climate change in a point over threshold model an example on ocean wave storm hazard in ne spain

Adv Geosci., 26, 113–117, 2010 www.adv-geosci.net/26/113/2010/ doi:10.5194/adgeo-26-113-2010 © Author(s) 2010 CC Attribution 3.0 License Advances in Geosciences Climate change in a Point-Over-Threshold model: an example on ocean-wave-storm hazard in NE Spain R Tolosana-Delgado1 , M I Ortego2 , J J Egozcue2 , and A S´anchez-Arcilla1 Universitat Universitat Polit`ecnica de Catalunya, Laboratori d’Enginyeria Mar´ıtima, Barcelona, Spain Polit`ecnica de Catalunya, Departament de Matem`atica Aplicada III, Barcelona, Spain Received: 15 February 2010 – Revised: 31 Mai 2010 – Accepted: 18 August 2010 – Published: 27 September 2010 Abstract A reparametrization of the Generalized Pareto Distribution is here proposed It is suitable to parsimoniously check trend assumptions within a Point-OverThreshold model of hazardous events This is based on considerations about the scale of both the excesses of the event magnitudes and the distribution parameters The usefulness of this approach is illustrated with a data set from two buoys, where hypotheses about the homogeneity of climate conditions and lack of trends are assessed Introduction Climatic change is a problem of general concern When dealing with hazardous events such as wind-storms, heavy rainfall or wave storms this concern becomes even more serious Climate change might mean an increase of human and material losses, and therefore efforts to detect it from limited data sets should be taken In this contribution, a hazard assessment of storm events in the northern Mediterranean Spanish coast is carried out, following a standard model for extremes such as heavy rainfall or wave storms An event is defined as the period during which a certain magnitude of the phenomenon (significant wave height in this case) exceeds a given reference threshold For this reason, this model is typically called Point-over-Threshold (POT) model (Embrechts et al., 1997): time-occurrence of these events is assumed to be Poisson distributed, and the magnitude exceeding the threshold for each event is modelled as a random variable with a Generalized Pareto Distribution (GPD) Independence is assumed, both between this magnitude and occurrence in time, and from event to event For this contribution, we focus on assess- ing the presence of a change on the magnitude parameters: the independence assumed ensures us that the occurrence and magnitude estimation can be done separately Scarcity of data arises as an additional difficulty, as hazardous events are usually rare Estimation of hazard parameters such as return periods may imply a great amount of uncertainty Bayesian methods (e.g Gelman et al., 1995) have been used successfully to deal with this unavoidable uncertainty of the results, and therefore a Bayesian estimation of GPD models (Egozcue and Tolosana-Delgado, 2002) seems appropriate The selection of proper scales for the description of phenomena also arises as an important issue A handful of phenomena are better described by a relative scale (e.g positive data where the null value is unattainable) and are thus suitably treated in a logarithmic scale: logarithmic scales has been used successfully for daily rainfall data and wave-height (Egozcue and Ramis, 2001; Pawlowsky-Glahn et al., 2005; Egozcue et al., 2005; S´anchez-Arcilla et al., 2008) 2.1 The Generalized Pareto Distribution Classical parametrization The Generalised Pareto Distribution (GPD) models excesses over a threshold (Pickands, 1975) If X is the magnitude of an event and x0 a value of the support of X, the excess over the threshold x0 is Y = X − x0 , conditioned to X > x0 Therefore, the support of Y is either an interval [0,ysup ] or, the positive real line In our case, X will be the natural logarithm of significant wave height measurements from buoys, and the threshold x0 = 5.2, as explained in the application section Correspondence to: R Tolosana-Delgado (raimon.tolosana@upc.edu) Published by Copernicus Publications on behalf of the European Geosciences Union FY (y|β,ξ = 0) = − exp − y β different sub-families of distributions GPD distributions with ξ < have limited support, with expectation and upper bound R Tolosana-Delgado et