Extreme Value distribution for singular

measures

Faranda, Davide

Department of Mathematics and Statistics, University of Reading;

Whiteknights, PO Box 220, Reading RG6 6AX, UK. d.faranda@pgr.reading.ac.uk

Lucarini, Valerio

Department of Meteorology, University of Reading;

Department of Mathematics and Statistics, University of Reading;

Whiteknights, PO Box 220, Reading RG6 6AX, UK. v.lucarini@reading.ac.uk

Turchetti, Giorgio

Department of Physics, University of Bologna.INFN-Bologna

Via Irnerio 46, Bologna, 40126, Italy. turchett@bo.infn.it

Vaienti, Sandro

UMR-6207, Centre de Physique Théorique, CNRS, Universités d'Aix-Marseille I,II,

Université du Sud Toulon-Var and FRUMAM

(Fédération de Recherche des Unités de Mathématiques de Marseille);

CPT, Luminy, Case 907, 13288 Marseille Cedex 09, France.

vaienti@cpt.univ-mrs.fr

Abstract

In this paper we perform an analytical and numerical study ofExtreme Value distributions in discrete dynamical systems that havea singular measure. Using the block maxima approach described inFaranda et al. [2011] we show that, numerically, the Extreme Valuedistribution for this maps can be associated to the Generalised Ex-treme Value family where the parameters scale with the informationdimension. As numerical analysis we consider di�erent low dimen-sional maps. In perfect fractals such as the middle third Cantor set

1

or the Sierpinskij triangle obtained using Iterated Function Systemsexperimental parameters show a very good agreement with respect tothe guessed theoretical values. In Multifractals and maps that have astrange attractor such as Lozi and Henon maps a slower convergence tothe Generalised Extreme Value distribution is observed when ensembleof initial conditions is considered. Netherless, within the uncertaintycomputed range, results are in excellent agreement with the theoreticalscaling proposed.

1 Introduction

Extreme Value Theory (EVT), developed for the study of stochastical se-ries of independent and identical distributed variables by Fisher and Tippett[1928] and formalized by Gnedenko [1943], has been successfully applied todi�erent scienti�c �elds to understand and possibly forecast events that occurwith very small probability but that can be extremely relevant from an eco-nomic or social point of view: extreme �oods [Gumbel, 1941], [Sveinsson andBoes, 2002], [P. and Hense, 2007], amounts of large insurance losses [Brodinand Kluppelberg, 2006], [Cruz, 2002]; extreme earthquakes [Sornette et al.,1996], [Cornell, 1968], [Burton, 1979]; meteorological and climate events [Fe-lici et al., 2007a], [Felici et al., 2007b],[Vitolo et al., 2009], [Altmann et al.,2006], [Nicholis, 1997], [Smith, 1989] has been modeled with this theory.When dealing with natural hazard it is crucial to de�ne the so called riskfactor: EVT is applied in this context to de�ne not only to derive the seismicrisk [Burton, 1979], [Martins and Stedinger, 2000], but also the �nancial riskafter instabilities in stock markets [Gilli and Këllezi, 2006], [Longin, 2000],[Embrechts et al., 1999].

To analyze the extreme value distribution in a series of data two mainapproaches can be applied: the Pick-over-threshold approach and the Block-Maxima approach. The former consists in looking at exceedance over highthresholds Todorovic and Zelenhasic [1970] and a Generalized Pareto distri-bution is used for modeling data obtained as excesses over thresholds [Smith,1984], [Davison, 1984], [Davison and Smith, 1990].The so called block-maxima approach is widely used in climatological and�nancial applications since it represents a very natural way to look at ex-tremes in �xed time intervals: it consists of dividing the data series of someobservable into bins of equal length and selecting the maximum (or the min-imum) value in each of them [Coles et al., 1999], [Felici et al., 2007a], [Katzand Brown, 1992], [Katz, 1999], [Katz et al., 2005].

2

Sandro, credo che tu possa modi�care al meglio la parte chesegue

Since dynamical systems theory can be used to understand features ofphysical systems like climate and forecast �nancial behaviors, many authorshave studied how to extend EVT to these �eld. When dealing with dynam-ical systems we have to know what kind of properties (i.e. stability, degreeof mixing, correlations decay) are related to Gnedenko's hypotheses and alsowhich observables we must consider in order to obtain an EV distribution.Furthermore, even if the convergence is achieved, we should evaluate howfast it is depending on all parameters and properties used. Empirical stud-ies show that in some cases a dynamical observable obeys to the extremevalue statistics even if the convergence is highly dependent on the kind ofobservable we choose [Vannitsem, 2007], [Vitolo et al., 2009]. For example,Balakrishnan et al. [1995] and more recently Nicolis et al. [2006] and Haiman[2003] have shown that for regular orbits of dynamical systems we don't ex-pect to �nd convergence to EV distribution.The �rst rigorous mathematical approach to extreme value theory in dy-namical systems goes back to the pioneer paper by P. Collet in 2001 [Collet,2001]. Collet got the Gumbel Extreme Value Law (see below) for certainone-dimensional non-uniformly hyperbolic maps which admit an absolutelycontinuous invariant measure and exhibit exponential decay of correlations.Collet's approach used Young towers [Young, 1999], [Young, 1998] and hissuggestion was successively applied to other systems. Before quoting them,we would like to point out that Collet was able to establish a few conditions(usually called D and D0) and which have been introduced by Leadbetteret al. [1983] with the aim to associate to the stationary stochastic processgiven by the dynamical system, a new stationary independent sequence whichenjoyed one of the classical three extreme value laws, and this law could bepulled back to the original dynamical sequence. Conditions D and D0 re-quire a sort of independence of the stochastic dynamical sequence in termsof uniform mixing condition on the distribution functions. Condition D wassuccessively improved by Freitas and Freitas [Freitas and Freitas, 2008], inthe sense that they introduced a new condition, called D2, which is weakerthan D and that could be checked directly by estimating the rate of decayof correlations for Hölder observables 1. . We notice that conditions D2 and

1We brie�y state here the two conditions, we defer to the next section for more detailsabout the quantities introduced. If Xn; n � 0 is a stochastic process, we can de�neMj;l � fXj ; Xj+1; � � � ; Xj+lg and we put M0;n = Mn. Moreover we set an and bn twonormalizing sequences and un = x=an + bn, where x is a real number, cf. next section forthe meaning of these variables. The condition D2(un) holds for the sequence Xn if for any

