MIXED GEOGRAPHICALLY WEIGHTED REGRESSION USING ADAPTIVE BANDWIDTH TO MODELING OF AIR POLLUTER STANDARD INDEX

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1 VOL., NO. 5, AUGUS 7 ISSN ARPN Journl of Engneerng nd Appled Scences 6-7 Asn Reserch Pulshng Networ (ARPN). All rghts reserved. MIXED GEOGRAPHICALLY WEIGHED REGRESSION USING ADAPIVE BANDWIDH O MODELING OF AIR POLLUER SANDARD INDEX Dw Isprynt, Hs Ysn, Bud Wrsto, Adul Hoyy nd Kuuh Wnrso Deprtement of Sttstcs, Dponegoro Unversty, Semrng, Indones Deprtment of Industrl Engneerng, Fculty of Engneerng, runojoyo Mdur Unversty, Indones E-Ml: dwsprynt@yhoo.com ABSRAC Ar polluton s one of the most concerned prolems on erth tody. It s closely relted wth nd mostly generted from the trnsportton nd ndustrlzton sectors, s well s from the envronmentlly degrdng effect of the urn physcl development. Ar polluton promotes the lower level of r qulty, whch n turn promotes the greter rs on helth, especlly tht of the humn eng. hs reserch ms to d the government n the polcy mng process relted to r polluton mtgton y developng stndrd ndex model for r polluter (Ar Polluter Stndrd Index - APSI) sed on the Mxed Geogrphclly Weghted Regresson (MGWR) pproch usng the dptve ndwdth. he dptve ndwdth ernel hs dfferent ndwdth vlue n ech oservton locton. Ae Informton Crteron-corrected (AICc) vlue s used to choose the most optmum ndwdth. he Monte Crlo Smulton s used to tests for regresson coeffcent non-sttonrty. In ths reserch, we lso consder seven vrles tht re drectly relted to the r polluton level, whch re the trffc velocty, the populton densty, the usness center spect, the r humdty, the wnd velocty, the r temperture, nd the re sze of the urn forest. Bsed on AICc nd MSE vlue t s now tht the MGWR model wth dptve squre ernel s the est ndwdth to nlyze ths model. Keywords: dptve squre ernel, r polluter, APSI, MGWR, Monte Crlo smulton.. INRODUCION Ar polluton s rel prolem tht thretens the envronment nd even threten humn lfe. It s chrcterzed y decrese n r qulty, especlly n the g ctes n recent yers. Fctors to e mjor source of r polluton n lrge ctes s trnsportton-engned vehcles, the exhust gs ndustres, populton densty, shoppng centers, r humdty, r temperture nd wnd speed, nd so on. Fctors whch my prevent or nht the emergence of r polluton s the presence of mny green res nd trees n cty prs (Atsh, 7) nd (Fhm, et l., ). Elements of pollutnt mjor ccordnce wth r polluton ndex (ISPU) s Cron Monoxde (CO), prtculte mtter (PM), sulfur doxde (SO) Ntrogen Doxde (NO) nd ozone (O3) (Fhm, et l., ). he elements of these pollutnts re extremely dngerous when exposed to humns for long tme nd contnuously. It s chrcterzed y ncresed dsese cused y r polluton, mong others, crdovsculr, sthm, llergc, mmunologcl dsorders nd cncer, oth developng nd developed countres (Fhm, et l., ). he study of r polluton showed prllel reltonshp etween the numer of people wth llnesses cused y r polluton wth n ncrese n ndustrlzton nd urnzton n developng countres where r polluton levels re very hgh (Eter, 6). he cuses nd effects of r polluton hs een lned to the locton, menng etween sptl locton wll e dfferent cuses nd effects. Demogrphclly, the potentl mpcts nd cuses of r polluton wll dffer etween regons, n ddton to the mpct nd cuses of r