirical time series recorded from different physical systems: an ionic current flow through a single channel in a biological membrane, an energy of solar flares, a seismic electric signal recorded during seismic Earth activity, and FX rate daily returns.

PACS numbers: 05.40.-a,05.10Gg,02.50.-r,05.45.Tp

I. INTRODUCTION delbrot, has been introduced in 1962 by Lamperti [19]. Historically oldest approach to self-similarity has been proposed by Kolmogorov in 1940, who introduced frac- The importance of Lévy stable distributions or pro- tional Brownian motion, which is a Gaussian self-similar cesses in physics and related areas has long been known process with stationary increments. For the details see [1–7]. They are also increasingly important in many other Mandelbrot and Van Ness [20] and references therein. fields of application. Consequently, a general trend nowa- The study of non-Gaussian self-similar processes with days is to put Lévy stable type anomalous diffusion on stationary increments was initiated by Taqqu [21]. He a similar footing with Brownian diffusion [8–18]. How- addressed the question what type of limiting distribu- ever, in what concerns our understanding of a structure tions is expected to appear if the stationary sequence of their self-similarity, the situation is vastly different for has a stronger dependence violating the validity of cen- the two types of models. tral limit theorem, and further developed a non-Gaussian At the level of the Langevin equation the Lévy mo- limit theorem originated by Rosenblatt. On the other tion is a generalization of the Brownian one which de- hand, the works of Sinai and Dobrushin in the field of scribes the motion of small macroscopic particles in a statistical physics appeared independently in the same liquid (or a gas) experiencing unbalanced bombardments time [22]. Since the self-similar processes such as Brown- due to surrounding atoms. The Brownian motion mim- ian, fractional Brownian, Lévy stable and fractional Lévy ics the influence of the ”bath” of surrounding molecules stable motion are stochastic processes that are invariant in terms of a mean-field, time-dependent stochastic force in distribution under suitable scaling of time and space, which is commonly assumed to be white Gaussian noise. so these processes are closely related to the notion of That postulate is compatible with the assumption of a renormalization in statistical and high energy physics. short correlation time of fluctuations (much shorter than the time-scale of the macroscopic motion) and the assumption of weak interactions with the bath. In con- A significant difference between Gaussian and Lévy trast, the Lévy motions describe results of strong colli- stable distributions is that the latter have heavy tails and sions between the test particle and the surrounding envi- their variance is infinite. This means that much larger ronment, and hence, lead to models of the bath that go jumps or flights are possible for Lévy stable distributions, beyond the standard ”close-to-equilibrium” Gaussian de- which causes their variance to diverge. Since many nat- scription. The already observed unusual statistical prop- ural processes follow Lévy stable distributions, [2, 23– erties of systems driven by them serve as a challenge 25], the necessity of modeling physical phenomena with for generalizations of thermostatics trying to explain the heavy tailed distributions is dramatically increasing in non-Gibbsian phenomena [2, 4, 5]. many fields of physics. In this paper we employ ergodic The notion of self-similarity, originally coined by Man- theory foundation of ubiquity of Lévy stable self-similar processes in physics and provide a catalog of models for anomalous diffusion [2–6]. To be more precise we de- velop tools for study of diffusion processes described by ∗Electronic address: [email protected] the following Langevin-type stochastic differential equation driven by Lévy stable noise served in many real phenomena there is a need to build efficient estimators of the self-similarity index [30, 31, 38]. α dXt = b(t, Xt)dt + σ(t, Xt)dZt , (1) A self-similar stochastic process is a process that is invariant under suitable translations of time and scale. α where dZt stands for the increments of Lévy α-stable We mention that the self-similarity is described by a α motion Zt ,0<α≤ 2, see [25]. real positive parameter H>0 called self-similarity index Section II starts with time and scale invariance of self- which provides information on the investigated time se- similar processes. Two basic examples: fractional Brow- ries structure, correlations and fractal properties. For ex- nian motion and fractional Lévy stable motion are de- ample, the Brownian motion is self-similar with H =1/2; scribed there in the form of stochastic integrals. Also a it has no memory and its increments have finite variance. connection with long-memory or long-range dependence It is well-known that if a process has purely random is established. In Section III we provide a theoretical jus- increments with infinite variance then the process can tification of the recently proposed BMW2 computer test be self-similar with index of self-similarity different from [26] introduced to detect the origins of the self-similarity 1/2. An example of H =1/α self-similar process is feature of a particular model. Thus it follows that the es- the Lévy stable motion with stationary and indepen- timated index of self-similarity can reflect a long-memory dent, identically distributed increments with symmetric effect or infinite variance impact of the process. In Sec- α-stable distribution [25, 37]. When one applies to that tion IV we discuss the integral representation of Lévy sta- process the R/S analysis, the obtained Hurst exponent ble self-similar processes in the language of nonsingular equals 1/2 since the estimator shows a lack of memory flows and exploit the connection with the Hopf decom- [26]. Thus the second origin of the self-similarity is the position. This gives a natural physical interpretation of process’ increments distribution what is, to our knowl- the BRW decomposition introduced in [27]. We identify edge, neglected by many authors. There is another ex- the three components of the BRW decomposition with ample of even more complicated process — the fractional mixed fractional motion, harmonizable and evanescent Lévy stable motion [25, 37] which has the memory prop- processes, respectively. The first process corresponds to erty and increments with infinite variance. In this case a dissipative part and two others to a conservative part the self-similarity index carries information on both, on of the dynamics given by the nonsingular flow represent- long-memory and increments distribution. Hence study- ingaLévy stable and self-similar process. A number of ing the process’ self-similarity one needs to have robust special examples is discussed in details in order to demon- statistical tools and clear algorithms to extract informa- strate that the proposed integral representation is user- tion on both of the factors. A simple hint is as follows: friendly and could provide new insights into the mech- if one wants to investigate the self-similarity property, anism underlying a range of natural phenomena. Fi- one needs to distinguish between the long-memory prop- nally, in Section V the obtained results are applied to erty and the process’ increments distribution properties. determine basic features of an empirical data series. We Otherwise a wrong conclusion can be drawn. In [26] we demonstrate this by studying four empirical time series provided an explicit algorithm distinguishing between the recorded from different physical systems: an ionic current origins of the self-similarity in the case of a given time se- flow through a single channel in a biological membrane, ries on the base of a simple simulation experiment (com- an energy of solar flares, a seismic electric signal recorded puter test). during seismic Earth activity, and FX rate daily returns.

