From Weakly Chaotic Dynamics to Deterministic Subdiffusion via Copula Modeling
aa r X i v : . [ phy s i c s . d a t a - a n ] A p r Journal of Statistical Physics manuscript No. (will be inserted by the editor)
From weakly chaotic dynamics to deterministic subdi ff usionvia copula modeling Pierre Naz´eAbstract
Copula modeling consists in finding a probabilistic distribution, calledcopula, whereby its coupling with the marginal distributions of a set of random vari-ables produces their joint distribution. The present work aims to use this techniqueto connect the statistical distributions of weakly chaotic dynamics and deterministicsubdi ff usion. More precisely, we decompose the jumps distribution of Geisel-Thomaemap into a bivariate one and determine the marginal and copula distributions respec-tively by infinite ergodic theory and statistical inference techniques. We verify there-fore that the characteristic tail distribution of subdi ff usion is an extreme value copulacoupling Mittag-Le ffl er distributions. We also present a method to calculate the ex-act copula and joint distributions in the case where weakly chaotic dynamics anddeterministic subdi ff usion statistical distributions are already known. Numerical sim-ulations and consistency with the dynamical aspects of the map support our results. Keywords
Weakly chaotic dynamics · deterministic subdi ff usion · copula modeling Transport phenomena behaving di ff erently from the usual Brownian motion havebeen detected by several experiments over the last century [1–8], in particular a typeof subdi ff usive process, where the mean squared displacement grows proportional to t α , with 0 < α <
1. This relatively simple characteristic made it an object of intenseresearch and theoretical models were created trying to describe it as well [9, 10]. Inthis work, we study this phenomenon using Geisel-Thomae map [11], a dynamicalsystem that reproduces a subdi ff usive process when one observes the trajectory ofa set of initial conditions. For being a spatially extended Pomeau-Manneville map[12], Geisel-Thomae map preserves the weakly chaotic dynamics characteristic of Pierre Naz´eUnicamp, Instituto de F´ısica “Gleb Wataghin”, DFCM, 09210-170, Campinas, SP, BrazilE-mail: p.naze@ifi.unicamp.br Pierre Naz´e that system [13], being therefore a deterministic subdi ff usion model with this spe-cific dynamical behavior. Techniques such as continuous-time random walk (CTRW)[14,15] and infinite ergodic theory [13] look carefully into the relation between thesephenomena, coming to a conclusion that the subdi ff usive process observed in suchsystem is a straightforward consequence of its weakly chaotic dynamics. In this work,we deepen our knowledge about their relation, connecting both phenomena by theircharacteristic statistical distribution. In order to do so, we present the technique ofcopula modeling.Two situations are possible in a study of statistical dependency across randomvariables: the random variables can be statistically independent of each other or not.The former case is characterized by the joint cumulative distribution function (CDF)being given by the product of the variables CDFs. The latter case, on the other hand,presents a non-trivial functional form, in which finding its joint CDF becomes a hardtask to be accomplished given that there is no specific model to be achieved in princi-ple. One way out to solve that problem is the fundamental Sklar theorem [16], whichsays that, given two random variables X and Y , their joint CDF J can be expressed,in a unique way, as J ( x , y ) = C ( F X ( x ) , F Y ( y )) , (1)where F X and F Y are the respective marginal CDFs of X and Y , and C is a copula,a joint CDF defined on the unit sized square with additional properties (see [17] formore details). Copula modeling consists then in dividing the statistical dependencyof random variables into marginal and copula distributions and using statistical meth-ods to infer them properly. Examples of copula distributions are very well known inthe literature (see [17–19] and references therein) and computationally accessible bycopula modeling software package [20].Here the central idea of this work. Consider X t as the random variable of jumpsexecuted by the particle until a time t , according to Geisel-Thomae map. Rewriting X t = R t − L t , where R t and L t are respectively the sum of the jumps done only tothe right and left senses, we express the jumps distribution associated to X t as anexpression of the joint distribution of R t and L t . We perform then a copula modeling,where these new random variables will work as the marginal distributions. Thus,as R t and L t are connected to the number of first-passages of Pomeau-Mannevillemaps, whose distribution characterize the weakly chaotic dynamics [21–24], they willobey the same statistical quantity. Therefore, the jumps distribution of deterministicsubdi ff usion is connected to the distribution of weakly chaotic dynamics.The article is organized as follows. In section 2 we present the Geisel-Thomaemap, its subdi ff usivity and the connection between the distributions of weakly chaoticdynamics and deterministic subdi ff usion. In section 3, the marginal distributions arecalculated exactly by techniques of infinite ergodic theory and the copula one inferredby statistical methods. We present also a method to calculate exactly the copula andjoint distribution of the system since the statistical distribution already mentionedare known. In section 4, we summarize what we have done, emphasize to physicscommunity the importance of copula modeling and discuss perspectives from thiswork. rom weakly chaotic dynamics to deterministic subdi ff usion via copula modeling 3 ff usion via copula modeling The Pomeau-Manneville map is a function T , defined on the unit interval, whoseexpression is given by T ( x ) = x + (2 x ) + α (mod 1) , (2)with α >
