A novel entropy recurrence quantification analysis
AA novel entropy recurrence quantification analysis
G. Corso, T. L. Prado,
2, 3
G. Z. dos S. Lima,
1, 4 and S. R. Lopes ∗ Departamento de Biof´ısica e Farmacologia, Universidade Federal do Rio Grande do Norte, Natal, RN, Brazil Laborat´orio Associado de Computao e Matemtica Aplicada,Instituto Nacional de Pesquisas Espaciais, S˜ao Jos´e dos Campos, SP, Brazil Instituto de Engenharia, Ciˆencia e Tecnologia, UniversidadeFederal dos Vales do Jequitinhonha e Mucuri, Jana´uba, MG, Brazil. Escola de Ciˆencias e Tecnologia, Universidade Federal do Rio Grande do Norte, Natal, RN, Brazil. Departamento de F´ısica, Universidade Federal do Paran´a, Curitiba, PR, Brazil.
The growing study of time series, especially those related to nonlinear systems, has challenged themethodologies to characterize and classify dynamical structures of a signal. Here we conceive a newdiagnostic tool for time series based on the concept of information entropy, in which the probabilitiesare associated to microstates defined from the recurrence phase space. Recurrence properties canproperly be studied using recurrence plots, a methodology based on binary matrices where trajec-tories in phase space of dynamical systems are evaluated against other embedded trajectory. Ournovel entropy methodology has several advantages compared to the traditional recurrence entropydefined in the literature, namely, the correct evaluation of the chaoticity level of the signal, the weakdependence on parameters, correct evaluation of periodic time series properties and more sensitivityto noise level of time series. Furthermore, the new entropy quantifier developed in this manuscriptalso fixes inconsistent results of the traditional recurrence entropy concept, reproducing classicalresults with novel insights.
PACS numbers: 05.45, 05.10.-a,07.05.Rm,05.45.Tp
I. INTRODUCTION
The development of nonlinear dynamics has influencedso many areas of science, from physics, chemistry andengineering to life science and ecology. From economicsto linguistics, and more recently neuroscience [1–6]. Inlast decades, the increasing mathematical knowledge ofthe complex structures of nonlinear systems has providedsuccessful tools to the understanding of irregular spaceand temporal behaviors displayed by collected data inall applied sciences. Time series analysis has turned tobe a key issue providing the most direct link betweennonlinear dynamics and the real world [7].In many cases, time series data are not extracted fromlinear systems. In these cases, linear data analyses bringunsatisfied results since linear time series analysis meth-ods are characterized by averaged quantities like meanvalues or variances, as well as (auto)-correlations, notcapturing the nonlinear aspects of the signal [7].On the other hand, nonlinear analysis tries to extractinformation from the underlying dynamics of the data. Inthis way, nonlinear techniques supply new tools for datadiagnostics using a whole set of quantities, such as di-vergence rates, predictability, scaling exponents and en-tropies in symbolic representation. All those methods arebased on more general phase space properties [8] such asrecurrence properties and others.Nowadays, nonlinear time series are a central issue inscience. An important characteristics of such time se-ries is the presence of natural and non trivial period- ∗ lopes@fisica.ufpr.br icities, characterized by repeating segments of the sig-nal in a rather complex way [9]. In the last decadesseveral sophisticated mathematical tools have been de-veloped to characterize these periodicities. We cite, forinstance, the development of wavelet analysis that haveenable scientists to explore in detail the time-frequencystructure of signals [10]. Another technique designed toanalyze the statistical periodicities in several scales wasthe detrended fluctuation analysis, a contemporary de-velopment of Hurst analysis [11–14].We explore in this work a new quantifier of nonlin-ear analysis of time series based on properties of phasespace recurrences. The modern concept of recurrencedates to Henri Poincar´e work [15] and it is a fundamentalattribute of dynamical systems. A modern visualizationmethod known as recurrence plot ( RP ) was introducedin [16], and is constructed from the recurrence matrix R ij defined as: R ij ( (cid:15) ) = Θ( (cid:15) − || x i − x j || ) , x i ∈ R , i, j = 1 , , · · · , M, (1)where x i and x j represent the dynamical state at