BayesianSpatialandTrendAnalysisonOzoneExtremeDatain ......2 Advances in Meteorology 2.3. Bayesian...

Research ArticleBayesian Spatial and Trend Analysis on Ozone Extreme Data inSouth Korea 1991ndash2015

Cheru Atsmegiorgis Kitabo

Department of Statistics College of Natural and Computational Sciences Hawassa University Hawassa Ethiopia

Correspondence should be addressed to Cheru Atsmegiorgis Kitabo cheruedenyahoocom

Received 11 April 2020 Accepted 12 November 2020 Published 27 November 2020

Academic Editor Herminia Garcıa Mozo

Copyright copy 2020 Cheru Atsmegiorgis Kitabo(is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Background Extreme events like flooding extreme temperature and ozone depletion are happening in every corner of the world(us the need to model such rare events having enormous damage has been getting priorities in most countries of the worldMethods (e dataset contains the ozone data from 29 representative air monitoring sites in South Korea collected from 1991 to2015 Spatial generalized extreme value (GEV) using maximum likelihood estimation (MLE) and two max-stable and Bayesiankriging models are the statistical models used for analysis Moreover predictive performances of these statistical models arecompared using measures like root-mean-squared error (RMSE) mean absolute error (MAE) relative bias (rBIAS) and relativemean separation (rMSEP) have been utilized Results From the time plot of ozone data extreme ozone concentration is increasinglinearly within the specified period (e return level of ozone concentration after 10 25 50 and 100 years have been forecastedand showed that there was an increasing trend in ozone extremes High spatial variability of ozone extreme was observed andthose areas around the territories were having extreme ozone concentration than the centers Moreover Bayesian Kriging broughtabout relatively the minimum RMSE compared to the other models Conclusion (e extreme ozone concentration has clearlyshowed a positive trend and spatial variation Moreover among the models considered in the paper the Bayesian Kriging has beenchosen as the better model

1 Background

Extreme events like flooding extreme temperature andozone depletion are happening in every corner of the world(us the need to model such rare events is being gettingprioritized in most countries of the world (ese days de-pletion of the stratospheric ozone layer by halogenatedchemicals is a global problem Since the issue was primarily amatter of scientific curiosity during the 1970s and early1980s recently it became an urgent policy question forgovernments in both developed and developing countries

(ere are various effects of ozone starting from humanhealth to increasing the chance of existence of natural ca-tastrophe Some of them are the effects on human andanimal health terrestrial plants aquatic ecosystems bio-geochemical cycles air quality materials climate changeand ultraviolet radiation According to Stedman et al the

effect on human health like eye diseases skin cancer andinfectious diseases is due to increased penetration of solarUV-B radiation [1] (e changes in plant form and sec-ondary metabolism due to UV-B could have an importantimplication for plant competitive balance plant pathogensand biogeochemical cycles

(e climate and environmental applications used mul-tivariate extreme distributions in order to take into accountthe extremal dependence [2ndash5] For the study of extremes ina spatial context max-stable processes are a relevantframework for modeling and inference Different parametricmodels have been later proposed by various authors [6ndash8](e extremal t process is a model of that family that extendsthe Schlather model [9 10] It has a supplementary pa-rameter that lets the extremal coefficient function vary in a

HindawiAdvances in MeteorologyVolume 2020 Article ID 8839455 11 pageshttpsdoiorg10115520208839455

wider range Max-stable processes have been used in ap-plications for modeling air pollution using ozone datarainfall temperature snowfall and snow depth [11ndash14]

Designing the methodology that brings accurate riskmeasures of these extreme events needed to minimize thecasualties on human being and reduce the damage on in-frastructure is indispensable (us the importance ofmodeling such events and getting ready to take precautionmeasures will have great importance

(e main goal of this paper is to model and compare thepredictive performance of the extreme values like spatialGEV with MLE estimation two max-stable spatial modelsand the other recent model Bayesian kriging In most papersBayesian kriging has not been considered for comparison(e comparative criterion considered for comparing thepredictive performance of Schlatherrsquos extremal t andBayesian kriging is the 5-fold cross validation (e paperconsiders 29 air monitoring sites found South Korea and thedaily ozone data have been collected from the year 1991 to2015 8 hoursrsquo maximum ozone concentration has beenselected from the daily maximum ozone concentration(en using this daily maximum ozone the monthlymaximum has been selected From the monthly maximumannual maximum have been selected Actually only ozoneseason months from May to September have been utilizedfor extracting data for the model fitting process

2 Methodology

21 Basics of Extreme Value eory (EVT) (ese days thestatistical modeling of univariate extremes has been wellassessed However these tools to model spatial extreme datahave been active research areas of the time

211 Univariate Extreme Value Models For each station inthis study the univariate extreme valuemodel has been fittedusing generalized extreme value (GEV) distribution usingBayesian perspective (e GEV defined as

G(z) exp minus 1 + ξz minus μσ

1113874 1113875minus (1ξ)

1113890 11138911113896 1113897 (1)

on z 1 + ξ((z minus μ)σ)gt 01113864 1113865 and with parameters satisfyingminus infinlt μltinfin σ gt 0 and minus infinlt ξ ltinfin

(e maximum value of ozone per site(Zti) sim GEV(μti σ ξ) and the parameters of the model arelocation (μ) scale (σ) and shape (ξ) for each 29 air mon-itoring sites Block maxima first approved in work by Fisherand Tippett approach has been utilized [15] Time series plotof the ozone concentration versus time has been plottedTrend has been observed (us the location parameter hasbeen expressed as a linear function of time

μti β0i + β1i(t) (2)

As the data contain ozone concentration in differentsites spatial dependence is inevitable

212 Generalized Extreme Value Distribution (e gener-alized extreme value (GEV) distribution has a locationparameter μ and a scale parameter σ and combines the three

types of extreme value distributions by use of a shape pa-rameter ξ It is usually represented by GEV(μ σ ξ) (edistribution function is as follows the case ξ lt 0 correspondsto theWeibull family for the case ξ 0 the limit of the GEVas ξ⟶ 0 is used which leads to the Gumbel family

G(z) exp minus exp minusz minus μσ

1113874 11138751113882 11138831113876 1113877 (3)