3

D0 allow immediately to get Extreme Value Laws for absolutely continuousinvariant measures for uniformly one-dimensional expanding dynamical sys-tems: this is the case for instance of the 1-D maps with constant densitystudied in Sect. 3 below. Another interesting issue of Collet's paper wasthe choice of the observables g's whose values along the orbit of the dynam-ical systems constitute the sequence of events upon which we successivelysearch for the partial maximum. Collet considered a function g(dist(x; �)) ofthe distance with respect to a given point �, with the aim that g achievesa global maximum at almost all points � in the phase space; for exampleg(x) = � log x. Using a di�erent g, Freitas and Freitas [Moreira Freitas andFreitas, 2008] were able to get the Weibull law for the family of quadraticmaps with the Benedicks-Carlesson parameters and for � taken as the criticalpoint or the critical value, so improving the previous results by Collet whodid not keep such values in his set of full measure.The latter paper [Moreira Freitas and Freitas, 2008] strongly relies on condi-tion D2; this condition has also been invoked to establish the extreme valuelaws on towers which model dynamical systems with stable foliations (hyper-bolic billiards, Lozi maps, Hénon di�eomorphisms, Lorenz maps and �ows).This is the content of the paper by Gupta, Holland and Nicol [Gupta et al.,2009]. We point out that the observable g was taken in one of three di�erentclasses g1; g2; g3, see Sect. 2 below, each one being again a function of thedistance with respect to a given point �. The choice of these particular formsfor the g's is just to �t with the necessary and su�cient condition on thetail of the distribution function F (u), see next section, in order to exist anon-degenerate limit distribution for the partial maxima [Freitas et al., 2009],[Holland et al., 2008]. The paper Gupta et al. [2009] also covers the easiercase of uniformly hyperbolic di�eomorphisms, for instance the Arnold Catmap which we studied in Sect. 3.2.Another major step in this �eld was achieved by establishing a connectionbetween the extreme value laws and the statistics of �rst return and hittingtimes, see the papers by Freitas, Freitas and Todd [Freitas et al., 2009], [Fre-itas et al., 2010b]. They showed in particular that for dynamical systems pre-serving an absolutely continuous invariant measure or a singular continuousinvariant measure � , the existence of an exponential hitting time statistics

integer l; t; n we have j�(X0 > un;Mt;l � un)� �(X0 > un)�(Mt;l � un)j � (n; t), where (n; t) is non-increasing in t for each n and n (n; tn) ! 0 as n ! 1 for some sequencetn = o(n), tn !1.

We say condition D0(un) holds for the sequence Xn if limk!1 lim supn nP[n=k]

j=1 �(X0 >un; Xj > un) = 0. Whenever the process is given by the iteration of a dynamical systems,the previous two conditions could also be formulated in terms of decay of correlationintegrals, see Freitas and Freitas [2008], Gupta [2010]

4

on balls around � almost any point � implies the existence of extreme valuelaws for one of the observables of type gi; i = 1; 2; 3 described above. Theconverse is also true, namely if we have an extreme value law which applies tothe observables of type gi; i = 1; 2; 3 achieving a maximum at �, then we haveexponential hitting time statistics to balls with center �. Recently these re-sults have been generalized to local returns around balls centered at periodicpoints [Freitas et al., 2010a]. We would like to point out that the equiv-alence between extreme values laws and the hitting time statistics allowedto prove the former for broad classes of systems for which the statistics ofrecurrence were known, for instance for expanding maps in higher dimension.

����������������In Faranda et al. [2011] we have shown that there exists an algorith-

mic way to study EVT for discrete time dynamical systems whit an ab-solutely continuous invariant measure and satis�es mixing properties if ablock-maxima approach is used. We have established the best conditionsto observe convergence to the analytical results highlighting deviations fromtheoretical expected behavior depending on the number of maxima and num-ber of block-observation.

In this work we test our algorithm for maps that do not have an abso-lutely continuous invariant measure but a singular one. First we will showthat for Iterated Functions Systems that generate perfect fractals there existsan extreme value distribution which parameters can be related to the fractaldimension of the set. When the IFS is used to generate a multifractal object,an Extreme Value distribution can be still observed and the parameters canbe again related with the so called Information dimension (see Sect. 3). Incase of strange attractor, such as Lozi and Henon, we will show that theconvergence to the extreme value distribution is highly sensitive to initialconditions and global properties such as informations on the dimension canbe obtained by ensemble average. In this sense, in a multifractal object, thealgorithm described here acts like a magnifying glass on the structure of theattractor around the chosen initial condition.The work is organized as follow: in section 2 we brie�y recall methods andresults of EVT for independent and identical distributed (i.i.d.) variablesand dynamical systems. In section 3 we porpoise a scaling for the parame-ters based on the information dimension. In section 4 we present numericalexperiments on di�erent kind of maps are presented.

5

2 De�nitions and EVT

Gnedenko [1943] studied the convergence of maxima of i.i.d. variables

X0; X1; ::Xn�1

with cumulative distribution (cdf) F (x) of the form:

F (x) = Pfan(Mn � bn) � xg

Where an and bn are normalizing sequences andMn = maxfX0; X1; :::; Xn�1g.It may be rewritten as F (un) = PfMn � ung where un = x=an + bn. Suchtypes of normalizing sequences converge to one of the three type of ExtremeValue (EV) distribution if necessary and su�cient conditions on parent dis-tribution of Xi variables are satis�ed [Leadbetter et al., 1983]. Let us de�nethe right endpoint xF of a distribution function F (x) as:

xF = supfx : F (x) < 1g (1)

then, it is possible to compute normalizing sequences an and bn using thefollowing corollary of Gnedenko's theorem :Corollary (Gnedenko): The normalizing sequences an and bn in the con-vergence of normalized maxima Pfan(Mn � bn) � xg ! F (x) may be taken(in order of increasing complexity) as:

� Type 1: an = [G( n)]�1; bn = n;

� Type 2: an = �1n ; bn = 0;

� Type 3: an = (xF � n)�1; bn = xF ;

where n = F�1(1� 1=n) = inffx;F (x) � 1� 1=ng (2)

G(t) =

Z xF

t

1� F (u)

1� F (t)du; t < xF (3)

In [Faranda et al., 2011] we have shown that this approach is equivalentto �t unnormalized data directly to a single family of generalized distributioncalled GEV distribution with cdf:

FG(x;�; �; �) = exp

(�

�1 + �

�x� �

�

���1=�

)(4)

6

which holds for 1 + �(x � �)=� > 0, using � 2 R (location parameter)and � > 0 (scale parameter) as scaling constants in place of bn, and an[Pickands III, 1968], in particular the following relations hold:

� = bn � =1

an:

� 2 R is the shape parameter also called the tail index: when � ! 0, thedistribution corresponds to a Gumbel type (also called Type 1 distribution).When the index is positive, it corresponds to a Fréchet (type 2 distribution);when the index is negative, it corresponds to a Weibull (type 3 distribution).

