Int. J. Mobile Network Design and Innovation, Vol. 3, No. 2, Abdullah Konak

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1 It. J. Mobile Network Desig ad Iovatio, Vol. 3, No. 2, A krigig aroach to redictig coverage i wireless etworks Abdullah Koak Iformatio Scieces ad Techology, Pe State Berks, P.O. Box 7009, Tulehocke Road, Readig, PA , USA Fax: koak@su.edu Abstract: This aer itroduces krigig as a ew tool to redict etwork coverage i wireless etworks. The roosed aroach aims to reduce the cost of active site surveys by estimatig etwork coverage at oits where o site survey data is available. The roosed aroach was comared with a radial basis fuctio artificial eural etwork usig several roblems. Keywords: krigig; sigal estimatio; coverage la; eural etworks. Referece to this aer should be made as follows: Koak, A. (2009) A krigig aroach to redictig coverage i wireless etworks, It. J. Mobile Network Desig ad Iovatio, Vol. 3, No. 2, Biograhical otes: Abdullah Koak is curretly a Associate Professor of Iformatio Scieces ad Techology at the Pesylvaia State Uiversity-Berks. He received his degrees i Idustrial Egieerig, BS from Yildiz Techical Uiversity, MS from Bradley Uiversity, ad PhD from Uiversity of Pittsburgh. Previous to this ositio, he was a Istructor i the Deartmet of Idustrial ad Systems Egieerig at Aubur Uiversity for two years. His curret research iterest is i the alicatio of oeratios research techiques to comlex roblems, icludig such toics as telecommuicatio etwork desig, etwork reliability aalysis/otimisatio, facilities desig, ad data miig. He is a member of IIE ad INFORMS. Itroductio Desigig a fixed ifrastructure wireless etwork, such as a cellular hoe etwork or wireless local area etwork, ivolves several decisios, icludig the selectio of trasmitter locatios ad frequecy/ower assigmets. Network erformace idicators such as data rate, trasmissio delay, acket loss, ad jitter deed o a roer etwork desig. All these key erformace idicators deed uo the distributio of sigal stregth over a targeted area of service. A wireless etwork should rovide 00% coverage with sigal stregth above a miimum threshold value over all its target area. To esure a accetable level quality of service for the users of a fixed ifrastructure wireless etwork desigers rely o site survey techiques ad/or sigal roagatio models. Site surveyig for a ew etwork deloymet usually starts with lacig ortable trasmitters at relimiary locatios. Sigal stregth ad service quality data is the gathered for a set of test oits. This survey data is used to modify trasmitter locatios to esure adequate coverage for users i the target area of service. The umber ad distributio of such test oits deeds uo the size of the service area as well as its hysical toology ad aticiated umber of users. Proer selectio of relimiary trasmitter locatios is also imortat for a effective site survey ad desig (Hills, 200). There are difficulties associated with data gatherig as well. Some arts of the target area might be iaccessible durig the survey. Chages i eviromet may affect quality of measuremets ad cause variatios (Zvaovec et al., 2003). Site survey ersoel should be exerieced i carryig out comlex site surveys ad correctly iterretig the results. Therefore, site surveyig is a very time cosumig ad labour itesive rocess. Several tools have bee develoed to aid site surveys ad automate the desig rocess based o re-existig site surveys. For examle, Rollabout (Hills ad Schlegel, 2004; Hills et al., 2004) is a rollig cart with a lato comuter that automatically collects data ad creates the coverage ma of a target area for wireless local etworks. Commercial site survey software such as Ekahau (Badma, 2006) rovides a array of effective tools to survey ad la wireless etworks. If ortable trasmitters are ot available for the gatherig of test oit data, modellig tools are utilised for redictig etwork coverage over a target area. I this aroach, the etwork coverage is simulated usig electromagetic wave roagatio models. Comrehesive surveys o electromagetic wave roagatio models for wireless etworks are give i Sarkar et al. (2003) ad Coyright 2009 Idersciece Eterrises Ltd.