A. Self-similar processes II. SELF-SIMILARITY OF FRACTIONAL LEVY´ STABLE MOTION As we mentioned above, the self-similar processes are the ones that are invariant under suitable translations of Over the past decade there has been much interest in time and scale. They are important in probability theory the asymptotic behavior of dynamical systems, in par- because of their connection to limit theorems and they ticular in detecting self-similar character of these sys- are of great interest in modeling heavy-tailed and long- tems and testing for the existence of so called ”long memory phenomena. In fact, Lamperti used the term memory” or ”long-range dependence”. It turns out that ”semi-stable” in order to underline that the role of self- the self-similar processes are very important mathemat- similar processes among stochastic processes is analogous ical objects which can be used to model many physical, to the role of stable distributions among all distributions. { } geophysical, hydrological, economical and biological phe- A process X(t) t≥0 is called self-similar [19] if for nomena (see [16–18, 22, 28–38] and references therein). some H>0,

The mathematical constructions were successfully used d H to model diffusion on fractals, currency and stock mar- X(at) = a X(t) for every a>0, (2) ket prices, ionic current flow through a single channel in d a biological membrane, turbulences, communication and where = denotes equality of all finite-dimensional distri- many others. Since the self-similarity property was ob- butions of the processes on the left and right. The process ∞ X(t) is also called an H-self-similar process and the pa- slowly that the sum k=−∞ r(k) diverges. We say that rameter H is called the self-similarity index or exponent. in this case the increment process exhibits long-memory If we interpret t as ”time” and X(t) as ”space” then (2) or ”long-range dependence” [30]. Moreover, formula (5) tells us that every change of time scale a>0 corresponds by the Wiener Tauberian theorem (see [39] Chapt. V.2) to a change of space scale aH . The bigger H,themore implies that the spectral density h(λ) of the stationary dramatic is the change of the space co-ordinate. process fGn has a pole at zero. A phenomenon often Notice that (2), indeed, means a ”scale-invariance” of referred to as ”1/f noise”. ∞ the finite-dimensional distributions of X(t). This prop- If 0 0, H Zα (t) t∈R is defined by the following integral represen- d tation (X(t + b) − X(b)) =(X(t) − X(0)) . (3)