time i and j , Θ is the Heaviside function, M is length of theanalyzed time series and (cid:15) is the threshold or vicinityparameter, consisting of a maximum distance betweentwo points in a trajectory such that both points can beconsidered recurrent to each other.In this way, the RP is a symmetric matrix of “ones”and “zeros” where an one (zero) intends for recurrent(non recurrent) points in phase space. The recurrenceanalysis technique is conceptually simpler than spectrallinear and nonlinear analysis like Fourier or Wavelets andnumerically easier to be performed since it does not haveto decompose the signal within a basis [10]. Instead, a r X i v : . [ s t a t . O T ] J u l the RP is computed using repetitions (or recurrences) ofsegments of the signal which produce a time mosaic of therecurrence signal, an imprinting of signal time patterns.Based on the statistical properties of the recurrenceplot, a large number of quantifiers have been developedto analyze details of a RP [17]. Many of them, deal withstatistical properties such as mean size, maximum size,frequency of occurrence of diagonal, vertical or horizontalrecurrence lines. An important class of recurrence quan-tifiers are those that try to capture the level of complex-ity of a signal. As an example, we mention the alreadyknown entropy based on diagonal lines statistics [18, 19].This quantity has been correlated with others dynami-cal quantifiers as, for example, the largest Lyapunov ex-ponent, since both capture properties of the complexitylevel of the dynamics [17].Nevertheless, at this point, it is important to men-tion that sometimes the diagonal entropy, (ENTR), asdefined in the literature [17] behaves in an unexpectedway, indeed, a quite known problem occurs in the dy-namics of the logistic map, namely ENTR decreases de-spite the increase of nonlinearity. In fact, to deal withthat, the literature has presented an adaptive methodto compute recurrences to conciliate the behavior of adecreasing ENTR with the increasing complexity of thelogistic map [20].Here we develop a new entropy recurrence quantifierbased on the formal definition of system entropy makinguse of all microstates displayed in a recurrence plot, notjust diagonal or horizontal lines. In order to introducethe new quantifier, we consider a recurrence plot and se-lect on it random samples of microstates, small squarematrices of N × N elements (we show results for matrixsizes up to 4 × RP it is possible to define an ensemble of microstates.As we shall see, the microstates reflect the dynamical re-current patterns of the time series. Indeed, the definitionof an entropy quantifier based on microstates frequencyprovides a good estimator that correctly capture the rela-tion between the entropy quantifier and the level of com-plexity and/or chaoticity, fixing the behavior displayedby the former diagonal entropy quantifier ENTR. As wewill show, making use of just captured data, it also pro-vide good results for all major dynamical changes thatoccurs in dynamical systems.The rest of this paper is organized as follows: in section2 we briefly show the recurrence technique and presentour methodology. In section 3 we apply the new method-ology to three well known time series: the first is a peri-odic harmonic (sine) signal perturbed by a white noise,the second is the time series of the discrete logistic mapand the last one is well known nonlinear flux describedby Lorenz equations. Finally in section 4 we discuss ourresults and point out future perspectives. II. METHODOLOGY
The methodology section is divided in three parts.First, we introduce the recurrence analysis techniquebased on the recurrence plot R ij , as defined in Eq. (1).In addition, we show some quantifiers used in the litera-ture to summarize the information of the RP that will beused in the manuscript to promote the adequate compar-ison with our method. In the next subsection we definea novel way to represent microstates extracted from therecurrence matrix, which is used to calculate our pro-posed entropy. Finally, in the last subsection we detailthe microstate set used to construct the new proposedentropy. A. Recurrence plots and recurrence quantificationanalysis
