Parameter Estimation for Bivariate Mixed Lognormal Distribution

Transcription

1 Parameter Estimatio for Bivariate Mixed Logormal Distributio Chig Yee Kog * Suhaila Jamaludi Fadhilah Yusof Hui Mea Foo * Departmet of Mathematics Faculty of Sciece Uiversiti Tekologi Malaysia 83 Skudai Johor Malaysia. * Correspodig chigyeekog87@gmail.com Abstract Bivariate mixed logormal distributio is a probability model used for represetig raifalls behavior at two moitorig statios. The paper discuss o the parameter estimatio for bivariate mixed logormal distributio i which all parameters are assumed to be ukow. Six cases were cosidered i the aalysis ad the parameters were estimated usig the maximum likelihood. The optimal model was selected based o the miimum Akaike s iformatio criterio (AIC) from selected model. The aalysis is ru by usig the raifall data observed for the time period of 33 years (975-7) from Arau Perlis with each of the other 7 earby moitorig statios ad 5 far distace statios. Amog the 7 statios studied 6 statios (87.5%) choose the same case model (M) as the miimum AIC procedures. Meawhile 4 of the far distace statios choose the case M as the best fit case model. Keywords: bivariate mixed logormal; akaike s iformatio criterio 4

2 . INTRODUCTION Raifall data is widely used i hydrological applicatio therefore may researches had carried out to estimate the characteristic of rai (Moro et al. Youg Zhag ad Sigh Mielke et al. Habib ad Krajewski[]-[5]). The raifall behaviour of two moitorig statios usually recogise bivariate mixed logormal distributio as a suitable probability model. The cocept of applyig mixed distributio i raifall data origi by Kedem[6] while Shimizu[7] eriched the usage of mixed distributio by implemetig it i bivariate distributio. Mixed distributio is a mixture of discrete distributio ad cotiuous distributio. The raifall characteristic is a combiatio of discrete distributio ad cotiuous distributio where discrete distributio represets the zero values for the days that do ot rai ad cotiuous distributio represets raifall amout o raiy days. The mai objective of this paper is to estimate all the parameters of bivariate mixed logormal distributio where all the parameters are assume to be ukow. The past research that have bee carried out i Malaysia mostly do ot cosidered the attribute of the raifall which there exist zeros value (days that do ot have rai). The importat of icluded the zeros for the study of characterizatio of raifall ca be aalyzed as suggested i Ha ad Yoo [8].. RAINFALL DATA The daily raifall data studied were obtaied from the Malaysia Meteorological Departmet ad Draiage ad Irrigatio Departmet which cosist of thirty three years time period (975-7). Thirtee rai gauge statios were chose for this study where statio Arau as the subject seve eighbourig statio ad five statio that located far away from Arau. Details of the statios are show i Table ad Figure.The distace betwee the statios ca be calculated by usig the Haversie formula suggested by Siott [9] ad the result is show i Table. 4

3 Table : The latitude ad logitude of the chose thirtee rai gauge statios. No. Statio Name Latitude Logitude Arau 6 o 5 48 N o 6 E. Abi Kg. Bahru 6 o 8 48 N o 6 48 E. Guar Nagka 6 o N o 8 E 3. Kodiag 6 o 3 36 N o 48 E 4. SIK 5 o N o E 5. Dispesari Kroh 5 o 4 36 N o E 6. Bkt Berapit 5 o 48 N o 8 48 E 7. Rumah Pejaga JPS. Parit Nibog 5 o 7 48 N o 3 36 E 8. Jam. Sg. Simpag Jl. Empat 4 o N o 55 E 9. Pitu Kawala Sembrog o 6 4 N o 4 E. Ldg. Beut Regam o 5 48 N 3 o 3 E. Ldg. Getah Kukup Potia o 5 4 N 3 o E. S. K. Kg. Aur Gadig o N 3 o 7 36 E Figure : The map of the locatio of the thirtee rai gauge statios. 43

4 3. METHODOLOGY The steps to estimate the parameter for bivariate mixed logormal distributio is first defied by Shimizu [7] ad applied i this study. The raifall data of two selected rai gauge statio will be restructured. The maximum likelihood estimatio (MLE) equatios of the parameters are calculated accordig to six cases model which are cosidered. The optimal model is selected based o the miimum Akaike s iformatio criterio (AIC) from the six case models. 3. Restructure Raifall Data at Two Statios The raifall data of two statios ca be categorized ito dataset i types of x y ad x ywhere x y x ad y are positive values. Uder homogeeous assumptio i time the sample size N ca be recostructed ito the four types of dataset without loss of geerality 3 x x x x 3 y y y y 3 3 where N r r x i i the umber r 3 each statio are show i Table. y j j xk yk k 3 r is a o-egative iteger. The values of r 3 ad r for 44

