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Physics Letters A 372 (2008) 4768–4774
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Physics Letters A
www.elsevier.com/locate/pla
Permutation entropy of fractional Brownian motion and fractional Gaussian noise
L. Zunino a,b,c,∗, D.G. Pérez d, M.T. Martín e, M. Garavaglia a,c, A. Plastino e, O.A. Rosso f,g
a Centro de Investigaciones Ópticas, C.C. 124 Correo Central, 1900 La Plata, Argentinab Departamento de Ciencias Básicas, Facultad de Ingeniería, Universidad Nacional de La Plata (UNLP), 1900 La Plata, Argentinac Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, 1900 La Plata, Argentinad Instituto de Física, Pontificia Universidad Católica de Valparaíso (PUCV), 23-40025 Valparaíso, Chilee Instituto de Física (IFLP), Facultad de Ciencias Exactas, Universidad Nacional de La Plata and Argentina’s National Council (CCT-CONICET), C.C. 727, 1900 La Plata, Argentinaf Centre for Bioinformatics, Biomarker Discovery and Information-Based Medicine, School of Electrical Engineering and Computer Science, The University of Newcastle,University Drive, Callaghan NSW 2308, Australiag Chaos & Biology Group, Instituto de Cálculo, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón II, Ciudad Universitaria,1428 Ciudad de Buenos Aires, Argentina
a r t i c l e i n f o a b s t r a c t
Article history:Received 24 September 2007Received in revised form 14 April 2008Accepted 6 May 2008Available online 17 May 2008Communicated by A.R. Bishop
PACS:02.50.-r02.50.Ey05.45.Tp89.70.Cf
Keywords:Permutation entropyFractional Brownian motionFractional Gaussian noise
We have worked out theoretical curves for the permutation entropy of the fractional Brownian motionand fractional Gaussian noise by using the Bandt and Shiha [C. Bandt, F. Shiha, J. Time Ser. Anal. 28 (2007)646] theoretical predictions for their corresponding relative frequencies. Comparisons with numericalsimulations show an excellent agreement. Furthermore, the entropy-gap in the transition between theseprocesses, observed previously via numerical results, has been here theoretically validated. Also, we haveanalyzed the behaviour of the permutation entropy of the fractional Gaussian noise for different timedelays.
© 2008 Elsevier B.V. All rights reserved.
1. Introduction
Entropic studies almost always assume that the underlyingprobability distribution is given. This is not at all the case if one isdealing with an input signal regarded as a time series. Part of theconcomitant analysis involves extracting the probability distribu-tion from the data and, rarely, a univocal procedure imposes itself.One recent and successful method is that introduced by Bandt andPompe [1]. The Bandt and Pompe method (BPM) for evaluatingthe probability distribution is based on the details of the attrac-tor reconstruction procedure. It is the only one among those inpopular use that takes into account the temporal structure of thetime series generated by the physical process under study. A no-table result from the Bandt and Pompe approach is a notorious
* Corresponding author at: Centro de Investigaciones Ópticas, C.C. 124 CorreoCentral, 1900 La Plata, Argentina.
E-mail addresses: [email protected] (L. Zunino), [email protected](D.G. Pérez), [email protected] (M.T. Martín),[email protected] (M. Garavaglia), [email protected](A. Plastino), [email protected] (O.A. Rosso).
0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2008.05.026
improvement in the performance of the information quantifiersobtained using the probability distribution generated by their al-gorithm [2–8]. Of course, one must assume with the BPM that thesystem fulfills a very weak stationary condition and that enoughdata are available for a correct attractor reconstruction. The per-mutation entropy is just the celebrated Shannon entropic measureevaluated using the BPM to extract the associated probability dis-tribution.
We are interested in the characterization of stochastic processesthrough this quantifier. In particular, we have chosen the fractionalBrownian motion and its noise, the fractional Gaussian noise, forthe analysis. The former is a ubiquitous non-stationary model formany physical phenomena which have empirical spectra of power-law type, 1/ f α , with 1 < α < 3. Thus, the characterization of theseprocesses has become of interest in different and heterogeneousscientific fields, like physics, biology, finance, telecommunicationsand music [9–12]. It should be stressed that both processes, fBmand fGn, were jointly introduced in the seminal work of Mandel-brot and Van Ness published in 1968 [13]. Moreover, many authorshave made use of the physical connection between fBm and fGnfor modelling and synthesis purposes [14–17].
