Universal behavior of extreme value statistics for selected observables of dynamical systems

Univ ersal b eha vior of extreme v alue statistics for select ed observ ables of dynamical systems V alerio Lucarini, ∗ Davide F aranda, and Jero en W outers Klimac ampus, University of Hambur g, Grindelb er g 5, 20144, Hambur g, Germany (Dated: September 17, 2018) The main results of the extreme v alue t heory developed for the in vestiga tion of t h e observ ables of dynamical systems rely , up to n o w, on th e Gned enko approac h. In t h is framew ork, ex tremes are basically identified with t h e block maxima of the time series of the c hosen observ able, in the limit of infinitely long blocks. It has b een prov ed that, assuming suitable mixing conditions for the un derlying dyn amical systems, the ex tremes of a sp ecific class of observ ables are distributed according to the so called Generalized Extreme V alue (GEV) distribution. Direct calculations sho w that in the case of qu asi-perio dic dynamics the blo ck max ima are not distributed a ccording to the GEV distribu tion. In this pap er w e show that, in order to obtain a universal b ehaviour of the extremes, the req uiremen t of a mixing dy namics can b e relaxed if the Pa reto approach is used, based up on considering the exceedances o ver a giv en th reshold. Requiring that the inv arian t measure locally scal es with a w ell defi ned exp onent - the local d imension -, w e show that the limiting distribution for the exceedances of th e observ ables previously studied with t he Gnedenko approac h is a Generalized Pareto distribution where the parameters dep end s only on the local dimensions and the v alue of the threshold. This result allo ws to ex t end the ex treme value th eory for dyn amical systems to th e case of regular motions. W e also provide connections with t he results obtained with the Gnedenko approach. In order to provide furt h er supp ort to our fin dings, w e p resen t the results of numerical exp eriments carried out considering the w ell-known Chirik ov standard map. I. INTRO DUCT ION Extreme v alue Theory was origina lly in tro duced b y Fisher and Tipp ett [1] and formalis e d b y Gnedenko [2], who sho wed that the distribution of the maxima of a s a m- ple of indep enden t identically distributed (i.i.d) sto chas- tic v ariables co n verges under very general co nditions to a member o f the so- called Gener alised Extr e me V alue (GEV) distribution. The attention of the scien tific com- m unity to the problem of understanding extreme v a lues theory is growing, also b ecause this theory is cr ucial in a wide class of applications for defining risk factors such as those r e la ted to insta bilities in the financia l markets and to natural hazards related to seismic, climatic a nd h ydro - logical extreme even ts. Even if the probability of extr e me even ts decre a ses with their magnitude, the damage that they ma y bring increase s rapidly with the magnitude a s do es the cost of pr o tection against them. F rom a theo- retical p oint o f view, extreme v alues of obser v ables are related to large fluctuations of the corresp onding under- lying system. An extensive account of r ecent results and relev a n t applica tio ns is given in [3]. The traditional (’Gnedenk o’) approach for the statisti- cal inference of extr emes is r elated to the or iginal results by Gnedenko [2]: we par tition the exp erimental time se- ries into bins of fixed length, we extr act the maximum of each bin, and fit the selected data to the GEV distribu- tion family us ing metho ds suc h as maximum likelihoo d estimation (MLE ) or L-moments. See [4] for a detailed ∗ Email: valerio. lucarini@zmaw .de ; Also at: Departmen t of Mathematics and Statistics, Univ ersity of Reading, Reading, UK. account of this metho dolog y . The selection of just one maximum in a fixed p erio d ma y lead to the loss of rele- v ant information on the lar g e fluctuations of the system, esp ecially w he n there are many la r ge v a lues in a given per io d [5]. This problem can b e taken care of b y con- sidering several of the larg est order statistics instead o f just the lar g est one. F or suc h maxima distr ibutions we exp ect co n vergence to the Generalize d Pareto Distribu- tion (GPD) in