Parameter Estimation of Geographically Weighted. Trivariate Weibull Regression Model

Transcription

1 Appled Mathematcal Sceces, Vol. 10, 016, o. 18, HIKARI Ltd, Parameter Estmato of Geographcally Weghted rvarate Webull Regresso Model Suyto Departmet of Statstcs, Uverstas Mulawarma, Samarda Idoesa Purhad, Sutko ad Irhamah Departmet of Statstcs, Isttut ekolog Sepuluh Nopember Surabaya, Idoesa Copyrght 016 Suyto et al. hs artcle s dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. Abstract I ths study, Geographcally Weghted rvarate Webull Regresso (GWWR model ad parameter estmato procedure are proposed. GWWR s trvarate Webull regresso model whch all of the regresso parameters deped o the geographcal locato, ad parameter estmato s doe locally at each locato the study area. he locato s expressed as a pot coordate two-dmesoal geographc space (lattude ad logtude. he trvarate Webull regresso model (WR s the jot probablty desty fucto model of trvarate Webull dstrbuto, whch the scale parameters deped o the covarates (depedet varables. he WR model s costructed from the jot survval fucto of trvarate Webull dstrbuto proposed by Lee ad We, whch the scale parameters are stated the regresso parameters wth detcal covarates ad o-detcal regresso parameters. he goal of ths study s to estmate the parameters of GWWR model by usg Maxmum Lkelhood Estmato (MLE method. he result showed that the maxmum lkelhood estmator of GWWR model was ot closed form ad t ca be obtaed by usg the Newto-Raphso teratve method. o demostrate the parameter estmato procedure, the proposed model was appled to the cotuous oegatve data o the real data. Based o the applcato to the real data, GWWR model was better tha global model (WR.

2 86 Suyto et al. Keywords: WR, geographcally weghted, geographcal locato, GWWR, MLE 1. Itroducto he Webull dstrbuto orgally depeds o three parameters those are the locato parameter, the scale parameter ad the shape parameter whch are symbolzed by, ad respectvely. By settg the locato parameter 0, t s obtaed a smple model whch s called the scale-shape verso of the Webull dstrbuto. I the Webull dstrbuto, the scale parameter or shape parameter ca deped o the supplemetary varables or covarates [14]. Clearly, the scale parameter of Webull dstrbuto ca be expressed the regresso parameters. Whe the scale parameter of the probablty desty fucto (PDF of Webull dstrbuto s expressed the regresso parameters, t s obtaed the ew PDF model called the Webull regresso model. Geerally, the PDF of the Webull dstrbuto ca be determed from oe of the terrelatoshp fuctos those are the cumulatve dstrbuto fucto (CDF, the survval fucto ad the hazard fucto. Utl ow, the referece dscussg the Webull regresso model s ot vast. Several researchers have dscussed the Webull regresso model those are O Qugley et al. [1] proposed a regresso model for survval tme studes. he study dscussed parameter estmato of the uvarate Webull regresso model ad the estmator of parameters was calculated by Fortrat program. Haagal [5] proposed a bvarate Webull regresso model for the cesored lfe tme samples. he model was derved from bvarate expoetal of Marshal-Olk wth detcal covarates. Haagal [4] dscussed a bvarate Webull regresso model for survval tme by extedg Freud s bvarate expoetal dstrbuto wth detcal covarates ad o-detcal regresso parameters. Parameter estmato for both model, he proposed the MLE method. he Webull dstrbuto ad Webull regresso are commoly used models relablty, lfe tme ad evrometal data aalyss. It wll be a terestg topc to be dscussed whe appled to spatal data. Spatal data s type of data cotag both attrbute ad locato formato. Spatal data s type of data whch dcates terdepedece betwee locato ad the data. I the spatal data, the value of a observato oe locato depeds o the value of the observato earby locatos []. Parameter estmato o the Webull regresso model appled to spatal data wll be a challegg problem, because terdepedece betwee data ad locato causes the estmator value of the regresso parameters at oe locato ad the other locatos are dfferet. herefore, applyg the classcal regresso or global model for the Webull regresso model whch s appled to the spatal data wll produce a vald model. So, t s eeded a approprate method whch ca be appled o the Webull regresso modelg to the spatal data, ad the proposed method ths study s Geographcally Weghted Regresso (GWR method.