2 66 A. Koak Zvaovec et al. (2003). Simulatio rovides a cost-effective way to aalyse alterative desig cofiguratios. I additio, simulatio ca be used to determie relimiary locatios of trasmitters before a site survey. However, the accuracy of a coverage redictio deeds uo the roagatio model ad a detailed ad accurate reresetatio of the target area. Most moder simulatio software is caable of readig mas or bluerits ad eables users to defie objects o the ma of a target area. I the case of idoor wireless etworks, however, creatig a realistic bluerit of the target area icludig walls, wall tyes, furiture, stairs ad elevator shafts ca be costly ad time cosumig. I additio, site survey results should be verified before imlemetatio. I this aer, a ovel krigig-based emirical aroach is roosed to estimate the coverage for a fixed ifrastructure wireless etwork. The mai objective is to create a accurate ad comlete etwork coverage ma of a target service area from a limited umber of test oit measuremets. Thereby, the cost of time cosumig site surveys ca be reduced. I additio, the roosed aroach ca be used to estimate the etwork coverage where samles could ot be take due to iaccessibility. Fially, the roosed aroach ca also be used to validate the accuracy of measuremets durig a site survey. There has bee limited work i the literature to estimate the sigal stregth for wireless etworks usig emirical aroaches. I Nasereddi et al. (2005), a radial basis fuctio artificial eural etwork (ANN) was roosed to redict the sigal-to-oise ratio, which is a imortat idicator for quality of service i wireless etworks. To redict the sigal-to-oise ratio at a oit, this ANN aroach utilises three iuts: the x-y coordiates (idices) of oit, the idex of the trasmitter with highest trasmitted ower at oit, ad the trasmissio ower. First the ANN is traied for kow oits, ad the the traied ANN is used to redict the sigal-to-oise ratio for ukow oits o the target area. Neskovic et al. (2000) roosed a backroagatio ANN to redict the wave roagatio i idoor eviromets. I this case, the iut of the ANN icludes the distace from the trasmitter to the oit, objects (e.g., walls) alog the straight lie draw from the trasmitter to the oit, ad toological iformatio about the target area. Therefore, this ANN aroach aims to relace hysical electromagetic wave roagatio models rather tha to redict etwork coverage from emirical data. Che ad Kobayashi (2002) roosed a liear regressio aroach to determie the arameters of wave roagatio models for wireless local area etworks (WLANs) based o the measured sigal stregths at test oits. The fitted regressio model was used to the estimate sigal stregths for ukow oits. Che ad Kobayashi (2002) reorted that the quality of the estimatio deeded o the uderlyig wave roagatio model. O the other had, the krigig aroach roosed i this aer does ot assume a uderlyig wave roagatio model ad estimatios are oly based o the samle field measuremets. 2 Ordiary krigig Krigig is a iterolatio techique based o the methods of geostatistics which is a subject cocered with satial data. Geostatistics assumes that there is a imlied coectio betwee the measured data value at a oit ad where the oit is located i sace (i.e., each data value is associated with a locatio i the sace). Krigig was develoed by Krige (95), from whom the ame krigig was derived, ad Mathero (963) to accurately redict ore reserves from the samles take over a miig field. Ordiary krigig is the most commoly used tye of krigig. I ordiary krigig, a ukow value z ˆu at oit u is estimated as a weighted-liear combiatio of kow samles as follows (Issaks ad Srivastava, 989): zˆ = wz () u i i i= where i= w =. i Krigig is used to determie the otimal weights, which roduce the miimum estimatio error, i equatio () usig the followig simultaeous equatios: wγ( h,) + + wγ( h, ) + λ = γ( h, u) (2) w γ( h, ) + + wγ( h, ) + λ = γ( h, u) w + + w = where γ ( h i, j ) is a semivariogram which is a fuctio of distace h i,j betwee oits i ad j, ad λ is the Lagrage multilier to miimise the krigig error. I geostatistics, a semivariogram reresets satial correlatios betwee oits i sace. Accordig to the geostatistics, as the distace h i,j betwee two oits i ad j icreases, the correlatio betwee them is exected to decrease. For examle, air ollutio samles take i close roximity are exected to be more correlated tha samles take distace aart. The otimal weights i equatio () are calculated as follows. γ, γ, γ, u w ( h ) ( h ) ( h ) = w γ( h, ) γ( h, ) γ( h, u) λ 0 Ordiary krigig assumes that the mea is costat i the local eighbourhood of each estimatio oit. As a result, the exected value of estimatio error at a ukow oit u is zero (i.e., Ez ( ˆu zu) = 0). The weights determied by equatio (3) are called otimal sice they miimise the variace of estimatio error (i.e., Var( zˆ u zu) ). Prior to determiig the weights usig equatio (3), a meaigful semivariogram fuctio should be selected. I ordiary krigig, a successful estimatio deeds o the choice of the semivariogram fuctio. Although there are a ifiite umber of ossible semivariogram fuctios, most (3)