The recurrence plot is a graphical binary representa-tion of the complex recurrence patterns extracted fromtime series [16] or spacial profiles [21, 22]. The recurrenceplot was firstly developed by Eckmann et. all. [16] andfurther explored by several authors. A good compilationin literature of this issue is found in [17]. Important char-acteristics of the recurrence plots are the presence of finitesize diagonal lines, indicating periodic signals or recur-rence segments and isolated points suggesting stochasticand/or chaotic signals. An accurate extraction of specificfeatures of a time series can be obtained by using a set oftools, developed initially by Zbilut and Webber [18, 19]as measures of signal complexity based on the recurrencematrix. These tools are called recurrence quantificationanalysis (
RQA ) or recurrence quantifiers.The
RQA studies different aspects of the recurrenceplot, from the density of recurrent (non recurrent) pointsto the statistics of vertical (horizontal) or diagonal lines[17]. In order to avoid problems with very large recur-rence plots, when analyzing long time series, it is conve-nient to divide the original time series into smaller sub-series (or windows) with size K such that K (cid:28) M , being M the size of the entire trajectory. For each window, weconstruct a recurrence matrix that is used to computethe recurrence quantifiers. The simplest RQA is the re-currence rate (RR) defined as the density of recurrentpoints in R ij .An important question in recurrence analysis isthe measure of diagonal lines that represent recur-rence segments of trajectories. Diagonal lines are R ij structures parallel to the lines of identity defined as R i + k,j + k = 1 ( i, j = 1 , , · · · M − (cid:96) ; k = 1 , , · · · , (cid:96) ), R i,j = R i + (cid:96) +1 ,j + (cid:96) +1 = 0, where (cid:96) is the length of thediagonal line. Two pieces of a trajectory following a di-agonal line undergo for a certain time (the length of thediagonal) a similar evolution, once they have visited thesame region of phase space at different times. This is thekey idea behind recurrence and thus a clearcut signatureof a deterministic behavior in the time series.Accordingly, P ( (cid:96) ) = { (cid:96) i ; i = 1 , , · · · K } is the fre-quency distribution of the lengths (cid:96) of diagonal lines.It is also described by P ( (cid:96) ) = (cid:80) K − (cid:96) − i,j =1 (1 − R i,j )(1 − R i + (cid:96) +1 ,j + (cid:96) +1 ) (cid:81) (cid:96)k =1 R i + k +1 ,j + k +1 , for K the maximumlength of the diagonal lines. The determinism quanti-fier is defined by DET= (cid:80) (cid:96) max (cid:96) = (cid:96) min (cid:96)P ( (cid:96) ) / (cid:80) Ki,j =1 R i,j with (cid:96) min ( (cid:96) max ) the minimal (maximal) diagonal line. It wasreported that the inverse of (cid:96) max (the so called DIV -divergence quantifier) is related with the largest positiveLyapunov exponent [16, 23].The vertical (horizontal) lines in R ij are associated tolaminar states, common in intermittent dynamics [17].The laminarity is another common quantifiers based onvertical (horizontal) lines in RQAs , and it is constructedin a quite similar form as the determinism. In fact, lam-inarity is defined as LAM= (cid:80) v max v = v min vP ( v ) / (cid:80) Ki,j =1 R i,j with v min ( v max ) the minimal (maximal) vertical (hori-zontal) lines. The frequency distribution of vertical (hori-zontal) lines can be written as P ( v ) = (cid:80) Ki =1 (cid:80) K − v − j =1 (1 − R i,j )(1 − R i,j + v +1 ) (cid:81) vk =1 R i,j + k +1 .It was reported the use of the distribution of diagonallines P ( (cid:96) ) for a different quantifier of recurrences, basedon the Shannon entropy [17]. The general equation forShannon entropy is as follows: S = − Q (cid:88) i =1 p ( i ) log p ( i ) , (2)where p ( i ) evaluates the probability of occurrence of aspecific state i , Q is the number of accessible states,and S captures, in this sense, how much informationresides on such collection of states. The Shannon en-tropy is an open tool that can be adapted to any prob-ability space p ( i ). The recurrence plot space of proba-bility assumes many different possibilities, for instancethe distribution of diagonals p ( (cid:96) ) = P ( (cid:96) ) / (cid:80) K(cid:96) =1 P ( (cid:96) ). Inthat context equation (2) assumes the form: ENTR= − (cid:80) (cid:96) max (cid:96) = (cid:96) min p ( (cid:96) ) ln p ( (cid:96) ). Despite the use of this tool in theliterature [19], it presents serious problems as reportedin [20]. While the entropy was primarily conceived as aquantification of disorder, this first approach, based onentropy applied to chaotic systems ( e.g. Logistic Map),provides a unsatisfactory result, sometimes indicating amore organized regime for an arising chaoticity levels.