tro duced by Pick ands I I I [6] and B alkema and De Haan [7] to mo del the exceedances ov er a given threshold. W e call this the ’Pareto’ a pproach. Also this approach has b een widely adopted for studying empiri- cally natural ex treme phenomena such as those related to wa ves, winds, temper atures, earthquakes and floo ds [8–10]. Both the Gnedenko and P ar e to appr oaches w ere orig- inally designed to study extreme v alues for series of i.i.d v ar ia bles. In this ca se it is well known that a strong connections exists be t ween the tw o metho dolog ies, as we hav e that if blo ck maxima ob ey the GE V distribution, then exceedances ov er some hig h threshold will ha ve an asso ciated GPD. Mo reov er, the sha pe par ameter of the GPD and that of the corresp onding GEV distribution are ident ical [11]. As a r e s ult, s e veral pr actical methods (e.g. Hill’s and Pick ands’ estimators) develop ed fo r estimat- ing the sha pe para meter of the GEV distr ibution of th e extremes of a given time series are actually ba sed upo n comparing the GPD fits at v a rious thresholds [1 2, 13]. In practical terms, it app ears that, while the Gnedenko and Pareto appro aches provide equiv alent information in the asymptotic limit o f infinitely long time series, the GPD statistics is mor e robust when realistic, finite time series are considered ( see, e.g., [14]). In rec en t y ear s, esp ecially under the influence of the 2 rapid developmen t of numerical modelling in the g eo- ph ysica l sciences and of its applications for the investiga- tion of the so cio -economic impacts of extreme ev ents, it has beco me of great relev ance to understand whether it is p o ssible to apply the extreme v alue theor y on the time series of observ ables of deterministic dynamical sys tems . Carefully devised numerical exper imen ts on clima te mod- els of v arious degree s of co mplexit y have shown tha t the sp eed o f conv ergence (if a n y) of the statistical pr o per ties of the extr e mes definitely dep ends on the chosen clima tic v ar ia ble of interest [4 , 15–17]. Several papers ha ve addressed this iss ue at a more gen- eral level. A firs t imp orta nt result is that when a dy- namical system has a re g ular (perio dic of quasi- per io dic) behaviour, we do no t expect, in general, to find conv er- gence to GEV distr ibutions for the extremes of any ob- serv able. These re s ults have been presented by Balakr- ishnan et al. [18], and more recently , b y Nicolis et al. [19] and by Haiman [20]. A different mathematical approach to extreme v alue theory in dyna mical systems was prop osed in the land- mark paper by Collet [21], whic h has pa ved the wa y fo r the r ecent r esults obtaine d in th e last few years [22–25]. The starting p oint of all of these inv estigatio ns has b een to asso ciate to the sta tio nary sto c hastic pr oc e ss giv en by the dynamica l system, a new stationary indep endent s e- quence which ob eys o ne of the cla ssical thre e extreme v alue laws intro duced by Gnedenko [2]. T he assump- tions which are necessar y to observe a GEV distribu- tion in dynamical systems rely o n the choice of suitable observ ables (sp ecific functions of the distance b etw een the o rbit and the initial condition, chosen to b e on the attractor) and the fulfillment of particular mixing con- ditions that guar a n tee the indep endence o f subsequent maxima. Recent studies hav e shown that the r esulting parameters of the GEV distr ibutions can b e expres sed as simple functions o f the lo cal (aro und the initial co ndi- tion) dim ensio n of the attra c to r, and detailed numerical inv e stigations hav e clarified the co nditio ns under which conv ergence to th e theor etical GEV distr ibutions can be satisfactorily ac hieved when considering finite time series [26–28]. In this pap er, w e wis h to attempt a unification of these t wo lines o f work by using the Pareto rather than the Gnedenko approach. W e choose the same clas s of ob- serv ables pr e sen ted in [22–28] a nd show that, assuming only that the lo cal mea sures scales with the lo cal dimen- sion [29], it is p ossible to obtain by direc t in tegra tion a GPD for the thresho ld exceedences when considering a g e neric or bit of a dynamical systems, without requir- ing any