(erefore we now have a single formulation for mod-eling block maxima data which overcomes the issue ofdistribution choice from the three types by allowing the datato choose the appropriate tail behavior for the model (euncertainty for this choice is now incorporated into themodel within the uncertainty for the parameter estimate of ξ


1113874 1113875minus (1ξ)

1113890 11138911113896 1113897 (4)

defined on z 1 + ξ((z minus μ)σ)gt 01113864 1113865 and with parameterssatisfying minus infinlt μltinfin σ gt 0 and minus infinlt ξ ltinfin (e case ξ gt 0corresponds to the Frechet family

(e yearly maximum values of the ozone concentrationper geographic zone i Z1i ZTi follows generalizedextreme value (GEV) distribution of the following form

Zti GEV μti σti ξ1113872 1113873 (5)

with the location parameter

μti β0i + β1i(t) for t 1 25 and i 1 2 29

(6)

where β0i is the intercept and β1i is the trend in t

22 Return Level Estimation for Block Maxima ApproachFinding the probability of occurrence for large events is themajor motivation for modeling extreme data (is demandsus to be able to predict the level of the process that we expectto exceed only once in the time period required which isknown as a return level

For yearly maximum data the value of the return levelcan be estimated from the GEV distribution function di-rectly If we let Zp be the (1P) year return level of theprocess then

G zp1113872 1113873 exp(minus p) (7)

where this expression comes from requiring the exceedancerate of zp to average once every (1P) years and consideringa constant rate of exceedance Accordingly the approximatevalue of the (1P) year return level for the GEV distributionis given as

zp

μ minusσξ

1 minus minus log(1 minus p)1113864 1113865minus ξ

1113876 1113877 for ξ ne 0

μ minus σ log minus log(1 minus p)1113864 1113865 for ξ 0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(8)

2 Advances in Meteorology

23 Bayesian Kriging Model Fitting for Annual MaximumOzone A spatial model with a simpler version that isconsidered in this paper utilizes a model with only one latentspatial process and no measurement errors

Level 1 Y(u) Xβ + σT(u)

Level 2 T(u) sim N 0 Ry(empty)1113872 1113873

Level 3 pr β σ2empty1113872 1113873

(9)

(e likelihood function is given by

L β σ2empty|Y1113872 1113873prop σ21113872 1113873(minus n2)

Ry(empty)11138681113868111386811138681113868

11138681113868111386811138681113868(minus 12)

expminus 12

(y minus Xβ)prime Ry(empty)1113872 1113873minus 1

(y minus Xβ)1113882 1113883

(10)

231 Posterior for the Mean Parameter β (e conjugateprior for the mean parameter β has been considered As-suming a normal conjugate prior for the mean parameter βwhich means

β|σ2lowastemptylowast1113872 1113873 sim N mβ σ2lowastVβ1113872 1113873 (11)

(e posterior distribution is given by

β|Y σ2lowastemptylowast1113872 1113873 sim N Vminus 1β + Xprimeσ2lowastX1113872 1113873

minus 11113874

Vminus 1β mβ + Xprimeσ2lowasty1113872 1113873σ2lowast V

minus 1β + Xprimeσ2lowastX1113872 1113873

minus 11113874 1113875 sim N 1113955βN σ2lowastV 1113954βN

1113874 1113875

(12)

(us normal distribution is a conjugate prior

232 Posterior for theModel Parameter for σ2 (eposteriordistribution for σ2 is obtained by

Pr σ2|y βlowastemptylowast1113872 1113873prop Pr σ21113872 1113873Pr y|βlowastt nσ2q hemptylowast1113872 1113873 (13)

where the prior distribution is the first term and the secondis the likelihood (e posterior distribution for all the threeprior distributions is a scaled-inverse-χ2 of the form and isgiven by

σ2|y βlowastemptylowast1113872 1113873 sim χ2ScI(υ Q) (14)

(e choice of prior distribution determines the pa-rameters (υ Q) In a particular case of the inverse-gammadistribution the scaled-inverse χ2 is the conjugate prior forσ2 (is conjugate distribution is specified by twohyperparameters

σ2|βlowastemptylowast1113872 1113873 sim χ2ScI nσ S2σ1113872 1113873 (15)

which corresponds to ((nσS2σ)σ2) sim χ2(nσ)

(e posterior distribution is

σ2|y βlowastemptylowast1113872 1113873 sim χ2ScI nσ + nnσS

2σ + n1113954σ2

nσ + n1113888 (16)

where 1113954σ2 (1n)(y minus Xβ)primeRminus 1y (y minus Xβ) is the maximum

likelihood estimator for σ2

233 Posterior for the Model Parameters β and σ2 (eposterior distribution is obtained by

Pr β σ2|yemptylowast1113872 1113873propPr β σ2|yemptylowast1113872 1113873Pr y|βt nσ2q hemptylowast1113872 1113873

(17)

(e posterior distribution is a normal-scaled-inverse-χ2ie the product of normal and scaled-inverse-χ2 densitiesfor both priors considered It is given as follows

β σ2|y ϕlowast1113872 1113873 sim N(b V)χ2ScI(υ Q) (18)

(e prior choice determines the parameters(b v V and Q)

24 Max-Stable Models A general representation of max-stable processes can be described by two components astochastic process X(s) and a Poisson process Π with theintensity dζζ2 on (0 infin) Let Xi(s)iisinN X(s) withE[X(s)] 1 and let ζ i isin Π be points of the Poisson process(en the spectral representation of Schlaterrsquos model is givenby

Z(s) maxige1

ζ iXi(s) s isin S (19)

(is is called Schlaterrsquos spectral representation A moreflexible class of max-stable processes by taking Xi(s) to beany stationary Gaussian random field with finite expectationwas suggested by Schlather [7] A stationary max-stableprocess with unit Frechet margins can be obtained by

Z(s) maxi

ζ i max 0 Xi(s)1113864 1113865 (20)

where μ E max(0 Xi(s))1113864 1113865ltinfin and ζ i1113864 1113865 denote the pointsof a Poisson process on (0 infin) with intensity measureμminus 1ζ minus 1dζ