L. Zunino et al. / Physics Letters A 372 (2008) 4768–4774 4769
In a previous effort [18], the normalized permutation entropy ofthe fractional Gaussian noise and fractional Brownian motion wasnumerically computed. A clear entropy-gap was observed in thetransition between these two stochastic processes, that does notdepend upon neither the length of the associated time series northe embedding dimension. Curiously enough, this is a new result.Previous approaches that employ probability distributions based ona wavelet description fail to detect such gap [19].
In this Letter we have worked out theoretical curves for theabove-mentioned normalized permutation entropy of the fractionalGaussian noise and fractional Brownian motion. To such an endwe have used theoretical results published recently by Bandt andShiha [20]. This allows the previously observed entropy-gap to benow conclusively classified as a real phenomenon, not a numericalartifact. Also, we have analyzed the behaviour of the normalizedpermutation entropy of the fractional Gaussian noise for differenttime delays. Finally, the curves we worked out using the Bandt andShiha results were compared with those obtained from numericalsimulations of the two stochastic processes under analysis.
The reminder of the Letter is organized as follows. In Section 2we describe the Bandt and Pompe probability distribution and itsassociated permutation entropy. In Section 3 we give a brief reviewof the two stochastic processes under analysis: the fractional Gaus-sian noise and fractional Brownian motion. The theoretical curvesand the comparison with their numerical simulations counterpartsare presented in Section 4. Discussions and conclusions are thesubject of the last section. Finally, in Appendix A we give somedetails concerning the Bandt and Shiha theoretical results that weuse throughout the Letter.
2. The Bandt and Pompe approach
Given a time series {xt : t = 1, . . . , M}, an embedding dimensionD > 1, and a time delay τ , consider the ordinal patterns of orderD [1,2,21] generated by
s �→ (xs−(D−1)τ , xs−(D−2)τ , . . . , xs−τ , xs). (1)
To each time s we are assigning a D-dimensional vector that re-sults from the evaluation of the time series at times s, s−τ , . . . , s−(D − 1)τ . Clearly, the greater the D value, the more informationabout the past is incorporated into the ensuing vectors. By theordinal pattern of order D related to the time s we mean the per-mutation π = (r0, r1, . . . , rD−1) of (0,1, . . . , D − 1) defined by
xs−rD−1τ � xs−rD−2τ � · · · � xs−r1τ � xs−r0τ . (2)
In order to get a unique result we consider that ri < ri−1 if xs−riτ =xs−ri−1τ . Thus, for all the D! possible permutations πi of order D ,the probability distribution P = {p(πi), i = 1, . . . , D!} is given bythe relative frequency
p(πi) = �{s | s � 1 + (D − 1)τ , s has ordinal pattern πi}M − (D − 1)τ
, (3)
where � is the cardinality of the set—roughly speaking, the num-ber of elements in it. To determine p(πi) exactly an infinite timeseries should be considered, taking M → ∞ in the above formula.This limit exists with probability 1 when the underlying stochasticprocess fulfills a very weak stationarity condition: for k � D , theprobability for xt < xt+k should not depend on t [1].
The advantages of the BPM reside in (a) its simplicity, (b) its ro-bustness, and (c) its invariance with respect to nonlinear monoto-nous transformations. Also, this method provides an extremely fastcomputational algorithm. It can be applied to any type of timeseries (regular, chaotic, noisy, or experimental) [1]. Remark thatfor the applicability of this approach we need not to assume thatthe time series under analysis is representative of a low dimen-sional dynamical system. Of course, the embedding dimension D
plays an important role for the evaluation of the appropriate prob-ability distribution, since D determines the number of accessiblestates, D!, and tells us about the necessary length M of the timeseries needed in order to work with a reliable statistics. In partic-ular, Bandt and Pompe suggest for practical purposes to work with3 � D � 7. Concerning this last point in all calculations reportedhere the condition M � D! is satisfied [6].