spec ia l mixing pr ope rties. The par ameters will depe nd only o n the choice of the thresho ld and, more impo rtantly , on the lo cal dimension. Note that Castillo and Hadi [5 ] had a lready p ointed out that in the case of p erio dic or quasi-p erio dic motion the Gnedenko ap- proach to the ev aluation of the extr e me v a lue sta tistics is inefficien t, basica lly b ecaus e in the limit of very lar ge blo c ks, w e tend to obser v e always the same maximum in all bins. T o s uppor t our analytical results w e provide nu merica l expe r imen ts that we carry out considering the classic Chiriko v standard map [30]. This pap er is orga n- ised as follows. In sectio n 2 w e reca pitulate the extreme v alue theor y for dynamical sy stems obtained using the Gnedenko approa c h. In section 3 we present our general results obta ined using the Pareto appro ach. In sec tio n 4 w e provide support to our inv estiga tion by examining the res ults of the n umerical simulations p erformed o n the standard map. In s e c tion 5 w e pre s en t our final r emarks and future sc ien tific p ers pectives. II. GNEDENK O APPR OA CH: G ENERALIZED EXTREME V ALUE DISTRIBUTIONS IN DYNAMICAL SYSTEMS Gnedenko [2] studied the conv ergence of maxima of i.i.d. v a riables X 0 , X 1 , ..X m − 1 with cum ulative distribution function (cdf ) F ( x ) = P { a m ( M m − b m ) ≤ x } where a m and b m are normaliz- ing s equences and M m = max { X 0 , X 1 , ..., X m − 1 } . Under general h yp othesis on the nature of the paren t distribu- tion of data, Gnedenko [2] show ed that the as y mptotic distribution o f maxima, up to an a ffine change of v ar i- able, b elongs to a single family of gener alized distribu- tion ca lled GEV distribution whose cdf can b e written as: F GE V ( x ; µ, α, κ ) = e − t ( x ) (1) where t ( x ) = (  1 + κ ( x − µ α )  − 1 /κ if κ 6 = 0 e − ( x − µ ) /α if κ = 0 . (2) This ex pr ession holds for 1 + κ ( x − µ ) /α > 0, using µ ∈ R (lo cation parameter) and α > 0 (scale parameter) as sc a l- ing constants in pla ce of b m , and a m [31], in particular, in F ar anda et al. [27] we ha ve shown that µ = b m and α = 1 / a m , where κ ∈ R is the shap e para meter (also called the tail index). When κ → 0, the distr ibution co r- resp onds to a Gumbel type (Type 1 dis tr ibution). When the index is pos itiv e, it corres ponds to a F r´ echet (Type 2 distribution); when the index is neg ative, it corresp onds to a W e ibull (T yp e 3 distributio n). Let us cons ider a dynamical system (Ω , B , ν , f ), where Ω is the in v ariant set in some manifold, usually R d , B is the Borel σ -alg ebra, f : Ω → Ω is a measura ble map and ν a n f - in v a riant Borel measur e. In order to adapt the e x treme v alue theor y to dynamical systems, follow- ing [22–25], w e consider the stationary stochastic pro ces s X 0 , X 1 , ... given by: X m ( x ) = g (dist( f m ( x ) , ζ )) ∀ m ∈ N (3) 3 where ’dist’ is a distance in the am bient space Ω, ζ is a given p oint and g is an obs erv able f unction. The partial maximum in th e Gnedenk o appro ach is defined a s: M m = max { X 0 , ..., X m − 1 } . (4) Defining r = dist( x, ζ ), we consider the three classes of observ ables g i , i = 1 , 2 , 3: g 1 ( r ) = − log( r ) (5) g 2 ( r ) = r − β (6) g 3 ( r ) = C − r β (7) where C is a co nstant and β > 0 ∈ R . Using the ob- serv able g i we obtain conv erg ence of the statistics of the blo c k maxima of their time series obtained by evolving the dynamical s y stem to the Type i distribution if one can prove t wo sufficien t conditions called D 2 and D ′ , which basically imply a sort of indep endence of the series of extremes resulting from the mixing of the underlying dynamics [22]. The conditions cannot b e simply related to the usual concepts of strong or w eak mixing, but are indeed not obeyed by o bserv able s of systems featuring a regular dynamics or power-la w de c ay of co r relations. A co nnection also exists b etw een the existence of ex- treme v a lue laws and the statistics of fir st return a nd hitting times, which provide infor mation on how fast the po in t star ting fro m the initial co ndition ζ comes back to a neighborho o d of ζ , as shown by F reitas et al. [23] and F reitas et al. [32]. In pa rticular, they prov ed that for dynamical systems