(e correlation ρ(middot) and μminus 1 21113937

1113968is used if the

random process is specified for a stationary isotropicGaussian random field Xi(s) with unit variance(is processZ(s) is called an extremal Gaussian process and the bivariatemarginal distributions are given by

Pr Z s1( 1113857le z1 Z s2( 1113857le z2( 1113857 expminus 12

1z1

+1z2

1113888 1113889 1 +

1 minus 2(ρ(h) + 1)z1z2

z1 + z2( 11138572

1113971

⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (21)

Advances in Meteorology 3

where h is the Euclidean distance between station s1 and s2For this paper the power exponential correlation functionhas been utilized Extremal dependence was discussed byvarious scholars in different disciplines [2ndash4 16] Variousparametric models have been suggested by different experts[6ndash8 17] (e extremal t process is a model of that familythat extends the Schlather model [9 10]

241 Specification of Parameters of Schlaterrsquos Model (ecovariate specification for the location and scale parametersfor fitting Schlaterrsquos model is

μ(s) β01 + β11latitude + β12longitude

σ(s) β02 + β21latitude2

+ β22longitude

Shape(ξ) sim constant

(22)

25 Model Choice Criteria Using 5-Fold Cross ValidationTo compare the precisions of predictions and forecastsobtained from the fitted models we used some validationcriteria [7 18ndash20] We used the root-mean-squared error(RMSE) mean absolute error (MAE) relative BIAS (rBIAS)and relative mean separation (rMSEP) (ese validationcriteria are defined as

RMSE

1m

1113944

m

i11113954zi minus zi( 1113857

2

11139741113972

MAE 1

M1113944

m

i11113954zi minus zi

11138681113868111386811138681113868111386811138681113868

rBIAS 1

mz1113944

m

i11113954zi minus zi( 1113857

rMSEP 1113936

mi1 1113954zi minus zi( 1113857

2

1113936mi1 Zp minus zi1113872 1113873

2

(23)

where m is the total number of observations that we need tovalidate zi is the data indexed by i 1113954zi is the prediction valueand Zi and Zp are the arithmetic mean of the observationsand predictions respectively

3 Results and Discussion

31 Results

311 Air Monitoring Sites Selected for the StudyTwenty-five yearsrsquo (from 1991 to 2015) extreme ozone datahave been collected from the 29 sites (e reason for con-sidering 1991 as the start of data collection was the existenceof large number of missing data in almost all 29 sites prior to1991 (e distribution of air monitoring sites in South Koreais displayed in Figure 1

312 Descriptive Outputs of Maximum OzoneConcentration (e minimum first quartile median thirdquartile and maximum are the descriptive measures used toportray the characteristics of extreme ozone concentrationsin each 29 sites and they are depicted in Figure 2

Relative to the standard ozone concentration in SouthKorea which is 006 ppm the distribution of maximumozone concentration in all air monitoring sites was above thestandards

Time series plot of the maximum ozone concentration ofall stations is displayed in Figure 3 As seen in Figure 3 wecan clearly observe a positive trend in the ozone concen-tration with respect to time (is indicates that the locationparameter of the generalized extreme value (GEV) distri-bution can be expressed as a linear combination of time

313 Station by Station Analysis by GEV We assumed thatthe maximum values of O3 per monitoring stationZ1 ZT are independent and follow a GEV distributionof the following form

Zt sim GEV μt σ ξ( 1113857

μt β0 + β1t t 1 2 T 29(24)

where β0 is an intercept parameter while β1 represents atrend in t for the location parameter and t denotes the orderin which the measurements were obtained Moreover σ andξ are constants in time and denote the scale and shapeparameters of the GEV distribution (e time series plot ofall sites in Figure 3 can be taken as a justification to use alinear trend for the location parameter of generalized ex-treme value distribution of maximum ozone concentration

314 Prior Distribution of Parameter of GEV (e prior isdefined by the marginal distributions μ sim N(0 102)σ sim logN(0 102) and ξ sim N(0 102) and the initial value forthe prior of trend parameter is set to be 0728 With the aboveprior specification a Markov chain θ0 θn with targetdistribution π(θ|x) can be generated (en chains of length10000 have been generated using the initial valuesθ0 (4 1 01 01) and the proposal standard deviations (002 01 01 01) Convergence of the posteriors of allparameters has been checked using trace plots andGeweke test

As it is depicted in Table 1 all the values of β1 for all thestations in the country are positive indicating that there is apositive trend in themaximum ozone concentration with time

38

37

36

35

34

126 127 128 129 130

Figure 1 (e distribution of the selected 29 air monitoring sites inSouth Korea


315 Spatial Generalized Extreme Value Analysis AssumingNo Spatial Dependence In this perspective the spatial ex-treme value analysis does not assume spatial dependenceamong stations in the study areas Using this specificationthe following likelihood is used for this analysis

l(Zψ) 1113944n

i11113944

k

j1logfGEV Zi xj1113872 1113873 μ Zj1113872 1113873 σ Zj1113872 1113873 ξ Zj1113872 11138731113872 1113873

(25)

316 Exploratory Plot of Location Parameter versus Geo-graphic Coordinates We used the exploratory plot in Fig-ure 4 to show the relation between the dependent variableozone extreme value with two covariates latitude and lon-gitude (e plot shows approximately linear relation

As seen in the above plot there seems to be a linear trendbetween the covariates latitude and longitude and the ozoneconcentration

317 Exploratory Plot of Scale Parameter versus GeographicCoordinates Similarly the relationship between the scaleparameter of the spatial GEV model and the geographiccovariates latitude and longitude has been displayed inFigure 5

318 Model Selection Criterion Using Takeuche InformationCriteria Based on the linear trend relation between theozone concentration with that of latitude and longitude thespatial generalized extreme value analysis has been fitted(erefore a various combinations of covariates have beenused to model a spatial GEV model Using this specification9 models have been fitted Takeuche information criteria(TIC) were used for selection Based on this criterion themodel having small TIC would be preferable TIC of thecombinations of covariates is displayed in Table 2