3 Parameter estmato of GW trvarate Webull regresso model 86 he GWR s a statstcal method used for aalyzg the spatal data local form of regresso [1]. Based o the GWR method, each regresso parameter depeds o the geographcal locato where the data s observed, ad therefore parameter estmato s doe locally at each locato the study area producg the local model. he geographcal locato s expressed as a vector coordate two-dmesoal geographc space (lattude ad logtude []. Based o the results of the prevous studes, the GWR was a effectve method for regresso modelg to spatal data. It was the smple techque ad useful geographcally-oreted method to explore spatal o-statoary. It ot oly ca explore the varato of the parameters, but sgfcace of the varato ca also be tested [11]. he GWR method ca be used to aalyze the spatal data varous felds ad to determe the effect of predctor order to respose both globally ad locally by cosderg elemet of geography or locato as a weghtg estmatg of the parameters [7]. Parameter estmato o the lear model of spatal multvarate usg the geographcal weghtg produced ubased, effcet ad cosstet estmator of the local models [6]. he GWR method ca estmate the regresso parameters at each locato the study area ad produced more accurate predctos for respose varable. Furthermore, the resduals of the GWR model have more desrable spatal dstrbuto tha those derved from global model [16]. Applyg GWR method o the multvarate Posso regresso model dcated that the local model or geographcally weghted multvarate Posso regresso (GWMPR was better tha global model (MPR based o the value of Akake Iformato Crtero (AIC [15]. Studes o Webull regresso both theory ad applcato are stll lmted to the uvarate ad bvarate cases usg global model. Meawhle, there are may problems spatal statstcs may applcatos o the varous felds volvg two or more resposes data from multvarate Webull dstrbuto wth the jot PDF ca deped o the covarates. Based o the prevous studes of Webull regresso model ad based o the dea of GWR method, t s eeded the Webull regresso model developmet ad ts applcato to spatal data. herefore ths study, the GWWR model s proposed, that s the jot PDF model of trvarate Webull dstrbuto wth scale parameters are expressed the regresso parameters depedg o the geographcal locato. he proposed model s derved from the jot survval fucto of trvarate Webull dstrbuto proposed by Lee ad We [10], whch the scale parameters are expressed the regresso parameters wth detcal covarates ad o-detcal regresso parameters. he study s focused o the costructo of GWWR model ad parameter estmato by usg MLE method. o demostrate parameter estmato procedure ad to evaluate the proposed model, t wll be appled to the real data that s Rver water polluto dcator Surabaya 01. Furthermore, the pla of ths study s orgazed as follows. I secto we troduce the relatoshp of the fuctos the trvarate Webull dstrbuto ad derve the WR model. he GWWR model s troduced secto, ad parameter estmato of GWWR model s dscussed secto 4. he applcato

4 864 Suyto et al. of the GWWR model to the real data s preseted secto 5 ad fally we coclude the study the secto 6.. rvarate Webull Regresso Model As t has bee metoed prevously, the survval fucto, CDF, PDF ad the hazard fucto are the ma fuctos the Webull dstrbuto. I the trvarate case, ther relatoshps ca be descrbed as follows. Suppose there are cotuous oegatve radom varables Y1, Y, Y assocated wth a typcal ut. Lawless [9] defed the jot survval fucto deoted by S( y1, y, y as S( y1, y, y P ( Yk yk k1, (.1 P( Y y, Y y, Y y 1 1 ad the jot CDF F( y1, y, y s defed by F( y1, y, y P ( Yk yk k1. (. P( Y1 y1, Y y, Y y he margal survval fucto ad the margal dstrbuto fucto wll be deoted by S( yk P( Y yk ad F( yk P( Y yk for k 1,,. Based o the probablty propertes, the relatoshp betwee the jot CDF gve by equato (. ad the jot survval fucto (.1 ca be wrtte as c F( y1, y, y 1 P ( Yk yk 1 P ( Yk yk k1 k1 1 S( y S( y S( y S( y, y S( y, y S( y, y S( y, y, y 1 (. he relatoshp betwee the jot survval fucto S( y1, y, y ad the jot F( y, y, y ca also be wrtte as CDF 1. S( y, y, y 1 F( y F( y, y F( y, y, y 1 k k1 k 1 k1 1k k 1 If the CDF expressed by equato (. s absolutely cotuous, by dfferetatg both sde of the equato (. wth respect to each varable, the t s obtaed the jot PDF f ( y1, y, y as follows f ( y, y, y 1 F y1 y y S y1 y y (,, (,,. (.4 y y y y y y 1 1