p ossessing an inv ariant measur e ν , the existence of an exp onential hitting time statistics on balls aro und ν -almost any po int ζ implies the exis - tence o f extreme v alue laws for one o f the observ ables of t yp e g i , i = 1 , 2 , 3 describ ed ab ove. T he co n verse is also true, namely if we hav e an extreme v a lue law which applies to the obser v ables of type g i , i = 1 , 2 , 3 achiev- ing a ma x im um at ζ , then w e hav e exp onential hitting time statistics to balls with cen ter ζ . Recently these re- sults hav e b een g e neralized to lo cal returns ar ound balls centered a t p erio dic points [24]. In F a randa et a l. [26, 27, 28] we analised b oth from an analytical and numerical point of view the extreme v alue distribution in a wide class of low dimensiona l maps. W e divided the time series of length k of the g i observ ables int o n bins ea c h containing the same num b er m o f ob- serv ations, and selected the maxim um (or the minim um) v alue in each o f them [33]. W e showed that at le ad- ing order (the formulas ar e asymptotically correct for m, k → ∞ ), the GE V parameter s in mixing maps can be written in terms of m (or equiv a le n tly n ) and the lo- cal dimension of the attractor D . W e ha ve: • g 1 -type o bs erv able: α = 1 D µ ∼ 1 D ln( k /n ) κ = 0 (8) • g 2 -type o bs erv able: α ∼ n − β D µ ∼ n − β D κ = β D (9) • g 3 -type o bs erv able: α ∼ n β D µ = C κ = − β D (10) Moreov er, we clearly show ed that o ther kind o f distribu- tions not belonging to the GEV family are obser v ed for quasi-p erio dic and p erio dic motio ns. II I. THE P A RETO APPRO ACH: GENERALIZED P A RETO DISTRIBUTIONS IN DYNAMICAL SYSTEMS W e define an exceeda nce a s z = X − T , which mea- sures by ho w m uch X exceeds the threshold T . As dis- cussed above, under the same co nditions under which the blo ck maxima of the i.i.d. sto ch as tic v aria bles X ob ey the GEV statistics, the exceeda nces z are asymp- totically dis tr ibuted according to the Generalised Pareto Distribution [11]: F GP D ( z ; ξ , σ ) =    1 −  1 + ξz σ  − 1 /ξ for ξ 6 = 0 , 1 − exp  − z σ  for ξ = 0 , (11) where the range of z is 0 ≤ z < ∞ if ξ ≤ 0 and 0 ≤ z ≤ σ /ξ if ξ > 0. W e c o nsider the same set up describ ed in the previous se ction and take in to account an o bserv a bles g = g (dist( x, ζ )) = g ( r ), such that g achiev es a ma xim um g max for r = 0 (finite or infinite) and is monotonically decr easing. W e study the exceedance ab ov e a threshold T defined as T = g ( r ∗ ). W e obtain an ex ceedence every time the dista nce b etw een the o r- bit of the dynamica l system and ζ is smaller than r ∗ . Therefore, we define the exceedances z = g ( r ) − T . By the Bayes’ theorem, we hav e that P ( r < g − 1 ( z + T ) | r < g − 1 ( T )) = P ( r < g − 1 ( z + T )) /P ( r < g − 1 ( T )). In terms of in v ariant meas ure of the system, we hav e that the probability H g,T ( z ) of observing an e xceedance of at least z given that an exceedence o ccurs is giv en by: H g,T ( z ) ≡ ν ( B g − 1 ( z + T ) ( ζ )) ν ( B g − 1 ( T ) ( ζ )) . (12) Obviously , the v alue of the previous expr e ssion is 1 if z = 0. In agreement with the conditions g iven on g , the e x pression contained in Eq. (12 ) monotonica lly de- creases with z and v anishes when the r adius is g iv en b y g − 1 ( g max ). Note that the cor resp onding cdf is g iven by F g,T ( z ) = 1 − H g,T ( z ). In order to address the problem of extremes, we hav e to consider small r a dii. At this rega rd we will in vok e, and assume, the existence of the following limit lim r → 0 log ν ( B r ( ζ )) log r = D ( ζ ) , for ζ chosen ν − a.e. , (13) where D ( ζ ) is the lo cal dimension of the attractor [2 9]. Therefore, w e r ewrite the fo llowing expression for the tail 4 probability of exceedance: H g,T ( z ) ∼  g − 1 ( z + T ) g − 1 ( T )  D . (14) where w e hav e droppe d the ζ dep endence of D to simplify the notation. By substituting g with spe cific observ able we ar e c onsidering, w e obtain explicitly the corresp ond- ing extreme v a lue distribution law. By choosing a n observ able of the form given by either g 1 , g 2 , or g 3 , we derive a s extr eme v alue distribution law one mem b er of the Gene r alised Pareto Distribution family given in Eq. (11). Results a re