(e second combination has been selected as goodmodel for both the location and scale parameters with theconsideration of the type of relationships displayed inFigures 4 and 5 and actually the smaller TIC value as wellTIC has been used as model selection criteria since Takeuchi[21] proposed Takeuchi information criterion (TIC) as arobust AIC in the spatial GEV model

319 Specification of Location and Scale Parameter (eresponse surface with the covariates latitude and longitudehas been fitted as follows

μ(s) β01 + β11lat + β12lon

σ(s) β02 + β21lat2

+ β22lon


(26)

0090

0085

0080

0075

0070

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Figure 2 Box plot of the distribution of ozone concentration in the 29 air monitoring sites

1991 1994 1997 2000 2003 2006 2009 2012 2015

010

009

008

007

006

O3 c

once

ntra

tion

leve

l

Year

Figure 3 Time series plots of ozone concentration levels in the 29 air monitoring sites in South Korea


Referring the results in Table 3 the model coefficient forlatitude covariate is negative indicating that there is aninverse relation between ozone concentration and latitude(e negative relation between ozone concentration dis-persion and latitude covariate has been observed

3110 Interpolation of Return Level for Spatial GEV (NoSpatial Dependence)

(1) Interpolation of 5-Year Return Level For annual maximadata the value of the return level can be estimated from theGEV distribution function directly

As depicted in Figure 6 different colors show the in-tensity of the extreme ozone concentrations in various airmonitoring sites of South Korea (e air monitoring sitesfound in territories having higher longitude have higher 5-year return level compared to other areas (e 5-year returnlevel is the ozone concentration that is expected to beexceeded once in 5 years

(2) Interpolation of 10 Yearsrsquo Return Level (e 10 yearsrsquoreturn levels (Figure 7) have been interpolated for the studyareas using the spatial generalized extreme value analysis Asimilar pattern has been observed as the 5 yearsrsquo return level(ose areas having smaller longitude have a lower returnlevel

(e higher return periods have higher return levels Toconfirm this relation between return periods and returnlevel we interpolated the 25 50 and 100 yearsrsquo return levels(ese return levels are the magnitude of the extreme ozoneconcentrations that are expected to be exceeded once in 2550 and 100 years respectively

(3) Interpolation of 25 Yearsrsquo Return Level (e areas in theupper left and lower right corner of the 25 years return levelplot have higher interpolated return level relative to theother level Similarly the areas at the center of the study areaSouth Korea depicted smaller 25 yearsrsquo return level More-over areas having longitude between 1277 and 1284irrespective of the latitude have smaller 25 yearsrsquo return levelcompared to other areas as depicted in Figure 8

(4) Interpolation of 50 Yearsrsquo Return Level Similarly theinterpolations of higher return periods like 50 years and 100years have been analyzed and accordingly the results aredisplayed in Figures 9 and 10 respectively As seen in the 510 and 25 yearsrsquo return levels the same pattern of inter-polated return levels for ozone concentration has beenobserved for both 50 and 100 yearsrsquo return periods

(5) Interpolation of 100 Yearsrsquo Return Level

3111 Model Parameter Estimation of Schlaterrsquos Model(e estimated parameter values based on Schlaterrsquos model isdisplayed in Table 4 (ese values of model parameters areused for interpolation of return levels of extreme ozone inthe specified period of time Moreover the standard error ofeach parameter of Schlaterrsquos model depicts the level ofdispersion of the possible model parameters in samplingdistributions

3112 Model Parameter Estimation of Extremal t Model(e estimated model parameter and the correspondingstandard error for each parameters value based on theextremal t model specifications have been displayed inTable 5

3113 Model Comparison Using 5-Fold Cross ValidationRandom classification of these 29 stations in to 5 folds hasbeen done Four of the folds have 6 air monitoring sites andthe last fold has 5 sites Using 5 yearsrsquo return levels as actualozone concentration and the interpolated ozone concen-tration as predicted values the 7 performance measures havebeen computed for each fold and we took the average of it(Table 6)

We used the root-mean-squared error (RMSE) as ameasure of predictive performance In Table 6 using the 5yearsrsquo return level the RMSE for the Bayesian kriging modelbrought the minimum RMSE(is indicates that it has goodprediction performance relative to the other models Similarresults have been found for the 10 25 50 and 100 yearsrsquoreturn levels

Table 1 (e slope (β1) and its standard deviation (SD) of the trendline fitted for Bayesian extreme value analysis at 5 level of sig-nificance in all 29 air monitoring sites

Stations β1 (SD)

Seoul 0014 (0719)Busan 0011 (0312)Daegu 0058 (0787)Incheon 0047 (0740)Anyang 0017 (0356)Songnam 0078 (0810)Uijeongbu 0062 (0570)Kuangmyeon 0029 (0450)Ansan 0018 (0630)Gimcheon 0069 (0490)Bucheon 0009 (0060)Chuncehon 0059 (0740)Kang-neung 0018 (0390)Wonju 0017 (0027)Changwon 0038 (0640)Gwangju 0027 (0623)Daejon 0083 (0590)Ulsan 0012 (0777)Suwon 0010 (0705)Cheongju 0005 (0479)Chinju 0027 (0014)Suncheon 0057 (0710)Jeonju 0077 (0640)Kungsan 0007 (0030)Iksan 0097 (0840)Yeosu 0128 (0940)Gwangyang 0076 (0690)Pohang 0004 (0150)Gumi 0061 (0736)


32 Discussion Quentin conducted a study with the titleldquoComparison of spatial extreme value models application toprecipitation datardquo [22] (is paper compares six modelsfrom max-stable process and other hierarchical modelsusing the return level But the uniqueness of paper mainlylies on the consideration of other means of model com-parison called k-fold cross validation in addition to returnlevel measure

0095

0090

0085

0080

0075

Mu

(gev

)

350 360 370Latitude

(a)

0095

0090

0085

0080

0075

Mu

(gev

)

127 128 129Longitude

(b)

Figure 4 (e relation between the location parameter of the spatial GEV model and the covariates latitude and longitude

Sigm

a (ge

v)

0024

0022

0020

0018

0016

0014

0012

0010350 360 370

Latitude

(a)

Sigm

a (ge

v)0024

0022

0020

0018

0016

0014

0012

00101270 1280 1290

Longitude

(b)