In order to adapt the extreme value theory to dynamical systems, we willconsider the stationary stochastic process X0; X1; ::: given by:

Xn(x) = g(dist(fn(x); �)) 8n 2 N (5)

where 'dist' is a Riemannian metric on , � is a given point and g is anobservable function, and whose partial maximum is de�ned as:

Mn = maxfX0; :::; Xn�1g (6)

The probability measure will be here the invariant measure � for thedynamical system. As we anticipated in the Introduction, we will use threetypes of observables gi; i = 1; 2; 3, suitable to obtain one of the three typesof EV distribution for normalized maxima:

g1(x) = � log(dist(x; �)) (7)

g2(x) = dist(x; �)�1=� (8)

g3(x) = C � dist(x; �)1=� (9)

where C is a constant and � > 0 2 R.These three types of functions are representatives of broader classes which arede�ned, for instance, throughout equations (1.11) to (1.13) in Freitas et al.[2009]. It is easy to understand where they arise. The Gnedenko corollarysays that the di�erent kinds of extreme value laws are determined by thedistribution of F (u) = �(X0 � u) and by the right endpoint of F , xF . Wewill see in the next section that the local invertibility of gi; i = 1; 2; 3 inthe neighborhood of 0 together with the Lebesgue's di�erentiation theorem(which basically says that whenever the measure � is absolutely continuouswith respect to Lebesgue with density �, the the measure of a ball of radius �

7

centered around almost any point x0 scales like ��(x0)), allow us to computethe tail of F and according to this tail one will have the three types of extremelaws. At this regard let us compare Th. 1.6.2. in Leadbetter et al. [1983]with formulas (1.11)-(1.13) in Freitas et al. [2009]: one see immediately howand why the assumptions for the gi translate into similar conditions for thethree di�erent types of asymptotic behavior for the distribution function F .

3 Extreme Value distributions for maps with

singular measures

In this section we will make a conjecture on the kind of distribution of Ex-treme Value if we consider maps with singular measures and satisfy the con-ditions D2 and D0, su�cient to get extreme valuers distributions: The mainproblem when dealing with maps that have singular density measure �(�) isin the computation of the integral:

�(B�(�)) =

ZB�(�)

�(�)d� (10)

where B�(�) is the d-dimensional ball of radius � centered in �.We have to know the value of this integral in order to evaluate F (u) and,therefore, the sequences an and bn.As shown in the previous section � is linked to the observable type: in allcases, since we substitute u = 1 � 1=n, � ! 0 means that we are interestedin n!1.To give a solution for this problem we �rst recall the di�erent notions offractal dimensions introduced in Grassberger [1985]:

� Haussdor� (capacity) dimension. The Haussdor� dimension of acompact metric space X is a real number D0 such that:

D0 = lim�!0+

ln(N(�))

ln(�)(11)

(if the limit exists), where N(�) is the number of elements forming a�nite cover of the relevant metric space and � is a bound on the diameterof the sets involved.

� Information dimension Let us de�ne the Information function to be

I= �MXi=1

Pi(�) ln(Pi(�))

8

where Pi(�) is the natural measure or probability that i-th element ispopulated, then the Information dimension is de�ned as:

D1 = lim�!0+

I

ln(�)(12)

� Correlation dimension Let us de�ne the correlation integral:

C(�) = limN!1

1

N2

1Xi;j=1

H(�� jxi � xjj)

where i 6= j and H is the Heaviside step function, then the correlationdimension is de�ned as:

D2 = lim�;�0!0+

lnhC(�)C(�0

iln���0

� (13)

if the limit above exists.

One interesting relationship holds between the dimensions introduced:

D2 � D1 � D0

and the equality holds if we deal with a perfect fractal [Grassberger, 1983].In this case we can de�ne the unique fractal dimension D.

For our analysis the dimensionD1 is the one that should produce the rightExtreme Value laws since it is related with the scaling of the balls B�(�) [].Whit D1 as the right scaling dimension, a �rst order approximation of theintegral in equation 10 is:

�(B�(�)) / (�;D1)�D1 (14)

where (�;D1) is, in general, a non-smooth function of the initial condi-tion �. This result is not rigorous, therefore we must check the agreementbetween numerical experiments and this kind of scaling. Besides, we shouldensure that � belongs to an invariant measure that can be also singular butatomless, in agreement with Freitas et al. [2010b].

Once we know that eq. 14 holds, it is possible to compute the expectedparameters for the three types of observables:

9

Case 1: g1(x)= -log(dist(x,�)). We proceed starting from 5 and 6 weknow that:

1� F (u) = 1� �(g(dist(x; �)) � u)

= 1� �(� log(dist(x; �)) � u)

= 1� �(dist(x; �) � e�u)

(15)

where, in the last line we use a sort of Lebesgue's Di�erentiation Theo-rem without a good mathematical justi�cation. In fact it is clear that ourExtreme Value distribution can be discontinuous. Nevertheless, we will shownumerically that this EV distribution may be thought as a product of the cor-respondent continuous GEV distribution times the characteristic functionsof dist(x; �). Then:

1� F (u) ' �(Be�u(�)) / �;D1e�uD1 (16)

To use Gnedenko corollary it is necessary to calculate uF

uF = supfu;F (u) < 1g

in this case uF = +1.Using Gnedenko equation 3 we can calculate G(t) as follows:

G(t) =

Z1

t

1� F (u)

1� F (t)du =

Z1

t

e�uD1

e�tD1du =

1

D1

Z1

td

e�v

e�tD1dv =

1

d(17)

According to the Leadbetter et al. [1983] proof of Gnedenko theorem wecan study both an and bn or n convergence as:

limn!1

n(1� Ff n + xG( n)g) = e�x

limn!1

nD1e�D1( n+xG( n)) = e�x (18)

then we can use the connection between n and normalizing sequences to�nd an and bn.