detailed b elow: • g 1 -type o bs erv able: σ = 1 D ξ = 0 ; (15) • g 2 -type o bs erv able: σ = T β D ξ = β D ; (16) • g 3 -type o bs erv able: σ = ( C − T ) β D ξ = − β D . (17) The prev ious expr essions show that there is a simple al- gebraic link b etw een the par ameters of the GPD a nd the lo cal dimensio n of the attractor around the p oint ζ . This implies that the statistics of extremes provides us with a new algor ithmic tool for estimating the lo ca l fine struc- ture of the attracto r. These results show that it is pos si- ble to der ive gener al pro per ties for the extreme v alues o f the o bs erv ables g 1 , g 2 , or g 3 independently on the qualita- tive prop erties of the underlying dynamics, b e the s ystem per io dic, quasi-p erio dic, or chaotic. Ther efore, b y taking the P ar eto instead of the Gnedenko approa c h, w e are able to o vercome the m ixing conditions (or the requirements on the prope r ties of the hitting time s tatistics) needed to derive a general extre me v alue theory for dyna mical sys- tems, as propo s ed in [21–25]. In [26–28] w e had prop osed that the rea son why a link b etw e en the extreme v alue the- ory and the lo cal prop erties of the in v ar iant measur e in the vicinity of the po in t ζ can be explained b y the fact that selecting the extremes o f the o bs erv ables g 1 , g 2 , or g 3 amounts to perfo r ming a zo om around ζ . In the c ase of the Gnedenko appro ach, such a picture is acc urate only if the dynamics is mixing (time and spatial selection cri- teria ar e equiv alent). Instead, in the case of the Pareto approach, this is literally what we are doing when writing Eq. (14), as we ar e remapping the r adius of the ball in a monotonic fashion though the inv erse o f the g - functions. A. Relationship be t ween the Gnedenko and P areto approac hes The rela tion b et ween GEV a nd GPD parameters have bee n alrea dy discussed in literature in cas e of i.i.d v ar i- ables [13, 14, 34, 3 5]. Coles [13] and Katz et al. [34] have pr ov en that the cdf of the GE V defined as F GE V ( z ; µ, α, κ ) can b e a symptotically wr itten a s that of GPD under a high enough thres hold as follows: F GE V ( z ; µ, α, κ ) ∼ F GP D ( z ; T , σ , ξ ) = = 1 −  1 − ξ  z − T σ  1 /ξ (18) where κ = ξ , σ = α + ξ ( T − µ ), and T = µ + σ ξ ( λ − ξ − 1), with ln( α ) = ln( σ ) + ξ ln( λ ). In the present case, we hav e to compa re Eq s. (8)-(10) for GEV with Eq s . (1 5 )-(17) for GPD, keeping in mind that the GEV r esults hold only under the mixing conditions discussed b efore. While it is immediate to chec k that κ = ξ , the other relations hips are v alid in the limit of large n , as ex p ected. IV. NUMERICAL INVESTIGA TION The standar d ma p [36] is an are a-preserv ing chaotic map defined on the bidimensiona l torus, a nd it is one of the most widely- studied examples of dynamical c hao s in ph ysics . The corres ponding mec hanical system is usually called a kicked ro tator. It is defined as: ( y t +1 = y t − K 2 π sin(2 π x t ) mo d 1 x t +1 = x t + y t + 1 mo d 1 (19) The dyna mics o f the map giv en in E q . (19 ) can b e regular or chaotic. F or K << 1 the motion follows quasi pe rio dic orbits for a ll initial co nditio ns , wherea s if K >> 1 the motion turns to b e chaotic and irr egular. An in teresting behavior is achiev ed when K ∼ 1: in this ca se we ha ve co existence o f regular a nd chaotic motions depending on the chosen initia l conditions [37]. W e perfo r m for v arious v a lues of K ra nging from K = 10 − 4 up to K = 1 0 2 an ensemble of 200 simula- tions, each characterise d b y a different initial co ndition ζ randomly taken on the bidimensional to rus, and we com- pute for each orbit the observ ables g i , i = 1 , 2 , 3. In each case, the ma p is iter ated until obtaining a statistics con- sisting 10 4 exceedances, where the threshold T = 7 · 10 − 3 and β = 3. W e hav e carefully ch eck ed tha t all the re- sults are indeed robust with respe c t to the choice of the threshold and of the v a lue of β . F or each o rbit, w e fit the statistics of the 10 4 exceedances v alues of the observ- ables to a GPD distr ibution, using a ML E estimation [5] implemented in the MA TLAB c  function gp dfit [3 8]. The results are shown in Fig. 1 for the infer r ed v alues of ξ and σ and