Figure 5 (e relationship between the scale parameter of the spatial GEV model and the covariates latitude and longitude

Table 2 Model selection criteria for spatial GEV model withvarious combinations of covariates using Takeuche informationcriteria (TIC)

Combinations Covariate combinations TIC values1 μ σ β01 + β11lat + β21lon 2738585

2 μ β01 + β11lat + β12lon 2741467σ β02 + β21lat2 + β22lon

3 μ β01 + β11lat + β21lon 2741862σ β02 + β1lat4 μ β01 + β11lat σ β02 + β22lon 2746124

5 μ β01 + β11lat 2742389σ β02 + β21lat + β22lon

6 μ β01 + β11lon 2743248σ β02 + β21lat + β22lon7 μ β01 + β11lat σ β02 + β12lat 27459958 μ β01 + β11lat σ β02 + β22lon 27459959 μ β01 + β11lon σ β02 + β2lon 2747262lon longitude lat latitude loc location μ location s scaleξ considered constant

Table 3 Spatial GEV response surface coefficients estimation andits standard deviation (SD) using maximum likelihood estimationwith covariates latitude and longitude and constant shapeparameterLocation parameterConstant (SD) Latitude (SD) Longitude (SD)minus 0051 (0001) minus 0001 (000628) 00013 (000185)Scale parameterConstant (SD) Latitude (SD) Shape (SD)0063 (000363) minus 0003 (000175) minus 04898 (156)SD standard deviation of the parameter


When using the 5 yearsrsquo return level as an observedvalue the RMSE for the Bayesian kriging models broughtabout the RMSE of 0063 and it is relatively the minimumcompared to 0077 for Schlaterrsquos model 0118 for the spatialGEV with MLE and 0083 for the extremal t model Sim-ilarly using the 25 yearsrsquo return level as observed value theRMSE for Schlaterrsquos spatial GEV (MLE) Bayesian krigingand extremal t are 0083 0130 0071 and 0089 respectively(e last return level considered as an observed value was the

50 yearsrsquo return level Using this value it has brought aboutan RMSE of 0089 0137 0077 and 0091 for Schlaterrsquosspatial GEV (MLE) Bayesian kriging and extremal tmodelsrespectively

Zahidetal conducted a study on spatial prediction andconcluded that Bayesian universal kriging fits better thanuniversal kriging [23] Based on the result from the study byCui Bayesian kriging was more precise as compared to thoseobtained with ordinary kriging [24] A simulation study

375

370

365

360

355

350

1270 1275 1280 1285 1290

Figure 6 Interpolations of the 5 yearsrsquo return level for extreme ozone concentration in 29 air monitoring sites versus longitude (x-axis) andlatitude (y-axis) in South Korea

375

370

365

360

355

350

1270 1275 1280 1285 1290

Figure 7 Interpolation of the 10 yearsrsquo return level for ozone concentration in 29 air monitoring sites versus longitude (x-axis) and latitude(y-axis) in South Korea

375

370

365

360

355

350

1270 1275 1280 1285 1290

Figure 8 Interpolations of the 25 yearsrsquo return levels for extreme ozone concentrations in 29 air monitoring sites versus longitude (x-axis)and latitude (y-axis) in South Korea


375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290

Figure 10 Interpolations of the 100 yearsrsquo return level for extreme ozone concentrations in 29 air monitoring sites versus longitude (x-axis)and latitude (y-axis) in South Korea

Table 4 Schlaterrsquos model parameter estimation using the linear correlation between location and scale parameter with that of latitude andlongitude (shape parameter considered constant)

Nugget (SD) Range (SD) Smooth (SD) μConstant (SD) μLon (SD)

0003 (00012) 0015 (00035) 61825 (293) minus 0049 (000053) minus 0001 (000083)μLat (SD) σCons (SD) σLon (SD) σLat2(SD) ξCons (SD)0001 (00071) minus 0064 (00061) minus 0001 (00036) 0001 (000083) minus 0270 (0034)Lon longitude Lat latitude Cons constant SD standard deviation μmean σ population standard deviation

Table 5 Extremal t max-stable model parameter estimation using the location and scale with geographic coordinates (shape parameterconsidered constant)


0197 (000053) 0546 (00018) 92897 (189) minus 0045 (000016) minus 0001 (000073)μLat (SD) σCons (SD) σLon (SD) σLat2(SD) ξCons (SD)0002 (00038) minus 0044 (0013) minus 0001 (0091) 0005 (0000045) minus 0222 (0026)Lon longitude Lat latitude Cons constant SD standard deviation μMean σ population standard deviation

Table 6 Predictive performance measures of various models using 5 yearsrsquo return level as observed maximum ozone value

Performance measures Schlatherrsquos model Spatial GEV (MLE) Bayesian kriging Extremal-t modelMSE 0006 0014 0004 0007RMSE 0077 0118 0063 0083MAE 0008 0089 0002 0009MAPE 9184 14338 9034 10336BIAS minus 0008 minus 0030 minus 0078 minus 0009rBIAS minus 0092 minus 0118 minus 0088 minus 0104rMSEP 1004 1182 1003 1009


highlighted that the methodology of conditional simulationsfor the extremal t process was effective for a large range ofα isin [1 6] values which characterizes the spatial asymptoticdependence [25]

(e finding from the analysis indicated that the Bayesiankriging model brought about the minimum predictiveperformance measure compared to other extreme valuemodels considered in this research For researchers inter-ested to extend this research I would suggest to consider thecriterion of amount of time consumed for estimating modelparameters and the CPU used for model comparison

4 Conclusion

In this paper spatial GEV using MLE estimation twomodels from max-stable process Schlatherrsquos and extremal tand the Bayesian kriging have been put in comparison withrespect to the predictive performance measures

(e station by station analysis using the Bayesian ap-proach depicted that all the 29 stations considered in thispaper brought about a positive trend in the annual extremeozone concentration(is would indicate that that the ozoneconcentration of all these air monitoring sites has an in-creasing trend with time From the interpolation of the 5 1025 50 and 100 yearsrsquo return level most stations found in theterritories will have a higher ozone concentration