By using equation 18:

n 'ln(n(�;D1))

D1

so that:

an = D1 bn = �1

D1ln(n) +

ln((�;D1))

D1

10

And, again, for the observable properties we expect:

� = 0

Case 2: g2(x)=dist(x,�)�1=�. We can heuristically proceed as for g1:

1� F (u) = 1� �(dist(x; �)�1=� � u)

= 1� �(dist(x; �) � u��)

= �(Bu��(�)) ' (�;D1)u��D1

(19)

in this case uF = +1.

n = F�1(1� 1=n) = �(n(�;D1))1=(�D1) (20)

and, as discussed in Faranda et al. [2011], using Beirlant [2004] choice ofnormalizing sequences we expect:

bn = c � n��

where c 2 R is a constant.Again we make a conjecture also on the value of �. In [Freitas et al.,

2009], for absolutely continuous invariant measure we had:

� = 1=�d

therefore, in this case we expect:

� = 1=�D1

Case 3: g3(x)=C-dist(x,�)1=�. Eventually we compute an and bn for theg3 observable class:

1� F (u) = 1� �(C � dist(x; �)1=� � u)

= 1� �(dist(x; �) � (C � u)�)

= �(B(C�u)�(�)) ' (�;D1)(C � u)�D1

(21)

in this case uF = C.

n = F�1(1� 1=n) = C � (n(�;D1))1=(�d) (22)

11

n = F�1(1� 1=n) = C � (n(�;D1))1=(�d) (23)

For type 3 distribution:

an = (uF � n)�1; bn = uF ; (24)

and, the same argument described for g2 can be applied here to obtain:

� = �1=�D1

4 Numerical Experiments

For the numerical experiments we have used a wide class of maps that havesingular measure, also considering the case of strange attractors such as theones observed iterating Lozi Map or Henon map. The algorithm used is thesame described in Faranda et al. [2011]: we chose an initial conditions � thatbelongs to the invariant measure, then we iterate the map and use the blockmaxima approach to obtain the GEV distribution.

In the previous section, we have claimed that this distribution is dis-continuous, therefore, to compute the experimental parameters a maximum-likelihood estimation (MLE) is not suitable since the MLE methods is consis-tent under the following conditions: identi�cation, compactness, continuityand dominance [Newey and McFadden, 1994]. In the case of singular mea-sures the distribution of extrema cannot be �tted with this methods as thecompactness conditions and the continuity conditions fail. For these reasonwe have chosen to use a di�erent method to obtain the �tted parameters thatis the L-moments estimation detailed in Hosking [1990]. This method doesnot imply any continuity or compactness of the distribution since it is purelydiscrete. The relationship between the moments and the parameters of theGEV distribution are described in Hosking [1990], while the 95% con�denceintervals has been derived using a bootstrap procedure. As comparison wehave checked that the results presented in Faranda et al. [2011] are com-parable with MLE and L-moments methods. We have found that both themethods give similar results even if L-moments has an uncertainty on theestimation of parameters generally bigger.

Eventually, the discontinuity of the Extreme Value distribution makesdi�cult to test the goodness of �t with the standard methods (�2 test,Kolmogorov-Smirnov test). On the other hand, we know that the data are

12

extrema and that mixing properties hold for all the maps used in the followingexperiments, so that the GEV model seems a reasonable a priori assumption,furthermore, among a wide class of continuous distribution, GEV distribu-tion is the one that minimize Kolmogorov Smirnov parameter (see Lilliefors[1967] for a description of the test).

To summarize the results of the previous section we will check the fol-lowing behaviors in respect to n. Since we keep the length of the seriesk = n �m �xed, similar relationships hold also with respect to m by substi-tuting n = k=m.For g1 type observable:

� =1

D� / �

1

Dln(n) � = 0 (25)

For g2 type observable:

� / �n1=(�D1) � / �n1=(�D1) � =1

�D1(26)

For g3 type observable:

� / n1=(�D1) � = C � = �1

�D1(27)

4.1 IFS for Cantor Set and Sieripinskij Triangle

A Cantor set can be obtained as an attractor of some Iterated FunctionSystems (IFS). An IFS is a �nite family of contractive maps ff1; f2; :::; fsgacting on a complete metric space and such a dynamics possesses a uniquecompact set K 2 X which is non-empty and invariant by the IFS:

K =s[

i=1

fi(K)

and one calls this compact set the limit set or the attractor of the IFS[Hutchinson, 1981], [Crovisier and Rams, 2006].The �rst example we consider is the middle one third Cantor set that is theattractor of the IFS ff1; f2g de�ned as:(

f1(x) = x=3

f2(x) = (x+ 2)=3(28)

where x 2 [0; 1] and, at each time step, with the same probability, f1 orf2 is iterated. Equivalently the previous IFS can be written as:

13

xt+1 = (xt + p)=3 (29)

where, at each time step, we extract randomly with equal probability pto be 0 or 2.

To obtain the so called Sierpisnkij triangle, we have �rst de�ned:

v =

������1 0�1 00 1

������ :then, we extract randomly at each time step, with equal probability p to

be the integer 1,2 or 3. And we iterate the following map:(xt+1 = (xt + vp;1)=2

yt+1 = (yt + vp;2)=2(30)

All the fractal dimension de�ned before are identical for this attractivesets and analytically computable with simple geometrical considerations:D = log(2)= log(3) for the Cantor set and D = log(3)= log(2) for the Sierpin-skij triangle [Sprott, 2003].

In the following experiments we have iterated the IFS system choosinginitial conditions � on the Cantor Set and on the Sierpinskij triangle.

First of all we have analised the histogram of the extrema for each ob-servable and some of them is shown for the cantor set in �gure 1. Thesehistograms are obtained iterating the map in equation 28 for 5 � 107 itera-tions, � = 1=3,� = 4, C = 10. Once computed the functions gi the series ofmaxima for each observable is computed taking each of them in bins contain-ing 5000 values of gi for a total of 1000 maxima. As claimed in the previoussection, this histograms are discontinuous and this is due to the structure ofthe Cantor set. Experimentally, we have tried to �t the empirical histogramto continuous distribution �nding the one that minimize the KolmogorovSmirnov test parameter. As expected the GEV distribution is the more re-liable among a wide class of distributions we have tested and therefore wecan conjecture that the observed histograms are given by the product of thecorrespondent continuous GEV distribution and the characteristic functionof the distance in the Cantor set, modulated by the observable function gi.

14

Figure 1: Histograms for IFS that generates a Cantor Set, � = 1=3, a) g1observable, b) g2 observable, c) g3 observable.

To check that e�ectively the parameters of GEV distribution obtained byL-moments estimation are related to the fractal dimension of the attractingCantor Set and the Sierpinskij triangle, we have considered an ensemble of 104

di�erent realizations of the eq. 28 and eq. `refIFSS; startingfromthesameinitialconditions:Tocheckthebehaviorwehavevariednandmkeepingfixedthelengthoftheseriesk=107.In [Faranda et al., 2011] we have shown that a good convergence is observedwhen n;m > 1000, therefore we will make all the considerations for (n;m)pairs that satisfy this condition.

