Spatial Extreme Value Using Bayesian Hierarchical Model For Precipitation Return Levels Prediction

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1 PROCEEDING OF 3 RD INTERNATIONAL CONFERENCE ON RESEARCH, IMPLEMENTATION AND EDUCATION OF MATHEMATICS AND SCIENCE YOGYAKARTA, 6 7 MAY 06 Spatal Extreme Value Usng Bayesan Herarchcal Model For Precptaton Return Levels Predcton Indra Tsan Hazhah, Sutkno, Dedy Dw Prastyo 3 Dept. of Statstcs, Insttut Teknolog Sepuluh Nopember, Surabaya, Indonesa Dept. of Statstcs, Insttut Teknolog Sepuluh Nopember, Surabaya, Indonesa 3 Dept. of Statstcs, Insttut Teknolog Sepuluh Nopember, Surabaya, Indonesa e-mal: ndratsan@gmal.com Abstract Extreme precptaton s a rare natural phenomenon categorzed as one of extreme clmates ndcator. It leads to natural dsasters such as floods and landsldes. Predcton of precptaton return level,.e. quantle whch has probablty /m of beng exceeded n m perod, become qute mportant to reduce the negatve mpact caused by ths extreme event. The lnk between observaton recorded on a partcular tme frame and quanttes of longer tme scales such as return level s provded by Extreme Value Theory (EVT) commonly used to learn the behavors of extreme events. Gven the observatons are recorded from several locatons, the extreme events at dfferent locatons are drven by geographcal and clmatologcally factors. Unfortunately, the data of these factors are not always avalable. In ths study, the spatal Bayesan herarchcal model (BHM) was employed to update the nformaton n the lkelhood that s not fully descrbed by those unobservable covarates. The proposed method was appled to predct the extreme precptaton return level n Lamongan dstrct, East Java, Indonesa. The Peak Over Threshold (POT) scheme was used to obtan extreme observatons. The pror dstrbuton was employed to update the lkelhood of Generalzed Pareto Dstrbuton as an asymptotc dstrbuton of exceedng resulted from POT procedure. The emprcal results showed that the return level got hgher for longer perods. Keywords: Extreme value theory, Peaks Over Threshold, Bayesan herarchcal model, spatal, return level. M 09 I. INTRODUCTION Extreme value theory (EVT) s a method that s commonly used to analyze extreme natural events. Such extreme events assocated wth the locaton are called spatal extreme value. The ntensve ran that usually happens durng the rany season can cause floods and landsldes. These dsasters are examples of negatves effect caused by extreme events. The studes to learn these extreme phenomena are qute mportant to predct and antcpate ther occurrence n the future. The EVT s used to analyze the heavy-taled dstrbuton of the extreme events. There are two approaches used EVT: () Block Maxma (BM) approach that has the Generalzed Extreme Value (GEV) dstrbuton and () Peaks over Threshold (POT) approach that follows the Generalzed Pareto Dstrbuton (GPD) dstrbuton. These research employed POT scheme. The characterstcs of the GPD are expressed by the three parameters of EVT,.e. threshold ( u ), scale ( ), and shape ( ). One of the most mportant thngs n the EVT s to calculate the value of return level,.e. the probablty of occurrence of a partcular level of extreme events n the comng perod. Knowledge of the return level s mportant for dsaster mtgaton and for the preparaton of long-term programs []. Ths paper dscusses the procedures to obtan the parameters estmates GPD and to calculate the return level usng Bayesan Herarchcal Model (BHM). The BHM s chosen because t has some advantages, for nstance, t able to accommodate geography and clmatology at each level of the herarchy structure. Ths paper s organzed as follows. Ths secton s followed by a secton for a method that brefly dscuss POT and BHM. The next part s used to descrbe the data and research methodology. The last part gves concluson and suggestons. M - 5