polluton cnnot use glol pproch, ecuse y usng glol pproch there re locl vrtons nvsle nfluence (Glert, ) & (Ronson, ). Sptl regresson method frequently used s Geogrphclly Weghted Regressson (GWR), whch s regresson method nvolvng the effect of the locton nto the predctor (Fothernghm, et l., ). In the lner regresson model generted only prmeter estmtor tht pply glolly, whle n the GWR models generted model prmeter estmtor tht s locl to ech oservton locton. Mxed geogrphclly Weghted Regresson (MGWR) s comnton of glol lner regresson model wth the GWR model. So tht the model wll e generted MGWR estmtor prmeters re glol nd some others re loclzed n ccordnce wth the locton of oservtons (Purhd nd Ysn, ). MGWR prmeter estmton s done y gvng dfferent weghtng for ech locton where the dt s collected. In determnng the vlue of the ernel functon cn e dvded nto two types of clcultons, nmely the fxed ernel nd dptve ernel ndwdth. Fxed ernel s the sme ndwdth t ll ponts of oservton locton, whle dptve ernel s the ndwdth tht hs dfferent vlues for ech oservton locton. he dptve ernel ws ppled n ths study n order to fx the ernel nd to get etter clculton ccurcy.. RESEARCH OBJECIVE hs study ms to model Ar Polluter Stndrd Index (APSI) wth MGWR usng the dptve ndwdth sed on AICc pproch. he cse study n ths reserch use Ar Polluter Stndrd Index APSI n Sury Cty n fve locton s the response vrle Y, whle the predctor vrles were the r temperture (X ), the wnd velocty (X ), the r humdty (X 3), the trffc velocty 4477

2 VOL., NO. 5, AUGUS 7 ISSN ARPN Journl of Engneerng nd Appled Scences 6-7 Asn Reserch Pulshng Networ (ARPN). All rghts reserved. (X 4), the re sze of the urn forest (X 5), the populton densty (X 6), nd the usness center spect (X 7) 3. MAERIAL AND MEHODS 3. Lner regresson Lner regresson s method tht models the reltonshp etween response vrles nd predctor vrles. Lner regresson model for p predctor vrles re generlly wrtten s follows: p y x () where =,,..., n ;,,..., p re the model prmeters nd,,..., n re error term wth men zero nd common vrnce. Estmton of regresson prmeters re done y Ordnry Lest Squres (OLS) method. estng of prmeter regresson model usng the F dstruton pproch nd the prtl use of the t dstruton pproch (Rencher, ). 3. Geogrphclly Weghted Regresson (GWR) GWR model represents the development of glol regresson model where the sc de s ten from the non-prmetrc regresson (Me et l., 6). hs model s loclly lner regresson tht tht produces the estmtor model prmeters tht re locl to ech pont or locton where the dt s collected. GWR model cn e wrtten s follows: p, u v u v y, x () where: y : oservton of respons y u, v : coordnt pont (longtude, lttude) u, v : p unnown functons of geogrphcl loctons u, v ; =,,...,p x : explntory vrle t locton u, v :error term wth men zero nd common vrnce Estmton of GWR model prmeter s usng the Weghted Lest Squres (WLS) method tht gves dfferent weghtng for ech oservton. So the estmtor of model prmeter for ech locton s: βˆ u, v X W u, v X X W u, v y (3) he GWR method provdes fesle wy for testng glol lner regresson reltonshp for sptl dt. hs mounts to test the followng hypotheses: H : u, v for ech,,,, p, nd,,, n (there s no sgnfcnt dfference etween the glol regresson model nd GWR) H :t lest there s u, v,,,,, p (there s sgnfcnt