0. The parameter α determines its dynamics: if α ≥
1, Pomeau-Mannevillemap is chaotic; if 0 < α <
1, the system is weakly chaotic. In this last case, thetrajectory of the particle passes through an intermittent regime, where it spends muchtime near the laminar region, located about the neutral point x =
0, and eventuallyvisits the turbulent one, located in the remaining part of the phase space. Because ofthis, the time evolution of dynamical observables occurs at a sublinear rate and itsconventional time average approaches to zero for long times [21]. To capture somechaotic aspects of the system, the time average is modified using the transformation t → t α in its normalization constant, leading us to a new type of ergodicity, wherethis new time average obeys an universal non-atomic distribution for a large class ofdynamical observables [25, 26]. -2-1.5-1-0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x G(x)Webcob
Fig. 1 (Color online) Cobweb diagram of a Geisel-Thomae map G . The chain of maps was generated bythe map f ( x ) = x + (2 x ) , being used an itinerary of 160 jumps beginning from x = . To construct Geisel-Thomae map, we consider initially a Pomeau-Mannevillemap T , with 0 < α <
1, defined on the interval [0 , / M , defined on ( − / , / M ( x ) = T ( x ), for x ∈ [0 , / M ( x ) = − T ( − x ), for x ∈ ( − / , G ( x + N ) : = M ( x ) + N , where N is a integer number, we constructGeisel-Thomae map G , now defined all over the real line. Summarizing such steps, Pierre Naz´e one has G ( x ) : = x + [2( x − N )] + α , x ∈ [ N , N + / x − [2( N − x )] + α , x ∈ ( N − / , N ] , (3)for all N integer. The FIG. 1 depicts a cobweb diagram of G .During the computational simulation of this model we consider a sample of n particles distributed uniformly over the interval ( − / , / t times by Geisel-Thomae map. The set of the n final points generatedby this procedure yields to a sample variance, which grows as a power-law withexponent 0 < α <
1, indicating that the model presents subdi ff usion [11]. This aspectis also revealed in the study of the PDF ρ ( x , t ) of the particle position x at time instant t , in which techniques from CTRW [15] or infinite ergodic theory [27] have shownthat such PDF is the solution of time-fractional di ff usion equation [28], given by ρ α ( x , t ) = √ Dt α M α | x |√ Dt α ! , (4)where M ν is the Mainardi function, given in turn by M ν ( z ) = π ∞ X n = ( − z ) n n ! Γ [ − n ν + (1 − ν )] , (5)where 0 < ν <
1, and D is the di ff usion constant.Thus, to show to the reader that Geisel-Thomae map exhibits subdi ff usion, Fig.2 shows comparisons of the PDF outlines generated by Eq. (4) with the histogramsof final position x at time t for di ff erent parameters α s. The agreement is very goodindeed. Finally, for the purposes of outlines comparisons, we remark that the di ff usionconstants were taken by inspection, not choosing any particular theoretical model toestimate it.To connect the statistical distributions of weakly chaotic dynamics and determin-istic subdi ff usion distribution, we consider the general situation where the randomvariable X t of the jumps executed until a time t is rewritten as X t = R t − L t , where R t and L t are respectively the jumps done only to the right and left senses. In thismanner, the probability distribution function (PDF) ρ ( x , t ) of X t is related to the jointPDF j t of R t and L t in the following way ρ ( x , t ) = Z ∞ j t ( | x | + z , z ) dz . (6)Substituting the copula C t and the marginal CDFs F R t and F L t in Eq. (1), and suchresult in its PDF version in Eq. (6), one has ρ ( x , t ) = Z ∞ c t ( F R t ( | x | + z ) , F L t ( z )) ρ R t ( | x | + z ) ρ L t ( z ) dz , (7)in which c t is the PDF of the copula C t and ρ R t , ρ L t are the PDFs associated to themarginal distributions. Eq. (7) is the main result of this work: applied to Geisel-Thomae map, it connects the statistical distributions of deterministic subdi ff usion rom weakly chaotic dynamics to deterministic subdi ff usion via copula modeling 5 ρ ( x , ) x α =0.3 Bilateral ML: D=0.34 ρ ( x , ) x α =0.4 Bilateral ML: D=0.35 ρ ( x , ) x α =0.5 Bilateral ML: D=0.31 ρ ( x , ) x α =0.6 Bilateral ML: D=0.27
Fig. 2 (Color online) Comparisons between the PDF outlines generated by Eq. (4) (red lines) with thehistogram produced by Geisel-Thomae map for α = . , . , . , .