5 Table : Distace betwee the twelve statios with Statio Arau ad the values of r r 3 No. Statio Name Distace betwee 3 the statios with Statio Arau (km). Guar Nagka Kodiag Abi Kg. Bahru SIK Dispesari Kroh Bkt Berapit Rumah Pejaga JPS. Parit Nibog 8. S. K. Kg. Aur Gadig Jam. Sg. Simpag Jl Empat. Pitu Kawala Sembrog Ldg. Beut Regam Ldg. Getah Kukup Potia Bivariate Model for Raifall Data Let X Y be the radom vector of raifall values at two rai gauge statio where the probability distributio satisfies P X Y X x Y Fx x X Y y Gy y X x Y y 3H x y x y P P P () where r ( r 3 ) ad 3 F ad G are uivariate positive cotiuous distributio fuctios ad H is a bivariate positive cotiuous joit distributio fuctio. The coditioal o raifall at both of two statios or either oe statio are P P X x X x Y Fx x Y y X Y y Gx y X x Y y X Y H x y x y P () 3.3 Six Cases for Parameter Estimatio There are may possibility cases of relatio amog the parameters of bivariate mixed logormal distributio whe assumig all of the parameters are ukow but oly the followig six cases are cosidered: 45

6 46 M: M: M3: M4: M5: M6: (3) These six cases are cosidered because the shape parameter affects the structural stability i a logormal distributio. 4. RESULT AND DISCUSSION The results of AIC values of each cases model are displayed i Table 3 with the bolded ad italic values idicated the lowest AIC values. Case M is the domiated best fitted case model amog other case models. Amog the seve earby statios six of the statios chose case M ad oly oe statio chose case M3. While for the five statios that located far from Arau statio four statios attaied the lowest AIC values for case M ad oe statio attaied case M6 as the lowest AIC values. The statios that do ot obtai case M as the best fitted case model is located i ilad compare to others statios icluded statios Arau which are earer to the west coast of Malaysia. The raifall of west coast of Malaysia is affected by the southwest mosoo durig May to August. Therefore there is a possibility the differece i raifall characteristic betwee the statios i ilad ad statios ear the west coast cause the differet result i choosig the best fit model.

7 Table 3: AIC values for each case model for thirtee rai gauge statios. No. Statio Name AIC of Case M M M3 M4 M5 M6. Guar Nagka Kodiag Abi Kg. Bahru SIK Dispesari Kroh Bkt Berapit Rumah Pejaga JPS. Parit Nibog S. K. Kg. Aur Gadig Jam. Sg. Simpag Jl. Empat Pitu Kawala Sembrog Ldg. Beut Regam Ldg. Getah Kukup Potia CONCLUSIONS The characteristic of raifall is a mixed distributio which cotai discrete distributio (days that do ot rai) ad cotiuous distributio (days that rai). It is promiet to aalyze the raifall data by icludig the zero values. The estimated parameters of bivariate mixed logormal distributio were determied by the miimum value of AIC values. I this study six of the seve earby statios ad four of the five far located statios of statio Arau have chose case M based o the miimum AIC procedures. The parameter estimated will be further used for the effect of zero measuremets. Ackowledgmet Authors are faithfully appreciate the geerously of the staff of Malaysia Meteorological Departmet ad Draiage ad Irrigatio Departmet for providig the daily raifall data for the usage of this paper. The work is fiaced by Zamalah Scholarship provided by Uiversiti Tekologi Malaysia ad FRGS vote 4F4 from the Miistry of Higher Educatio of Malaysia. 47

8 REFERENCES [] Moro V.; Robertso A.W.; ad Ward M.N. (6). Seasoal Predictability ad Spatial Coherece of Raifall Characteristics i the Tropical Settig of Seegal. Mothly Weather Review. 34(): [] Youg K.C. (99). A Three-Way Model for Iterpolatig for Mothly Precipitatio Values. Mothly Weather Review. (): [3] Zhag L. ad. Sigh V.P (7). Bivariate raifall frequecy distributios usig Archimedea copulas. Joural of Hydrology. 33(-): [4] Mielke P.W.; Williams J.S.; ad Wu S.C. (977). Covariace Aalysis Techique Based o Bivariate Log-Normal Distributio with Weather Modificatio Applicatios. Joural of Applied Meteorology. 6(): [5] Habib E. ad Krajewski W.F. (). Estimatio of Raifall Iterstatio Correlatio. Joural of Hydrometeorology. : [6] Kedem B.; Chiu L.S.; ad North G.R. (99). Estimatio of Mea Rai Rate' Applicatio to Satellite Observatios. Joural of Geophysical Research. 95(D): [7] Shimizu K. (993). A Bivariate Mixed Logormal Distributio with a Aalysis of Raifall Data. Joural of Applied Meteorology. 3(): 6-7. [8] Ha E. ad Yoo C. (7). Use of mixed bivariate distributio for derivig iter-statio correlatio coefficiets of rai rate. Hydrological Processes. (): [9] Siott R.W. (984). Virtues of the Haversie. Sky ad Telescope. 68():