al.: β Climate change in a Point-Over-Threshold model ysup = − , (3) ξ The associated probability density functions for the first β E[y] (4) case=is1 − ξ , ≤ y < +∞ (1) for ξ = 114 1.0 density examples 0.2 0.0 0.4 f(y) 0.6 0.8 − −1 4 10 Y Gumbel bu ei W β t he éc Fr ll parameter space 1 −1.0 −0.5 0.0 0.5 1.0 ξ Fig.1.1.Examples Examples of of GPD GPD densities Fig densities(upper (upperdiagram) diagram)covering coveringallalldodomainsof of attraction, attraction, and and their mains their representation representationininthe theparameter parameterspace space (lower diagram), diagram), numbered numbered correspondingly (lower correspondingly This Thislower lowerdiagram diagram shows the the classical classical parametrization, parametrization, the shows thedomains domainsofofattraction, attraction,and and theproposed proposed reparametrization: reparametrization: the the therays raysare areiso-µ iso-µlines lines(increasing (increasing values clockwise), and the hyperbolas are iso-ν lines (increasing µµvalues clockwise), and the hyperbolas are iso-ν lines (increasing ν values upwards; thus Gumbel domain corresponds to ν → −∞) ν values upwards; thus Gumbel domain corresponds to ν → −∞) The associated probability density functions for the first case TheisGPD cumulative function function is − ξ1−−1 ξξ (y|β,ξ)) = = 1− 1+ FfYY(y|β,ξ +β y y β β ξ ≤ yy < 0, yβsup = +∞, distributions the Fr´echet domain of attraction, and for ξ = (exponential The distributions scale parameter distribution β,belong a positive value case), haveofanthe infinite supportisand to the The shape parameter, ξ , is real-valued, and it defines Gumbel domain of attraction Figure displays several rep-three different sub-families of distributions GPDand distributions resentations of GPD, as densities (upper diagram) in the with ξ 0, ysuplimitation = +∞, distributions belong tive scale, admitting physical features, are ele- to the Fr´ echetwill domain of attraction, and for =other (exponential ments which be considered constant On ξthe hand, distributions an infinite support and belong to the if case), climate change has have occurred (is occurring), there should domain attraction.ofFigure displays several beGumbel a change of theofparameters the GPD distribution (ξ representations of GPD, as densities (upper in the and β), maintaining the physical sense of thediagram) describedand pheparameter space (lower diagram) nomena In particular, if limited phenomena are described, a GPD-Weibull domain of attraction should be chosen as a 2.2 statement, A new parametrization priori in order to include this limited feature to the model (Egozcue et al., 2006) This may be easily controlled canparameterisation reasonably assume a climate change scebyOne a new of the that, GPD in distribution: nario, the description model of the variable of interest −β should not µchange, parameters could reflect ; νmodel = ln(−ξ · β) , = ln but the ξ the change: thus, modeling excesses of magnitudes over a threshold by a Generalised Pareto Distribution, using a relawhere µ is a new location parameter, informing about the tive scale, a physical(Eq limitation features, are eleupper boundadmitting of the distribution 3), and ν is a shape ments which will be considered constant On the other parameter The classical parameters can be retrieved with hand, if climate change has occurred (is occurring), there should − µ distribution (ξ ν + parameters µ be a change of the of the νGPD ; −ξ = exp β = exp and β), maintaining2 the physical sense of2the described phenomena In particular, if limited phenomena are described, The lower diagram of Figure displays the parameter space a GPD-Weibull domain of attraction should be chosen as a of the GPD with the two families of parameters: the classical priori statement, in order to include this limited feature to the parameters are represented as a cartesian coordinate system, model (Egozcue et al., 2006) This may be easily controlled whereas the proposed parameters form a hyperbolic coordiby a new parameterisation of the GPD distribution: nate system −β ; ν = ln(−ξ · β) µ = ln ξ estimation Bayesian where µ is a new location parameter, informing about the In a Bayesian estimation process (e.g Gelman et al., 1995; upper bound of the distribution (Eq 3), and ν is a shape paEgozcue and Tolosana-Delgado, 2002), the observable varirameter The classical parameters can be retrieved with β = exp y β for ξ = Adv Geosci., 26, 113–117, 2010 , ≤ y < +∞ (1) ν +µ ; −ξ = exp ν −µ The lower diagram of Fig displays the parameter space of the GPD with the two families of parameters: the classical parameters are represented as a cartesian coordinate system, whereas the proposed parameters form a hyperbolic coordinate system www.adv-geosci.net/26/113/2010/ R Tolosana-Delgado al.: Climate change in a Point-Over-Threshold model Tolosana, Ortego,et Egozcue and Sanchez-Arcilla: Climate change in a Point-Over-Threshold model Bayesian estimation able is assumed to follow a parametric model of unknown pa- 6.2 6.4 5.2 5.4 5.6 5.8 400 500 300 Hs (cm) 200 6.2 6.4 5.2 5.4 5.6 5.8 400 500 300 nificant wave-height data (S´anchez-Arcilla et al., 2008) in log-scale, simultaneously at two stations, the buoys of Roses Application and Tortosa Figure shows their location along the Catalan andare the illustrated sample of events intensities The same figTheseCoast issues using and a set of 18 years of sigure also shows the diagram of expected excess over a threshnificant wave-height data (S´anchez-Arcilla et al., 2008) in old, used for identifying x0 , the threshold of analysis Note log-scale, simultaneously at two stations, the buoys of Roses that there are some occurrence gaps in the Roses series (beand Tortosa Figure shows their location along the Catalan fore 1994 and between 1997 and 2001 approximately), but Coastthis anddoes the not sample of events and intensities The same figaffect computations regarding event magnitude ure alsoIfshows the diagram of expected a threshthe possible trends were due to aexcess global over climate change, old, used for identifying x , the threshold of analysis Note one should expect them0 to be consistently reflected at several that there are some occurrence gaps in the Roses series nearby, homogeneous locations For this reason, we(beanalfore 1994 and between two 1997 and 2001 approximately), butare yse simultaneously locations, selected because they this does not affect computations regarding event magnitude If the possible trends were due to a global climate change, one should expect them to be consistently reflected at several www.adv-geosci.net/26/113/2010/ 43.0 42.0 41.0 Latitude R T Longitude { µ =a 0set, of ν18=years H0 : {These µ = issues , are ν =illustrated 0}, H1 :using 0} of sig- 2005 40.0 R T µ(t) = µH0 0+: {∆µ µ · t,= ν(t) == ν00}, + νH· 1t,: {∆µ = , ∆ν = 0} , ∆ν Then, the climate change hypothesis can be checked by assessing change on ν: the Application 2000 expected excess over threshold within this framework: abrupt change in a point of time, change as a function time or other), etc µ(t) = µof0 + ∆µ ·(linear, t, ν(t)logistic = ν0 + ∆ν · t, For hazardous phenomena with a physical upper limit, the Then, thechoice climateischange hypothesis canchange be checked by asparsimonious to consider a linear on ν with change constant, on ν: time,sessing whilst the µ remains 1995 within this framework: abrupt change in a point of time, change as a function of time (linear, logistic or other), etc Assessing the climate change local scalethe For hazardous phenomena withhypothesis a physical at upper limit, parsimonious choice is to consider a linear change on ν with Several models about parameter changes can be assessed time, whilst µ remains constant, 2005 200 Finally, estimation of the parameters is derived from π(µ,ν), either as the mostMlikely value (maximum posterior estimaì fYvalue (yi |à,) (à,) ∝ πas0 (µ,ν) tion), the expected or as any other desired statistic i=1 directly from the estimated grid posteThese are computed riorestimation probabilities Finally, of the parameters is derived from π(µ,ν), either as the most likely value (maximum posterior estimation),4as Assessing the expected orchange as any hypothesis other desired statistic the value climate at local scale These are computed directly from the estimated grid posterior probabilities Several models about parameter changes can be assessed 2000 Tortosa Hs (cm) i 1995 M comes into the playground: a posterior (yi |µ,ν) for the (à,) (à,) ì fYdistribution parameters is derived by