∞ 1 1 H H− α H− α B. Fractional Brownian motion Zα (t)= (t − u)+ − (−u)+ dZα(u), (6) −∞

The fractional Brownian motion (fBm) {BH (t)}t≥0 has where Zα(u) is a symmetric Lévy α-stable motion the integral representation (Lsm) [25, 37]. The integral is well defined for 0 Gaussian process with such properties for lows from the above integral representation and the fact 0 stable process with stationary increments (this is true random variables. The situation is quite different when for 0 <α<1only). H =1 /2, namely the Yj’s are dependent and the time The increment process corresponding to the fractional series has the autocovariance function of the form Lévy stable process is called a fractional stable noise 2H−2 (fsn). By analogy with the case α =2,wesaythatfsn r(k) ∼ VarY1 H(2H − 1)k , as k →∞. (5) has the long-range dependence when H>1/α and the The autocovariance function r(k) tends to 0 as k →∞for negative dependence when H<1/α.IfH =1/α the in- all 0 Brownian noise is studied by virtue of the autocovari- to 1/2 for the surrogate data independently on the ance function. Since in the α-stable case the second mo- initial values. ment is infinite one has to use another measure of depen- • dence, e.g. the codifference τ(j) which equals the covari- If the self-similarity results only from the process’ ance when α = 2 [37]. For most, but not all, values of α increments infinite variance (e.g. Lévy stable mo- and H, τ decreases as jαH−α for large j. This is analo- tion) then the estimator values should be the same gous to the behaviour of the autocovariance function in for the original and surrogate data. the Gaussian case α = 2. Finally, we note that there is • The self-similarity resulting from both origins (e.g. no long-range dependence when 0 <α 1 because H is fractional Lévy stable motion) should be observed constrained to lie in the interval (0, 1). as a partial change in the estimators values. For simulations of the above self-similar processes we need specific computer generators. Two of such algo- In order to justify the above test the behavior of the rithms for generation of fractional Gaussian noise(fGn) estimator was investigated on simulated time series. The and fractional stable noise (fsn) are described in detail in calculations were performed for two cases: fractional [26] and [43]. Lévy stable motion for α =1.8 and the self-similarity index H taking values {0.6, 0.7, 0.8, 0.90} (Fig. 1 depicts sample paths of corresponding fractional stable noises for III. TESTING OF SELF-SIMILARITY H =0.6andH =0.9) and Lévy stable motion with the self-similarity index H taking values {0.6, 0.7, 0.8, 0.9}. In this section we use the absolute value (AV) method The index of stability α in the latter case ranges from for estimation of the self-similarity index H.The α =10/9toα =5/3. method is based on calculating mean value from the pro- We note that in order to simulate fractional Lévy sta- cess realizations and studying its scaling with a sample ble motion we directly applied its integral representa- length [31]. A time series of length N one divides into tion given by (18) for t =1/100, 2/100,...,9999/100, 100 subseries of length m and calculates the first absolute (10000 observations). The fractional Lévy stable motion moment was also considered and simulated in [43]. The AV estimator calculated for every given case of N/m 1 the fractional Lévy stable motion and Lévy motion is AV (m) = X(m)(k) −X , (8) N/m presented in Fig. 2. The AV estimator gives information k=1 on both, the memory and distribution of the investigated where X(m) is an m-th subseries and X is the overall process. The values of the estimator should form the line A = H −1. It is true in both studied cases, cf. circles and series mean. The obtained statistics scales with the window size and dashed lines in Fig. 2. The estimator values for the surrogate data obtained from fractional Lévy stable motion the absolute value exponent A equals HAV − 1, where and Lévy motion are markedly different. In the case of HAV is the self-similarity index the fractional Lévy stable motion values of the estimator AV (m) ∝ mHAV −1. (9) are close to 1/1.8 − 1 ∼−0.44, cf. plus signs and dotted line in Fig. 2. The values obtained for the Lévy stable Notice, that this estimator gives information on the self- motion do not change after shuffling and are pretty close similarity index. If the variance of the time series is infi- to the value 1 subtracted from the self-similarity index nite the estimator also works correctly, so it can be used value H, cf. plus signs and dashed line in Fig. 2. to investigate, for example, the Lévy stable motion. To investigate memory of a studied process we apply the estimator to an original data set obtained as a real- IV. RELATION TO THE BRW ization of the process given by (6) and to the surrogate DECOMPOSITION data. Surrogate data refers to data that preserve cer- tain linear statistic properties of the experimental time In this section we exploit the connection between the- series, without the deterministic component [44]. It is ory of self-similar Lévy stable processes and ergodic the- commonly used to determine the memory of a process ory of nonsingular flows. We use the integral representa- by means of the local dispersion and nonlinear predic- tion of an H-self-similar symmetric α-stable (SαS)pro- tion methods. The surrogate data can be obtained by cess {Xt}t>0 of the form several different ways [44, 45]. In this paper we obtain it by random shuffling of the original data positions. 2 According to [26] we have he following BMW com- Xt = ft(u)dZα(u),t>0, (10) puter test. S H 1/α • If the self-similarity results from the process mem- where the kernel ft(u)=t [at(u)f(φt(u))]mt(u) is ory only (e.g. fractional Brownian motion) then represented by means of the tools used in ergodic theory 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