In �gures 2-4 the results of the computation for the IFS that generatesthe Cantor Set ( plots on the left) and the Sierpisnkij triangle (plots on theright) are presented. In all the cases considered the behavior well repro-duce the theoretical expected trend described in equations 25-27. The initial

15

condition here shown is � = 1=3 for Cantor Set and � ' (0:02; 0:40) forSierpinskij triangle, but similar results hold for di�erent initial conditions ifchosen on the attractive sets. The black line is the mean value over di�erentrealizations of the map, while the black dotted lines represent one standarddeviation.

For g1 observable function, according to equation 25, we expect to �nd� = 0 in both cases and this is veri�ed by experimental data shown in Figure2a). For the scale parameter a similar agreement is achieved in respect to thetheoretical parameters � = log(3)= log(2) ' 1:5850 for the Cantor Set and� = log(2)= log(3) ' 0:6309 for Sierpinskij shown in �gure 2b) with a greenline. Eventually, Location parameter � shows a logarithm decay with n asexpected from equation 25. A linear �t of � in respect to log(n) is shownwith a red line in �gure 2c). The linear �t angular coe�cients K� of equation25 well approximates 1=D in both cases: its value is jK�j = 1:57 � 0:01 forCantor and jK�j = 0:63� 0:01 in the Sierpinskij triangle.The agreement between the conjecture and the results are con�rmed evenfor g2 type and g3 type observable functions shown in �gures 3 and 4 respec-tively. In this case we have experienced some problems in the convergencefor the Cantor map has using � = 2 and � = 3. The problem is possiblydue to the fact that L-moments method works better if � 2 [�0:5; 0:5] whilefor � � 3 the shape parameter j�j > 0:5 for the Cantor map. For this rea-son results are shown using � = 4 both for Sierpinskij triangle and CantorIFS, for all the experiments presented the constant value in g3 will be C = 10.

For both observables there is strong agreement between the experimentaland theoretical � values. In �gures 3b), 3c), 4b) a log-log scale is used tohighlight the behavior described by eq. 26 and 27. We can check that theangular coe�cient of a linear �t approximate the expected theoretical valueK = 1=�D. In table 4.1 we compare these results with the theoretical valuesand within the error (that is here represented as one standard deviation ofthe ensemble of realization) we �nd a good agreement.

Eventually, computing g3 as observable function we expect to �nd a con-stant value for � = C = 10 while � has to grow with a power law in respectto n as expected comparing with equation 27.

Other tests have been done computing the statistics using parameter� = 5; 6; 7; 8 for g2 and g3 observables. Also in this cases, no deviation fromthe behavior described in eq. 25-27 has been found.

16

Parameter Cantor SierpinskijTheoretical jK�j ' 0:3962 jK�j ' 0:1577�(g2) jK�j = 0:393� 0:004 jK�j = 0:157� 0:004�(g2) jK�j = 0:394� 0:005 jK�j = 0:16� 0:01�(g3) jK�j = 0:389� 0:006 jK�j = 0:154� 0:008

Table 1: Angular coe�cients computed for Sierpisnkij and Cantor IFS.

Figure 2: g1 observable. a) � VS log10(n); b) � VS log10(n); c) � VS log(n).Cantor set, Right: Sierpinskij triangle. Dotted lines represent one standarddeviation, red lines represent a linear �t, green lines are theoretical values.

17

Figure 3: g2 observable a) � VS log10(n); b) log10(�) VS log10(n); c) log10(�)VS log10(n). Left: Cantor set, Right: Sierpinskij triangle. Dotted linesrepresent one standard deviation, red lines represent a linear �t, green linesare theoretical values.

18

Figure 4: g3 observable. a) � VS log10(n); b) log10(�) VS log10(n); c) log10(�)VS log10(n). Cantor set, Right: Sierpinskij triangle. Dotted lines representone standard deviation, red lines represent a linear �t, green lines are theo-retical values.

19

4.2 IFS with non-uniform weights

Let us now consider the case of a multifractal measure, that can be generateby the following IFS:

fk(x) = ak + �kx x 2 [0; 1] k = 1; 2; :::; s (31)

and each fi is iterated with probability wi. A non-overlap condition holds:

0 � a1 � a1 + �1 � a2 � :::as�1 + �s�1 � as � as + �s � 1

The dimensional spectrum Dq associated to the measure can be derivedusing the correlation integrals [Yamaguti and Prado, 1995], [Santoro et al.,2002], [Barnsley, 2000]. In the limit q ! 1 we obtain an explicit expressionfor the Information dimension:

D1 =w1 logw1 + :::+ ws logws

w1 log �1 + :::+ ws log �s(32)

In this analysis we have considered the following IFS:(f1(x) = x=3 with weight w

f2(x) = (x+ 2)=3 with weight 1� w(33)

and we have changed the weight w between 0.35 and 0.65 with 0:01 step.For w = 0:5 we obtain the same results shown in the previous section, whilefor di�erent weights we can check the expression 32.

In case of a multifractal, the single initial condition cannot give informa-tion on overall quantity such as the Information Dimension, that is related tothe global properties of the IFS considered. As pointed out in the introduc-tion, the Extreme Value distribution for the single initial condition � give usinformation about the structure of the attractor in a small ball centered in�, therefore, to obtain global properties of the attractor, we have to averageon di�erent initial conditions.

In Figure 5 we present the dimension D1 computed using relationship25-27. In particular We can compute the dimension from eq. 25 as:

D(�(g1)) =1

< �(g1) >(34)

We can infer dimension also from eq. 26, eq. 27 as:

D(�(g2; g3)) =1

�j < �(g2; g3) > j(35)

20

and in all expressions above the brackets < : > indicate an average ondi�erent initial conditions. For the rest of the numerical computations weset � = 3. The parameters have been computed using 1000 di�erent initialconditions on the support of the attractor, and the block-maxima approachis here used with n = m = 1000. The error bar are computed using thestandard error propagation rules.

The agreement between the theoretical dimension and the experimentaldata is evident for all the weights and for all the observable considered. Theuncertainty increases when w is much di�erent from 0.5. This is due tothe fact that as soon as we change the weight to be di�erent from 0.5 theparameters spread increase to take in account the local properties of theattractor.

The best agreement and less uncertainty is achieved considering the di-mension as computed from g1 observable. This is possibly due to the slowerconvergence for g2 or g3 observables to the respective theoretical distribu-tions: in g1 we modulate the distances with a logarithm function while in g2and in g3 power laws are used. Netherless, all the data show the right trend.

Figure 5: Di�erent estimation ofD1 dimension obtained using Extreme Valuedistribution. IFS in equation 33. STO RIFACENDO LA FIGURA CON LECORREZIONI AL METODO.