B. A new entropy of the Recurrence Plot
In this paper we developed a novel way to extract infor-mation from the recurrence matrix. To properly definean entropy we introduce a new concept of microstatesfor a RP that are associated with features of the dy-namics of the time series. These microstates are evalu-ated by matrices of dimension N × N that are sampledfrom the RP . The matrices can assume several configura-tions as can be seen in Fig. 1 for the particular situation N = 2. The total number of microstates for a given N is N ∗ = 2 N . The microstates are populated by ¯ N randomsamples obtained from the recurrence matrix such that¯ N = (cid:80) N ∗ i =1 n i , where n i is the number of times that amicrostate i is observed.For P i = n i / ¯ N , the probability related to the mi-crostate i , we define an entropy of the RP associatedwith the probabilities of occurrence of a microstate as S ( N ∗ ) = − i = N ∗ (cid:88) i =1 P i ln P i . (3)A clear advantage of new methodology to compute theentropy using Eq. (3) over the former diagonal entropyis the possibility of computation of information over allpossible microstates. In addition, is possible to estimateanalytically the maximum value of entropy S ( N ∗ ) corre-sponding to the case in which all microstates are equallyprobable. For this case P i = 1 /N ∗ . In this situation,we have S ( N ∗ ) = ln N ∗ . The analogous case the mini-mum value of the entropy corresponds to the situation inwhich all sampling matrices are at the same microstate.In this case S ( N ∗ ) is trivially zero. We will show in thenext topic that our approach does not suffer the inconve-niences of the traditional recurrence entropy quantifier,ENTR, for example, those explored in [20]. C. The microstates grammar
The microstates of the recurrence matrix have closerelation with patterns of the time series dynamics. Forsimplicity we analyze in detail the case N = 2.For a 2 × C iN , 0 ≤ i ≤ N × N matrix, for N = 2 is 2 = (cid:80) N i =0 C iN .The first and last lines of Fig. 1 show the trivial situa-tions with all cells are non-recurrent or fully recurrentrespectively. The second line, class C N , corresponds toone occupied cell with C = 4 distinct configurations.The fourth line, class C N , illustrates the situation withthree recurrent cells and also with C = 4 different pat-terns. The middle line of figure shows the most commonsituation, where two cells are recurrent and two are non-recurrent. The combination of possibilities is C N = 6.The N matrix patterns of Fig. 1 are the total set ofmicrostates used to sample and compute the entropy ofthe R ij .We notice that one of the most explored microstateemployed in standard recurrence analysis is the ( )pattern associated to diagonal lines. The quantifier LAMis related to horizontal and vertical lines whose code pat-terns are ( , , and ). All these fourcases are degenerated microstates since all have the samemeaning. In the following paragraphs we analyze in de-tail the dynamical information of the microstates. Figure 1. (Color online) All possible microstates of N × N, N = 2 matrix sample along with the correspondent binaryrecurrent construction. In this setup exists 16 different com-binations that can be grouped according to the amount ofrecurrent points. The microstates classes are designated by C i , where i corresponds to the number of recurrent elementsin each microstate. Firstly, consider the classes of microstates C N and C N in which all points are non-recurrent and recurrentrespectively. Both situations are typical microstates ob-tained using a (unappropriated) very small or large valueof vicinity parameter (cid:15) . Using an appropriate (cid:15) they willexist, but hardly will be the most frequent microstate.When we analyze the dynamical interpretation of themicrostates we realize the presence of degenerated mi-crostates. All microstates in the class C N are conceptu-ally the same. These microstates do not exist (ideally)in periodic motion or in fixed points dynamics, althoughthey are frequent in chaotic and stochastic series. Themicrostates associated with the C N class are not prop-erly degenerated, since all of them have their own dy-namical characteristics, but in standard analyses theyare less frequent than other microstates, because of theirintricate combination of recurrences.In the recurrence analysis context, the most relevantmicrostates are all in the C N class. For a more ac-curate analysis, observe Fig. (2). The group of verti- cal/horizontal recurrences are related to laminar statesand are characterized by recurrence quantifiers as LAM.All vertical and horizontal elements have exactly thesame information (degenerated microstates), in this casea trajectory recurs to a given position in phase space af-ter some times steps later, and stay nearby for a givenamount of time steps. This phenomenon is associated tothe symmetry recurrence matrix. Figure 2. (Color online) Scheme of the structural microstatesobtained for N = 2. Differently from Fig. 1 we depict dy-namical possible temporal (or spatial) patterns for the caseof N = 2 for vertical, horizontal and diagonal microstates. Inside the class C N , the diagonal microstates are themost important group, examples of temporal