perturbing i=1 the prior distribution by the data likelihood (Eq 2) according to the parametric model, Roses rameters In the case presented, the event magnitudes above In a Bayesian estimation process Gelman et al., 1995; the threshold follow a GPD with(e.g the proposed reparametrizaEgozcue and Tolosana-Delgado, 2002), the observable tion, Y ∼ GP D(µ,ν) These parameters are given avariprior able is assumed to follow a parametric model of unknown paprobability distribution π (µ,ν), encoding the knowledge rameters In the case looking presented, thedata eventInmagnitudes above available before at the practical computer the threshold follow with the proposed reparametrizaapplications, thisaisGPD typically a uniform distribution on a disspanning the rangeparameters of a priori credible values of the tion, crete Y ∼ grid GP D(µ,ν) These are given a prior parameters Then, the set of excesses the {yi ,iknowledge = 1, ,M } probability distribution π 0data (µ,ν), encoding comes into the playground: a posterior distribution for the available before looking at the data In practical computer parameters bya perturbing the prior distribution applications, thisisisderived typically uniform distribution on a disdata likelihood (Eq according to values the parametric crete by gridthespanning the range of a2) priori credible of the model, Then, the data set of excesses {y ,i = 1, ,M} parameters 115 3 threshold (log−wave height) Fig data series series(upper (upperplots), plots),location location Fig.2.2 Significant Significant wave wave height height data ofofthe and diagram diagramofofexpected expectedexcesses excesses thebuoys buoys(middle, (middle, right right plot) plot) and asasa afunction (lower plot) plot) This Thisisisused usedtotochoose choose functionof of the the threshold threshold (lower thethreshold, threshold, as as (under (under the hypothesis the hypothesis that thatexcesses excessesare areGPD GPDdisdistributed)the thefunction function should should be aa line tributed) line above aboveit.it.Dashed/black Dashed/blacklines lines denoteRoses, Roses,and and solid/red solid/red ones denote ones Tortosa Tortosa both prone to the samelocations kind of storms, mostly N-NW E nearby, homogeneous For this reason, weoranaldominated (Mestral and Llevant regimes, respectively) We yse simultaneously two locations, selected because they are assume thattothe have amostly different valueor at E both prone theparameters same kindmight of storms, N-NW both stations, but that they should evolve consistently, dominated (Mestral and Llevant regimes, respectively) We R the parameters assume µthat (t) = µ0 , µT (t) might = µ0 +have bµ , a different value at both R stations, but that they T should evolve consistently, ν (t) = ν0 + aν · t, R ν (t) = ν0 + aν · t + bν , T µ (t) = µ0 ,estimation µ (t) =ofµ0all+these bµ , parameters (initial A Bayesian joint ν R (t) = ν0 + aν · t, ν T (t) = ν0 + aν · t + bν , Adv Geosci., 26, 113–117, 2010 Adv Geosci., 26, 113–117, 2010 8e+40 density 2e+40 0e+00 2e+40 0e+00 0.2 −0.05 0.8 bµ 0.00 0.05 aν 0.8 0.6 bµ 0.4 0.2 density 0.6 0.6 0e+00 1e+40 2e+40 3e+40 4e+40 5e+40 6e+40 8e+40 0.4 0.4 µ0 6e+40 0.2 4e+40 6e+40 8e+40 4e+40 density 6e+40 6e+40 4e+40 density 2e+40 0e+00 ν0 4e+40 the local differences bµ ,bν between Tortosa and Roses) is secondary result, the method may provide esticarriedAs outa using simple R routines, with flatalso prior distribumates of hazard-related parameters, like return periods, probtions within grids defined in Table The maximum posteabilities of(most exceedance upper bounds of of excesses (as we rior estimates likely and value of the vector parameters are fitting the data within the Weibull domain) One according to the joint posterior distribution) are includedmust in nevertheless bear in mind that these parameters are all exthe same table The marginal posterior distributions of the tremely uncertain, especially for data series so short as those parameters are shown in Fig 3, together with a visual assessused here Figure shows an example of this uncertainty, by ment of the hypotheses of zero parameter according to the depicting the data set together with kernel density estimates position the posterior with distribution respect to the zero value of theofexcess upper bound