Abbreviations

EVT Extreme value theoryGEV Generalized extreme valueMAE Mean absolute errorMLE Maximum likelihood estimationrBIAS Relative biasrMSEP Relative mean separationRMSE Root-mean-squared errorSD Standard deviationTIC Takeuche information criteriaUV-B Ultraviolet B

Data Availability

All the data used in this study are available from the cor-responding author upon request

Additional Points

(e use of national data that can represent South Korea andthe utilization of the long period (25 years) extreme ozoneconcentration data are among the strengths of this study(e use of national data can help us to make inference to thenational level Although the data source where I extractedthe data is a reliable source the use of secondary data by itselfhas some limitation

Conflicts of Interest

(e author declares no conflicts of interest

Acknowledgments

(e author is grateful to Prof Jong Tae Kim for locating thesource of the data for ozone extreme observations in all airmonitoring sites selected for this research from onlinesource

References

[1] D H Stedman J J Margitan and D H Stedman ldquoAtomicchlorine and the chlorine monoxide radical in the strato-sphere three in situ observationsrdquo Science vol 198 no 4316pp 501ndash503 1981

[2] S G Coles and J A Tawn ldquoStatistical methods for multi-variate extremes an application to structural designrdquo AppliedStatistics vol 43 no 1 pp 1ndash48 1994

[3] L De Haan and J de Ronde ldquoSea and wind multivariateextremes at workrdquo Extremes vol 1 no 1 p 7 1998

[4] M Schlather and J A Tawn ldquoA dependence measure formultivariate and spatial extreme values properties and in-ferencerdquo Biometrika vol 90 no 1 pp 139ndash156 2003

[5] L Fawcett and D Walshaw ldquoSea-surge and wind speed ex-tremes optimal estimation strategies for planners and engi-neersrdquo Stochastic Environmental Research and RiskAssessment vol 30 no 2 pp 463ndash480 2015

[6] R L Smith J A Tawn and H K Yuen ldquoStatistics of mul-tivariate extremesrdquo International Statistical ReviewRevueInternationale de Statistique vol 58 no 1 pp 47ndash58 1990

[7] M Schlather ldquoModels for stationary max-stable randomfieldsrdquo Extremes vol 5 no 1 pp 33ndash44 2002

[8] Z Kabluchko M Schlather and L de Haan ldquoStationary max-stable fields associated to negative definite functionsrdquo Annalsof Probability vol 37 no 5 pp 2042ndash2065 2009

[9] P J Ribeiro and P J Diggle ldquoBayesian inference in Gaussianmodel-based geostatisticsrdquo Technical report ST-99-08 Lan-caster University Lancashire England 2011

[10] T Opitz ldquoExtremal t processes elliptical domain of attractionand a spectral representationrdquo Journal of MultivariateAnalysis vol 122 pp 409ndash413 2013

[11] A C Davison R Huser and E (ibaud ldquoGeostatistics ofdependent and asymptotically independent extremesrdquoMathematical Geosciences vol 45 no 5 pp 511ndash529 2013

[12] A C Davison and M M Gholamrezaee ldquoGeostatistics ofextremesrdquo Proceedings of the Royal Society A MathematicalPhysical and Engineering Sciences vol 468 no 2138 2012

[13] Y Lee S Yoon M S Murshed et al ldquoSpatial modeling of thehighest daily maximum temperature in Korea via max-stableprocessesrdquo Advances in Atmospheric Sciences vol 30 no 6pp 1608ndash1620 2013

[14] J Blanchet and A C Davison ldquoSpatial modeling of extremesnow depthrdquo e Annals of Applied Statistics vol 5 no 3pp 1699ndash1725 2011

[15] R A Fisher and L H C Tippett ldquoLimiting forms of thefrequency distribution of the largest or smallest member of asamplerdquo Mathematical Proceedings of the Cambridge Philo-sophical Society vol 24 no 2 pp 180ndash190 1928

[16] L Fawcett and D Walshaw ldquoEstimating return levels fromserially dependent extremesrdquo Environmetrics vol 23 no 3pp 272ndash283 2012

[17] L De Haan ldquoA spectral representation for max-stable pro-cessesrdquo Annals of Probability vol 12 no 4 pp 1194ndash12041984

[18] Atkinson and Lloyd What should be done in a science as-sessment in protecting the ozone layer lessons models and


prospects Atmospheric and Climate Sciences vol 6 no 1pp 327ndash345 1998

[19] Z Kabluchko M Schlather and L de Haan ldquoStationary max-stable fields associated to negative definite functionsrdquo eAnnals of Probability vol 37 no 5 pp 2042ndash2065 2009

[20] H Zhang ldquoInconsistent estimation and asymptotically equalinterpolations in model-based geostatisticsrdquo Journal of theAmerican Statistical Association vol 99 no 465 pp 250ndash2612009

[21] K Takeuchi ldquoDistribution of information statistics and cri-teria for adequacy of modelsrdquoMathematical Sciences vol 153pp 12ndash18 1976

[22] Q Sebille A L Fougeres and C Mercadier A comparison ofspatial extreme value models Application to precipitation dataPhD thesis Institut Camille Jordan Villeurbanne France2016

[23] E Zahidetal ldquoStatistical modeling of spatial extremesrdquo Sta-tistical Science vol 27 no 2 pp 161ndash186 2009

[24] H Cui ldquoBayesian inference in Gaussian model-based geo-statisticsrdquo Technical report ST-99-08 Lancaster UniversityLancashire England 1995

[25] A Verdin B Rajagopalan W Kleiber and C Funk ldquoABayesian kriging approach for blending satellite and groundprecipitation observationsrdquoWater Resources Research vol 51no 2 pp 908ndash921 2014


wider range Max-stable processes have been used in ap-plications for modeling air pollution using ozone datarainfall temperature snowfall and snow depth [11ndash14]

Designing the methodology that brings accurate riskmeasures of these extreme events needed to minimize thecasualties on human being and reduce the damage on in-frastructure is indispensable (us the importance ofmodeling such events and getting ready to take precautionmeasures will have great importance