21

4.3 Baker, Henon and Lozi maps

The Baker mapThe Baker map is a 2-dimensional map de�ned as:

xt+1 =

( axt mod 1 if yt < �

1=2 + bxt mod 1 if yt � �

yt+1 =

(yt�

mod 1 if yt < �yt��1��

mod 1 if yt � �(36)

we have used � = 1=3; a = 1=5 and b = 1=4.Rigorous analytical results are available for the estimation of the infor-

mation dimension [Kaplan and Yorke, 1979]. For our parameter values, theanalytical expected value is D1 ' 1:4357.

We have performed the same analysis detailed in section 4.1, the onlydi�erence is related to the choice of the initial conditions: since the map isdeterministic we have extracted 1000 random initial conditions on the sup-port of the attractor and computed the GEV parameters with L-momentsmethod as detailed before and � = 4 for g2 and g3. The results are shownin �gures 6-8 the black continuous lines will represent the parameter averageover di�erent initial conditions and the black dotted lines the standard devi-ation of the distribution of the parameters.

Looking at � parameters behavior we notice that the expected theoreticalvalues are within one standard deviation of the results of the �t for all threeobservable. The agreement seems to be better when we increase n even ifthis correspond to a decrease of m in our set up. This is true also for �(g1)shown in Figure 6b) that approaches 1=D1 ' 0:6965 value for bigger n val-ues. The angular coe�cient of the linear �t for �(g1) shown in the semilogplot in Figure 6c) is jK�j = 0:67� 0:02 in fair good agreement with 1=D1.

Log-log plots of the parameters against n are shown in �gures 7b), 7c) and8b), and the value of the computed angular coe�cient are reported in Table2. The agreement with expected value is good enough for all the parametersand better for �(g2). Eventually, Figure 8c) shows that �(g3) approachesC = 10.

22

Figure 6: g1 observable. a) � VS log10(n); b) � VS log10(n); c) � VS log(n).Baker map. Dotted lines represent one standard deviation, red lines representa linear �t, green lines are theoretical values.

23

Figure 7: g2 observable a) � VS log10(n); b) log10(�) VS log10(n); c) log10(�)VS log10(n). Baker map. Dotted lines represent one standard deviation� redlines represent a linear �t, green lines are theoretical values.

24

Figure 8: g3 observable. a) � VS log10(n); b) log10(�) VS log10(n); c) log10(�)VS log10(n). Baker map. Dotted lines represent one standard deviation, redlines represent a linear �t, green lines are theoretical values.

25

The Henon and Lozi mapsThe Henon map is a 2-dimension map de�ned as:

xt+1 = yt + 1� ax2tyt+1 = bxt

(37)

while in the Lozi map ax2t is substituted with ajxtj :

xt+1 = yt + 1� ajxtjyt+1 = bxt

(38)

The fractal dimensions for these maps have been computed in Grassberger[1983], Badii and Politi [1987], Lozi [1978] and they are dependent on thechoice of parameters value a and b. We have computed the extreme valuestatistics for a = 1:4, b = 0:3 for the Henon map so that:

D1 = 1:25826� 0:00006

For the Lozi map a = 1:7 and b = 0:5, such that the dimension is:

D1 ' 1:40419

As in the previous cases the GEV distribution is computed with L-moments methods varying n and m and averaging the distribution parame-ters over 1000 di�erent initial conditions lying on the attractor. Results arepresented in �gures 9-11, Left plots refer to Henon map, while on the Rightthe results refer to Lozi map.Regarding �, the results are satisfactory for all the observable functions, witha very good agreement between the averaged parameters and the theoreticalones, even if the parameter distributions has a bigger spread that indicatesa slower convergence towards the expected values. The experimental valuesof �(g1) , shown in Figure 9b, approach 1=D1 ' 0:7947 for Henon map and1=D1 ' 0:7121 for Lozi map, as expected. The same theoretical values shouldbe obtained for the angular coe�cient computed from the semilog plot of�(g1) represented in Figure 9c) and numerically we obtain jK�j = 0:81�0:01for Henon and jK�j = 0:716� 0:008 for Lozi.

The other angular coe�cients related to g2 and g3 observables for theplots shown in �gures 10b), 10c), and 11b) are presented in Table 2. Within2 standard deviation they are comparable with the theoretical ones. Theconstant C = 10 is approached quite well as �(g3) (Figure 11c)).

26

Par. Baker Henon LoziTheor. jK�j ' 0:1743 jK�j ' 0:1986 jK�j ' 0:1780�(g2) jK�j = 0:171� 0:004 jK�j = 0:202� 0:003 jK�j = 0:179� 0:002�(g2) jK�j = 0:20� 0:02 jK�j = 0:18� 0:02 jK�j = 0:182� 0:005�(g3) jK�j = 0:13� 0:03 jK�j = 0:22� 0:01 jK�j = 0:175� 0:002

Table 2: Angular coe�cients computed for Baker, Henon and Lozi maps.

The slower convergence for these maps may be related to the di�cul-ties experienced computing the dimension with all box-counting methods, asshown in Grassberger [1983],Badii and Politi [1987]. In that case it has beenproven that the number of points that are required to cover a �xed frac-tion of the attractor support diverges faster than the number of boxes itselffor this kind of non uniform attractor. In our case the situation is similarsince we consider balls around the initial condition �. As pointed out in theprevious subsection, the best result for the dimension is achieved using theparameters provided by g1 observable since the logarithm modulation of thedistance exalts proper extrema while weights less possible outliers.

27

Figure 9: g1 observable. a) � VS log10(n); b) � VS log10(n); c) � VS log(n).Left: Henon map, Right: Lozi map. Dotted lines represent one standarddeviation, red lines represent a linear �t, green lines are theoretical values.

28

Figure 10: g2 observable a) � VS log10(n); b) log10(�) VS log10(n); c) log10(�)VS log10(n). Left: Henon map, Right: Lozi map. Dotted lines represent onestandard deviation� red lines represent a linear �t, green lines are theoreticalvalues.

29

Figure 11: g3 observable. a) � VS log10(n); b) log10(�) VS log10(n); c)log10(�) VS log10(n). Left: Henon map, Right: Lozi map. Dotted linesrepresent one standard deviation, red lines represent a linear �t, green linesare theoretical values.

30

5 Final Remarks

The importance of EVT is evident in very di�erent disciplines where there isa great interest in modeling events that occur with small probability as theire�ects can be relevant for economical activities and/or from a social pointof view. In geophysical applications is crucial to have a tool to understandand forecast climatic extrema and events such as strong earthquakes and�oods. The mathematical models used to study them present a rich structureand their attracting sets are very often strange attractors. In such senseis extremely important to develop an extreme value theory for dynamicalsystem with singular measures.