or spatialpatterns of theses microstate are observed Fig. 2 panels(e) and (f). Note also that the diagonal microstates arenot degenerated. For these microstates there is one thatrecurs with the same derivative signal, and another onethat has a contrary signal of the derivative. Even though,the main information of these microstates is related to re-current trajectories in phase space that develops nearby,and contain the basic idea of the recurrence quantifierDET.Using the illustrative case N = 2, let us consider thedynamics according to the collected data. For periodicsignals, just diagonal microstates are observed as shownin Fig. 2(e). A chaotic signal shall have a sample of mi-crostates, such as C N , C N and C N quite frequent. Onthe other hand, a smaller portion of diagonal, vertical orhorizontal microstates as described in Fig. 2 will not bevery frequent. Moreover, some residual C N microstatesare also expected. Finally a stochastic signal should havea more proportional distribution of microstates. Themost important result in this work regards the evalua-tion of all these complex behaviors using a comprehensivequantifier: the Shannon entropy.A similar but more refined and accurate analysis canbe done for N >
2. Indeed, despite the larger compu-tational effort required to compute the entropy for thiscase, much more confident results can be obtained. Nev-ertheless, we have to be aware of the exponential increaseof degenerated microstates for larger values of N . III. RESULTS
To explore in details the results obtained by the novelentropy, we apply this tool to a four illustrative data: awhite noise, a sine function superposed by white noise,the logistic map signal with and without noise and theLorenz equations time series. We test the entropy against (cid:15) , the vicinity parameter, and the microstate sizes N .Moreover, we compare the novel entropy against otherwell studied recurrence quantification methods. Finally,we discuss the structural advantages of our approach. A. White noise data
We start our analysis in a random time series with nocorrelation, namely, a white noise signal. Fig. 3 depictsthe entropy S as function of the threshold (cid:15) . We employthree values of N = 2 , , (cid:15) leads the entropy to a conceivable interval of validityfor S . This result puts in evidence the resilience of ourmethodology against this parameter. The cases (cid:15) → (cid:15) → (cid:15) , there will be just non-recurrent or isolated pointsin the RP . In this case, microstates of classes C N , and C N , dominate the distribution and the entropy resultsto be small. The opposite regime, for (cid:15) → C N and C N , and again,the entropy will result in a minimal value. In this way,intermediary values of (cid:15) will produce a richer distribu-tion of microstates among classes and a more confidententropy output.We also use the white noise to construct a simple modelfor maximal entropy in the recurrence plot methodology.The lack of correlation of the white noise implicates ina theoretical maximum entropy. This result is straight-forward, but we have to take into accountborder effectsof the recurrence space in the methodology to computea correct result of S . In this simple model is possible toextract the exact value of (cid:15) for which the entropy is amaximum. Consider the random data signal x used tocompute S in Fig. 3 being 0 (cid:54) x (cid:54)
1. It is clear that, forthe vicinity parameter (cid:15) = 1, RR is maximum and equalto 1, independently of the point where (cid:15) is centered. Inthis case all points in phase space are recurrent.When the vicinity parameter is less than 1, let us say (cid:15) = 0 . ε S S(2 )S(2 )S(2 )0.293 Figure 3. (Color online) Entropy S as a function of the recur-rent threshold ( (cid:15) ) for 3 different microstate sizes and appliedto a white pseudo-random data series. Note that the curveshape is insensible on the amount of possible microstates andthat the range of validity of the threshold is quite large, span-ning from 0 . (cid:46) (cid:15) (cid:46) .
45. This information stems from thefact that the entropy should reach its maximum for this typeof data portant, due to the boundaries of the phase space. Forexample, suppose we are computing the recurrence pointsof the first value in data. In this case, there is no left re-currence points and a (cid:15) = 0 . (cid:15) = 0 . . (cid:15) needs to be consid-ered in all points of the data, even close to the borderswhere it will capture less recurrent points. We explorein more detail this situation in Fig. 4 where we considerthe position where (cid:15) is computed in phase space.Fig. 4 depicts a graphical view of the recurrence spaceborder effect. The figure illustrates the percentage ofphase space recurrence versus the position for what (cid:15) iscomputed for different values of (cid:15) . We emphasize thatthis uniformity can be assumed only for a white noise,for simplicity we use a normalized phase space. In thispicture RR will be the averaged sampling obtained takinginto account the position of (cid:15) over all phase space. Thisestimation is performed computing the area below eachtrapezoidal curve as shown. In Fig. 4, we highlight aparticular case for which (cid:15) = 0 .