should b e compared with E qs. (15)-(17). When K ≪ 1, we obtain that the estimates of ξ and σ are compatible with a dimension D = 1 for all the initial conditions: we have that the ensemble spread is neg lig i- ble. Similar ly , for K ≫ 1, the estimates for ξ and σ agr ee remark ably well with having a lo cal dimension D = 2 for all the initial conditions. In the transition regime, whic h o ccurs fo r K ≃ 1, the ensem ble spread is muc h hig her, bec ause the scaling prop erties of the measur e is differ- ent among the v arious initial conditions. As exp ected, 5 the ensemble av erage s of the pa rameters change mo no- tonically fr om the v alue p ertaining to the regular r egime to that p ertaining to the c haotic regime with incr e asing v alues of K . Ba sically , this measur es the fact that the so-called r egular islands s hrink with K . Note that in the case of the o bserv a ble g 1 , the es timate of the ξ is ro- bust in a ll regimes, even if, as exp ected, in the trans itio n betw een low and high v alues of K the ensemble spread is larg er. These re s ults can also b e co mpared with the analysis presented in F ara nda et al. [26], where w e used the Gnedenk o approach. In that case, the v alues ob- tained in the regula r regions were inconsistent with the GEV findings, the very reaso n b eing that the dynamics was indee d not mixing. Here, it is clear that the statis- tics can b e c o mputed in all cases, a nd we have a p ow erful metho d for dis criminating regular fro m chaotic b ehaviors through the analysis o f the inferr ed lo cal dimension. V. CONCLUSIONS The growing attention of the s cien tific co mm unity in understanding the behavior of extreme v alues hav e led, in the r ecent pa st, to the development o f an extreme v alue theory for dynamica l systems. In this fra mew ork , it has bee n shown that the sta tistics of ex treme v alue can b e linked to the statistics of return in a neighbor hoo d of a certain initial co nditions by choosing sp ecial observ ables that dep end on the distance betw een the itera ted tra jec- tory and its s ta rting po in t. Until now, rig orous results hav e b een obtained assuming the existence of an in v ari- ant mea s ure for the dynamical sy stems and the fulfill- men t of indep endence re q uirement o n the se r ies of ma x- ima achiev ed by imp o sing D ′ and D 2 mixing conditions, or, alter natively , assuming a n exp onential hitting time statistics [22–25]. The par a meters of the GE V distri- bution obtained c ho osing a s observ ables the fu nction g i , i = 1 , 2 , 3 defined a bove dep end on the lo ca l dimension of the attractor D and numerical algorithms to p erform sta- tistical inference can b e set up for mixing systems having bo th absolutely contin uous and singular inv ariant mea- sures [27, 28, 39, 40]. Instea d, when considering systems with regular dynamics, the statistics of the blo ck maxima of any observ able do es not conv erge to the GEV family [18, 19]. T aking a complement ar y p oint of v ie w, in this pap er we have studied the statistics of exceedances for the s ame class of obse r v ables and derived the limiting distributions assuming only the existence of an inv ar iant measure and the p oss ibilit y to define a lo cal dim ensio n D ar o und the po in t ζ of interest. T o prove that the limiting distribution is a GPD w e did not use any further conditions. In partic- ular no assumptions on the mixing natur e of the maxima sequence ha ve b een made. T his means that a GPD lim- iting distribution holds for the statistics of exceedance of ev ery kind of dyna mical systems and it dep ends only on the thres hold v alue a nd on the lo ca l dimension once we choose the o bs erv ables g i , i = 1 , 2 , 3 . Other functions can conv erge to the limiting behaviour of the GPD family if they asymptotically b ehave like the g i ’s (compare the discussion in [2 3]). Nonetheless, this requires , ana lyti- cally , to per form separ ately the limit for the threshold T going to g max and that for the r adius of the ball going to zero. In practical terms , this requires, p otentially , muc h stricter selec tio n criteria for the exceedances when finite time serie s ar e considered. W e note that, as the parameter ξ is inv ersely prop or- tional to D , one ca n