In this work we have extended the results presented in Faranda et al.[2011] to the case of dynamical systems with singular measures. Our mainresults is that there exist an extreme value distribution for this kind of sys-tems that is related to the GEV distribution when observable functions ofthe distance between the iterated orbit and the initial conditions are chosen.In this case the distribution of extrema can be seen as the product of a GEVdistribution and the characteristic function of these distances modulated bythe observable function. Furthermore, the parameters of the distribution areintimately related to the fractal dimension for perfect fractals and to the In-formation Dimension for multifractals. We have tested our conjecture withnumerical experiments on di�erent low dimensional maps which invariantmeasure is a perfect fractal (such as the middle third Cantor set or the Sier-pinskij triangle), multifractal, strange attractors such as Lozi and Henon. Inall cases considered there is agreement between the theoretical parametersand the experimental ones.In multifractals, that present a rich structure highly dependent on the initialcondition chosen, the algorithm described with the selection of maxima actslike a magnifying glass on the neighborhood of the initial condition. In thisway we have both a powerful tool to study and highlight the �ne structureof the attractor, but, on the other hand we can obtain global properties av-eraging on di�erent initial conditions.

To conclude we have provided a tool to study the extreme value distribu-tion in dynamical systems with singular measures extending the results forabsolutely continuous invariant measures presented in Faranda et al. [2011].From a practical point of view we have now an algorithmic way to look at thestatistics of extrema that can be used in principle in di�erent kind of deter-ministic dynamical systems. In future work we intend to apply the statisticson geophysical models.

31

6 Acknowledgments

S.V. was supported by the CNRS-PEPS Project Mathematical Methods ofClimate Models , and he thanks the GDRE Gre�-Me� for having supportedexchanges with Italy. V.L. and D.F. acknowledge the �nancial support of theEU FP7-ERC project NAMASTE �Thermodynamics of the Climate System".

References

E.G. Altmann, S. Hallerberg, and H. Kantz. Reactions to extreme events:Moving threshold model. Physica A: Statistical Mechanics and its Appli-cations, 364:435�444, 2006.

R. Badii and A. Politi. Renyi dimensions from local expansion rates. PhysicalReview A, 35(3):1288, 1987.

V. Balakrishnan, C. Nicolis, and G. Nicolis. Extreme value distributionsin chaotic dynamics. Journal of Statistical Physics, 80(1):307�336, 1995.ISSN 0022-4715.

M.F. Barnsley. Fractals everywhere. Morgan Kaufmann Pub, 2000. ISBN0120790696.

J. Beirlant. Statistics of extremes: theory and applications. John Wiley &Sons Inc, 2004. ISBN 0471976474.

E. Brodin and C. Kluppelberg. Extreme Value Theory in Finance. Submittedfor publication: Center for Mathematical Sciences, Munich University ofTechnology, 2006.

P.W. Burton. Seismic risk in southern Europe through to India examinedusing Gumbel's third distribution of extreme values. Geophysical Journalof the Royal Astronomical Society, 59(2):249�280, 1979. ISSN 1365-246X.

S. Coles, J. He�ernan, and J. Tawn. Dependence measures for extreme valueanalyses. Extremes, 2(4):339�365, 1999. ISSN 1386-1999.

P. Collet. Statistics of closest return for some non-uniformly hyperbolic sys-tems. Ergodic Theory and Dynamical Systems, 21(02):401�420, 2001. ISSN0143-3857.

C.A. Cornell. Engineering seismic risk analysis. Bulletin of the SeismologicalSociety of America, 58(5):1583, 1968. ISSN 0037-1106.

32

S. Crovisier and M. Rams. IFS attractors and Cantor sets. Topology and itsApplications, 153(11):1849�1859, 2006. ISSN 0166-8641.

M.G. Cruz. Modeling, measuring and hedging operational risk. John Wiley& Sons, 2002. ISBN 0471515604.

A.C. Davison. Modelling excesses over high thresholds, with an application.Statistical Extremes and Applications, pages 461�482, 1984.

AC Davison and R.L. Smith. Models for exceedances over high thresholds.Journal of the Royal Statistical Society. Series B (Methodological), 52(3):393�442, 1990. ISSN 0035-9246.

P. Embrechts, S.I. Resnick, and G. Samorodnitsky. Extreme value theoryas a risk management tool. North American Actuarial Journal, 3:30�41,1999. ISSN 1092-0277.

D. Faranda, V. Lucarini, G. Turchetti, and S. Vaienti. Numerical conver-gence of the block-maxima approach to the Generalized Extreme Valuedistribution. Arxiv preprint arXiv:1103.0889, 2011.

M. Felici, V. Lucarini, A. Speranza, and R. Vitolo. Extreme Value Statisticsof the Total Energy in an Intermediate Complexity Model of the Mid-latitude Atmospheric Jet. Part I: Stationary case.(3337K, PDF). Journalof Atmospheric Science, 64:2137�2158, 2007a.

M. Felici, V. Lucarini, A. Speranza, and R. Vitolo. Extreme value statisticsof the total energy in an intermediate complexity model of the mid-latitudeatmospheric jet. Part II: trend detection and assessment. Journal of At-mospheric Science, 64:2159�2175, 2007b.

RA Fisher and LHC Tippett. Limiting forms of the frequency distributionof the largest or smallest member of a sample. In Proceedings of the Cam-bridge philosophical society, volume 24, page 180, 1928.

A.C.M. Freitas and J.M. Freitas. On the link between dependence and in-dependence in extreme value theory for dynamical systems. Statistics &Probability Letters, 78(9):1088�1093, 2008. ISSN 0167-7152.

A.C.M. Freitas, J.M. Freitas, and M. Todd. Hitting time statistics and ex-treme value theory. Probability Theory and Related Fields, pages 1�36,2009.

A.C.M. Freitas, J.M. Freitas, and M. Todd. Extremal Index, Hitting TimeStatistics and periodicity. Arxiv preprint arXiv:1008.1350, 2010a.

33

A.C.M. Freitas, J.M. Freitas, M. Todd, B. Gardas, D. Drichel, M. Flohr,RT Thompson, SA Cummer, J. Frauendiener, A. Doliwa, et al. Ex-treme value laws in dynamical systems for non-smooth observations. Arxivpreprint arXiv:1006.3276, 2010b.

M. Gilli and E. Këllezi. An application of extreme value theory for measuring�nancial risk. Computational Economics, 27(2):207�228, 2006. ISSN 0927-7099.