5 Parameter estmato of GW trvarate Webull regresso model 865 For trvarate case, the hazard fucto ca be defed as P ( yk Yk yk yk ( Yk yk k1 k1 f ( y1, y, y h( y1, y, y lm. y y y y S( y, y, y 0 y y 0 1 Lee ad We [10] costructed the jot survval fucto for the trvarate Webull dstrbuto of radom varable Y1, Y ad Y whch has a expresso gve by equato (.5 as follows a k a yk S( y1, y, y exp k1, (.5 k wth 0a 1 ; 0 y,, for k 1,,. he parameter a represets the k k k degree of depedece the assocato of Y1, Y ad Y, k s the scale parameter ad k s the shape parameter for k 1,,. Based o the ts expresso, the jot survval fucto dsplayed (.5 s a cotuous fucto, ad therefore the jot PDF ca be determed by usg the relatoshp (.4 ad t has the form k 1 1 a k yk a a f0( y1, y, y Q0 A0 exp[ A0 ] k1 a k, (.6 k wth A y k 0 k1 k k a ad a a 0 ( 1( ( Q a a a a a A a A. Parameter estmato of the jot PDF (.6 ca use the MLE method. As a exteso of the PDF model stated (.6, the scale parameters ca deped o the covarates [9], [14], that s the scale parameters ca be expressed the regresso model. Because the scale parameters of PDF (.6 are postve value, to make easer, we ca defe a relatoshp as follows k log βx, (.7 k where β [ 0 1 ] k k k kp s p 1 dmesoal vector of regresso parameters wth kh for k 1,, ; h 0,1,, p, ad x [1 X1 X X ] p s p 1 dmesoal vector of covarates or depedet varables. By substtutg the scale parameters stated the relatoshp (.7 to the equato (.6, t ca be obtaed the jot PDF model expressed term of regresso parameter, that s ( / 1 1( 1,, k k a exp[ k ] a f y y y y 1 1 exp[ a k βx k Q A A1 ], (.8 k1 a a wth

6 866 Suyto et al. / 1 ( k a exp[ k a a A yk βx k ] ad Q1 a( a 1( a a ( a 1 A1 a A1. k1 a he jot PDF whch has expresso (.8 s called trvarate Webull regresso model (WR, ad based o the GWR method, WR s the global model of GWWR model.. Geographcally Weghted rvarate Webull Regresso Geographcally Weghted rvarate Webull Regresso (GWWR s the WR model whch all of regresso parameters deped o the geographcal locato. I ths study, we wll propose two models for GWWR, frstly s GWWR model wth all of parameters deped o the geographcal locato, ad secodly s GWWR model wth oly regresso parameters depedg o the geographcal locato, meawhle the parameter of degree of depedece (a ad shape parameters ( k are costat at each locato whch are assumed equal to the value of jot PDF parameters of populato dstrbuto. th Suppose u ( u1, u deotes a vector of pot coordate for locato where the data s observed for 1,,,, wth u 1 s lattude ad u s logtude, the from WR model (.8, t ca be developed to the ew WR model wth all of parameters deped o the geographcal locato whch s called GWWR model. Here, the parameters are assumed to be fuctos of the locato o whch the observatos are obtaed. Suppose, all of parameters of WR model (.8 deped o the geographcal locato, the GWWR model at locato wth coordate u for the frst model has a expresso as follows k ( u ( ( / ( 1 ( k u a u k u f ( y1, y, y yk exp[ β k ( u x ] k1 a( u a( u, (.1 wth a( u a( u A Q A exp[ ] ( ( / ( ( k u a u k u A yk exp[ β k ( u x ]; k1 a( u a( u ( ( [ ( 1][ ( ] ( [ ( 1] a u ( Q a u a u a u a u a u A a u A, ad x [1 X X X ] s vector of the value of observato for covarates at 1 p locato. For the secod model of GWWR, the parameter a ad k are assumed costat at each locato, the the secod model ca be derved from the frst model of GWWR (.1 wth a( u a ad k ( u k for k 1,,. hs model has ( p 1 parameters whch all of parameters are regresso parameters depedg o the geographcal locato. he parameters for secod model of GWWR ca be wrtte the vector form as θ1( u [ β1 ( u β ( u β ( u ]