The normalized permutation entropy is just the normalizedShannon entropy associated to the probability distribution P ={p(πi), i = 1, . . . , D!}
HS [P ] = S[P ]/Smax =[−
D!∑i=1
p(πi) ln(
p(πi))]
/Smax, (4)
where Smax = ln D!, (0 �HS � 1)—S stands for Shannon entropy.
3. Fractional Brownian motion and fractional Gaussian noise
Fractional Brownian motion (fBm) is the only family of pro-cesses which is Gaussian, self-similar,1 and endowed with station-ary increments—see Ref. [19] and references therein. The normal-ized family of these Gaussian processes, {B H (t), t > 0}, is the onewith B H (0) = 0 almost surely, i.e., with probability 1, E[B H (t)] = 0(zero mean), and covariance given by
E[
B H (t1)B H (t2)] = 1
2
(t2H
1 + t2H2 − |t1 − t2|2H)
, (5)
for t1, t2 ∈ R. Here E[·] refers to the average computed witha Gaussian probability density. The power exponent 0 < H < 1is commonly known as the Hurst parameter or Hurst exponent.These processes exhibit memory for any Hurst parameter exceptfor H = 1/2 as one realizes from Eq. (5). The H = 1/2-case cor-responds to classical Brownian motion and successive motion in-crements are as likely to have the same sign as the opposite,there is no correlation among them. Thus, Hurst’s parameter de-fines two distinct regions in the interval (0,1). When H > 1/2,consecutive increments tend to have the same sign so that theseprocesses are persistent. For H < 1/2, on the other hand, consecu-tive increments are more likely to have opposite signs, thus theseprocesses are anti-persistent. Fractional Brownian motions are con-tinuous but non-differentiable processes (in the classical sense). Asa non-stationary process, they do not possess a spectrum definedin the usual sense; however, it is possible to define a generalizedpower spectrum of the form
ΦB H ( f ) ∝ 1
| f |α , (6)
with the exponent α = 2H + 1, 1 < α < 3.The fractional Gaussian noise is the process {W H (t), t > 0} ob-
tained from the fBm increments (for discrete time), i.e.,
W H (t) = B H (t + 1) − B H (t). (7)
This is a stationary Gaussian process with mean zero and covari-ance given by
ρ(k) = E[W H (t)W H (t + k)
]= 1
2
[(k + 1)2H − 2k2H + |k − 1|2H ]
, k > 0. (8)
1 Self-similar stochastic processes are invariant in distribution under suitable scal-ing of time and space. Formally, a (stochastic) process X(t) is self-similar withindex H if, for any c > 0,
X(t)d= cH X
(c−1t
),
whered= is equality in distribution.
4770 L. Zunino et al. / Physics Letters A 372 (2008) 4768–4774
Fig. 1. Theoretical curves for the normalized permutation entropy HS of the frac-tional Gaussian noise with embedding dimension D = 3 (top) and D = 4 (bottom),and time delay τ = 1 as a function of the parameter α. The mean and standarddeviation values obtained via numerical simulation are included for comparisonpurpose (red lines). (For interpretation of the references to colour in this figurelegend, the reader is referred to the web version of this Letter.)
Note that for H = 1/2 all correlations at nonzero lags vanish and{W 1/2(t), t > 0} thus represents just white noise. The asymptoticbehavior of ρ(k) follows by Taylor expansion (see Ref. [22], pp.50–53):
ρ(k)
H(2H − 1)k2H−2→ 1, (9)
as k → ∞. Thus, for 1/2 < H < 1 the ensuing correlations are notsummable,
∑∞k=−∞ ρ(k) = ∞. This property is often referred to as
long-range dependence (long memory). On the other hand, for 0 <
H < 1/2 the correlations are summable,∑∞
k=−∞ ρ(k) = 0, and theprocess has short-range dependence (short memory). The powerspectrum associated to the fractional Gaussian noise is also givenby Eq. (6) but with α = 2H − 1, −1 < α < 1.
