Joural of heoretcal ad Appled Iformato echology 5 th October 06. Vol.9. No. 005-06 JAI & LLS. All rghts reserved. ISSN: 99-8645 www.jatt.org E-ISSN: 87-395 PARAMEER ESIMAION OF GEOGRAPHICALLY WEIGHED MULIVARIAE t REGRESSION MODEL HARMI SUGIARI, PURHADI, 3 SUIKNO, 4 SANI WULAN PURNAMI Departmet of Statstcs, Uverstas erbuka, agerag Selata, Idoesa 34 Departmet of Statstcs, Isttut ekolog Sepuluh Nopember, Surabaya, Idoesa E-mal: harm.sugart@gmal.com, purhad@statstka.ts.ac.d, 3 sutko@statstka.ts.ac.d, 4 satwula08@gmal.com ABSRAC he use of ordary lear regresso model spatal heterogeety data ofte does ot sutable wth the data pots, especally the relatoshp betwee respose varable ad explaatory varables. herefore, the geographcally weghted t regresso (GWtR s used to overcome spatal heterogeety term. he model s a exteso of geographcally weghted regresso (GWR whch the respose varable follows multvarate t dstrbuto. he am of ths study s to obta the estmator of geographcally weghted multvarate t regresso (GWMtR model wth kow degrees of freedom. he maxmum lkelhood estmato (MLE method wll be appled to maxmze a weghted logarthm lkelhood fucto. Based o the EM algorthm, the estmator of geographcally weghted multvarate t regresso model ca be determed. Keywords: Maxmum Lkelhood Estmato (MLE, EM Algorthm, Geographcally Wegted Regresso, Multvarate t Model. INRODUCION he lear regresso model are ofte used to descrbe the relatoshp betwee the respose varable ad the depedet varable. he estmato of lear regresso model by the ordary least suares (OLS method ca be used whe the error has ormal dstrbuto. A umber of studes of multvarate t lear regresso were performed. A study of uvarate lear regresso ad multvarate lear regresso are developed to obta a robust lear regresso model whe data follow the heavy tal dstrbuto. he maxmum lkelhood estmato (MLE method ca be appled to fd the mea ad covarace matrx of estmator robust. I addto, t s suggested to treat the degrees of freedom of t dstrbuto as a kow parameter whe the sample sze s small ad to estmate t whe the sample sze s large []. Although, MLE method ad Bayesa method have a weakess whe the data have ull probablty uder samplg model assumpto, the ferece of multvarate t lear regresso wth ukow degrees of freedom was developed. Feradez ad Steel [] suggested that Bayesa aalyss s based o a set of observatos wth the accuracy of the tal data. MLE method s ot recommeded wthout further study of the propertes of the local maxma whe t s used to fd the estmator of model wth errors multvarate t dstrbuto ad ukow degrees of freedom. Lu ad Rub [3] developed a maxmum lkelhood method usg EM algorthm to estmate the parameters of the multvarate t regresso model wth kow ad ukow degrees of freedom. he algorthm has aalytcally ute smple ad has stable mootoe covergece to a local maxmum lkelhood estmate. Whe the relatoshp betwee the respose varable ad the depedet varables s explaed by observg geographc factors for each locato, the use of spatal regresso models s more sutable tha the ordary lear regresso model whch s costraed by the assumpto of spatal data [4]. Spatal data s the measuremet data that cotas a locato formato, so the observato at oe locato depeds o the observato at other earby locato. herefore, the use of the GWR model serves to overcome the spatal heterogeety. 45