7 Parameter estmato of GW trvarate Webull regresso model 867 wth β ( [ 0( 1( ( ] k u k u k u kp u for k 1,,. Meawhle, for GWWR model (.1, t has 7 p parameters whch wll be estmated, amely oe parameter of degree of depedece, three shape parameters ad ( p 1 regresso parameters. For smplfy, the parameters of GWWR model at locato gve (.1 ca be wrtte the vector expresso as θ( u [ a( u γ ( u θ ( u ], (. 1 wth γ( u [ 1( ( ( ] u u u. Furthermore, the ma problem ths study s how to estmate the GWWR model parameters. 4. Parameter Estmato of GWWR Model Parameter estmato of GWWR model ths study wll be doe by usg the MLE method. he tal step of the MLE method s defg the lkelhood fucto. Suppose, there are the radom samples ( y1 j, y j, y j of resposes take from the populato of trvarate Webull dstrbuto whch has jot PDF (.6 ad ( X1j, Xj,, X pj for j 1,,, are the observato values of covarates. Suppose, the pot coordate of all locatos the study area s kow, the based o the jot PDF (.1, the lkelhood fucto at locato whch has coordate u ca be defed as L( θ( u y ; x f ( θ( u y, x wth j j j j j1 A, k ( u ( ( / ( 1 ( k u a u k u = ykj exp[ β k ( u x j ], (4.1 j1 k1 a( u a( u j1 a( u a( u j j Aj Q A Q ad x are gve (.1. exp[ ] he maxmum lkelhood (ML estmator of parameter θu ( ca be performed by maxmze the lkelhood fucto (4.1, or equvaletly by maxmze the atural logarthm of lkelhood fucto (log-lkelhood, because the maxmum of both lkelhood fucto ad ts atural logarthm s attaed at the same pot, that s at pot θu ˆ( whch s called the ML estmator. he atural logarthm of lkelhood fucto (4.1 ca be wrtte as L * ( θ ( u y ; x log( L ( θ ( u y ; x. (4. j j j j Based o the GWR method, parameter estmato o the GWWR model s part, ad depedet o the weghtg fucto or kerel selected. he weghtg fucto s to place dfferet emphases o the dfferet observatos geeratg the estmated parameters.

8 868 Suyto et al. he observatos whch are closer to a locato geerally exert more fluece o the parameter estmates at locato tha those farther away. Whe parameters at locato are estmated, more emphases should be placed o the observatos whch are close to locato. he weghts at each locato are take as a fucto of dstace from to other locato the studed geographcal rego. wo of the weghtg fuctos whch are the most commo choce practce are the adaptve Gaussa ad b-square fucto [11]. Suppose, the weght placed to the pot whch has coordate u for the model at locato wth coordate u s w j, the by usg the adaptve Gaussa fucto t ca be calculated as follows 1 dj wj exp, j 1,,,, (4. b( u ad t ca also be calculatedby usg adaptve b-square fucto whch has form d j 1, f dj b( u wj b( u, j 1,,,, (4.4 0, f dj b( u wth d j s Euclda dstace from locato badwdth for parameter estmato of model at locato j to j, ad b( u s called a [1], []. Based o the equato (4. ad (4.4, that the weghtg fucto deped o the choce of badwdth, ad therefore, the problem s how to select approprate badwdth o the parameter estmato of GWWR model. Because f the badwdth s too large the the weghts ted to oe for all pars of pots so that the estmated parameters become uform at each locato, whch result the local model becomes equvalet to global model. Coversely, as the badwdth becomes smaller, the parameter estmates wll creasgly deped o observatos close proxmty to locato ad hece wll have creased varace []. here are umbers of crteros that ca be used for badwdth selecto, whch oe of crteros s geeralzed cross-valdato crtero (GCV. he formula for the GCV score s ( f ˆ f ( b( u 1 GCV( b ( v, (4.5 wth s the sample sze, f ˆ ( b( u s the ftted value of f ad v s the umber of parameters the model. he optmum badwdth s b0 { b0( u, 1,,, } whch s selected as desrable the value of badwdth so that GCV( b0 s the mmum value of GCV.