1, showing explicitly thatit does not correspond to 10% or 20% of the phase spaceas recurrent, but an intermediate amount. The generalexpression for the trapezoidal areas as a function of (cid:15) ispresented in Eq. 4. Finally, since we expect RR=0.5for a maximal entropy of the white noise case, Eq. 4gives us the optimum value of the threshold, namely (cid:15) ≈ . . In fact, despite some numerical inaccuracy, Fig.3 confirms this maximum for different values of N .RR = 2 (cid:15) − (cid:15) . (4) R e c u rr e n t P h a s e Sp a c e Phase Space ε =0.10 + ε =0.10 ε =0.10for larger values of ε Figure 4. Graphical representation of the recurrence rate astrapezoidal areas in the (normalized) recurrence phase spacefor the white noise signal. The trapezoidal shape is expectedsince points on the left (right) borders of the phase space donot have recurrences on the left (right) sides. Here, the y axis gives us the recurrence percentage of the phase space fordifferent values of (cid:15) . B. The sine function superposed by white noise
To further explore the methodology we proceed witha function that continuously change from periodic tostochastic behaviors, depending on a single parameter p . We explore the function y = y ( t ) defined by: y ( t ) = sin( ωt ) + p rand ( t ) (5)where p is a parameter that controls the random per-turbation of the model, rand ( t ) is an uniform randomfunction such that 0 < rand ( t ) < ω = 0 .
033 forconvenience. The limit of small p in Eq. (5) makes y atrivial deterministic sine function, whereas for large p therandom function dominates and y behaves like a whitenoise.In this context the entropy evaluated from the mi-crostates quantifies the change from an orderly system( eg. sine function) to a disorderly system ( eg. tendencyto white noise). In Fig. 5 we show the entropy as a Figure 5. (Color online) Entropy quantifier S as a functionof the random parameter p for N = 2, 3 and 4 for the sine-random signal, Eq. 5, using (cid:15) = 0 .
14. It is noticeable theincrease of S as p increases reaching a maximum for largevalues of p as predicted by Eq. 4. While the absolute valueof entropy changes as a function of N , the general shape ofthe curve is preserved function of p for three microsates lengths, N = 2, 3 and4. We observe in the graphic a continuous and smoothincrease of entropy with p . Around p ≈ . p →
0, shows a minimal entropyregime. For sake of comparison we indicate the theoreti-cal maximal entropy for each N : S max ( N ∗ = 2 ) = 2 . S max ( N ∗ = 2 ) = 6 .
24, and S max ( N ∗ = 2 ) = 11 . S max corresponding to the adequate N . In this way,the S max is a good candidate to a model benchmark thatdoes not depend on the specific analyzed dynamics, butonly on the number of microstates.In Section II (C) we explored the case of microstateswith length N = 2, and showed that some microstatesare degenerated. When we increase N the number of de-generated states increases dramatically. New microstatesappear. As a trivial example, we cite microstates relatedto larger diagonal lines that represent system states withmore deterministic behavior, since for this case, a trajec-tory tends to repeat the dynamics of a previous trajectorythat visit the same region. The consequence of a largeramount of microstates is the accentuated fluctuation inthe entropy, see for instance Figs. 3, 5 and 6. C. The logistic map
To explore further our methodology we test the en-tropy S in a system that presents chaotic regimes. Weemploy the logistic map [25, 26] defined by the followingequation x n +1 = r x n (1 − x n ) . (6)The parameter r controls the non-linearity of the system.The r is responsible for the bifurcation cascade route tochaos and windows of periodic behavior that can be seenin the bifurcation diagram Fig. (7)-(f). Figure 6. (Color online) Entropy quantifier S for the logisticmap versus the parameter r . Each curve correspond to a dif-ferent microstate size N . There is a greater resolution on theperiodic windows and chaotic regimes for larger N . Clearly, S for N ≥
3, ( S (2 ) and N ≥ S (2 )) capture the increaseof the complexity of the map dynamics as r increases. Never-theless, more pronounced results are obtained for N = 4. Allsimulations were computed using (cid:15) = 0 . In Fig. (6) we show the entropy S as a function of r for N = 2, N = 3 and N = 4. While the general aspectof these curves are similar for all N , the resolution andthe main concept of the increasing of the entropy S as r grows is more pronounced for N = 4. Larger N impliesin a larger number of microstates allowing for a more pre-cise evaluation of the system states. On the other hand,it leads to a larger sampling effort and computationaltime. The main effect of the (cid:15) size can be observed in thedoubling period cascading. The entropy jump of valuedoes not occur synchronously with the actual r param-eter where a bifurcation