−0.2 −0.2

−0.4 −0.4

−0.6 −0.6

−0.8 −0.8

−1 −1 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000

FIG. 1: A sample path of the fractional L´evy stable noise for (left panel) H =0.6andα =1/1.8, and (right panel) H =0.9 and α =1/1.8.

0 0

−0.05

−0.1 −0.1

−0.15

−0.2 −0.2

−0.25

−0.3 −0.3

Absolute value exponent A −0.35 Absolute value exponent A

−0.4 −0.4

−0.45

−0.5 −0.5 0.5 0.6 0.7 0.8 0.9 1 0.5 0.6 0.7 0.8 0.9 1 Index of self−similarity H Index of self−similarity H

FIG. 2: Values of the AV exponent A = HAV − 1 for the original time series (circle) and the surrogate data (plus sign) of (left panel) the fractional L´evy stable noise and (right panel) the L´evy stable noise.

[46]. Here {φt}t>0 is a nonsingular multiplicative flow It was demonstrated in [27] that every SαS self-similar on the phase space (S, µ), {at}t>0 is a cocycle for this process {Xt}t>0 admits a unique BRW decomposition flow taking values in {−1, 1}, mt = d(µ(φt))/dµ and f is into three independent parts α-integrable with respect to µ [27]. d (1) (2) (3) The stochastic process Xt defined in (10) can be inter- {Xt}t>0 = {Xt }t>0 + {Xt }t>0 + {Xt }t>0, (12) preted, similarly as in Section II, as a weighted average of the Lévy stable motion Zα(t) over the infinite past where the first process on the right-hand side is a mixed with the weight given by the kernel ft.Fortheexact fractional motion (mfm), the second is harmonizable, and definition we refer the interested reader to [25, 37]. Let the third one is an H-ss evanescent process. us point out that the stochastic integral (10) is equiva- The Hopf decomposition in ergodic theory of the phase lent to the diffusion without drift (i.e. b =0)andwith space S of the integral representation (10) into invariant diffusion coefficient (σ = ft), see (1). parts C and D, such that the flow φt is conservative The self-similarity property of the above integral with on C and dissipative on D, generates a decomposition parameter H, follows directly from 1/α-self-similarity of of {Xt}t>0 into two independent SαS H-ss processes C D C the process Zα(t) and the following property of the kernel {Xt }t>0 and {Xt }t>0.Theclass{Xt }t>0 generated by conservative flows consists of harmonizable processes H− 1 u (2) (3) fct(u)=c α ft . (11) { } { } c Xt t>0 and evanescent processes Xt t>0. The pro- D cess {Xt }t>0 is a mfm and one can choose a minimal (see Fig. 3, top left). D representation of {Xt }t>0 of the form (14) below. Fur- Model 3. Let 0 0 is a fractional motion (fm) if and only Put β = H − 1/α. Then a fractional SαS motion [49] if {φt}t>0 restricted to D is ergodic. This is a promising {Xt}t>0 is defined by the kernel link between the BRW decomposition of any SαS H-ss − β − − β process and the theory of dynamical systems in statisti- ft(s)=I[s<0][(t s) ( s) ] β cal physics. The ergodic theory of SαS stationary pro- + I[0 0, (13) t This is quite opposite to the relation between mixed and the usual moving averages. It is clear that a Lévy sta- where f is α-integrable with respect