Within the scaling processes, the majority of workers havemostly selected two types of them, namely, the ones that are self-similar with stationary increments and the long-range dependentprocesses. Because of self-similarity, the former cannot be station-ary. The latter, on the other hand, are associated with stationary
Fig. 2. Theoretical curves for the normalized permutation entropy HS of the frac-tional Brownian motion with embedding dimension D = 3 (top) and D = 4 (bottom)as a function of the parameter α. The mean and standard deviation values obtainedvia numerical simulation are included for comparison purpose (red lines). (For in-terpretation of the references to colour in this figure legend, the reader is referredto the web version of this Letter.)
processes. There is a close relationship between both types. In-deed, the increments of any finite-variance, self-similar processwith stationary increments have long-range dependence, as long as1/2 < H < 1. In particular, fractional Gaussian noise (fGn), whichis the increment process of the fBm, has long-range dependencewhen 1/2 < H < 1 [14].
4. Theoretical results
The Bandt and Pompe probability distribution associated to dif-ferent stochastic processes has been recently analyzed by Bandtand Shiha [20]. They provide us with theoretical expressions forthe relative frequencies p(πi) for stationary Gaussian process andfractional Brownian motion with arbitrary time delay τ , and D = 3and D = 4.2 For further details about these results see Appendix A.
2 Theoretical relative frequencies for patterns of order D � 5 require the general-ization of Lemma 1 of Ref. [20]. See also Ref. [23].
L. Zunino et al. / Physics Letters A 372 (2008) 4768–4774 4771
Fig. 3. Top: Theoretical curves for the normalized permutation entropy HS of thefractional Gaussian noise (−1 < α < 1) and fractional Brownian motion (1 < α < 3)with embedding dimension D = 3 and time delay τ = 1 as a function of the pa-rameter α. Bottom: Enlargement near the transition between the two processes(α → 1). The numerical simulation results are included for comparison purpose (redlines). (For interpretation of the references to colour in this figure legend, the readeris referred to the web version of this Letter.)
We compute now the normalized permutation entropy by usingthese Bandt and Shiha results. Figs. 1 and 2 depict the normalizedpermutation entropy of the fractional Gaussian noise and fractionalBrownian motion, respectively, for D = 3 and D = 4, and time de-lay τ = 1. It should be remarked that for the fBm the results areindependent of the time delay τ because of the self-similarityproperty of this process. Observe that the theoretical curves aresimilar for the two embedding dimensions, D = 3 (top) and D = 4(bottom) in both cases, fGn (Fig. 1) and fBm (Fig. 2).
In order to validate the analytical curves we compare themwith numerical simulations of fBm and fGn time series. To this endwe adopt the Davies–Harte algorithm [24], as recently improvedby Wood and Chan [25], which is both exact and fast. Actually,this method simulates fractional Gaussian noise and get samplesof fractional Brownian motion by evaluating cumulated sums ofthe sequential data points obtained and by setting B H (0) = 0. Foreach value of α = 1 within the interval [−0.8,2.8] with step 0.2
Fig. 4. Top: Theoretical curves for the normalized permutation entropy HS of thefractional Gaussian noise (−1 < α < 1) and fractional Brownian motion (1 < α < 3)with embedding dimension D = 4 and time delay τ = 1 as a function of the pa-rameter α. Bottom: Enlargement near the transition between the two processes(α → 1). The numerical simulation results are included for comparison purpose (redlines). (For interpretation of the references to colour in this figure legend, the readeris referred to the web version of this Letter.)
we simulated 10 realizations with M = 215 data points in eachtime series. All of them start with a different initial condition.Note that for α = 1 (the limit between processes and noises) wedo not have a numerical simulation. The mean values plus thecorresponding standard deviations of the normalized permutationentropy are plotted as functions of the parameter α. We find thatthe agreement between theoretical curves and numerical simula-tions is excellent.