exp ect that each time we ana lyse systems of intermediate or high dimensionality , the dis- tributions for g 2 and g 3 observ ables will be virtually in- distinguishable from what obtained co nsidering the g 1 observ able: the ξ = 0 is in this sense an attr acting mem- ber of the GPD family . This may also explain, at lea st qualitatively , why the Gum b e l ( k = 0 for the GEV fa m- ily) distribution is so efficient in describing the extremes of a large v ariety of natural phenomena [3]. The universality of this approach allows to resolve the debate on whether there exists or not a g eneral way to ob- tain informa tio n ab out extreme v alues for quasi-p erio dic motions raised in Ba lakrishnan et al. [18] and Nicolis et al. [ 19]. This is due to the fact that considering several of the larg e st order statistics instead of just the large s t one we can study orbits wher e numerous exceeda nce a re observed in a given blo ck, as it happ ens for s ystems with per io dic or quasi- per iodic b ehaviors. As an example, one may consider a system with m ultiple commensurable fre- quencies: choo sing a block length lar ger or equal to the smallest commo n per io d, we sele c t alwa ys the same v alue for any considere d observ able. On the other hand for mixing maps w e find, as exp ected, an asymptotic eq uiv- alence o f the results obtained via the Gnedenk o and via the Pareto a pproach. The Pareto appr oach provides a way to reconstruct lo- cal pr ope r ties of in v ariant measures: once a thr e s hold is chosen and a suitable exceeda nce s tatistics is recorded, we can compute the lo cal dimensio n for different initial conditions taken on the attracto r . This is true also in the o pp osite direction: if the knowledge of the exact v alue for the lo cal dimension is av ailable, once we chose a small enough radius (threshold), it is pos sible to compute a pri- ori the pro per ties of ex tremes without doing a n y further computations. In fact, the expressio n for the parameters Eqs. (15)-(17) do not co n tain a n y dep endence on the prop erties of the dynamics except the local dimensio n. Besides the analytical results, we hav e proved that Pareto approach is easily accessible for numerical in vesti- gations. The algorithm used to perfo r m numerical simu - lations is versatile and computationally accessible: unlike the GE V a lgorithm that requires a very high n umber of iterations to obtain unbiased statistics, using the P ar eto approach w e can fix a priori a v alue for the threshold and the num b er of maxima necess ary to construct the statis- tics. With the simulations car ried out o n the standard maps, w e o btain meaning ful res ults with a muc h s ma ller statistics with resp ect to what o bserved when considering the Gnedenko approach. 6 W e hop e tha t the pres en t con tribution ma y pro vide a to ol that is not o nly useful for the analysis of extr e me even ts itself, but als o for characterising the dynamica l structure of attractors b y giving a robust way to c o m- pute the lo cal dimensions , with the new pos s ibilit y of emb ra cing also the c a se of quasi- p erio dic motions. 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Processes Geophys 18 , 7 (2011). 7 −3 −2 −1 0 1 2 −0.2 −0.1 0 0.1 0.2 log10(K) ξ (g 1 ) a) −3 −2 −1 0 1 2 0 0.1 0.2 0.3 0.4 log10(K) ξ (g 2 ) b) −3 −2 −1 0 1 2 −0.5 −0.4 −0.3 −0.2 −0.1 log10(K) ξ (g 3 ) c) −3 −2 −1 0 1 2 0 0.5 1 1.5 log10(K) σ (g 1 ) d) −3 −2 −1 0 1 2 0 1 2 3 log10(K) σ (g 2 ) e) −3 −2 −1 0 1 2 0.02 0.04 0.06 0.08 0.1 log10(K) σ (g 3 ) f) FIG. 1. GPD parameters for t he observ ables g i , i = 1 , 2 , 3 computed ove r orbits of the standard map, for v arious v alues of the constant K . F or each v alue of K , results refer to an esnsem ble of 200 randomly chosen initial conditions ζ . The notation p ( g i ) indicates the p arameter p computed using t h e extreme v alue statistics of the observ able g i . a) ξ ( g 1 ) VS K , b) ξ ( g 2 ) VS K , c) ξ ( g 3 ) VS K , d) σ ( g 1 ) VS K , e) σ ( g 2 ) VS K , f ) σ ( g 3 ) VS K . Blac k solid li nes: ensemble-a verag e v alue. Blac k dotted lines: ensem ble spread ev aluated as one stand ard deviation of the ensemble. Green lines: theoretical v alues f or regular orbits. Red lines: theoretical v alues for c haotic orbits.