3 A krigig aroach to redictig coverage i wireless etworks 67 commoly used semivariogram models, such as liear, exoetial, ad sherical models rovide good results for most data sets. For examle, the exoetial semivariogram model is give as follows (Bailey ad Gatrell, 996): 0 hi, j= 0 γ ( hi, j) = (4) C0 + ( C C0 )ex( 3 hi, j/ R) hi, j> 0 where C 0 is the ugget effect, C is the still arameter, ad R is the rage arameter. Figure illustrates a examle of the exoetial semivariogram fuctio. R defies the distace beyod which the correlatio betwee two oits is assumed to be essetially zero. The ugget effect reresets variability at distaces smaller tha the tyical samle sacig i the data set. Still is the maximum value of the semivariogram fuctio. Selectig a good semivariogram fuctio requires a careful study of the data set ad subjective judgemet. Geeral guidelies for a good semivariogram selectio are give i Bailey ad Gatrell (996). Figure A exoetial semivariogram fuctio Krigig has certai advatages over other iterolatio techiques. Krigig is a otimal iterolatio method because it roduces a ubiased estimate with miimum variace. A imortat cocer i iterolatio is to choose the best set of available samle oits to be iterolated to estimate a ukow oit. The stregth of krigig lies i the fact that it defies a otimal set of oits to iterolate by adjustig their weights. Notice that ot oly the distaces betwee kow ad ukow oits, but also the distaces betwee kow oits are cosidered i equatio (2). As a result, clustered samle oits cotaiig redudat iformatio are give less weight i estimatio. Aother advatage of krigig is that every estimate has a corresodig krigig stadard deviatio. Thus, a reliability ma of redictios ca be roduced. Oce the weights ad λ are calculated usig equatio (3), the variace of a estimatio z ˆu ca be calculated as follows: 2 zˆ w u i hi, u i= σ = γ( ) + λ (5) Note that the variace i equatio (5) is the variace of a idividual estimatio, ot the variace of radom variable z. ˆu 3 Imlemetatio Let set P be a set of kow test oits where quality of service measures such as sigal stregth or sigal-to-oise ratio (SNR) are measured durig a site survey; ad for each oit i P, let z i ad (x i, y i ) deote the measured value ad the xy-coordiates of the oit, resectively. The roblem is to estimate the quality of service at a oit u where o measuremet was take. We roose a rocedure based o the ordiary krigig to estimate a quality of service measure at a ukow oit u as follows: Defie a eighbourhood of oit u o the xy lae ad idetify the surveyed oits i this eighbourhood. Let V(u) be the set of oit u ad oits i its eighbourhood. 2 Defie ad calculate the Euclidia distaces ad semivariogram fuctios betwee the oits i V(u). 3 Calculate the weights usig equatio (3). 4 Estimate z u usig () ad calculate the estimatio variace usig equatio (5). I Ste, set V(u) ca be determied i various ways. For examle, all kow oits withi a redefied radius of oit u ca be defied as the eighbourhood of oit u. I fact, the set of all kow oits ca eve be used to estimate z u. I this aer, two differet methods are used to determie set V(u). I the first aroach, set V(u) icludes -earest eighbours of oit u. I the comutatioal exerimets, the effect of o the estimatio error is studied. The secod aroach ivolves the Delauay triagulatio. The mai advatage of usig the Delauay triagulatio to determie set V(u) is that it searches for atural eighbours (early equiagular) of oit u i set P. I additio, it imroves comutatioal effort sice oly a limited umber of kow data oits are cosidered i equatio (2) (Hessami et al., 200) ad the comuter imlemetatio of Delauay triagulatio is straightforward. The Delauay triagulatio is used to determie set V(u) as follows. First, oit u is triagulated with set P, ad the as a result of this triagulatio, the oits that are directly coected to oit u are idetified as the eighbours of oit u. Set V(u) icludes both oit u ad its eighbours. Figure 2 illustrates two differet examles. I case (a), oit u is located iside the covex hull of kow data oits (solid circles). Therefore, estimatig z u is a iterolatio. I case (b), however oit u is outside of the covex hull of kow data oits. Therefore, estimatig z u is a extraolatio. I both cases, set V(u) icludes oit u ad the kow oits which are directly coected to oit u as show i Figure 2.