Note how in theFor case instance, regarding the time trend, we can conclude that the of Tortosa the spread of the upper bound may be comparahypothesis of no trendof(athe 0) isitself strongly thusbecause there ν =data ble to the spread Thislikely, happens is noTortosa evidence in favor of have a change the shape of theand GPD measurements more in negative ν values, thus (i.e fall in the relative likelihood of strong vs medium storms) nearer to the Gumbel domain (exponential distribution) However, if there is a change inPosed time,initother is more probably a than Roses measurements words, in Roses positive one, of the order ofbear +0.02 units of ν/year the observed excesses evidence an upper boundary near to the data the actually observed Onprovide the contrary, Asquite a secondary result, method may also estiTortosa buoy measurements point to return a larger upper probbound, mates of hazard-related parameters, like periods, withof more uncertainty, the fitted GPD is more similar abilities exceedance andi.e upper bounds of excesses (as weto a distribution with no upper limit, like the exponential form are fitting the data within the Weibull domain) One must for ξ = of Eq (1) This is in agreement with the fact that nevertheless bear in mind that these parameters are all exRoses buoy is placed on a quite sheltered bay, whereas Tortremely uncertain, especially for data series so short as those is open to the Llevant winds: thus usedtosa here.buoy Figure shows an Mestral exampleand of this uncertainty, by one should expect potentially larger measurements in Tortosa depicting the data set together with kernel density estimates than in Roses of the excess upper bound distribution Note how in the case of Tortosa the spread of the upper bound may be comparable to the spread of the data itself This happens because Tortosa measurements have more negative ν values, and thus Table Prior and posterior characterization The prior distribution fall nearer to the Gumbel domain (exponential distribution) is uniform on a 5-dimensional grid, with 21 equally-spaced nodes than along Roseseach measurements Posed in other words, in Roses axis between the minimum and the maximum values the observed excesses bear evidence of an upper boundary reported The time increment aν is measured in ν units per year posterior is computed on the same support, and has quiteThe near to thedistribution data actually observed On the contrary, its maximum value at the vector indicated as “maxpost” Tortosa buoy measurements point to a larger upper bound, with more uncertainty, i.e the fitted GPD is more similar to µ0 ν0 aν bµ bν a distribution with no0.15 upper−9.00 limit, like the exponential form minimum −0.079 0.15 −5.00 for ξ = 0maximum of Eq (1).0.50 This is+9.00 in agreement the fact that +0.079 with 0.70 +5.00 Roses buoy is placed on a quite whereas Tormaxpost 0.225 -1.667 sheltered 0.0197 bay, 0.306 -1.250 tosa buoy is open to the Mestral and Llevant winds: thus one should expect potentially larger measurements in Tortosa than in Roses −5 −4 −3 −2 −1 density the same table The marginal posterior distributions of the parameters areµshown inνFigure 3, atogether a visual bwith bν asν µ 0 sessment of the hypotheses of zero parameter according to minimum −9.00 with −0.079 −5.00 the position 0.15 of the posterior respect0.15 to the zero value maximum 0.50 +9.00 +0.079 0.70 +5.00 that For instance, regarding the time trend, we can conclude maxpost 0.225of no −1.667 0.306 −1.250 the hypothesis trend (aν0.0197 = 0) is strongly likely, thus there is no evidence in favor of a change in the shape of the GPD (i.e in the relative likelihood of strong vs medium A storms) BayesianHowever, joint estimation of aall these inparameters if there is change time, it is (inimore tial values µ ,ν , common time trend in shape ν , and 0 probably a positive one, of the order of +0.02only unitsaν/year 2e+40 µ0and ,ν0 ,posterior commoncharacterization time trend in shape onlydistribution aν , and the Tablevalues Prior The prior local differences b ,b between Tortosa and Roses) is carµ ν grid, with 21 equally-spaced nodes is uniform on a 5-dimensional outaxis using simplethe R routines, distributions alongried each between minimumwith and flat the prior maximum values