6. The di ff usion constants was chosenby inspection and the number of initial conditions was n = and weakly chaotic dynamics, as it will be demonstrated in the next section when themarginal distributions will be calculated.We remark one more time that our approach is completely general. If one is study-ing a system – which does not need to be a map – that produces any kind of anomalousdi ff usion, and whose PDF of jumps distribution is unknown, the copula modeling canbe applied in the same manner. The advantage of Geisel-Thomae map, as we are go-ing to see, is to find the marginal distribution exactly. Even if it is not possible findit exactly, methods of statistical inference could be applied to find approximations ofthe marginal distributions [29]. G and a time of iteration t , consider two random variables, R t and L t , such that the former is the absolute value of the sum of the displacementsof jumps executed to the positive sense and the latter to the negative one: R t : = t − X k = ϑ R ( G k ( X )) , L t : = t − X k = ϑ L ( G k ( X )) , (8) Pierre Naz´e where the following observables are defined as ϑ R ( x ) : = ( G ( x ) − x ) H ( G ( x ) − x ) , (9)and ϑ L ( x ) : = ( x − G ( x )) H ( x − G ( x )) , (10)being H the step Heaviside function, G k the k -th iteration of G and X the uniformdistribution defined over [ − / , / ϑ R ( x ) = [2( x − N )] + α , x ∈ [ N , N + / , x ∈ ( N − / , N ] , (11)and ϑ L ( x ) = , x ∈ [ N , N + / N − x )] + α , x ∈ ( N − / , N ] . (12)We first remark that the marginal distribution must be identical for R t and L t .The initial conditions uniformly distributed over [ − / , /
2] produce a symmetrywhereby, if we have a particle in the position x , we must have another in − x . More-over, by the map symmetry, the position of the particle must be such that G t ( x ) = − G t ( − x ) at any time t . Thus the displacements will be di ff erent only by the sign,which will be vanished by the absolute values of the observables ϑ R and ϑ L . There-fore if the PDFs of R t and L t are respectively ρ R t and ρ L t , we must have ρ R t ( x , t ) = ρ L t ( x , t ); (13)for now on we are only referring to ρ R t .Before proceeding in our analysis, we briefly discuss the Aaronson-Darling-Kac(ADK) theorem, a fundamental result for determining the marginal distributions ofGeisel-Thomae map. Consider a map T conservative, ergodic, measure-preservingtransformation on its phase space A and µ is its invariant measure. The ADK theoremsays that a suitable time average of an observable ϑ ∈ L + ( µ ) converges in distributionto a random variable ξ α , which is scaled by the ensemble mean of the same variable.That is 1 a t t − X k = ϑ ( T k ( X )) d → ξ α Z A ϑ d µ, (14)where ( a t ) | ∞ t = is the return sequence, X a random variable and ξ α the normalizedMittag-Le ffl er distribution of order α , with 0 < α <
1. For more information, one cansee [13, 22–26].Returning to our discussion, the similarity between Geisel-Thomae map G and thePomeau-Manneville map T suggests that the former is also an weakly chaotic mapand it would be appropriate to use ADK theorem in the observable ϑ R to determine ρ R t . However, we cannot proceed with such idea, because the domain of G is thereal line, which is not covered by the usual theory [25]. To circumvent this aspect,Akimoto and Miyaguchi [13] have pointed out that G could be reduced to an weaklychaotic map for the purpose we have in mind. Observable given in Eq. (11) has theproperty of evaluating not from the value of x , but only from its di ff erence with its rom weakly chaotic dynamics to deterministic subdi ff usion via copula modeling 7 nearest integer. In that case, if a particular map ¯ G produces the same di ff erences forsome observable ¯ ϑ R and obeys to the ADK theorem, so the same theorem is valid for G and the observable ϑ R . Analyzing the structure of our chain, a possible way to usethis fact is considering¯ G ( x ) : = x + (2 x ) + α , x ∈ [0 , / x − [2(1 − x )] + α , x ∈ (1 / ,