2 ISBN II. LITERATURE REVIEW In ths part s dscussed how the procedure was done to model the spatal extreme value by usng the POT approach that follows GPD combned wth BHM. A. Peaks Over Threshold (POT) POT s one of the methods to dentfy the extreme value by usng the so-called threshold (u) as a reference value. The data above the threshold wll be dentfed as extreme values. Fgure shows how to collect extreme data usng POT. The values of, and are larger than the threshold (u) therefore these sx data are consdered as extreme values that wll be used n further analyss u F FIGURE. ILLUSTRATION OF EXTREME DATA COLLECTION USING POT The hgher threshold s the hgher probablty that the extremes wll approach the GPD []. The probablty densty functon (pdf) of GPD s formulated as follow: f x u x u, 0 exp x u, 0 () where 0 x u f 0 and u x u u f 0 parameter and shape parameter (the tal ndex), respectvely.. The and n () represent scale B. Parameter Estmaton of Generalzed Pareto Dstrbuton Parameter estmaton of GPD can be done usng several ways. One of them commonly used s a Maxmum Lkelhood Estmaton (MLE) that maxmzes the lkelhood functon of y, y,, y n, where y x u. Therefore, the equaton () s rewrtten n () as follows: f,, y The log-lkelhood functon of () s expressed as: y, 0 exp y, 0 n y ln L, y, y,, y n ln ln n () (3) M - 5

3 PROCEEDING OF 3 RD INTERNATIONAL CONFERENCE ON RESEARCH, IMPLEMENTATION AND EDUCATION OF MATHEMATICS AND SCIENCE YOGYAKARTA, 6 7 MAY 06 The frst dervatve of log-lkelhood wth resct to each parameters are: ln L n y n y ln y (4) ln L n y n y The soluton for shape parameter estmate usng MLE s not close form. The Newton-Raphson was employed to solve ths ssue. C. Determnaton of the Threshold The extreme values were obtaned once the threshold (u) was determned as a reference value. The determnaton of the threshold ams to fnd the optmum balance n order to obtan the mnmum error. In ths paper, the Mean Resdual Lfe Plot (MRLP) was used to determne the threshold wth the followng steps.. Frst, makng the MRP wth the coordnates of the ponts based on the followng equaton, n u u, x u : u x n maks u (6) where n u s the number of observatons above the threshold. The lnear functon of u s: u uu u 0 0 E X u X u where s extreme threshold value and s the scale parameter changes. (5), (7). Chosen value threshold pont when the plot began to form a lnear pattern around the value of u. D. Autocorrelaton Functon (ACF) In ths paper, the ACF was employed to detect f the extreme values larger than the threshold has a stochastc nature. Stochastc nature s random nature that can only be explaned by a dstrbuton [3]. Therefore, when the ACF s not sgnfcant or statonary, t s necessary to declusterng data that take the extreme values of the hghest n the range (r) determned n advance usng the extremal ndex at (8) [4]. N T mn, f max T : N N N T E. Clusterng the Locaton N T N N T T mn, f max T : N Groupng of locatons used to obtan shape parameter that estmated from statonary process. In ths paper, f the slope values that are not statonary, then the groupngs of locatons wth non-herarchcal clusterng method,.e. k-mean cluster, was employed. The number of groups was determned usng the average value of slhouette coeffcent calculated as n (9) [5]: S j B j k O j O j B j k mn, (9) max,, (8) M - 53