withinThe grids in Table The maximum posterior esreported timedefined increment aν is1.measured in ν units per year timates (most likelyisvalue of the vector parameters acThe posterior distribution computed on the sameofsupport, and has cording value to theatjoint posterior distribution) are included in its maximum the vector indicated as “maxpost” 8e+40 R Tolosana-Delgado et al.: Climate change in inaaPoint-Over-Threshold Point-Over-Threshold model Tolosana, Ortego, Egozcue and Sanchez-Arcilla: Climate change model 0e+00 116 −2.0 −1.0 bν 0.0 0.2 0.4 0.6 µ0 Fig 3.3 Marginal Marginal posterior posterior distributions Fig distributionsfor forthe themodel modelparameparameters, compared with the joint maximum posterior estimate (Table 1, ters, compared with the joint maximum posterior estimate (Table dashed line) and the hypothesis of zero parameter (solid line) The 1, dashed line) and the hypothesis of zero parameter (solid line) posterior density map show contour curves of logπ(µ0 ,bµ ) This is The posterior density map show contour curves of logπ(µ0 , bµ ) used to obtain estimates of µT Note the stripes in the lower T white This to obtain of µcorrespond Note the whiteposterior stripes in and is leftused margins of thisestimates figure: they to zero the lower and left margins of this figure: they correspond to zero probability posterior probability Conclusions Conclusions Assessing the scale of available data as well as model pa- Assessing the scale of available data as models well as of model rameters allows to parsimoniously check evolu-parameters allows to parsimoniously check models of evolution of these parameters with time For point-over-threshold tion of these parameters withwave time.height, For point-over-threshold (POT) models of significant this general prin(POT) models of significant wave height, ciple suggests to treat log-transformed data,this andgeneral fit themprina reparametrized Pareto Distribution restricted to a ciple suggests toGeneralized treat log-transformed data, and fit them the Weibull domain: the new parameters are the upper bound to reparametrized Generalized Pareto Distribution restricted of the distribution as the location parameter, and shape paramthe Weibull domain: new parameters are athe upper bound parameterization two advantadges: densities ofeter the This distribution as locationhas parameter, and a shape paramalways have a bounded domain for any densities physieter This parameterization has(as twoexpected advantadges: cal process), and checks on the evolution of the distribution always have a bounded domain (as expected for any physishape can be and donechecks independent the upperof bound cal process), on theofevolution the distribution This is applied to an 18-year long data set of significant shape can be done independent of the upper bound wave height from two different buoys, in the same region This is applied to an 18-year long data set of significant but sufficiently far away to consider them roughly indepenwave height from two different buoys, in the same region but dent If a climate change is present, this should be reflected sufficiently far away to consider them roughly independent as a consistent trend in the shape parameter of both series IfResults a climate is present, thisin should reflected as a showchange no significant trend extremebestorm magniconsistent trend in the shape parameter of both series Retudes during the last 18 years Thus, there is no evidence sults show no significant trend in extreme storm magnitudes in this (rather short) data set that climate change is recently during the last 18 years Thus, there is no evidence in this (rather short) data set that climate change is recently modifying distributional properties of the magnitude of extreme www.adv-geosci.net/26/113/2010/ 2000 7.6 7.2 6.6 1000 6.2 Hs (cm) 500 500 Edited by: J Salat Reviewed by: one anonymous referee 2000 2005 200 5.2 200 5.6 Roses Tortosa 1995 References 0.000 0.003 0.006 Fig Data set of events, compared with several results that share the same scale The left panel shows the data (black: Roses, red: Tortosa), together with the time evolution of the expectation of the fitted GPD (Eq 4, thick solid line), as derived from the maximum posterior parameter estimates (Table 1) The most likely posterior estimates (from the same Table, thick dashed line) and 95% confidence interval upper boundary (dashed line) for the upper bound of the excesses (Eq 3) are also displayed The marginal posterior density of these excess upper bound for Roses and Tortosa are displayed separately in the right panel Note the higher uncertainty in Tortosa than in Roses storms in the Catalan coast This does not deny climate change as a whole, given the shortness of the series and the inherent uncertainties of the GPD model A comparison of both stations suggest that the measurements in Tortosa are (relatively) more compatible with a Gumbel domain (i.e an exponential law for the excesses of log-significant waveheight) than those in Roses: though both stations fall within the Weibull domain (bounded distributions), measurements from Tortosa show significantly larger, more uncertain estimates of the upper bound of the distribution This is tentatively related to the sheltered position of the Roses buoy The uncertainty on this upper bound estimates is extremely large This would also happen with other hazardrelated parameters like return periods and exceedance probabilities The set of tools used (GPD with bounded domain for log-waveheight point-over-threshold exceedances within a Bayesian approach) has the additional advantadge to fairly portray this uncertainty www.adv-geosci.net/26/113/2010/ 117 Acknowledgements This research has been supported by the Spanish Ministry of Education and Science under two projects: “Ingenio Mathematica (i-MATH)” Ref No CSD2006-00032 and “CODA-RSS” Ref MTM2009-13272; and by the Ag`encia de Gesti´o d’Ajuts Universitaris i de Recerca of the Generalitat de Catalunya under the project Ref: 2009SGR424 The first author ackowledges also funding within the program “Juan de la Cierva” of the Spanish Ministry of Education and Science (ref “JCI-2008-1835”) 1000 2000 R Tolosana-Delgado et al.: Climate change in a Point-Over-Threshold model Egozcue, J and Ramis, C.: Bayesian hazard analysis of heavy precipitation in eastern spain, Int J Climatol., 21, 1263–1279, 2001 Egozcue, J and Tolosana-Delgado, R.: Program BGPE: Bayesian Generalized Pareto Estimation, Barcelona, Spain, CD-ROM, ISBN 84-69999125, 2002 Egozcue, J J., Pawlowsky-Glahn, V., and Ortego, M I.: Waveheight hazard analysis in eastern coast of spain bayesian approach using generalized pareto distribution Adv Geosci 2, 25– 30, 2005 Egozcue, J J., Pawlowsky-Glahn, V., Ortego, M I., and TolosanaDelgado, R.: The effect of scale in daily precipitation hazard assessment, Nat Hazards Earth Syst Sci., 6, 459–470, doi:10.5194/nhess-6-459-2006, 2006 Embrechts, P., Klăuppelberg, C., and Mikosch, T.: Modelling extremal events for insurance and finance, Berlin, Germany, Springer-Verlag, 663 pp., 1997 Gelman, A., Carlin, J., Stern, H., and Rubin, D.: Bayesian data analysis, New York, NY, USA, Wiley, 526 pp., 1995 Pawlowsky-Glahn, V., Tolosana-Delgado, R., and Egozcue, J J.: Scale effect in hazard assessment application to daily rainfall, Adv Geosci., 2, 117–121, 2005 Pickands, J I (1975) Statistical inference using extreme order statistics Ann Statist., 3, 119–131 S´anchez-Arcilla, A., G´omez-Aguar, J., Egozcue, J J., Ortego, M I., Galiatsatou, P., and Prinos, P.: Extremes from scarce data The role of bayesian and scaling techniques in reducing uncertainty J Hydr Res., 46(extra 2), 224–234, 2008 S´anchez-Arcilla, A., Gonz´alez-Marco, D., and Bola˜nos, R.: A review of wave climate and prediction along the spanish mediterranean coast, Nat Hazards Earth Syst Sci., 8, 1217–1228, 2008 Adv Geosci., 26, 113–117, 2010 ... coordinate system www.adv-geosci.net/26/113/2010/ R Tolosana-Delgado al.: Climate change in a Point- Over- Threshold model Tolosana, Ortego,et Egozcue and Sanchez-Arcilla: Climate change in a Point- Over- Threshold. .. lowerdiagram diagram shows the the classical classical parametrization, parametrization, the shows thedomains domainsofofattraction, attraction,and and theproposed proposed reparametrization: reparametrization:... expectation and upper parameter (lower diagram) bound 2.2 A new parametrization β (3) ysup = − , ξ One can reasonably assume that, in a climate change sceβ nario, the description model of the variable