In Figs. 3 and 4 the entropy-gap between fGn and fBm forD = 3 and D = 4, respectively, becomes notorious. Thus, the nu-merical simulation results found in a previous paper [18] are nowsupported by theoretical curves that we got by applying Bandt andShiha’s theoretical results.
Separate consideration should be given to a fBm with H → 0(α → 1). Here the normalized permutation entropy is maximal(HS → 1), mimicking the result obtained for a white noise (α = 0).The latter case agrees with intuitive ideas concerning the notion
4772 L. Zunino et al. / Physics Letters A 372 (2008) 4768–4774
Fig. 5. Top: Theoretical curves for the normalized permutation entropy HS of the fractional Gaussian noise (−1 < α < 1) with embedding dimension D = 4 and differenttime delays as a function of the parameter α with their corresponding numerical simulations (red lines). (For interpretation of the references to colour in this figure legend,the reader is referred to the web version of this Letter.)
of noise—for a totally random process, or ideal noise, we haveHS = 1. The new result here is that a fBm with H → 0 behaves in
such a manner, with a permutation entropy HS → 1 if the Hurstparameter approaches zero. We do expect, as stated, such a be-
L. Zunino et al. / Physics Letters A 372 (2008) 4768–4774 4773
Fig. 6. Theoretical curves for the normalized permuation entropy HS of the frac-tional Gaussian noise with embedding dimension D = 4 and different time delay τas a function of the parameter α.
havior for white noise but not for an fBm with small enough H .We conjecture that the entropy-gap originates in this anomalousfBm-behavior.
Fractional Gaussian noise results depend on the time delay con-sidered. In order to ascertain the effect that this parameter has onthe theoretical curves, Fig. 5 displays the normalized permutationentropy for the fGn with D = 4 and different time delays τ , to-gether with their corresponding numerical simulation results. Notethat the agreement with the simulation gets worse as τ increases.It should be stressed that the theoretical results are valid for anyτ > 0 because we are considering continuous stochastic processes.Obviously, numerical simulation comparisons are only possible forτ ∈ N because the inherent discrete character of these numericalsimulations.
We can recognize two different behaviors. For −1 < α < 0(short memory noises) HS → 1 for τ � 2—see Fig. 6, top. However,for 1 < τ < 2 the normalized permutation entropy shows a differ-ent behaviour—see Fig. 6, bottom. The curves exhibit a minimumfor a particular α value, that increases as τ grows. As a conse-quence, an implicit relationship between α and the time delay τgets established. Given τ , there exists a value of α for which the
permutation entropy is minimal. On the other hand, for 0 < α < 1(long memory noises) the curves obtained are similar, indepen-dently of the τ value, and always decrease with α. Note that HS
values increase as the time delay τ grows.
5. Conclusions
We have worked out theoretical curves for the normalized per-mutation entropy of two well-known and widely used stochasticprocesses, the fractional Gaussian noise and fractional Brownianmotion. The entropy-gap in the transition between these pro-cesses, previously observed on the basis of numerical simulations,has been theoretically validated. This gap was not observed in aprevious approach that employed a wavelet probability distribu-tions [19]. Thus, we conclude that the Bandt and Pompe methodfor evaluating the probability distribution is the source of thisentropy discontinuity. For fGn the permutation entropy curves de-pend of the time delay considered. Moreover, clear different be-haviors are recognized for short and long memory noises. The fBmresults are, however, independent of the time delay. Since it is aself-similar process the relative frequencies (Eq. (3)) do not de-pend on the value of the time delay. We propose to consider thisindependence as a test for self-similarity. This conclusion requiressomewhat more elaboration and will be the subject of future work.
Acknowledgements
This work was partially supported by Consejo Nacional deInvestigaciones Científicas y Técnicas (CONICET) (PIP 0029/98;5687/05 and 6036/05), Argentina, Comisión Nacional de Inves-tigación Científica y Tecnológica (CONICYT) (FONDECYT projectNo. 11060512), Chile, and Pontificia Universidad Católica de Val-paraíso (PUCV) (Project No. 123.788/2007), Chile. O.A.R. gratefullyacknowledges support from Australian Research Council (ARC) Cen-tre of Excellence in Bioinformatics, Australia.