to the Lebesgue mea- ble process may have many integral representations with sure. A SαS process with such representation is called different kernels defined on various measure spaces. How- the fractional motion. A superposition of independent ever, we can identify one property, common to all such fm processes of type (13) is called the mixed fractional representations, which characterizes mfm’s. Let {Xt}t>0 motion and it has the form be a SαS H-ss process with an arbitrary representation H− 1 u (10). Then X is a mfm if and only if gt(w, u)=t α g w, ,t>0. (14) t ∞ −αH−1 α t |ft(s)| dt < ∞ µ − a.e. (19) We will give a few examples of fm and mfm processes. 0 We start from d-dimensional case. Model 1 ∈ α Rd Observe first that this condition is equivalent to .Forf L ( ), let ∞ −αHt α | t | ∞ − −∞ e fe (s) dt < µ a.e. By [47] and the Lam- H− d s d perti transformation this gives the result. ft(s)=t α f ,s∈ R ,t>0, (15) t The class generated by conservative flows consists of harmonizable processes and processes of a third kind Rd and M be a SαS random measure on with the (evanescent). An H-ss SαS process {Xt}t>0 is said Lebesgue control measure. It is easy to check that a to be harmonizable if its kernel satisfies the condition: { t}t>0 SαS process X with such representation is H- ft1t2 (s)f1(s)=ft1 (s)ft2 (s)fort1,t2 > 0, see [27]. ss. We will show that {Xt}t>0 is a mfm. Indeed, A stochastic process whose minimal representation d let W = Sd be the unit sphere in R equipped with (10) contains a conservative flow without fixed points is the uniform probability measure ν and let g(w, u)= called evanescent. This class is not well understood at d−1 1/α (cdu ) f(uw), (w, u) ∈ Sd × (0, ∞), where cd = present. The following result in [27] is useful to verify d/2 2π /Γ(d/2) is the surface area of Sd.Usingpolarcoor- whether or not a process is evanescent. Let {Xt}t>0 dinates, we get for every a1,...,an ∈ R, t1,...,tn > 0, be a SαS H-ss process with an arbitrary representation (10). Then {Xt}t>0 is evanescent if and only if α ∞ −αH−1 α { ∈ | t | ∞} { ∈ | ajftj (s)| ds µ s S : 0 t f (s) dt < =0andµ s Rd } S : ft1t2 (s)f1(s)=ft1 (s)ft2 (s)fort1,t2 > 0 =0. ∞ H− d uw Finally we give examples of the harmonizable and | α |α d−1 = cd ajtj f u duν(dw) evanescent processes. Sd 0 tj ∞ Model 4. 1 Let H− α u α | j | = a tj g w, duν(dw), (16) − 1 tj H+is exp(is) 1 −(H−1+ ) Sd 0 ft(s)=t |s| α , (20) is which proves the claim. (see Fig. 3, bottom left). It is easy to check that stochas- Comparing the kernel from the above example with the d −1 tic process with such kernel is harmonizable. general form (10) we get that S = R \{0}, φt(s)=t s, −d Model 5. Let f1(s)=f(s), and d(µ(φt))/dµ = t . The following H well-known H-ss processes are special cases of the above ft(s)=I[0 0 is defined by the (see Fig. 3, bottom right). Then X(t) defined by the kernel above kernel does not have a corresponding harmonizable nor mixed moving average component, so provides an ft(s)=log|t/s − 1| (17) example of an evanescent component. 8 1.5