1] mod 1 , (15)with initial conditions ¯ x ∈ [0 , and the observable¯ ϑ R ( x ) = (2 x ) + α , x ∈ [0 , / − x )] + α , x ∈ (1 / , . (16)We observe that Eq. (15) is well defined for α ∈ R . ¯ G is an weakly chaotic map withfinite domain and obeys therefore the ADK theorem. Thus, the evaluation of ρ R t willlead to the same results. Considering Eq. 14 and using ϑ = ϑ R and a t ∼ t α /β for largetimes, one has R t : = t − X k = ϑ R ( G k ( X )) d ∼ t α β ξ α , (17)which lead us to ρ R t ( x ) ∼ β t α ρ ξ α (cid:18) β xt α (cid:19) , (18)for large t , in which ρ ξ α is the PDF of ξ α and β is a constant in time, in principledependent on α . In another words, the positive displacement R t obeys a Mittag-Le ffl erdistribution which uniformizes itself as time grows up according to a power-law.FIG. 3 shows an example of a comparison between Eq. (18) and the respectivedata for fixed α and a and di ff erent values of t . The match between them is great. For10 initial conditions uniformly distributed in [ − / , / t = × to build histograms statistically significant. Similar numerical simulationswith variations in a and α confirm ADK theorem as well. We also notice that β doesnot depend on time t .Note the reader that finding Mittag-Le ffl er distributions in both random variables R t and L t is quite reasonable. Observing the structure of Geisel-Thomae map, one cansee the presence of fixed points in all integers. In studies of Pomeau-Manneville maps,such fixed points are singularities in the invariant measure of the map, which producessmall intervals around itself, called laminar regions, where the particle spend mostpart of the time of its trajectory. In this context, plenty of works has studied that thenumber of first-passage times N t to the particle to return to laminar region obeys aMittag-Le ffl er distribution [21–24]. Thus, as the particle has two senses to return tosome laminar region, and the displacements R t and L t are proportional respectively The correspondence between the initial conditions from each map is: x = ¯ x , for x ≥ x = ¯ x − .
5, for x <
0. Again, we note that the only thing important here is the di ff erence between successivejumps. Note that ¯ G (cid:16) (cid:17) = <
1, which is necessary for having only two branches on [0 , / / , ρ R t ( x ) x t=100000t=200000t=300000t=400000t=500000 Fig. 3 (Color online) Comparison between histograms of the positive displacement performed by theexperiment with Eq. (18). The di ff usion process was generated by Eq. (3) with α = .
5. Each one of the2 . × particles uniformly distributed in the interval [ − / , /
2] is iterated t times. The red crosses, bluesquares, green circles, grey triangles and black losangles represent respectively the histograms for times t = × , 2 × , 3 × , 4 × and 5 × . The red, blue, green, grey and black lines (up to downat x =
0) are Eq. (18) calculated for β = .
52 and respective t already given. The theoretical curves matchalmost perfectly our data. The number β was evaluated by inspection. to N t in the right or left senses, it is quite natural to conclude that both variables mustobey Mittag-Le ffl er distributions as well.Before proceeding in finding the copula distribution, we have to guarantee thatthe random variables R t and L t present a non-trivial statistical dependency betweenthem. This is accomplished if the copula of the system is not equal to the productcopula C ( u , v ) = uv . (19)FIG. 4 shows a comparison between pseudo-observations data and random numberscreated using product copula distribution as the generator. 2500 points are used forthe former and 2500 points to the latter. The disagreement is evident. Similar resultsoccur for α = . , . , . C θ ( u , v ) that better de-scribes the jump distribution PDF ρ α ( x , t ) of Geisel-Thomae map via statistical in-ference. In all hypothesis tests performed, it was used software R , version 3.3.2, withthe package copula , version 0.999-14 [20]. All the data generated was made with atime of iteration t = and the analyzed cases were α = . , . , . , . R Because of such long time of iteration, it is considered that the initial conditions are mutually inde-pendent of each other, condition necessary to apply copula modeling [17].rom weakly chaotic dynamics to deterministic subdi ff usion via copula modeling 9 F L t F R t Empirical dataRandom numbers
Fig. 4 (Color online) Comparison between the scatter plot of pseudo-observations (black full circles) andrandom numbers (red empty squares) generated for product copula. The disagreement is evident. It wasused α = .