Joural of heoretcal ad Appled Iformato echology 5 th October 06. Vol.9. No. 005-06 JAI & LLS. All rghts reserved. ISSN: 99-8645 www.jatt.org E-ISSN: 87-395 Fgure : Bvarate t Dstrbuto wth v Some studes o models of classcal GWR model has bee dscussed, a error follows a ormal dstrbuto. Although Lu, et al. [5] proposed o-eucldea dstace (o-ed metrcs calbrated wth Eucldea dstace (ED, road etwork dstace ad travel tme metrcs GWR model, but the error stll follows a ormal dstrbuto. A early revew of the GWtR model has bee doe by Sugart, et al. [6]. he study dcates that the parameters of the model GWtR ca be estmated by maxmum lkelhood method. he method maxmze the weghted logarthm lkelhood fucto of the respose varable ad get the value of the estmator usg the Newto-Raphso terato. I ths paper we propose the maxmum lkelhood estmator of geographcally weghted multvarate t regresso model wth kow degrees of freedom usg EM algorthm.. MULIVARIAE t DISRIBUION Base o Nadarajah ad Dey [7], vector y s sad to have the -varate t dstrbuto wth degrees of freedom, mea vector µ, ad scale matrx, f ts probablty desty fucto (pdf s gve by: ( ( θ Γ D f( y + ( π Γ ( θ ( y µ ( y µ D < µ <, > 0, ad > 0., < y <, ( A llustrato of bvarate t dstrbuto wth, 0.4 µ 0, ad s show Fgure. 0.4 he bvarate t dstrbuto has haver tal tha Normal dstrbuto. It ca be show whe degrees of freedom 30 s creased as Fgure. Probablty Desty v 30, r 0.4 0. 0.5 0. 0.05 0 Fgure : Bvarate t Dstrbuto wth v30 he expectato ad varace of vector y ca be expressed by: E( y µ ; var ( y ; > 3. MULIVARIAE t REGRESSION Multvarate t regresso model s developed based o multvarate regresso model wth respose varables that follow multvarate t dstrbuto. Suppose respose varables (,,, assocated wth p depedet Y Y K Y varables ( X, X, K, X p ca be wrtte as: y Β x + ε ;,,..., ( ( p+ ( p+ ( Y β0 β L β p Y 0 p ; β β L β y Β ; M ( p+ M M O M Y β 0 β β L p ε X ε x ; ε p+ M M X p ε ( 0 X - - ( Vector y follows the-varate t dstrbuto wth degrees of freedom, mea vector Β ad scale - - X 0 x 46
Joural of heoretcal ad Appled Iformato echology 5 th October 06. Vol.9. No. 005-06 JAI & LLS. All rghts reserved. ISSN: 99-8645 www.jatt.org E-ISSN: 87-395 matrx. he probablty desty fucto (pdf of vector y s gve by: ( ( θ Γ D f( y + ( π Γ ( θ Β Β D (3 ( y x ( y x s Mahala-obs dstace. he expectato ad varace of vector y ca be deoted by: E( y Β x ; var ( y ; > Based o euato (3, the lkelhood fucto ca be stated as l( θ f( y ( + D Γ Γ ( θ + ( π he maxmum lkelhood estmato of parameter multvarate t regresso ca be obtaed by the maxmzed of logarthm lkelhood fucto ( Γ l l( θ l l ( π Γ ( θ (4 (5 + D l + Furthermore, assumg that the lkelhood fucto s dfferetable, the estmator ca be foud by solvg the smultaeous euatos below. l l( θ l l( θ 0 ad 0 (6 Β Dfferetatg wth respect to Β ad, respectvely s gve by l l( θ x ( y Β x + Β + D( θ ( [ ] + x ( y Β x [ + D( θ ] ω x ( y Β x l l( θ tr σ j + ε σj D ( θ + tr σ j + ω ε ε σj ε + ω + D θ, ε y Β x, ad σ j s the symmetrc matrx that has oes row colum j ad row j colum, ad zeros else. Sce the euato (6 does ot result closed form soluto, the estmator of Β ad ca be determed by the teratvely process. Lu ad Rub [3] used the EM algorthm to obta the maxmum lkelhood estmator that cossts of E- step (Estmato step followed by M-step (Maxmzato step. he E-step of the EM algorthm ams to obta the codtoal expectato of the complete data suffcet statstcs whe gve the observed values. he M-step volves weghted least suares estmato of Β ad. hus, the EM algorthm terates successvely utl the covergece s reached. At terato wth put ( r Β ad ( r, the E-step of EM algorthm calculate the expectato of l lkelhood complete ad the expected suffcet statstcs, respectvely as follows: ω ad yˆ r XB ˆ r ( r + D ( θ ωxx ω xx ( r ( ˆ ωxy ω x y r+ r+ r r ( r ˆ ( ˆ ωyy ω y y + S S S he M-step of EM algorthm wll calculate Βˆ ˆ ( SωXX Sω XY S S S S ωyy ω XY ω XX ω XY (7 (8 47