9 Parameter estmato of GW trvarate Webull regresso model 869 A smlar method of dervg the optmum badwdth s to mmze the Akake Iformato Crtero (AIC. he AIC for GWR s defed as v AIC( b log( ˆ log( v, (4.6 where ˆ s the estmated stadard devato of the error term wth ˆ MSE. 1 MSE s the mea square error defed by: MSE [ ˆ f f ( b( u ]. he AIC 1 has the advatage of beg more geeral applcato, because t ca be used Posso ad logstc GWR []. o estmate the parameters of GWWR by usg the MLE method ca be performed by maxmze log-lkelhood fucto (4. placed the weghtg of geographcal locato w. Based o the lkelhood fucto (4.1, the loglkelhood fucto (4. placed the weght j w j ca be formulated as follows 4 * j j j j q j q1 L( θ( u w w L ( θ( u y ; x L ( θ ( u w, (4.7 wth k ( u k ( u k ( u L1 ( θ( u wj wj log ( 1log( ykj βk x j (4.8 j1 k1 a( a( a( u u u L ( θ( u w w ( a( u log( A ; (4.9 j j j j1 a( u j j j j1 L ( θu ( w w A ; (4.10 L4 ( θu ( w w log( Q. (4.11 j j j j1 he ML estmator θu ˆ( maxmzg the fucto (4.7 ca be determed by solvg the system of equato L( θu ( wj 0, (4.1 θu ( wth 0 s a vector of zeros. he system of equato (4.1 s kow as lkelhood equato, ad the vector o the left had sde of equato (4.1 s 7 p dmesoal gradet vector or score vector. Let, the gradet vector g( θ( u L( θ( u w / θ( u be a scalar vector whch has geeral form as where j g( θu ( L( θ( u w L( θ( u w L( θ( u w j j j a( u γ ( u θ1 ( u, (4.1

10 870 Suyto et al. L( θ( u w L( θ( u w L( θ( u w L( θ( u w ad j j j j γ ( u 1( u ( u ( u L( θ( u wj L( θ( u wj L( θ( u wj L( θ( u wj, wth θ1 ( u β1 ( u β ( u β ( u L( θ( u wj L( θ( u wj L( θ( u wj L( θ( u wj for k 1,,. βk( u k 0( u k1( u kp ( u Cosderg o the expresso of the fuctos (4.8-(4.11, the lkelhood equato (4.1 s a system of terdepedet olear equatos whch does ot have the closed form soluto. Hece, to solve t ca apply the Newto-Raphso teratve method, ad the ML estmator of θu ( ca be estmated by the roots of lkelhood equato (4.1. o obta the ML estmator usg the Newto- Raphso algorthm ca take the formula ˆ ( q1 ˆ ( q 1 ˆ ( q ˆ ( q θ ( u θ ( u H ( θ ( u g( θ ( u, for q 0,1,,, (4.14 ˆ ( q ˆ ( q wth g( θ ( u s the gradet vector ad H( θ ( u s the Hessa matrx ( whch are evaluated at ˆ q θ ( u, that s the value of estmator of θu ( after terato. H( θu ( s the symmetrc matrx of the secod order partal dervatves of L( θu ( w wth respect to combato all of the compoets of vector j θu ( whch has sze (7 p (7 p ad t has geeral form as L( θ( u wj L( θ( u wj L( θ( u w j a ( u a( u γ ( u a( u θ1 ( u L( θ( u wj L( θ( u wj L( θ( u wj H( θu (. (4.15 γ( u a( u γ( u γ ( u γ( u θ1 ( u L( θ( u wj L( θ( u wj L( θu ( w j θ1( u a( u θ1( u γ ( u θ1( u θ1 ( u (0 he terato (4.14 s started from a tal value θˆ ( u, ad t s stopped utl ( q 1 th terato whe ( q1 ( q ( ( th q θˆ u θˆ u, wth s the small postve ˆ ˆ( q 6 umber (for example 10, ad θ( u θ ( u. Based o the expresso of the log-lkelhood fucto (4.7, to calculate the gradet vector (4.1 ad the Hessa matrx (4.15 drectly do ot smple, hece to make easer, they ca be splt to four parts, so that 4 4 Lq ( θu ( wj g( θ( u gq( θ( u, (4.16 q1 q1 θu ( ad the Hessa matrx s formulated as