B. Gnedenko. Sur la distribution limite du terme maximum d'une sériealéatoire. The Annals of Mathematics, 44(3):423�453, 1943.

P. Grassberger. Generalized dimensions of strange attractors. Physics LettersA, 97(6):227�230, 1983. ISSN 0375-9601.

P. Grassberger. Generalizations of the Hausdor� dimension of fractal mea-sures. Physics Letters A, 107(3):101�105, 1985. ISSN 0375-9601.

EJ Gumbel. The return period of �ood �ows. The Annals of MathematicalStatistics, 12(2):163�190, 1941. ISSN 0003-4851.

C. Gupta. Extreme-value distributions for some classes of non-uniformlypartially hyperbolic dynamical systems. Ergodic Theory and DynamicalSystems, 30(03):757�771, 2010. ISSN 0143-3857.

C. Gupta, M. Holland, and M. Nicol. Extreme value theory for hyperbolicbilliards. Lozi-like maps, and Lorenz-like maps, preprint, 2009.

G. Haiman. Extreme values of the tent map process. Statistics & ProbabilityLetters, 65(4):451�456, 2003. ISSN 0167-7152.

M. Holland, M. Nicol, and A. Török. Extreme value distributions for non-uniformly hyperbolic dynamical systems. preprint, 2008.

J.R.M. Hosking. L-moments: analysis and estimation of distributions usinglinear combinations of order statistics. Journal of the Royal StatisticalSociety. Series B (Methodological), 52(1):105�124, 1990. ISSN 0035-9246.

J.E. Hutchinson. Fractals and self-similarity. Indiana Univ. Math. J, 30(5):713�747, 1981.

J. Kaplan and J. Yorke. Chaotic behavior of multidimensional di�erenceequations. Functional Di�erential equations and approximation of �xedpoints, pages 204�227, 1979.

34

RW Katz. Extreme value theory for precipitation: Sensitivity analysis forclimate change. Advances in Water Resources, 23(2):133�139, 1999. ISSN0309-1708.

R.W. Katz and B.G. Brown. Extreme events in a changing climate: vari-ability is more important than averages. Climatic change, 21(3):289�302,1992. ISSN 0165-0009.

R.W. Katz, G.S. Brush, and M.B. Parlange. Statistics of extremes: Modelingecological disturbances. Ecology, 86(5):1124�1134, 2005. ISSN 0012-9658.

MR Leadbetter, G. Lindgren, and H. Rootzen. Extremes and related proper-ties of random sequences and processes. Springer, New York, 1983.

H.W. Lilliefors. On the Kolmogorov-Smirnov test for normality with meanand variance unknown. Journal of the American Statistical Association,62(318):399�402, 1967. ISSN 0162-1459.

F.M. Longin. From value at risk to stress testing: The extreme value ap-proach. Journal of Banking & Finance, 24(7):1097�1130, 2000. ISSN0378-4266.

R. Lozi. Un attracteur étrange (?) du type attracteur de Hénon. Le Journalde Physique Colloques, 39(C5):5�5, 1978. ISSN 0449-1947.

E.S. Martins and J.R. Stedinger. Generalized maximum-likelihood gener-alized extreme-value quantile estimators for hydrologic data. Water Re-sources Research, 36(3):737�744, 2000. ISSN 0043-1397.

A.C. Moreira Freitas and J.M. Freitas. Extreme values for Benedicks�Carleson quadratic maps. Ergodic Theory and Dynamical Systems, 28(04):1117�1133, 2008. ISSN 0143-3857.

W.K. Newey and D. McFadden. Large sample estimation and hypothesistesting. Handbook of econometrics, 4:2111�2245, 1994. ISSN 1573-4412.

N. Nicholis. CLIVAR and IPCC interests in extreme events'. In WorkshopProceedings on Indices and Indicators for Climate Extremes, Asheville,NC. Sponsors, CLIVAR, GCOS and WMO, 1997.

C. Nicolis, V. Balakrishnan, and G. Nicolis. Extreme events in deterministicdynamical systems. Physical review letters, 97(21):210602, 2006. ISSN1079-7114.

35

Friederichs P. and A. Hense. Statistical downscaling of extreme precipitationevents using censored quantile regression. Monthly weather review, 135(6):2365�2378, 2007. ISSN 0027-0644.

J. Pickands III. Moment convergence of sample extremes. The Annals ofMathematical Statistics, 39(3):881�889, 1968.

R. Santoro, NMMaraldi, S. Campagna, and G. Turchetti. Uniform partitionsand a dimensions spectrum for lacunar measures. Journal of Physics A:Mathematical and General, 35:1871, 2002.

R.L. Smith. Threshold methods for sample extremes. Statistical extremesand applications, 621:638, 1984.

R.L. Smith. Extreme value analysis of environmental time series: an appli-cation to trend detection in ground-level ozone. Statistical Science, 4(4):367�377, 1989. ISSN 0883-4237.

D. Sornette, L. Knopo�, YY Kagan, and C. Vanneste. Rank-ordering statis-tics of extreme events: application to the distribution of large earthquakes.Journal of Geophysical Research, 101(B6):13883, 1996. ISSN 0148-0227.

J.C. Sprott. Chaos and time-series analysis. Oxford Univ Pr, 2003. ISBN0198508409.

O.G.B. Sveinsson and D.C. Boes. Regional frequency analysis of extreme pre-cipitation in northeastern colorado and fort collins �ood of 1997. Journalof Hydrologic Engineering, 7:49, 2002.

P. Todorovic and E. Zelenhasic. A stochastic model for �ood analysis. WaterResources Research, 6(6):1641�1648, 1970. ISSN 0043-1397.

S. Vannitsem. Statistical properties of the temperature maxima in an in-termediate order Quasi-Geostrophic model. Tellus A, 59(1):80�95, 2007.ISSN 1600-0870.

R. Vitolo, PM Ruti, A. Dell'Aquila, M. Felici, V. Lucarini, and A. Sper-anza. Accessing extremes of mid-latitudinal wave activity: methodologyand application. Tellus A, 61(1):35�49, 2009. ISSN 1600-0870.

M. Yamaguti and C.P.C. Prado. A direct calculation of the spectrum ofsingularities f ([alpha]) of multifractals. Physics Letters A, 206(5-6):318�322, 1995. ISSN 0375-9601.

36

L.S. Young. Statistical properties of dynamical systems with some hyperbol-icity. Annals of Mathematics, 147(3):585�650, 1998. ISSN 0003-486X.

L.S. Young. Recurrence times and rates of mixing. Israel Journal of Mathe-matics, 110(1):153�188, 1999.

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