4 68 A. Koak I Ste 2, it is assumed that oly the distaces betwee the surveyed oits are kow about the surveyed area. Therefore, the semivariogram fuctio is defied as the Euclidea distace betwee two oits i ad j as follows: 2 2 ( hi, j) max{, ( xi xj) ( yi yj) } γ = + Comared to the revious ANN aroaches, the roosed krigig aroach does ot require ay iformatio about the area surveyed ad does ot assume a wave roagatio model. The oly iut to the krigig model is the set of kow oits. Figure 2 Examles of the Delauay triagulatio to determie set V(u) 3 Comutatioal exerimets The erformace of the roosed krigig aroach is tested to estimate the sigal-to-oise ratio (SNR), which is a imortat measure of the quality of service i wireless etworks, usig simulated data. I Nasereddi et al. (2005), a radial basis fuctio ANN is used to estimate SNR. Comrehesive surveys o ANN ad radial basis fuctios are give i Cheg ad Titterigto (994) ad Buhma (2003). I order to comare the results with this ANN aroach, traiig ad testig sets i this aer were geerated i the similar way as described i Nasereddi et al. (2005). I Nasereddi et al. (2005), two test roblems, the 4-tower ad 27-tower roblems give i Bartolacci et al. (2004), are used to test their ANN aroach. It should be oted that the 4-tower roblem actually icludes 5 trasmissio towers. I Bartolacci et al. (2004), the roblem is amed as 4-tower sice two of the towers are very close to each other. I additio to these two roblems, a 45-tower roblem (Bartolacci et al., 2004) was also studied to cosider the effect of icreasig the umber of trasmitters i the same target area. I all roblems, the target area is 0 km 0 km, ad relative trasmissio ower of each tower is radomly geerated betwee oe ad five. The traiig ad testig data sets were geerated as follows: The target area of each roblem is discretised by dividig the x ad y axis ito M grids. 2 The SNR at each itersectio oit () of the grid lies is calculated as follows: SNR S P = 0log 0 N P S where P is the received ower of the servicig tower, N ad P is the ower of the oise at oit. The servicig tower at oit is determied as the tower with the highest received ower at the oit, ad P is the total received ower from the other towers at oit. 3 For each roblem, the samle data is radomly artitioed ito traiig ad testig sets usig three differet ratios as 75%, 50%, ad 25% (75% artitioig meas that a oit is radomly selected for traiig with a robability of 0.75 ad for testig with 0.25). Table summarises the test roblems alog with their umbers of test ad traiig oits ad the results of comutatioal exerimets. The test roblems are amed usig their umber of towers, test/traiig artitioig ratio, ad umber of grids. For examle, 27T-25%-20 is the 27-tower roblem where a oit is selected for traiig with 0.25 robability (for testig with 0.75) ad the target area is divided ito grids. Sice i Nasereddi et al. (2005) the target area is divided ito grids, this case was rimarily ivestigated. For the krigig aroach, the oits i the traiig sets rereset the oits with kow SNR values, ad the objective is to estimate the SNR for the oits i the testig sets. Both the triagulatio ad -earest eighbour aroaches were tested. I the -earest eighbour aroach, = 2,..., 20 were used to study the effect of eighbourhood size o the estimatio accuracy. Usig the same traiig set, a radial basis fuctio ANN was traied for each test roblem, ad the ANN model was also tested o the same testig set with the krigig aroach. Therefore, the abilities of the ANN ad krigig aroaches to redict the SNR ca be comared usig the idetical traiig ad testig oits. It should be oted that the ANN aroach does ot use actual xy-coordiates of the oits, but their resective grid idices. For examle, the actual coordiates of the southwest corer oit i Figure 3 are (94, 489). This oit i the ANN model is rereseted usig the idex of (, ). I additio to this iformatio, the ANN model uses the idex ad the ower level of the tower servig to each oit as iut. I this aer, the radial basis fuctio ANN was imlemeted usig MATLAB s Neural Network Toolbox-Radial Basis Fuctio. For each roblem, 20 differet ANN models were traied by settig the sread N