5. Similar results occur for α = . , . , . o ff ers six possibilities to analyze: Clayton, Frank, Gumbel-Hougaard, normal, Plack-ett and t -Student (hereafter taking the degrees of freedom ν =
4) families (see themathematical expressions at the end of this subsection). The specific hypothesis testused was the Cram´er-Von Mises test, using the multiplier bootstrap as a method toestimate the p -values [29]. It was used the maximum pseudolikelihood (MPL) as amethod to estimate the parameters θ that characterize the copula family . Hypothesis test
Copula θ p -value(%)Clayton 10.55 10 − Frank 42.82 0.023Gumbel-Hougaard 9.19 0.697Normal 0.967 10 − Plackett 442.64 0.033 t -Student 0.988 0.661 Table 1
Table presenting the outcome from the command line gofCopula for the Clayton, Frank,Gumbel-Hougaard, normal, Plackett and t -Student cases taken under null hypothesis for α = . θ is thevalue of the parameter estimated by MPL method and the p -values were evaluated by multiplier bootstrap,with the exception of Frank case, where parametric bootstrap was performed. The sample size n = m = α = . , . , . Under various possibilities to execute such test, the options chosen here seem the most appropri-ate. Multiplier bootstrap is more computationally e ffi cient than the parametric bootstrap [30], and theAnderson-Darling test, which is an option to the Cram´er-Von Mises test, is on debate in the scientificcommunity about the values of parameters that compound it [31], and we decided not to analyze our dataunder such perspective. Lastly, previous tests with inversion of Kendall’s τ and inversion of Pearson’s ρ asestimative methods of the parameters did not give any new insight.0 Pierre Naz´eEvaluations of θ and χ Copula Clayton Frank
Gumbel-Hougaard
Normal Plackett t -Student α θ χ θ χ θ χ θ χ θ χ θ χ Table 2
Table presenting the values of θ and χ for Clayton, Frank, Gumbel-Hougaard, normal, Plackettand t-Student families for the cases α = .
3, 0,4, 0.5, 0.6. Gumbel-Hougaard copulas present the betteragreements.
TABLE 1 provide the outcomes from the hypothesis tests performed taking Clay-ton, Frank, Gumbel-Hougaard, normal, Plackett and t-Student copulas under null hy-pothesis for α = .
5. It was used the command gofCopula , a sample size n = m = α = . , . , .
6. FIG. 5 illustrates the results obtained in TABLE 1.It depicts a comparison between the scatter plot for pseudo-observations and randomnumbers created for (a) Clayton, (b) Frank, (c) Gumbel-Hougaard, (d) normal, (e)Plackett and (f) t -Student copula distributions used as generators. On each graphic,750 points were used in the scatter plot for pseudo-observations and 750 points forthe random numbers plot. All the random numbers plots, in some way or another, donot fit the pseudo-observations data. Describing respectively the figures of the worstand best case scenario from TABLE 1, the Clayton copula fits the region near theleft tail, but disagrees completely near the right one, and Gumbel-Hougaard copula,although fits almost perfectly on the tails, is spread in the remaining part of the plot.Although the result could be possibly better with more options of copula familiesat our disposal, note the reader that hypothesis test must be the first technique used todetermine the copula distribution, because it says whether the analyzed copula fam-ily is suitable or not. In the case which is not, we have to use other methods to finda better approximation. In our case, we evaluate the best model for the six familiesobserving their agreements with the jumps distribution. Thus, we perform χ tests[29] considering the points of the histogram generated by Geisel-Thomae map as theinput of the theoretical part and those points evaluated by Eq. (7) as the input of theexperimental one. It was used n = β = . , . , . , .
00 (verified byinspection) respectively for the cases α = . , . , . , .
6, where we take the numberof points for each χ test respectively 30, 60, 100, 140 points. The parameter estima-tion was evaluated by the command fitCopula and the histograms were constructedwith unitary bins, meaning that we only observe the dynamics coarse behavior [15].TABLE 2 presents the results. For the cases α = . , . , . , .
6, Gumbel-Hougaardcopula presents respectively χ = . , . , . , . ff usion treated here can be character- rom weakly chaotic dynamics to deterministic subdi ff usion via copula modeling 11 F L t F R t (a) Clayton copula for θ = 10.55 Empirical dataRandom numbers 0 1 0 1 F L t F R t (b) Frank copula for θ = 42.82 Empirical dataRandom numbers 0 1 0 1 F L t F R t (c) Gumbel copula for θ = 9.19 Empirical dataRandom numbers 0 1 0 1 F L t F R t (d) Normal copula for θ = 0.967 Empirical dataRandom numbers 0 1 0 1 F L t F R t (e) Plackett copula for θ = 422.6 Empirical dataRandom numbers 0 1 0 1 F L t F R t (f) t-Student ( ν =4) copula for for θ = 0.988 Empirical dataRandom numbers Fig. 5 (Color online) Comparison between the scatter plot of pseudo-observation (black full circles) andrandom numbers (red empty squares) generated for (a) Clayton, (b) Frank, (c) Gumbel-Hougaard, (d)normal, (e) Plackett and (f) t -Student copula families for α = .