dfference etween the glolregresson model nd GWR). he test sttstc gven y Leung, et l. (): F test n p y I H I L I L y y I H I H y Reject H f Ftest F, df, df where df, df np tr,, I H I L I L, x X Wu, vx X Wu, v x X Wu, vx X Wu, v L nd H XX X X. x X W X X W u, v u, v n n n n n estng prmeters re prtlly done wth the hypothess s follows: H : u, v H: u, v where,,, p est sttstc gven y: ˆ ( u, v ) ˆ c where ˆ y I L I L y nd c s the -th dgonl element of the mtrx CC where C X W u, v X X W u, v. Reject H or n other words the u, v sgnfcnt prmeters of the model when where df (4) s t, df, 4478

3 VOL., NO. 5, AUGUS 7 ISSN ARPN Journl of Engneerng nd Appled Scences 6-7 Asn Reserch Pulshng Networ (ARPN). All rghts reserved. nd tr,, I L I L. 3.3 Monte Crlo tests for regresson coeffcent nonsttonrty For mxed GWR, dffcultes rse when decdng whether reltonshp should e fxed glolly or llowed to vry loclly. Here Fothernghm et l. () dopt stepwse procedure; where ll possle comntons of glol nd loclly-vryng reltonshps re tested, nd n optml mxed model s chosen ccordng to mnmze AIC vlue. hs pproch s comprehensve, ut computtonlly expensve, nd s utlsed n the GWR 4. executle softwre (Ny, et. l., 9). Alterntvely, Monte Crlo pproch cn e used to test for sgnfcnt (sptl) vrton n ech regresson coeffcent (or reltonshp) from the sc GWR, where the null hypothess s tht the reltonshp etween dependent nd ndependent vrle s constnt (Fothernghm et l., ). he procedure s nlogous to tht presented for the locl egen vlues of GW PCA, where for the sc GWR the true vrlty n ech locl regresson coeffcent s compred to tht found from seres of rndomsed dt sets. If the true vrnce of the coeffcent does not le n the top 5% tl of the rned results, then the null hypothess cn e ccepted t the 95% level; nd the correspondng reltonshp should e glolly-fxed when specfyng the mxed GWR. Oserve, tht f ll reltonshps re vewed s non-sttonry, then the sc GW regresson should e preferred. Conversely, f ll reltonshps re vewed s sttonry, then the stndrd glol regresson should e preferred (Lu, et l., 4). Advnces on the mxed GWR model, where the reltonshps cn e llowed to vry t dfferent rtes cross spce cn e found n Yng et l. (). 3.4 Mxed Geogrphclly Weghted Regresson (MGWR) A ey ssumpton for ths sc GWR s tht the locl coeffcents vry t the sme scle nd rte cross spce. However, some coeffcents (nd reltonshps) my e expected to hve dfferent degrees of vrton over the study regon. In prtculr, some coeffcents (nd reltonshps) re vewed s constnt (or sttonry) n nture, whlst others re not. For these stutons, Mxed GWR cn e specfed (Fothernghm, et l., ). hs sem-prmetrc model trets some coeffcents s glol (nd sttonry), whlst the rest re treted s locl (nd non-sttonry), ut wth the sme rte of sptl vrton. In vector-mtrx notton, the MGWR model cn e rewrtten s (Lu, et l., 4). yx X ε (5) where y s the vector of dependent vrles; X s the mtrx of glolly-fxed vrles; s the vector of glol coeffcents; X s the mtrx of loclly-vryng vrles; nd s the mtrx of locl coeffcents. If we defne the ht mtrx for the glol regresson prt of the model, s S; nd tht for the GW regresson prt, s S; then equton (4) cn e rewrtten s yˆ yˆ y ˆ where yˆ S y nd yˆ S y. hus the clrton procedure cn e refly descred n the followng sx steps: Step-: Supply n ntl vlue