11 Parameter estmato of GW trvarate Webull regresso model Lq ( θu ( w j H( θ( u Hq ( θ( u, (4.17 q1 q1 θ( u θ ( u where H ( θu ( for q 1,,,4 are the symmetrc matrces whch has sze q (7 p (7 p respectvely. Specal for g4 ( θu ( ad H4 ( θu (, based o the expresso of equato (4.11, the compoets of vector g4 ( θu ( ca be formulated as wj Qj g4( θu (, (4.18 j1 Qj θu ( ad elemets of the matrx H4 ( θu ( ca be obtaed by usg the form w j Q j Q j Q j H4( θu ( Qj. (4.19 j1 Q j ( ( ( ( θ u θ u θu θ u Because the gradet vector ad the Hessa matrx of log-lkelhood (4.7 are ow kow, the the Newto-Raphso algorthm (4.14 ca be appled to calculate θu ˆ( that maxmze the fucto (4.7. By the same procedure, parameter estmato for the secod model of GWWR ca be doe by MLE method. Based o the log-lkelhood fucto (4.7, the compoets of gradet vector for the secod model of GWWR are the frst order partal dervatves of log-lkelhood fucto L( θ1( u wj wth respect to all of compoets of vector θ1 ( u, ad t has geeral form as L( θ1( u wj L( θ1( u wj L( θ1( u wj L( θ1( u wj g( θ1( u θ1( u β1 ( u β ( u β ( u ad the Hessa matrx whch has sze (p1 (p 1 has form L( θ1( u wj L( θ1( u wj L( θ1( u w j β1( u β1 ( u β1( u β ( u β1( u β ( u L( θ1( u wj L( θ1( u wj L( θ1( u wj H ( θ1( u. β( u β1 ( u β( u β ( u β( u β ( u L( θ1( u wj L( θ1( u wj L( θ1( u wj β( u β1 ( u β( u β ( u β( u β ( u Based o the GWR method, WR model s a specal model for GWWR model whch the parameter s assumed to be costat at each locato. So, the estmator of WR model parameters ca be obtaed va parameter estmato of GWWR model by usg MLE method wth spatal weghtg placed to all pars of pots s 1 j w. Whe w 1 for, j 1,,,, the the estmator value of j GWWR model parameters at each locato becomes detcal producg the global model amely the WR model.

12 87 Suyto et al. Note that, based o the asymptotc propertes of ML estmator, θu ˆ( s asymptotcally ubased estmator ad has asymptotc ormal dstrbuto that s asym 1 ˆ( ( (,[ ( ( ] θ u N θ u I θ u, wth I( θ( u E[ H( θ( u ] s formato matrx or Fsher matrx. he verse of the observed of observed formato matrx s the observed varace-covarace matrx of ML estmator θu ˆ(, that s ( ˆ 1 Var θu ( = [ ( ˆ I θu ( ]. Sce H( θu ˆ( s the scalar matrx, the I( θˆ( u ( ˆ H θ( u. he other propertes of the ML estmator s varat, amely f θu ˆ( s ML estmator of θu ( the f ( θu ˆ( s the estmator of f ( θu (, ad also S( θu ˆ( s estmator of survval fucto S( θu (, for f (. ad S (. are cotuous ad cotuously dfferetable fucto [14]. Furthermore, to acheve the goal of ths study, the aalyss s performed as the followg step. (1 Parameter estmato for the populato dstrbuto of trvarate Webull f gve (.6. dstrbuto wth the jot PDF s 0 ( estg goodess of ft for dstrbuto of trple of respose data (Y 1,Y,Y. o test the goodess of ft for multvarate dstrbuto ca be used a multvarate Kolmogorov-Smrov test proposed by Fasao et al. [] ad Justel et al. [8]. ( Parameter estmato of WR model (global model ad GWWR model for model A ad B. Model A s GWWR model wth all of parameters deped o the geographcal locato, ad model B s GWWR model wth oly regresso parameters depedg o the geographcal locato, meawhle the parameter of degree of depedece ( a ad shape parameters ( k are costat for all of locatos whch are equal to the value of jot PDF parameters of the populato dstrbuto. 5. Applcato he am of the applcato s to evaluate the performace of parameter estmato the proposed GWWR model by usg MLE method. Parameter estmato wll be doe for model A ad B of WR ad GWWR model. Furthermore, based o the result of parameter estmatos of WR ad GWWR model, we ca determe the best model. he proposed model was appled to the real data that s Rver water polluto dcator Surabaya 01, ad the data was secodary data from Lfe Evromet Departmet of Surabaya. he ut of observatos was the rvers Surabaya whch have the same water stream. he sze of sample was 7 pots take from 9 locatos alog the water stream July, September ad November 01. he study area was geographcally located betwee '5.1" '1.87" south lattude ad '07.1" '8.7 east logtude.