5. The parameters were estimated bymaximum pseudolikelihood method. For each figure, it was used 750 points. All the graphics presentsome disagreements between the plots, which corroborates the results presented in TABLE I. ized by a Gumbel-Hougaard copula coupling Mittag-Le ffl er distributions – a featureof its dynamics. The fact that such copula, which is an extreme value one, is de-scribing a tail distribution is not fully understood yet and it will be subject for futureresearch. We also remark that other parametrized families graphics do not present anyparticular region of agreement.FIG. 7 shows the graphic between the estimated parameter θ for each copula fam-ily and the parameter α . For all the cases, the parameters are proportional to α . Inthis way, as the statistical dependency between the random variables increases withthe parameters (see the mathematical expressions in the end of this subsection), thesame occurs as α is increased. Such result is consistent with the dynamics of our sys- ρ ( x , ) x α =0.3 Gumbel-Hougaard ρ ( x , ) x α =0.4 Gumbel-Hougaard ρ ( x , ) x α =0.5 Gumbel-Hougaard ρ ( x , ) x α =0.6 Gumbel-Hougaard
Fig. 6 (Color online) Comparisons between the PDFs generated by Gumbel-Hougaard case (red circles)with the histogram produced by Geisel-Thomae map for α = . , . , . , .
6. The parameter estimationwas done by MPL method with n = tem. As α →
0, the branches of Geisel-Thomae map approach to the diagonal, whichmeans, for the dynamical point of view, that the system stays near the neutral pointsfor much time than before. In this way, if the system begins in the right branch of anycell map, most part of the time it will jump only to the right sense. Then the passagefor regions where the system would jump to the left sense is almost negligible. R t is practically una ff ected by L t and therefore they are statistically independent in thelimit α →
0. The same idea is valid considering initial conditions beginning in theleft cell branches.Finally, for completeness, the analyzed copula families are enumerated below.
1. Clayton copula: C θ ( u , v ) = [max { u − θ + v − θ − , } ] − θ , (20)where θ ∈ [ − , ∞ ) \{ } . Independency occurs in the limit θ →
2. Frank copula: C θ ( u , v ) = − θ log " + ( e − θ u − e − θ v − e − θ − , (21)where θ ∈ R \{ } . Independency occurs in the limit θ → rom weakly chaotic dynamics to deterministic subdi ff usion via copula modeling 13 θ α ClaytonFrankGumbel-HougaardNormalPlackettT-Student θ α NormalT-Student
Fig. 7 (Color online) Dependency between the estimated θ for Clayton (yellow crosses), Frank (light bluetriangle), Gumbel-Hougaard (red circles), normal (blue squares), Plackett (gray inverted triangle) and t -Student (green asterisk) and α . The inset graphic is a zoom of the plots of Normal and t -Student family.All parameters grows as α is increased. For the point of view of statistical dependency across R t and L t ,they become more dependent as α increases.
3. Gumbel-Hougaard copula:C θ ( u , v ) = exp (cid:20) − (cid:16) ( − ln u ) θ + ( − ln v ) θ (cid:17) θ (cid:21) , (22)where θ ≥
1. Independency occurs at θ =
4. Normal copula:C ρ ( u , v ) = p π (1 − ρ ) F − ( v ) Z −∞ F − ( u ) Z −∞ e (cid:18) − x + y − ρ xy − ρ (cid:19) dxdy , (23)where ρ is the Pearson’s correlation coe ffi cient and F − is the inverse CDF ofunivariate normal distribution. Independency occurs to ρ =
5. Plackett copula:C θ ( u , v ) = + ( θ − u + v ) − p [1 + ( θ − u + v )] − θ ( θ − uv θ − , (24)where θ >
0. Independency occurs at θ =
6. t-Student ( ν = ) copula:C ρ ( u , v ) = Γ (3)4 Γ (2) p π (1 − ρ ) t − ( v ) Z −∞ t − ( u ) Z −∞ + x + y − ρ xy − ρ ) ! − dxdy , (25)where ρ is the Pearson’s correlation coe ffi cient, t − is the inverse CDF of univari-ate t -Student distribution for ν =