for y ˆ, sy s y ˆ, prctclly y regressng X on y usng ordnry lest squre (OLS). Step-: Set. Step-3: Set yˆ ˆ S yy Step-4: Set yˆ ˆ S yy Step-5: Set Step-6: Return to Step 3 unless yˆ yˆ y ˆ converges to y ˆ. 3.5 Adptve Kernel Functons Weghtng functon tht used to estmte the prmeters n the MGWR model s the squre dptve ernel functons (Fothernghm et l., ), whch cn e wrtten s follows: w u, v j dj h, f dj h, f dj h where dj denotes the dstnce etween the locton u, v to locton u j, v j nd h re non negtve prmeters re nown nd re usully clled smoothng u,. So prmeter (ndwdth) for locton v Wu, v dg w u, v, wu, v,, wnu, v nd one of the methods tht used to select the optmum ndwdth s the Ae Informton Crteron-corrected (AICc). 3.6 Selecton of the est model he method tht s used to select the est model s Ae Informton Crteron (AIC) whch s defned s follows: n tr( S) AIC ln ˆ c n nlnn n tr ( S) where: ˆ : he estmtor of stndrd devton of the error (6) 4479

4 VOL., NO. 5, AUGUS 7 ISSN ARPN Journl of Engneerng nd Appled Scences 6-7 Asn Reserch Pulshng Networ (ARPN). All rghts reserved. S : Mtrx projecton where yˆ Sy he est model selecton s done y determnng the model wth the smllest AIC vlue (Ny, et l., 5). 3.7 Methods he procedure to modelng Ar Polluter Stndrd Index (APSI) wth MGWR pproch usng the dptve ndwdth descred n the followng steps: ) Descre the dt s prelmnry to determne the spred of Ar Polluter Stndrd Index (APSI). ) Perform the glol lner regresson model c) Perform the sc GWR model d) Monte Crlo tests for regresson coeffcent nonsttonrty e) Select the glol regresson prt nd the GWR prt sed on Monte Crlo test f) Perform the MGWR model g) Compre the glol regresson model, GWR nd MGWR 4. RESULS Usng the GWmodel R Pcge (Lu, et l., 6) sed on AICc pproch, the optmum dptve ndwdth for ech locton re shown n le-. le-. Adptve ndwdth usng Bsqure Kernel. Locton Bndwdth SUF SUF SUF SUF SUF hen, usng ths ndwdth we estmte the GWR model. he goodness of fts for GWR model cn e stted y the followng hypothess:,,,, q, nd,,, n (GWR model s not sgnfcntly dfferent from the Regresson model) H : u, v H : t lest one u, v (GWR model s sgnfcntly dfferent from the regresson model) le-. Goodness of fts of GWR Model. Source of error Sum of squres Degree of freedom F p- vlue Improvement , GWR Regresson le- shows tht the F test sttstcl vlue s (p-vlue =.). Usng the sgnfcnce vlue of 5%, we must reject H, nd conclude tht the GWR model wth dptve ndwdth s sgnfcntly dfferent from the regresson model. herefore, we cn further conclude tht the GWR model s more proper to model the Ar Polluter Stndrd Index (APSI). hs mens tht the locton element s nfluentl n the APSI modelng. he sttstcs of locl prmeter n GWR model wth dptve squre ernel shown n le-3. le-3. Sttstc of GWR model wth Adptve Bsqure Kernel. Coeffcent Mn Mx Medn he next step s perform the monte-crlo smulton to test the regresson coeffcent nonsttonrty. hs test conducted to select the glol regresson prt nd the GW regresson prt. le-4 shows tht the r temperture (X ), the wnd velocty (X ), nd the r humdty (X 3) re the glol regresson prt. Menwhle, four other predctor vrles re the GWR prt ecuse these vrles hve the p-vlue less thn.5. le-4. Monte Crlo test of regresson coeffcent non-sttonerty wth dptve squre ernel. Coeffcent p-vlue.*.9.3,9.*.*.*.* Note: * sgnfcnt t 5% 448