(e main goal of this paper is to model and compare thepredictive performance of the extreme values like spatialGEV with MLE estimation two max-stable spatial modelsand the other recent model Bayesian kriging In most papersBayesian kriging has not been considered for comparison(e comparative criterion considered for comparing thepredictive performance of Schlatherrsquos extremal t andBayesian kriging is the 5-fold cross validation (e paperconsiders 29 air monitoring sites found South Korea and thedaily ozone data have been collected from the year 1991 to2015 8 hoursrsquo maximum ozone concentration has beenselected from the daily maximum ozone concentration(en using this daily maximum ozone the monthlymaximum has been selected From the monthly maximumannual maximum have been selected Actually only ozoneseason months from May to September have been utilizedfor extracting data for the model fitting process

2 Methodology

21 Basics of Extreme Value eory (EVT) (ese days thestatistical modeling of univariate extremes has been wellassessed However these tools to model spatial extreme datahave been active research areas of the time

211 Univariate Extreme Value Models For each station inthis study the univariate extreme valuemodel has been fittedusing generalized extreme value (GEV) distribution usingBayesian perspective (e GEV defined as


1113874 1113875minus (1ξ)

1113890 11138911113896 1113897 (1)

on z 1 + ξ((z minus μ)σ)gt 01113864 1113865 and with parameters satisfyingminus infinlt μltinfin σ gt 0 and minus infinlt ξ ltinfin

(e maximum value of ozone per site(Zti) sim GEV(μti σ ξ) and the parameters of the model arelocation (μ) scale (σ) and shape (ξ) for each 29 air mon-itoring sites Block maxima first approved in work by Fisherand Tippett approach has been utilized [15] Time series plotof the ozone concentration versus time has been plottedTrend has been observed (us the location parameter hasbeen expressed as a linear function of time

μti β0i + β1i(t) (2)

As the data contain ozone concentration in differentsites spatial dependence is inevitable

212 Generalized Extreme Value Distribution (e gener-alized extreme value (GEV) distribution has a locationparameter μ and a scale parameter σ and combines the three

types of extreme value distributions by use of a shape pa-rameter ξ It is usually represented by GEV(μ σ ξ) (edistribution function is as follows the case ξ lt 0 correspondsto theWeibull family for the case ξ 0 the limit of the GEVas ξ⟶ 0 is used which leads to the Gumbel family

G(z) exp minus exp minusz minus μσ

1113874 11138751113882 11138831113876 1113877 (3)

(erefore we now have a single formulation for mod-eling block maxima data which overcomes the issue ofdistribution choice from the three types by allowing the datato choose the appropriate tail behavior for the model (euncertainty for this choice is now incorporated into themodel within the uncertainty for the parameter estimate of ξ


1113874 1113875minus (1ξ)

1113890 11138911113896 1113897 (4)

defined on z 1 + ξ((z minus μ)σ)gt 01113864 1113865 and with parameterssatisfying minus infinlt μltinfin σ gt 0 and minus infinlt ξ ltinfin (e case ξ gt 0corresponds to the Frechet family

(e yearly maximum values of the ozone concentrationper geographic zone i Z1i ZTi follows generalizedextreme value (GEV) distribution of the following form

Zti GEV μti σti ξ1113872 1113873 (5)

with the location parameter

μti β0i + β1i(t) for t 1 25 and i 1 2 29

(6)

where β0i is the intercept and β1i is the trend in t

22 Return Level Estimation for Block Maxima ApproachFinding the probability of occurrence for large events is themajor motivation for modeling extreme data (is demandsus to be able to predict the level of the process that we expectto exceed only once in the time period required which isknown as a return level

For yearly maximum data the value of the return levelcan be estimated from the GEV distribution function di-rectly If we let Zp be the (1P) year return level of theprocess then

G zp1113872 1113873 exp(minus p) (7)

where this expression comes from requiring the exceedancerate of zp to average once every (1P) years and consideringa constant rate of exceedance Accordingly the approximatevalue of the (1P) year return level for the GEV distributionis given as

zp

μ minusσξ

1 minus minus log(1 minus p)1113864 1113865minus ξ

1113876 1113877 for ξ ne 0

μ minus σ log minus log(1 minus p)1113864 1113865 for ξ 0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(8)






(9)



Ry(empty)11138681113868111386811138681113868

11138681113868111386811138681113868(minus 12)

expminus 12


(y minus Xβ)1113882 1113883

(10)





minus 11113874




1113874 1113875

(12)











2σ + n1113954σ2

nσ + n1113888 (16)





(17)





Z(s) maxige1



Z(s) maxi

ζ i max 0 Xi(s)1113864 1113865 (20)






1z1

+1z2

1113888 1113889 1 +

1 minus 2(ρ(h) + 1)z1z2

z1 + z2( 11138572

1113971

⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (21)






+ β22longitude


(22)


RMSE

1m

1113944

m

i11113954zi minus zi( 1113857

2

11139741113972

MAE 1

M1113944

m


11138681113868111386811138681113868111386811138681113868

rBIAS 1

mz1113944

m

i11113954zi minus zi( 1113857

rMSEP 1113936

mi1 1113954zi minus zi( 1113857

2

1113936mi1 Zp minus zi1113872 1113873

2

(23)



31 Results







μt β0 + β1t t 1 2 T 29(24)




38

37

36

35

34

126 127 128 129 130




l(Zψ) 1113944n

i11113944

k


(25)








σ(s) β02 + β21lat2

+ β22lon


(26)

0090

0085

0080

0075

0070

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29


1991 1994 1997 2000 2003 2006 2009 2012 2015

010

009

008

007

006

O3 c

once

ntra

tion

leve

l

Year

















Stations β1 (SD)




0095

0090

0085

0080

0075

Mu

(gev

)

350 360 370Latitude

(a)

0095

0090

0085

0080

0075

Mu

(gev

)


(b)


Sigm

a (ge

v)

0024

0022

0020

0018

0016

0014

0012

0010350 360 370

Latitude

(a)

Sigm

a (ge

v)0024

0022

0020

0018

0016

0014

0012

00101270 1280 1290

Longitude

(b)













375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290



375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290













4 Conclusion



Abbreviations


Data Availability


Additional Points




Acknowledgments


References

































(9)