4 ISBN where denotes the average dstance the members j wth other members from dfferent cluster and descrbes the average dstance between the member j to other members n the same cluster. F. Bayesan Herarchcal Model (BHM) The purpose of usng BHM s to accommodate geographcal and clmatologcal nformaton nto the dstrbuton of parameters and to update the parameters through the data that were known as a posteror dstrbuton. The posteror dstrbuton was determned by the followng Bayes theorem. z k, j, c k, j y k, j, (0) p θ, θ Z s p Z s θ p θ θ p 3 θ, () where c k, js the constant transformaton of resultng the cluster, z k, j, s the result of the transformaton n the cluster k, staton j, and observaton. The s jont probablty (or lkelhood), s condtonal where ϕ, and are scale and shape parameters n lkelhood. The and are parameters n layer process. pror dstrbuton, and s pror dstrbuton. The parameters n () are θ, and θ, III. DATA AND RESEARCH METHODOLOGY Ths secton dscusses how to obtan estmates of the parameters and return level for the modelng of extreme precptaton events usng BHM. The softwares used n ths work are openbugs, Mcrosoft Excel, and R. The proposed method was appled to model ranfall data observed n sx locatons n Lamongan dstrct, East Java, Indonesa. A. Data and Pre-Processng Pre-processng the data used to dentfy the mssng value, outlers, and the observatons that do not ft the requrement. The Mcrosoft Excel was used for pre-processng and open-source software R was used to calculate descrptve statstcs such as hstograms and normalty plot, see [6]. TABLE. DATA STRUCTURE Year Month Staton S S S Observaton u v u v u v 98 x, x, x, 98 x, x, x, x,n x,n x,n B. Samplng Extreme Value of the Data. Installng EVA packages n R used to analyze extreme value.. Actvatng extreme toolkt and create MRLP for a specfed nterval. 3. Identfyng the mean resdual lfe plot, fnd upper and lower bounds that are stable on the plot. Use the lower and upper lmts of the modfed parameters to make the plot of scale and shape parameters. Next, select the approprate threshold values based on the results of the modfed parameter plot. 4. Collectng extreme data above the threshold value. 5. The data obtaned were tested wth ACF plot. When there s no sgnfcant lag (statonary), then test the GPD usng Anderson-Darlng test [3]. If t s not statonary yet, then the declusterng must be done. 6. A groupng staton usng the k-mean method and fll t wth the value Slhouette as n (9) to obtan a proper number of the cluster. C. Parameter Estmaton wth BHM. Complng the Drected Acyclc Graph (DAG) to see the relatonshp between the data, parameters of the model, and the pror dstrbuton usng openbugs software [7].. Determnng the ntal value for each parameter to be estmated. M - 54

5 Autocorrelaton Shape Mean Excess Modfed Scale PROCEEDING OF 3 RD INTERNATIONAL CONFERENCE ON RESEARCH, IMPLEMENTATION AND EDUCATION OF MATHEMATICS AND SCIENCE YOGYAKARTA, 6 7 MAY Determnng pror for,,,0,, and as the parameters n layer process and lkelhood. D. Applcaton to Real Data The steps descrbed before were appled to ranfall data n Lamongan dstrct observed n seven locatons. The data used s daly ranfall data span from 98 to 03. The frst step done s determnng the threshold value. Input the data nto Mcrosoft Excel CSV (MC-DOS) and comple t nto extreme toolkt already actvated by typng lbrary (extremes) n R, as shown Fgure. The MRLP, modfed shape, scale and all about extreme value usng extremes toolkt were produced as dsplayed n Fgure 3. FIGURE. EXTREME TOOLKIT Mean Resdual Lfe Plot: runnng V Threshold u FIGURE 3. MRLP (LEFT) AND MODIFIED SCALE AND SHAPE PARAMETER PLOT (RIGHT) As shown n Fgure 3, the threshold lmt (u) 60 began to show nstablty. Creatng the ACF plot and conductng Anderson Darlng s test (usng EVA package n R) determne f the extreme data comples the dstrbuton,.e. GPD. If the test ACF nsgnfcant and Anderson darlng s test show that the GPD s satsfed, then the next step s dong declusterng to take samples agan. But, f the test does not meet the GPD, t s necessary to re-determne the threshold. Threshold Autocorrelaton Functon for V6 (wth 5% sgnfcance lmts for the autocorrelatons) Lag LOCATIONS STATISTIC P-VALUE REMARKS BABAT NOT REJECT H0 BLAWI NOT REJECT H0 KARANGBINANGUN NOT REJECT H0 KEDUNGPRING NOT REJECT H0 LAMONGAN NOT REJECT H0 PANGKATREJO NOT REJECT H0 SUKODADI NOT REJECT H0 FIGURE 4. ACF PLOT (LEFT) AND THE RESULT OF ANDERSON DARLING TEST (RIGHT) M - 55