4 1

2 0.5 0

−2 0 10 0 −4 1 0 5 2 −6 1 0 −10 3 2 −5 t 3 0 4 −5 s 5 t 4 s 10 5 5 −10

8 10

6 5 4

2 0

0 −5 −2 10 2 −4 −10 0 5 0 1 1 2 2 0 4 0 3 −5 s 6 −1 s t t 4 8 5 −10 10 −2

FIG. 3: The kernel of the integral representation of (top left) a log-fractional SαS motion, (top right) a fractional SαS motion for H − 1/α =0.1; (bottom left) the real part of the kernel of the integral representation of the complex-valued SαS harmonizable process for H =0.8andH − 1/α =0.1, (bottom right) the kernel of the integral representation of the evanescent process.

V. EMPIRICAL EVIDENCE satellite. The captured energy was transmitted by X-rays emitted during blasts on a solar surface from the 1st of January 1997 to the 31st of August 2002 [53, 54]. The The above formalism can be easily applied to deter- time of the blast was recorded with an accuracy of 1min mine basic features of an empirical data series. Below we and the relative error of the energy measurement reads demonstrate this by studying four empirical time series −2 10 . The total analyzed sample consists of moments recorded from different physical systems: an ionic current and energy of 13 015 flares. A part of the time series is flow through a single channel in a biological membrane, presented in Fig. 4. an energy of solar flares, a seismic electric signal recorded The seismic signal is a record of an electrical field during seismic Earth activity, and FX rate daily returns. of the Earth surface called SES (Seismic Electric Sig- The ionic current was recorded from cell attached nal) [55]. The data were recorded on the 18th of April patches of adult locust (Schistocerca gregaria) extensior 1995 in Grevena-Kozani in Greece at the sampling fre- tibiae muscle fibres [32, 51, 52]. The potassium current quency 1Hz. The time series consists of 2 201 obser- (see Fig. 4) through a high conductance locust potas- vations covering about 37min. The measurement error sium channel (BK channel) was obtained by the patch reads 10nV/km. The data are especially interesting since clamp technique with sampling frequency 10kHz and at the SES activity usually precedes an earthquake [55]. a voltage of 100mV. The sample presents a time series, The time series is presented in Fig. 4. consisting of 250 000 points and covering, therefore, 25s The USD/CHF FX rate was recorded from 1985-05- of recording. The error of measurements of ionic current 20 to 1991-04-12 with daily resolution. The data pri- is equal to 1pA. marily comes from a set of financial data released by The solar flares energy data were recorded by UHURU Olsen & Associates for the Second International Confer- −5 16 x 10 3

2.5 12

2 10 ] 2 8 1.5 I [pA] E [J/m

6 1

0.5 2

0 0 3200 3210 3220 3230 3240 3250 3260 3270 3280 3290 3300 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 5 t [ms] t [min] x 10 3 0.06

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E [mV/km] −0.04

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Daily change of USD/CHF −0.06

0 −0.08

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−1 −0.12 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 0 500 1000 1500 t [s] Number of observation (day)

FIG. 4: A part of patch clamp recording of the single BK channel ionic current (top left); a part of solar ﬂares energy time series (top right); Seismic electric signal recorded on the 18th of April 1995 in Grevena-Kozani in Greece (bottom left); and the USD/CHF FX rate daily changes from 1985-05-20 to 1991-04-12 (bottom right).