13 Parameter estmato of GW trvarate Webull regresso model 87 he resposes (depedet varables were bochemcal oxyge demad (BOD, chemcal oxyge demad (COD ad dssolved oxyge (DO, whch were symbolzed by Y1, Y ad Y respectvely. he covarates (depedet varables were speed of water stream (X 1, trate cocetrato (X ad phosphate cocetrato (X. he best model was evaluated based o the AIC ad mea square error (MSE value, meawhle to obta optmum badwdth for GWWR model was used GCV crtero. he result of applyg GWWR model to the real data s summarzed as follows. he tal step of the applcato s estmatg jot PDF parameters of the populato dstrbuto whch has expresso (.6. Parameter estmato was doe by MLE method, ad the estmator value of jot PDF parameters of trvarate Webull dstrbuto (dstrbuto of populato s dsplayed able 1. able 1 he Value of Jot PDF Parameters of rvarate Webull Dstrbuto a he secod step of the aalyss s to test the goodess of ft of the resposes dstrbuto. Based o the vestgato of dstrbuto usg a multvarate Kolmogorov-Smrov test, t was cocluded that the resposes data ( y1, y, y for 1,,,7 follow a trvarate Webull dstrbuto wth value of jot PDF parameters s preseted able 1. he result of calculato of statstc ( j ( j D max D ca be see able, where the statstc D s defed by 1j 6 D sup F ( z, z, z z z z, wth F s the emprcal CDF. ( j j j j j j j j Z 1 1 able he Value of he Statstcs ( j D ad D (1 D ( D ( D (4 D (5 D (6 D D ( j Sx statstcs D dsplayed able were evaluated wth eght volumes of three-dmesoal space defed for each pot ( z j 1, j, j z z, 1,,,7 by ( Z j 1 z j 1, j j, j j,,( j j 1 1, j j, j j Z z Z z Z z Z z Z j j j z, where ( Z 1, Z, Z for j 1,,,6 s the j th permutato of varable ( Z1, Z, Z, wth Z1 F( Y1 ; Z F( Y Y1 ad Z F( Y Y1, Y are..d uform (0,1. Based o the vestgato of dstrbuto, the statstc D was N(, wth 0.516, 0.971, ad the percetle of dstrbuto of D was computed by Mote Carlo smulato samplg from depedet uform (0,1.

14 874 Suyto et al. For sgfcat level 0.0, t was obtaed D ( wth replcato, ad the exact value of the 98 th percetle of D was Because D D t ca be ferred that the respose data was take from trvarate (0.98 Webull dstrbuto wth PDF parameters was preseted able 1. he last step of the applcato s parameter estmato for global model (WR whch has expresso (.8 ad GWWR model (.1 for model A ad B. Parameter estmato for model A ad B of GWWR model was doe locally producg 7 local models respectvely, cossted of 9 local models July, 9 local models September ad 9 local models November 01. I ths study, we preset the descrptve statstcs of the estmator values of 7 GWWR models parameters for model A ad B. We also preset the result of parameter estmatos of model A ad B July at two locatos respectvely, amely the model at Surabaya rver Kedurus (P1 wth pot coordate (7 0 19'1.87" LS ad ' 6.57" EL ad the model at Mas rver Kebo Rojo brdge (P5 wth pot coordate (7 0 14'5.1" LS ad '.66" EL. he descrptve statstcs of the estmator values of 7 GWWR models parameters for model A ad B s preseted able, ad the result of parameter estmatos of WR ad GWWR for model A ad B July 01 at locato P1 ad P5 s dsplayed able 4. able he Descrptve Statstcs of he Estmator Values of 7 GWWR Models Parameters for Model A ad B Parameter Model A Model B Mea SD M Max Mea SD M Max a

15 Parameter estmato of GW trvarate Webull regresso model 875 Note: Sg (- able meas empty, SD s stadard devato, M s mmum value ad Max s maxmum value. able 4 he Estmator Value of WR ad GWWR Model Parameters for Model A ad B July 01 at Locato P1ad P5 Model A Model B Parameter GWWR GWWR WR WR P1 P5 P1 P5 a * -* -* * -* -* * -* -* * -* -* Badwdth AIC MSE GCV Note: sg -* able 4 s to mark that the value of parameter s costat for each locato whch s equal to the value of jot PDF parameters of the populato dstrbuto preseted able 1, ad sg (- meas empty. he spatal weghtg for each locato was calculated by usg the adaptve Gaussa weghtg fucto, ad optmum badwdth was evaluated by usg GCV crtero. he optmum badwdths for model A at locato P1 ad P5 were ad receptvely wth GCV value was