6 ISBN The ACF plot for Blaw staton s dsplayed n Fgure 3 (left). The plot shows that the extreme observatons are ndependent. The Anderson Darlng test wth null hypothess assumes that the extreme observaton follows GPD. Ths test was employed usng R software wth command gpdad(y) from EVA package. The observaton n all locatons follows GPD as P-values are larger than Type-I error 5%. The next step s estmatng the scale and shape parameter by means of MLE. The results are summarzed n Table. TABLE. ESTIMATES OF PARAMETERS LOCATIONS Scale Shape BABAT BLAWI KARANGBINANGUN KEDUNGPRING LAMONGAN PANGKATREJO SUKODADI Once the parameters estmators were obtaned, the next step s applyng BHM (n ths work usng openbugs software) to the data. The condtonal pror s updated by nvolvng the locaton nformaton as follows: s MVN, θ θ μ Σ p ~ () 0 lats longs (3) exp( s s ' ) (4) The mean accommodate the locaton coordnate represented by lattude and longtude whereas the varance-covarance matrx s accommodate the dstances among locatons. The equaton (3) and (4) are the mean and varance of the process layer () to fnd the posteror dstrbuton n(). Fgure 5 shows the posteror densty of scale parameters. FIGURE 5. POSTERIOR DENSITY OF SCALE PARAMETERS TABLE 3. PARAMETERS ESTIMATOR OBTAINED FROM BHM APPROACH Param. mean Sd MC_error val.5pc medan val97.5pc Start sample Beta Beta alpha alpha alpha scale[] scale[] scale[3] scale[4] M - 56

7 PROCEEDING OF 3 RD INTERNATIONAL CONFERENCE ON RESEARCH, IMPLEMENTATION AND EDUCATION OF MATHEMATICS AND SCIENCE YOGYAKARTA, 6 7 MAY 06 scale[5] scale[6] scale[7] Table 3 shows that the parameters 0, and are not sgnfcant because there s zero value wthn credble nterval whle the others are sgnfcant. The sgnfcant parameters can be used n the model. Thus, t can be concluded that the lattude and longtude varable are sutable for spatal models but not sutable as an effect of predctor models. The model of layer process wth equal to zero s expressed n (5): s MVN 0, θ θ Σ p ~ (5) Havng obtaned a sgnfcant model, the next step s to estmate the scale and shape parameter as reported n Table 4. TABLE 4. ESTIMATOR OF SCALE PARAMETERS AND RETURN LEVEL OBTAINED FROM BHM APPROACH LOCATIONS Return Level (Years) n (x>u) u scale Shape Pr(X>u) BABAT BLAWI KARANGBINANGUN KEDUNGPRING LAMONGAN PANGKATREJO SUKODADI The predctons of return level for 5, 0, 30, 40 and 00 years were summarzed n Table 4. In 08 (fve years return level), the predcted heavest ranfall daly n Babat Staton s 30.9 mm. The return level ncreased for twenty years become mm contnuously ncreased for longer perods. Ths also happened n other sx locatons. IV. CONCLUSION AND SUGGESTIONS Ths paper showed the procedure for obtanng estmates of GPD parameters and calculatng the return level usng a Bayesan Herarchcal models appled to ranfall data n Lamongan. The locaton nformaton represented by longtude and lattude varable were consdered as an nput varable n the model. The emprcal result shows that these two varables are not sgnfcant n the mean model. The return level got hgher for longer perods n all statons. These emprcal fndngs suggest that the future research should explore other clmatology varables as nput n the model. ACKNOWLEDGMENT The authors would lke to thank BMKG Surabaya for gvng permsson to use the data employed n ths paper. REFERENCES [] Cooley, D., Nychka, D., and Naveau, P. Bayesan Spatal Modelng of Extreme Precptaton Return Levels. New York : Journal of the Amerca Assocaton, 007. Vol. 0. [] Gll, M., and Kellez, E. An Applcaton of Extreme Value Theory for Measurng Rsk. Amsterdam: Elseber Scence, 003. [3] We, W. Tme Seres Analyss: Unvarate and Multvarate. New York: Addson-Wesley Publshng Co. 006 [4] Ferro, C., and Seger, J. Inference for Clusters of Extreme Values. Journal R. Stat. Socety, 65, [5] Handoyo, R., Ruman, and Nasuton, S. M. Orygnaly n Indonesa Perbandngan Metode Clusterng Menggunakan Metode Sngle Lnkage dan K - Means Pada Pengelompokan Dokumen. JSM STMIK Mkrosk, 5 (), [6] Evertt, Bran S., and Hothorn, Torsten. A handbook os Statstcal Analyses Usng R. London: Chapman & Hall/CRC, 006. M - 57

8 ISBN [7] Ntzoufras, I. Bayesan Modelng Usng WnBUGS. Greece: WILEY, 009. M - 58