ence on High Frequency Data in Finance, Zürich, April Data set Hurst DFA HAV 1-3, 1998. These data sets are quotations of foreign cur- Original time series rencies and metals available from international vendors Ionic current 0.84 ± 0.08 0.89 ± 0.07 0.88 ± 0.08 like Reuters, Knight-Ridder and Telerate. The analyzed Solar flares 0.69 ± 0.05 0.68 ± 0.07 0.86 ± 0.06 time series consists of 1 480 observations and the daily SES 0.92 ± 0.06 0.94 ± 0.10 0.89 ± 0.07 FX rate changes are presented in Fig. 4. . ± . . ± . . ± . The procedure presented in Section III (see also [26]) FX rate 0 62 0 10 0 51 0 05 0 57 0 08 was applied to the above time series. The obtained values Surrogate data of the parameters are listed in Table I. Ionic current 0.54 ± 0.05 0.50 ± 0.04 0.48 ± 0.05 Comparing the values of the different estimators for Solar flares 0.52 ± 0.04 0.48 ± 0.05 0.69 ± 0.06 the original data series and for the surrogate data one SES 0.56 ± 0.07 0.48 ± 0.07 0.49 ± 0.07 can estimate the components of the self-similarity index FX rate 0.60 ± 0.10 0.50 ± 0.05 0.54 ± 0.07 corresponding to the memory of the time series (d)and to the tails properties of the time series values distribu- TABLE I: Values of the Hurst and DFA exponents and the tion (α). The values of the components are presented self-similarity index HAV for the original time series and the in Table II. Let us observe that only for the solar flares surrogate data for the four different data sets. data the self-similarity results from both origins suggest- ing that proper model should be based on the fractional Lévy stable motion with H ∼ 0.86 and α ∼ 1.45, see 2 BMW computer test in Section III. position of the self-similarity index into the two compo- It is clearly seen that the procedure provides a decom- nents representing memory and the tails of the process, Data set d α Modeling process language of their deterministic kernels ft. Ionic current 0.36 ± 0.08 2.00 ± 0.22 fBm On the level of observed physical time series this leads Solar flares 0.19 ± 0.07 1.45 ± 0.14 fLsm to a challenging open problem how to see the ”shape” SES 0.43 ± 0.09 2.00 ± 0.34 fBm of the self-similarity from the data? To be more precise, FX rate 0.00 ± 0.05 1.80 ± 0.20 Lsm or Bm how to determine the type of the model from the given time series data. Details of these ideas are, however, the TABLE II: Values of the self-similarity index components subject of the current work and are beyond the scope (d, α), and the type of modeling processes for the four an- of this paper. If one knows how to recognize the form alyzed time series. of the kernel ft from the data (by a shape recognition procedure), then from our result it follows that this time series corresponds to a specific part of the BRW decomposition. For instance, mfm is identified if and only if the see formula (7). Nevertheless, it is not clear how to ap- kernel ft satisfies condition (19). ply the algorithm for the integral kernel (11) estimation. The main findings is that always a deterministic kernel The difficulty mainly arises from a lack of any additional determines the index of self-similarity H via formula (11). limitations for the class of functions to which can belong This provides a catalog of all possible self-similar models the kernel. So, one can not reduce the problem of finding for anomalous diffusion driven by Lévy stable noise. We the kernel to a parameter or parameters estimation. In- also study an explicit algorithm distinguishing between stead, one has to deal with a non-parametric estimation the origins of the self-similarity of a given time series on problem. However, having the d and α values one can thebaseoftheBMW2 computer test [26]. It turns out already correctly classify the studied time series, identify that it suffices to compare the behavior of AV estimator the process Zα(t) in the integral (10) and reduce the class of H for the original time series and the surrogate data. of possible kernels. More studies are needed to elucidate We provide here a theoretical justification of this algo- the recognition of the kernel. rithm for self-similar models using general formula (10). This is illustrated in Section V for four sets of empirical data recorded from different physical systems. VI. CONCLUDING REMARKS

In this paper we demonstrate that all self-similar mod- VII. ACKNOWLEDGMENTS els driven by Lévy stable noise are completely described by formula (10) in the form of a stochastic integral with respect to the Lévy symmetric stable motion and a de- We are grateful to Prof. P. N. R. Usherwood and terministic kernel ft. The classical Hopf decomposi- Dr. I. Mellor from the University of Nottingham (UK), tion in ergodic theory implies decomposition (12) into and to Dr. Z. Siwy from the Silesian University of Tech- three independent components: mfm, harmonizable and nology (Poland) for providing us with the experimen- evanescent. The first component corresponds to conser- tal data of ion current through high conductance locust vative dynamics, when two others to dissipative dynam- potassium channel. We are grateful to Prof. P. Varotsos ics. Next, we provide the five typical H-self-similar mod- from the University of Athens (Greece) for providing us els and characterize the three possible components in the with the SES data.

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