Joural of heoretcal ad Appled Iformato echology 5 th October 06. Vol.9. No. 005-06 JAI & LLS. All rghts reserved. ISSN: 99-8645 www.jatt.org E-ISSN: 87-395 4. GEOGRAPHICALLY WEIGHED MULIVARIAE t REGRESSION Geographcally weghted multvarate t regresso (GWMtR model s developed based o geographcally weghted t regresso (GWtR model whch proposed by Sugart et al. [6]. he model descrbes relatoshp betwee respose varables y ad p depedet varables x by cosderg the locato factor that expressed as vector coordate two dmesoal of geographc space. Hece, the estmator obtaed GWMtR model s a local estmator for each pot of observato. Let u ( u, u s a coordate two dmesoal geographc space (lattude ad logtude, the expectato of GWMtR model ca be defed as follows: ( y E Β u x ;,,..., (9 ( ( p+ ( p+ Y Y X y M ; x M ; ( ( p+ Y X p Β ( u ( p+ β0( u β( u L βp( u β ( u β ( u L β ( u 0 p M M O M β0 β βp u u L u ad vector y t ( ( u x, ( u, Β have the - varate t dstrbuto ad probablty desty fucto s ( Γ ( u ( θ( u Γ D f( y + ( π (0 ( θ Β ( Β D u y u x u y u x 5. PARAMEER ESIMAION OF GEOGRAPHICALLY WEIGHED MULIVARIAE t REGRESSION Maxmum lkelhood estmato of parameter GWMtR model ca be obtaed by the maxmzed of weghted logarthm lkelhood fucto (0. ( l l θ u w( u l f( y ( Γ w( u l ( π Γ ( θ( u D w( u l + ( w u s a weghted fucto for locatou ( ad ( θ Β Β D u y u x u y u x here are some types of weghtg fuctos that ca be used to descrbe the relatoshp betwee the observatos at the locato to other locatos. he oe obvous choce s the Gaussa fucto [8] d w( u exp h ( + d u u v v d s Eucldea dstace betwee locato ad locato wth badwdth h. Lu, et al. [5] propose o-eucldea dstace (o-ed metrcs calbrated wth Eucldea dstace (ED, road etwork dstace ad travel tme metrcs GWR model. he results dcate that GWR calbrated wth a o-eucldea metrc ca ot oly mprove model ft, but also provde addtoal ad useful sghts about relatoshps wth data set. However, Lu, et al. [9] propose a back-fttg approach to calbrate a GWR model wth parameter-specfc dstace metrcs. he results show that the approach ca provde both more accurate predctos ad parameter estmates, tha that foud wth stadard GWR. I order to select a approprate badwdth GWR, there are a umber of crtera that ca be used, oe of them s geeralzed cross valdato crtero (GCV whch s descrbed Matsu, et al. [0] trace{ [ yyˆ ( h ] [ yyˆ ( h ]} GCV v v trace (3 ( X W ( u X X W ( u 48
Joural of heoretcal ad Appled Iformato echology 5 th October 06. Vol.9. No. 005-06 JAI & LLS. All rghts reserved. ISSN: 99-8645 www.jatt.org E-ISSN: 87-395 y ˆ( h s the ftted value of y ( h usg a badwdth of h, W( u s weghted matrx that cossts weghtg fucto euato (. Estmato of GWMtR's parameter model s obtaed through the followg euato. ll ( θ( u ll ( θ( u 0 ad 0 (4 Β u u Dfferetatg wth respect to Β ( u ad ( u respectvely s gve by ( ( u ( ( u ( u ( w D ( θ( u ll θ u + u ε x Β +, ( u ll θ w ( u tr ( u σ j w ( u ω ( u ε Φ( u ε (5 ε y Β u x ω ( ( + ( u + D ( θ( u ( u σj Φ u u u Sce the euato (4 does ot result closed form soluto, the estmator of are determed by the teratvely process. u Β u ad A jot teratve process to solve (4 s gve by EM algorthm. he EM algorthm s developed to maxmze the weghted l lkelhood observed through the weghted l lkelhood complete. Let y τ has multvarate ormal dstrbuto ad τ has Gamma dstrbuto wth desty fucto, respectvely as follows: f( yτ ( π exp e ( u e( u ; τ τ τ f( τ Γ τ exp ( u x e ( u y Β f ( y, τ ( π τ + Γ τ exp + D( θ u (6 herefore, the τ y follows Gamma dstrbuto ( θ( u + D wth parameter:, desty fucto ca be expressed by τ τ exp ( + D θ u f( τy D( + θ u Γ he expectato of τ y ca be deoted as: wth ts (7 E( τy (8 + D( θ( u From (6, the weghted l lkelhood complete s gve by ( θ w f( τ ll u u l y, c w u l ( π (9 w( u τ ( ( + D θ u w l + u Γ τ Based o the codtoal expectato of τ y, the E-step of EM algorthm wll calculate the expectato of weghted l lkelhood complete. Becauseτ ukow, the use of E( τ y s easer tha the of E( τ. Hece, at terato wth put ( u ( ( u, ( u Θ Β, the E-step of EM ( r ( r ( r algorthm wll calculate the expectato of weghted l lkelhood complete ad the expected suffcet statstcs, respectvely as follows: Jot desty fucto betwee y τ ad τ ca be expressed by 49