16 876 Suyto et al. Meawhle for model B, the optmum badwdths of GWWR model at locato P1 ad P5 were ad respectvely wth GCV value was Based o the result of parameter estmatos dsplayg able 4, t dcates that the estmator value for some parameters of GWWR model dffer from the global model (WR, but the ferece ca be take after hypothess testg. able 4 shows that the AIC ad MSE value of GWWR model are smaller tha WR model, so t ca be cocluded that GWWR model s better tha WR model. Furthermore, ths study, model B of GWWR model wth parameter of degree of depedece ad shape parameters are costat s preferable to model A of GWWR model wth all parameters deped o the geographcal locato, because t has smaller MSE ad AIC value. 6. Cocluso GWWR model s WR model whch all of regresso parameters deped o the geographcal locato. Parameter estmato o the GWWR model s based o the GWR method, that s, parameter estmato s doe locally, depedet o the spatal weghtg fucto or kerel selected. he parameter estmato method o the GWWR model was maxmum lkelhood estmato. he result showed that the lkelhood equato was a system of terdepedet olear equatos whch does ot have the closed form soluto, therefore the maxmum lkelhood estmator was obtaed by usg the Newto-Raphso teratve method. o evaluate the proposed GWWR model, t was appled to the real data that s Rver water polluto dcators Surabaya 01.Based o the result ths study, GWWR model s better tha global model (WR, ad the best model of GWWR s GWWR model wth parameter of degree of depedece ad shape parameters are costat at each locato. Refereces [1] C. Brusdo, A. S. Fothergham, ad M. E. Charlto, Geographcally Weghted Regresso: A Method for Explorg Spatal Nostatoarty, Geographcal Aalyss, 8 (1996, [] G. Fasao ad A. Fracesch, A multdmesoal verso of the Kolmogorov-Smrov test, Mo. Not. of the Royal Astr. Soc., 5 (1987, [] A. S. Fothergham, C. Brusdo, ad M. Charlto, Geographcally Weghted Regresso: he Aalyss of Spatally Varyg Relatoshps, Joh Wley & Sos, Eglad, 00.

17 Parameter estmato of GW trvarate Webull regresso model 877 [4] D. D. Haagal, A Bvarate Webull Regresso Model, Ecoomc Qualty Cotrol, 0 (005, [5] D. D. Haagal, Parametrc Bvarate Regresso Aalyss Based o Cesored Samples: A Webull Model, Ecoomc Qualty Cotrol, 19 (004, [6] S.Har, Purhad, M. Mashur, ad S. Suaryo, Lear Model Parameter Estmator of Spatal Multvarate Usg Restrcted Maxmum Lkelhood Estmato, Joural of Mathematcs ad echology, 4 (010, [7] S. Har, Purhad, M. Mashur, ad S. Suaryo, Statstcal est for Multvarate Geographcally Weghted Regresso Model Usg the Method of Maxmum Lkelhood Rato est, Iteratoal Joural of Appled Mathematcs & Statstcs, 9 (01, [8] A. Justel, D. Pea, ad R. Zamar, A multvarate Kolmogorov-Smrov test of goodess of ft, Statstcs & Probably Letters, 5 (1997, [9] J. F. Lawless, Statstcal Model ad Methods for Lfetme Data, Joh Wley & Sos, Hoboke New Jersey, [10] C. K. Lee ad M. J. We, A Multvarate Webull Dstrbuto, Pak. Joural Stat. Operat. Res., 5 (009, [11] Y. Leug, C. L. Me, ad W. X. Zhag, Statstcal est for Spatal No- Statoarty Based o the Geographcally Weghted Regresso Model, Evromet ad Plag A, (000, [1] J. O'Qugley ad A. Roberts, WEIBULL: A Regresso Model for Survval me Studes, Computer Programs Bomedce, 1 (1980, [1] Purhad ad H. Yas, Mxed Geographcally Weghted Regresso Model (Case Study: the Percetage of Poor Households Mojokerto 008, Europea Joural of Scetfc Research, 69 (01, [14] H. Re, he Webull Dstrbuto, A Hadbook, CRCPress, alylor & Fracs Group, USA, 009. [15] ryato, Purhad, B. W. Otok, ad S. W. Puram, Parameter Estmato of Geographcally Weghted Multvarate Posso Regresso, Appled Mathematcal Sceces, 9 (015,

18 878 Suyto et al. [16] L. Zhag, J. H. Gove, ad L. S. Heath, Spatal resdual aalyss of sx modelg techques, Ecologcal Modellg, 186 (005, Receved: February 5, 016; Publshed: March 15, 016