4. Independency occurs at ρ = J t can be calculated exactly once we know thejumps distribution ρ ( x , t ). We observe first that ρ ( x , t ) can be expressed as ρ ( x , t ) = Z ∞ ∂∂ z − ρ ( | x | + z , t ) ρ ( z , t ) ρ (0 , t ) ! dz . (26)Then we equal such expression with Eq. (7) and isolate the copula PDF c t . Passinginto variables ( u , v ), we have c t ( u , v ) = − ρ ( y , t ) ∂ x ρ ( x , t ) + ρ ( x , t ) ∂ y ρ ( y , t ) ρ (0 , t ) ρ R t ( x ) ρ L t ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = F − Rt ( u ) y = F − Lt ( v ) , (27)where F − R t and F − L t are respectively the inverse CDF of R t and L t . In the case ofGeisel-Thomae map, ρ ( x ) will be Eq. (4), whose partial derivative ∂ x ρ α ( x , t ) is easilycomputable. Furthermore, based on the approach of [32], we can express ρ R t ( x ) = ρ L t ( x ) = α x − (1 + /α ) g α ( x − /α ) , (28)where g α ( x ) is one-sided L´evy PDF , which in turn can be expressed analytically byMikusinski’s integral representation. The inverse CDF of R t and L t can be in principlecalculable by this same representation. Indeed, we have F − R t ( x ) = log ( f ( x )) − α , (29)where f ( x ) is the solution of the following integral equation Z π f ( x ) w ( φ ) d φ = π (1 − x ) , (30)with w ( φ ) = sin (1 − α ) φ sin φ sin αφ sin φ ! α/ (1 − α ) . (31)At this point is easy to see the necessity of inference tests to find an approximate cop-ula, since the exact is hard to be analytically computatable. We remark also that Eq.(27) is symmetrical, that is, c t ( u , v ) = c t ( v , u ), as we have verified in the cumulativecase in FIG. 4 and FIG. 5 . Finally, using Sklar theorem in its PDF version in Eq.(27), the jointly PDF of the variables R t and L t will be given by j t ( x , y ) = − ρ ( y , t ) ∂ x ρ ( x , t ) + ρ ( x , t ) ∂ y ρ ( y , t ) ρ (0 , t ) . (32)In the case of Geisel-Thomae map, where the jumps distribution ρ α ( x , t ) could bedetermined by CTRW method, we have j t ( x , y ) = − ρ α ( y , t ) ∂ x ρ α ( x , t ) + ρ α ( x , t ) ∂ y ρ α ( y , t ) ρ α (0 , t ) . (33)In other words: the joint distribution of two random variables is determined if oneknows the distribution of their di ff erence. rom weakly chaotic dynamics to deterministic subdi ff usion via copula modeling 15 We presented in this work the connection between the statistical distributions ofweakly chaotic dynamics and deterministic subdi ff usion. Considering Geisel-Thomaemap, such relation was established by Sklar theorem, where the jumps distributionwas decoupled into Mittag-Le ffl er distributions and a Gumbel-Hougaard copula fordi ff erent subdi ff usion parameters α . We presented also a method to calculate the ex-act copula distribution of the system, although under the condition that the statisticaldistributions of weakly chaotic dynamics and deterministic subdi ff usion are known.In the end, we observed that the copula parameters, which measure the statisticaldependency between the marginal distributions, is proportional to the subdi ff usionparameter α , being consistent therefore with the dynamics of the system.If in the past copula modeling was a technique hard to be put on practice, becauseit was practically impossible to analyze a considerable amount of data of an statisti-cal experiment, at the present moment studies can be performed easily by the recentdevelopment of software packages. Besides that, new techniques in copula modelingare in constant development and the community behind all these advances increas-ingly grows. In this manner, physical phenomena in which statistical dependency isa fundamental subject for their understanding can be investigated by this new per-spective. Some examples that could be addressed in this manner are the hypothesis ofmolecular chaos in Boltzmann equation or the map families defined by Gaspard andWang in [21].Finally, it is important to stress that copulas are not just empty mathematicalfunctions waiting to describe problems in a redundant way, but carries importantproperties that help to understand the studied phenomena. For example, Gumbel-Hougaard copula has its roots in the extreme value theory, which means that it issuitable to model dependency between extreme events, such as a possible flood in thecity by the water level of the rivers that surround it [33]. On the other hand, knowingthat Gumbel-Hougaard copula has appeared naturally in a problem, suggests that thedependency across the variables obeys at some extent the extreme value theory. Inthis manner, as we have obtained that Gumbel-Hougaard copula describes the tail ofjumps distribution, this probably implies some deeper connection between extremevalue copulas and tail distributions. This aspect will be subject for future research. Acknowledgements
I thank Marcus V. S. Bonanc¸a, Alberto Saa and Roberto Venegeroles for reading themanuscript; A. Suzuki and M. L. Viola for discussions about copula modeling. Thiswork was financed by CNPq and CAPES.