Ry(empty)11138681113868111386811138681113868

11138681113868111386811138681113868(minus 12)

expminus 12


(y minus Xβ)1113882 1113883

(10)





minus 11113874




1113874 1113875

(12)











2σ + n1113954σ2

nσ + n1113888 (16)





(17)





Z(s) maxige1



Z(s) maxi

ζ i max 0 Xi(s)1113864 1113865 (20)






1z1

+1z2

1113888 1113889 1 +

1 minus 2(ρ(h) + 1)z1z2

z1 + z2( 11138572

1113971

⎛⎝ ⎞⎠⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (21)






+ β22longitude


(22)


RMSE

1m

1113944

m

i11113954zi minus zi( 1113857

2

11139741113972

MAE 1

M1113944

m


11138681113868111386811138681113868111386811138681113868

rBIAS 1

mz1113944

m

i11113954zi minus zi( 1113857

rMSEP 1113936

mi1 1113954zi minus zi( 1113857

2

1113936mi1 Zp minus zi1113872 1113873

2

(23)



31 Results







μt β0 + β1t t 1 2 T 29(24)




38

37

36

35

34

126 127 128 129 130




l(Zψ) 1113944n

i11113944

k


(25)








σ(s) β02 + β21lat2

+ β22lon


(26)

0090

0085

0080

0075

0070

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29


1991 1994 1997 2000 2003 2006 2009 2012 2015

010

009

008

007

006

O3 c

once

ntra

tion

leve

l

Year

















Stations β1 (SD)




0095

0090

0085

0080

0075

Mu

(gev

)

350 360 370Latitude

(a)

0095

0090

0085

0080

0075

Mu

(gev

)


(b)


Sigm

a (ge

v)

0024

0022

0020

0018

0016

0014

0012

0010350 360 370

Latitude

(a)

Sigm

a (ge

v)0024

0022

0020

0018

0016

0014

0012

00101270 1280 1290

Longitude

(b)













375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290



375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290













4 Conclusion



Abbreviations


Data Availability


Additional Points




Acknowledgments


References

































+ β22longitude


(22)


RMSE

1m

1113944

m

i11113954zi minus zi( 1113857

2

11139741113972

MAE 1

M1113944

m


11138681113868111386811138681113868111386811138681113868

rBIAS 1

mz1113944

m

i11113954zi minus zi( 1113857

rMSEP 1113936

mi1 1113954zi minus zi( 1113857

2

1113936mi1 Zp minus zi1113872 1113873

2

(23)



31 Results







μt β0 + β1t t 1 2 T 29(24)




38

37

36

35

34

126 127 128 129 130




l(Zψ) 1113944n

i11113944

k


(25)








σ(s) β02 + β21lat2

+ β22lon


(26)

0090

0085

0080

0075

0070

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29


1991 1994 1997 2000 2003 2006 2009 2012 2015

010

009

008

007

006

O3 c

once

ntra

tion

leve

l

Year

















Stations β1 (SD)




0095

0090

0085

0080

0075

Mu

(gev

)

350 360 370Latitude

(a)

0095

0090

0085

0080

0075

Mu

(gev

)


(b)


Sigm

a (ge

v)

0024

0022

0020

0018

0016

0014

0012

0010350 360 370

Latitude

(a)

Sigm

a (ge

v)0024

0022

0020

0018

0016

0014

0012

00101270 1280 1290

Longitude

(b)













375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290



375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290













4 Conclusion



Abbreviations


Data Availability


Additional Points




Acknowledgments


References






























l(Zψ) 1113944n

i11113944

k


(25)








σ(s) β02 + β21lat2

+ β22lon


(26)

0090

0085

0080

0075

0070

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29


1991 1994 1997 2000 2003 2006 2009 2012 2015

010

009

008

007

006

O3 c

once

ntra

tion

leve

l

Year

















Stations β1 (SD)




0095

0090

0085

0080

0075

Mu

(gev

)

350 360 370Latitude

(a)

0095

0090

0085

0080

0075

Mu

(gev

)


(b)


Sigm

a (ge

v)

0024

0022

0020

0018

0016

0014

0012

0010350 360 370

Latitude

(a)

Sigm

a (ge

v)0024

0022

0020

0018

0016

0014

0012

00101270 1280 1290

Longitude

(b)













375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290



375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290













4 Conclusion



Abbreviations


Data Availability


Additional Points




Acknowledgments


References











































Stations β1 (SD)




0095

0090

0085

0080

0075

Mu

(gev

)

350 360 370Latitude

(a)

0095

0090

0085

0080

0075

Mu

(gev

)


(b)


Sigm

a (ge

v)

0024

0022

0020

0018

0016

0014

0012

0010350 360 370

Latitude

(a)

Sigm

a (ge

v)0024

0022

0020

0018

0016

0014

0012

00101270 1280 1290

Longitude

(b)













375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290



375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290













4 Conclusion



Abbreviations


Data Availability


Additional Points




Acknowledgments


References






























0095

0090

0085

0080

0075

Mu

(gev

)

350 360 370Latitude

(a)

0095

0090

0085

0080

0075

Mu

(gev

)


(b)


Sigm

a (ge

v)

0024

0022

0020

0018

0016

0014

0012

0010350 360 370

Latitude

(a)

Sigm

a (ge

v)0024

0022

0020

0018

0016

0014

0012

00101270 1280 1290

Longitude

(b)













375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290



375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290













4 Conclusion



Abbreviations


Data Availability


Additional Points




Acknowledgments


References
































375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290



375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290













4 Conclusion



Abbreviations


Data Availability


Additional Points




Acknowledgments


References





























375

370

365

360

355

350

1270 1275 1280 1285 1290


375

370

365

360

355

350

1270 1275 1280 1285 1290













4 Conclusion



Abbreviations


Data Availability


Additional Points




Acknowledgments


References































4 Conclusion



Abbreviations


Data Availability


Additional Points




Acknowledgments


References





























BayesianSpatialandTrendAnalysisonOzoneExtremeDatain ......2 Advances in Meteorology 2.3. Bayesian...

Documents

Transcript of BayesianSpatialandTrendAnalysisonOzoneExtremeDatain ......2 Advances in Meteorology 2.3. Bayesian...