Joural of heoretcal ad Appled Iformato echology 5 th October 06. Vol.9. No. 005-06 JAI & LLS. All rghts reserved. ISSN: 99-8645 www.jatt.org E-ISSN: 87-395 ω r ( u E τ Θ ( u ( r + D ( θ( u ( r ωyy w( u ω ( u yy ( u ( u τ y y Θ ( u S w E + w ( r ( u ( u ( r ωxy ( u ( u τ x y Θ ( u S w E w ( u ω ( r ωxx w( u ω ( u x x ( u ( u τ x x Θ ( u S w E u x y he M-step of EM algorthm wll calculate ( ω Sω ˆ ( r + ( r + ( r + u S XX u XY u Β ˆ (0 u S u S u S u S u ( ( YY ω ω XY ω XX ω XY ( he estmator of Β ( u ad ( u ca be foud by update the E-step ad M-step teratvely utl the covergece of algorthm s reached. Lage, et al. [] used the Fsher Iformato matrx to determe the estmator of the asymptotc Β. he Fsher varace-covarace matrx of u Iformato matrx ca be expressed as ll ( θ( u KΒΒ E Β( u Let ( + g, [ ] z u ε +z [ ] z zj ε u ε j he euato (5 ca be wrtte as ( θ( u ( u ( w ( ( D ( θ( u ( u g ll + u u εx Β + u z x w ( Hece, the cotrbuto of the curret observato Β s euato ( to fd estmator of (3 l ( θ( u Β ( u u l [ ] w u g u z x Based o Lage, et al. [], whe declared the dmeso of ( u, the the expectato of some fucto ca be expressed z [ ] z E [ ] tr u z u z z E z + z + + + ( ( E z ( + z ( + ( 3 E z + z + + + ( ( Furthermore, the Fsher Iformato matrx ca be foud through ( θ( Β ll u z z ( u z x [ ( u ] x u z z E w E g w ( u E( z g x tr( [ ( u ] x w E g tr ([ ] ( u ( z u xx w ( u tr ( + xx Hece, the Fsher Iformato matrx ca be expressed as K ΒΒ l ( θ( u Β( u ( u tr j ( l E w + + u x x (4 he, the cosstet estmator of the asymptotc Β ca be varace-covarace matrx of expressed by u 50
Joural of heoretcal ad Appled Iformato echology 5 th October 06. Vol.9. No. 005-06 JAI & LLS. All rghts reserved. ISSN: 99-8645 www.jatt.org E-ISSN: 87-395 ( cov ˆ Β u K ΒΒ + ( ˆ w tr j ( u u xx [7] S. Nadarajah ad D. Dey, "Multtude of multvarate t dstrbutos," Statstcs: A Joural of heoretcal ad Appled Statstcs, vol. 39, o., pp. 49-8, 005. (5 6. CONCLUCION GWMtR model s a exteso of geographcally weghted regresso (GWR whch respose varables follow multvarate t dstrbuto. I ths model, the respose varables wll be predcted by depedet varables for each locato. herefore, there are may estmator of coeffcets regresso that deped o the locato the data are observed. Parameter estmato of GWMtR model ca be doe by maxmum lkelhood estmato (MLE method wth EM algorthm. REFERENCES: [8] C. Brusdo, A. Fothergham ad M. Charlto, "Geographcally Weghted Regresso: A Method for Explorg Spatal Nostatoarty," Geographcal Aalyss, o. 8, pp. 8-98, 998. [9] B. Lu, P. Harrs, M. Charlto ad C. Brusdo, "Calbratg a Geographcally Weghted Regresso Model wth Parameter- Specfc Dstace Metrcs," Proceda Evrometal Sceces, vol. 6, p. 0 5, 05. [0] H. Matsu, Y. Arak ad S. Kosh, "Multvarate Regresso Modelg for Fuctoal Data," Joural of Data Scece, vol. 6, pp. 33-33, 008. [] K. Lage, R. Lttle ad J. aylor, "Robust Statstcal Modelg Usg he t Dstrbuto," Joural of the Amerca Statstcal Assocato, vol. 84, pp. 88-896., 989. [] C. Feradez ad M. F. Steel, "Multvarate Studet-t Regresso Models: Ptfalls ad Iferece," he Netherlads, 997. [3] C. Lu ad D. B. Rub, "ML Estmato of the t Dstrbuto Usg EM ad ts Extesos, ECM ad ECME," Statstca Sca, vol. 5, pp. 9-39, 995. [4] L. Asel, Spatal Ecoometrcs: Methods ad Models, Sprger, 988. [5] B. Lu, M. Charlto, P. Harrs ad A. Fothergham, "Geographcally weghted regresso wth a o-eucldea dstace metrc: a case study usg hedoc house prce data," Iteratoal Joural of Geographcal, pp. -, 04. [6] H. Sugart, Purhad, Sutko ad S. Puram, "Peaksr Parameter utuk Model Geographcally Weghted t Regresso (GWtR," Koferes Nasoal Matematka XVII, Surabaya, 04. 5