References
1. Richardson, L. F. and Proctor, D.: Di ff usion over Distances Ranging from 3 km. to 86 km. Quar. J.Roy. Met. Soc. (222), 149-151. (1927)6 Pierre Naz´e2. Bohm, D.: The Characteristics of Electrical Discharges in Magnetic Fields. In: Guthrie, A., andWakerling, R. K. (eds.). pp. 1-12. McGraw-Hill, New York (1949)3. Koenig, S. H. and Brown III, R. D.: H CO as Substrate for Carbonic Anhydrase in the Dehydrationof HCO − . Proc. Nat. Acad. Sci. (USA) (9), 2422-2425. (1972)4. Bernasconi, J., Beyeler, H. U., Str¨assler, S. and Alexander, S.: Anomalous frequency-dependentconductivity in disordered one-dimensional systems. Phys. Rev. Lett. , 819 (1979)5. Cardoso, O. and Tabeling, P.: Anomalous di ff usion in a linear array of vortices. Europhys. Lett. (3),225 (1988)6. Solomon, T. H., Weeks, E. R. and Swinney, H. L.: Observation of anomalous di ff usion and L´evyflights in a two-dimensional rotating flow. Phys. Rev. Lett. (24), 3975 (1993)7. Dieterich, P., Klages, R., Preuss, R. and Schwab, A.: Anomalous dynamics of cell migration. Proc.Nat. Amer. Soc. , 459-463 (2008)8. Abe, S. and Suzuki, N.: Anomalous di ff usion of volcanic earthquakes. Europhys. Lett. , 59001(2015)9. Metzler, R. and Klafter, J.: The random walk’s guide to anomalous di ff usion: a fractional dynamicsapproach. Physics reports, (1), 1-77 (2000)10. Anomalous transport: foundations and applications. In: Klages, R., Radons, G. and Sokolov, I.M.(eds.). John Wiley & Sons (2008)11. Geisel, T. and Thomae, S.: Anomalous Di ff usion in Intermittent Chaotic Systems. Phys. Rev. Lett., (22), 1936-1939 (1984)12. Pomeau, Y. and Manneville, P.: Intermittent transition to turbulence in dissipative dynamical systems.Com. Math. Phys. (2), 189-197 (1980)13. Akimoto, T. and Miyaguchi, T.: Role of infinite invariant measure in deterministic subdi ff usion.Phys. Rev. E , 030102(R) (2010)14. Montroll, E.W. and Weiss, G.H.: Random walks on lattices. II. Journal of Mathematical Physics, (2), 167-181 (1965)15. Zumofen, G. and Klafter, J.: Scale-invariant motion in intermittent chaotic systems. Phys. Rev. E ,851 (1993)16. Sklar, A.: Distribution functions in n dimensions and their margins. Publ. Inst. Statist. Univ. Paris ,229-231 (1959)17. Nelsen, R.: An Introduction to Copulas. Springer, New York (2006)18. Copula Theory and its Applications. In: Durante, F., Haerdle, W., Jaworski, P., and Rychlik, T. (eds.).Springer-Verlag, Berlin Heidelberg (2010)19. Trivedi, P. K. and Zimmer, D. M.: Copula modeling: an introduction for practitioners. Foundationsand Trends in Econometrics (1), 1-111 (2007)20. https://cran.r-project.org/web/packages/copula/
21. Gaspard, P. and Wang,X.-J.: Sporadicity: between periodic and chaotic dynamical behaviors. Proc.Nat. Acad. Sci. (13), 4591-4595 (1988)22. Akimoto, T. and Aizawa, Y.: New aspects of the correlation functions in non-hyperbolic chaoticsystems. J. Kor. Phys. Soc. , 254 (2007)23. Saa, A., and Venegeroles, R.: Pesin-type relation for subexponential instability. J. Stat. Mech.: Theo.and Exp. (03), P03010 (2012)24. Naz´e, P. and Venegeroles, R.: Number of first-passage times as a measurement of information forweakly chaotic systems. Phys. Rev. E (4), 042917 (2014)25. Aaronson, J.: An introduction to infinite ergodic theory. American Mathematical Society, Province(1997)26. Zweim¨uller, R.: Ergodic properties of infinite measure-preserving interval maps with indi ff erentfixed points. Erg. Theo. & Dyn. Sys. (5), 1519-1549 (2000)27. Miyaguchi, T. and Akimoto, T.: Ergodic properties of continuous-time random walks: Finite-sizee ff ects and ensemble dependences. Phys. Rev. E, (3), 032130 (2013)28. Mainardi, F., Luchko, Y. and Pagnini, G.: The fundamental solution of the space-time fractionaldi ff usion equation. Frac. Calc. App. Anal., (2), 153-192 (2001)29. DeGroot, M. and Schervish, M. J.: Probability and Statistics. Pearson, Boston (2011)30. Kojadinovic, I., Yan, J., and Holmes, M.: Fast large-sample goodness-of-fit tests for copulas. Stat.Sin., 841-871 (2011)31. Genest, C., Huang, W., and Dufour, J. M.: A regularized goodness-of-fit test for copulas. J. Soc.Fran. Stat., (1), 64-77 (2012)32. Saa, A. and Venegeroles, R.: Alternative numerical computation of one-sided L´evy and Mittag-Le ffl er distributions. Phys. Rev. E, (2), 026702. (2011)33. Gumbel, E. J. and Goldstein, N.: Analysis of Empirical Bivariate Extremal Distributions. J. Amer.Stat. Assoc.59