occurs. In fact it jumps in a r value a little bigger than the true r value of the bifurca-tion. It occurs due to the fact that before the observedjump, due to a finite value of (cid:15) , the quantifier can notdistinguish the orbit post bifurcation when compared tothe orbit before the bifurcation. It is a common effect ofthe recurrence analysis.To test the advantages of our methodology, we com-pare our results for N = 4, S (2 ) against some wellknown recurrence quantifiers. The results are depictedin Fig. (7) (a)-(e). To clarify the dynamics, 7(f) dis-plays the bifurcation diagram for the logistic map. Thequantifiers RR (panel (c)) and LAM (panel (b)) are al-most insensible to the growing complexity of the logisticmap as r increases. The quantifier ENTR is extremelyerratic and does not transmit the real concept of entropyfor the system, since it should reflex the increasing com-plexity of the map for larger values of r . This point,in particular, was explored in details in the literature and it is pointed as a drawback of the ENTR quantifier[20]. We should mention also the perturbing fact thatENTR increases in periodic windows and does not showprecisely the period doubling cascading window, occur-ring, for example, for 3 . < r < . S quantifier for N ≥ S (2 )), based on the microstatesobtained from the RP , depicts quite well what is goingon in the system dynamics. It also reflects correctly theincreases of complexity in period doubling cascade. Fi-nally, S (2 ) properly capture the increase of complexityof the dynamics as r increases. Furthermore, we shouldemphasize that S has a really weak dependency of on (cid:15) ,differently of other recurrence quantifiers. Figure 7. (Color online) Recurrence quantifier analysis ob-tained from DET, LAM, ENTR, and the new defined recur-rence entropy S for N ≥ S (2 )) for the logistic map,panels (a)-(e). Bifurcation diagram is plotted in panel (f). S was computed using (cid:15) = 0 . M = 1000 after transienttime. Some informations obtained for the new entropy S issimilar to the others quantifiers. Nevertheless, note that, dif-ferently from the other quantifiers, S capture the increase ofcomplexity due to the increase of r in chaotic regions. In Fig. 8 we explore the logistic map dynamics per-turbed by random noise. In this case the system losesmany features that it had for the deterministic case, like
Figure 8. (Color online) Recurrence quantifier analysis ob-tained from DET, LAM, ENTR, and the new defined recur-rence entropy S for N ≥ S (2 )) for the logistic map per-turbed by white noise. The noisy bifurcation diagram is plot-ted in panel (f). In each iteration of the map, white noise per-turbations are added in the dynamics of the map, correspond-ing to 0 .
5% of the maximum amplitude of the map. S wascomputed using (cid:15) = 0 . M = 1000 after a transient time.The new entropy S is quite robust against noise. Differentfrom the other quantifiers, S captures the increase of complex-ity due to period doubling bifurcations even in the presenceof noise as can be observed in the intervals 3 . < r < . . < r < . the numerous periodic windows within the chaotic regionand the perfect periodicity in the well know periodic re-gions [24]. In this context, all quantifiers roughly bringsthe same information, but we notice that both, DET and S (2 ) preserve respectively the reduction/increase in thequantifier while the chaos develop further given the in-crease of r . An important point must be mentioned: Ob-serve that, for the large noisy periodic window, occur-ring in the interval 3 . < r < .
9, the entropy S (2 ) ismuch more sensitive to changes occurring in the noisy dy-namics, when compared to RR, DET, LAM and ENTR. S (2 ) shows a clear increase of magnitude in the wellknown doubling period cascading interval. It is impor-tant to state that technically S (2 ) preserves the samequantitatively and qualitatively advantages against theother quantifiers as have been shown in the case withoutnoise. D. The Lorenz equations
In order to illustrate our results for the new methodol-ogy to obtain the system entropy applied to a continuouschaotic system. We present results for S using microstateof size N = 4. The results of the new entropy quantifierare applied to the classical Lorenz equations [25]. The Lorenz model is a well studied continuous dynamical sys-tem that present all sort of nonlinearities and even chaos[25, 26]. These equations are a reduction from seven tothree differential equations originally developed to modela convection motion in atmosphere [25]. The three equa-tions that resume the model are written as:˙ x = − σ ( x + y ) , ˙ y = x ( r − z ) − y, ˙ z = xy − bz, (7)with three free parameters: the Rayleigh number r , thePrandtl number σ and the quantity b . The system be-haves in a periodic way for the set of parameters ( σ = 10, b = 8 /
3) and 0 < r (cid:46) .