Appendix A
For a Gaussian process with stationary increments and embed-ding dimension D = 3, Bandt and Shiha (see Ref. [20], pp. 656–659) found that p(π123) = p(π321) = u/2 and p(π132) = p(π213) =p(π231) = p(π312) = (1 − u)/4. In the case of a stationary Gaussianprocess with covariance ρ(τ ) they shown that p(π123) is given by
p(π123)(τ ) = 1
πarcsin
(1
π
√1 − ρ(2τ )
1 − ρ(τ )
). (A.1)
Thus, for fractional Gaussian noise we should replace Eq. (8) inthe last expression. Since the fractional Brownian motion is a self-similar process, the relative frequencies p(πi) do not depend onthe value of τ . Moreover, Bandt and Shiha shown that
p(π123)(τ ) = 1
πarcsin
(2H−1) for all τ , (A.2)
with H the Hurst parameter.For embedding dimension D = 4 they also found the relative
frequencies for these processes. For a stationary Gaussian processand arbitrary τ > 0,
p(π1234)(τ ) = p(π4321)(τ ) = 1
8+ 1
4π(arcsinα1 + 2 arcsinα2),
p(π3142)(τ ) = p(π2413)(τ ) = 1
8+ 1
4π(2 arcsinα3 + arcsinα4),
p(π4231)(τ ) = p(π1324)(τ ) = 1
8+ 1
4π(arcsinα4 − 2 arcsinα5),
p(π2143)(τ ) = p(π3412)(τ ) = 1 + 1(2 arcsinα6 + arcsinα1),
8 4π
4774 L. Zunino et al. / Physics Letters A 372 (2008) 4768–4774
p(π1243)(τ ) = p(π2134)(τ )
= p(π3421)(τ )
= p(π4312)(τ )
= 1
8+ 1
4π(arcsinα7 − arcsinα1 − arcsinα5),
p(π1423)(τ ) = p(π4132)(τ )
= p(π3241)(τ )
= p(π2314)(τ )
= 1
8+ 1
4π(arcsinα7 − arcsinα4 − arcsinα5),
p(π3124)(τ ) = p(π1342)(τ )
= p(π4213)(τ )
= p(π2431)(τ )
= 1
8+ 1
4π(arcsinα3 + arcsinα8 − arcsinα5),
p(π1432)(τ ) = p(π4123)(τ )
= p(π2341)(τ )
= p(π3214)(τ )
= 1
8+ 1
4π(arcsinα6 − arcsinα8 + arcsinα2), (A.3)
where
α1 = 2ρ(2τ ) − ρ(τ ) − ρ(3τ )
2[1 − ρ(τ )] , α2 = 2ρ(τ ) − ρ(2τ ) − 1
2[1 − ρ(τ )] ,
α3 = ρ(2τ ) + ρ(3τ ) − ρ(τ ) − 1
2√[1 − ρ(2τ )][1 − ρ(3τ )] , α4 = ρ(τ ) − ρ(3τ )
2[1 − ρ(2τ )] ,
α5 = 1
2
√1 − ρ(2τ )
1 − ρ(τ ), α6 = ρ(τ ) + ρ(3τ ) − ρ(2τ ) − 1
2√[1 − ρ(τ )][1 − ρ(3τ )] ,
α7 = ρ(τ ) + ρ(2τ ) − ρ(3τ ) − 1
2√[1 − ρ(τ )][1 − ρ(2τ )] ,
α8 = ρ(τ ) − ρ(2τ )√[1 − ρ(τ )][1 − ρ(3τ )] . (A.4)
For fractional Brownian motion, Bandt and Shiha obtained thesame formula but with
α1 = 1 + 32H − 22H+1
2, α2 = 22H−1 − 1,
α3 = 1 − 32H − 22H
2 · 6H, α4 = 32H − 1
22H+1,
α5 = 2H−1, α6 = 22H − 32H − 1
2 · 3H,
α7 = 32H − 22H − 1
2H+1, α8 = 22H − 1
3H. (A.5)
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