5 VOL., NO. 5, AUGUS 7 ISSN ARPN Journl of Engneerng nd Appled Scences 6-7 Asn Reserch Pulshng Networ (ARPN). All rghts reserved. Bsed on the result of monte-crlo smulton to test the regresson coeffcent non-sttonrty, the MGWR model ws conducted. he model prmeter of MGWR model hs shown n le-5 dn le-6. le-5. Prmeter of Glol Regresson prt. Coeffcent X X X3 Vlue le-6. Prmeter of GWR prt. Coeffcent Mn Mx Medn he selecton of the est model s done y usng the AICc crteron. le- 6 shows the comprson of the glol regresson model wth the GWR model nd MGWR model ether y usng the squre dptve ernel functon. le-7 shows tht the MGWR model s the est model for modellng Ar Polluter Stndrd Index - APSI n Sury Cty ecuse t hs the smllest MSE nd AICc. le-7. Model comprsons. Model AICc MSE Regresson 3, ,58. Bsc GWR 3,586.6,64.3 Mxed GWR* 3,585. 7,48. Note: *Best Model 5. CONCLUSSIONS he MGWR model of the r polluton s lso nfluenced sgnfcntly y the locton fctor (geogrphcl fctor). herefore, n ths study, the GWR model s sutle to model the r polluter. he locl nfluence of GWR for oservton shows tht the fve sgnfcnt nfluencng predctor vrles re the r temperture (X ), the wnd velocty (X ), the r humdty (X 3), the trffc velocty (X 4), the populton densty (X 6). he other predctor vrles re re sze of the urn forest (X 5) nd the usness centre spect (X 7) not sgnfcnt n the model GWR. ACKNOWLEDGEMEN We would le to gve thn to Drectorte of Reserch nd Pulc Servces, he Mnstry of Reserch, echnology nd Hgher Educton Repulc of Indones for ther support. hs reserch ws funded y PUP Reserch Grnt 7. REFERENCES Atsh F. 7. he Deterorton of Urn Envronments n Developng Countres: Mtgtng the Ar Polluton Crss n ehrn, Irn. Ctes. 4(6): Eter. 6. Ar Polluton Induced Asthm nd Aletertons n Cytone Pttern. Allergy Asthm Immunol. 5(): Fhm M., Dhrm B., Fetrryn D. nd Bsoro. Asoss ntr polus udr dengn IgE totl serum dn tes fl pru pd pols llu lnts. Jurnl Penyt Dlm. pp. -9. Fothernghm A.S., Brunsdon C. nd Chrlton M.. Geogrphclly Weghted Regresson, Jhon Wley & Sons, Chchester, UK. Glert A. nd Chrorty J.. Usng Geogrphclly Weghted Regresson for Envronmentl Justce Anlyss: Cumultve Cncer Rss from Ar oxcs n Flord. Socl Scence Reserch. 4: Leung Y., Me C.L. nd Zhng W.X.. Sttstcl est for Sptl Nonsttonrty Bsed on the Geogrphclly Weghted Regresson Model, Envronment nd Plnnng. 3(5): Lu B., Hrrs P., Chrlton M. nd Brunsdon C. 4. he GWmodel R pcge: Further opcs for Explorng Sptl Heterogenety usng Geogrphclly Weghted Models Geo-sptl Informton Scence. 7(): 85-, Lu B., Hrrs P., Chrlton M., Brunsdon C., Ny. nd Golln I. 6. Pcge GWmodel. Me C.L., Wng N. & Zhng W.X. 6. estng the mportnce of the explntory vrles n mxed geogrphclly weghted regresson model. Envronment nd Plnnng. 38: Ny., Fothernghm A.S., Brunsdon C. nd Chrlton M. 5. Geogrphclly Weghted Posson Regresson for Dsese Assocton Mppng. Sttstcs n Medcne. 4(7): Ny.; Chrlton M.; Fothernghm A.S.; Brunsdon, C. 9. How to use SGWRWIN (GWR4.). Ntonl Centre for Geocomputton, Ntonl Unversty of Irelnd Mynooth. Purhd nd Ysn, H.. Mxed Geogrphclly Weghted Regresson Model (Cse Study: he Percentge of Poor Households n Mojoerto 8). Europen Journl of Scentfc Reserch. 69():

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