06 [25]. For r beyond 24 .
06 itstarts to display chaotic behavior with sporadically ap-pearance of periodic windows.An example of the application of our methodology tocontinuous classical chaotic system is depicted in Fig.(9). As observed, the transition to the chaotic regimeis associated with a dramatic increase in the entropy.Another important aspect is that, after the transitionto chaotic states, the entropy display a smooth increasesimilar to the one depicted in the maximum Lyapunovexponent due to the growing level of system chaoticity.This fact is easily observed by computing the Lyapunovexponent spectrum for the Lorenz system. Note thatthe maximum entropy for N = 4 is S (2 ) = ln(2 ) ≈ .
09, but the maximum entropy should exist only in thecondition of a random system, putting in evidence a cleardifference between the random signal and a deterministicchaotic signal, as expected.
IV. DISCUSSION AND FINAL REMARKS
This work has explored a new tool to study recurrencepatterns in time series. The patterns are evaluated withinthe framework of Recurrence Plots RP . Our methodbrings a novel quantifier that analyzes the microstatesobtained from sampled matrices extracted from RP . Ina broad sense, the manuscript has studied the diversityof the computed microstates in the RP .To quantify the diversity of accessed microstates wehave computed a proper Shannon entropy for the sys-tem. To demonstrate the validity of our method, wehave applied the concept to a random signal, a simplemodel composed of a sine signal superposed by whitenoise, a discrete chaotic system (the logistic map) and acontinuous system exemplified by the Lorenz equations.Moreover, we have tested the methodology for diverse re-currence vicinity size (cid:15) and microstates sizes. In addition,we have compared our results with standard recurrencequantifiers indexes: the recurrence rate, the laminarity,the determinism and the traditional entropy of recurrentdiagonals as defined in the literature so far.The main advantage of employing the Shannon entropybased in microstates, as proposed here, is the fact that it Figure 9. The entropy S computed for the Lorenz equations.We use here N = 4, (cid:15) = 0 . M = 1000 and ¯ N = 10 microstate samples. In (a) we present the bifurcation dia-gram while in (b) we show the entropy as a function of theparameter r in Eq. 7. The time series was computed withtime step h=10 − and evolved for 10 time steps to avoidtransient dynamics. As should be expected, the quantifier S subtly increases when the chaotic attractor born and, moreimportant, predicts quite well the expanding level of chaotic-ity of the Lorenz model as the r parameter is variated. is intrinsic to the meaning of an entropic quantifier. Simi-lar to any other entropies like the Boltzmann entropy, theKolmogorov-Smirnov entropy, the Shannon entropy, ournew methodology to compute the recurrence entropy in-creases with the complexity of the system. At this point,we should mention that the more traditional diagonal en-tropy computed as a recurrence quantification does nothave this propriety. In fact, this drawback of the quan-tifier ENTR, has been taken into consideration in theliterature [20] but, the solution found to solve the prob-lem involve some structural changes in the meaning ofrecurrence. Our new methodology to compute the en- tropy does not use any changes in the definition of recur-rence. The entropy we introduced in this work naturallyincreases with signal complexity.Another important issue is the computational effort toestimate a quantifier. Using microstates sampling, theamount of data to be analyzed is proportional to thenumber of possible microstates. For instance when we use N = 3 a useful sampling can be as low as a few hundredsof microstates. While in other methods, the totality ofthe RP must be analyzed. In a situation with a standard10 data points, the matrix would reach 10 points thatshould be evaluated to compute a proper quantificationof diagonals, verticals or simple density of recurrences.In addition, it is well known within recurrence plottechniques that the value of vicinity size, (cid:15) , and also theminimal sizes of diagonal, l min , or vertical lines, v min , af-fects the value of the quantifiers [16, 17]. The methodproposed in this work shows great robustness againstchanges of (cid:15) . The new methodology for the entropy ob-tains stable results using microstates with sizes N ≥ ACKNOWLEDGMENTS
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