5 A krigig aroach to redictig coverage i wireless etworks 69 arameter of the radial basis layer from 0 to 2.0 by 0. itervals. For each case, the ANN model with the smallest estimatio error was used i the comariso. Figure 3 illustrates the locatio of the traiig ad testig oits for the 4-tower roblem with 75% artitioig o the cotour lot of the SNR. I this examle, kow oits, which are idetified by (+) i the figure, are used to trai the ANN ad to build the krigig models i order to estimate the SNRs for ukow oits, which are idetified by ( ). After iitial exerimets, the oits which are very close to the locatios of trasmitters were excluded sice surveyig such test oits is ot of ractical iterest. I additio, the ANN aroach did ot erform well i redictig such oits due to the extremely high sloe of the SNR fuctio withi a few metres aroud the trasmitters. I a real life case, samles ca be take more frequetly or i a more systematic maer durig a site survey. However, the mai objective i this comutatioal study is to comare the erformaces of the ANN ad krigig aroaches o simulated data ad demostrate advatages ad disadvatages of each aroach. Figure 3 meters The coverage ma of the 4-tower roblem ad the traiig set for 75% artitioig meters Note: + deotes the oits where the SNRs are samled (traiig oits) ad deotes test oits. To comare the ANN ad krigig aroaches, two statistics are used: the coefficiet of determiatio (R 2 ) ad the - value of a aired t-test. Table resets the R 2 values achieved by the krigig aroaches ad the ANN model to estimate the SNRs at ukow oits. R 2 rovides some iformatio about the goodess of fit of a redictio model, ad it is frequetly used i comarig alterative estimatio models. A aired t-test was erformed to comare the absolute error of estimatio ( z ˆ z ) at each ukow oit by the ANN ad the -earest eighbour krigig aroach with = 8. The tested hyothesis is give as H 0 MAE K = MAE ANN H MAE K MAE ANN where MAE K ad MAE NN are the mea absolute errors of the -earest eighbour krigig aroach ad ANN aroaches, resectively. Note that the umber of testig oits i each case is suitably large eough to use the aired t-test due to the cetral limit theorem. Table Test roblem The umber of test ad traiig oits for test roblems ad summary of comutatioal exerimets # test/ # traiig Triagulatio krigig R 2 - earest krigig R 2 ANN R 2 - value 4T-25% / T-50% / T-75%-20 82/ T-25% / T-50%-20 64/ T-75%-20 87/ T-25% / < T-50%-20 36/ T-75%-20 7/ T-25% / T-50% / T-75%-30 52/ T-25% / T-50% / T-75% / Note: For the -earest eighbour aroach, the reorted R 2 is for = 8. If the R 2 values i Table are carefully aalysed, several atters ca be observed. Firstly, both ANN ad krigig aroaches erformed oorly for the cases with a limited umber of traiig oits (i.e., cases with 25% ad 50% testig/traiig set artitioig). It is also kow that radial basis fuctio ANNs work best with a large umber of traiig samles. For each roblem grou, the erformace of the ANN aroach sigificatly imroved from 25% to 75% testig/traiig set artitioig. Whe the target area is divided ito grids, the miimum distace betwee two oits is about 500 metres. Therefore, the SNR at a ukow oit is estimated usig oits at least a kilometre aart. Particularly for the cases with the 25% ad 50% testig/traiig set artitioig, traiig oits are very sarely ad soradically distributed over the target areas. For examle i 45T-25%-20, oly 7 traiig oits are used to redict the SNR at 247 testig oits. This is the mai reaso for the low levels of R 2 i these cases. As see i Table, the estimatio accuracy for the 45-tower roblem was sigificatly imroved whe the target area is divided ito smaller grids, such as ad For roblem 45T-75%-40, the krigig with -earest eighbour ad ANN aroaches rovided R 2 values of 0.95 ad 0.93, resectively.

6 70 A. Koak Secodly, the krigig aroach outerformed the ANN aroach for the cases with limited umber of traiig oits, which is also imortat for the cotributio of this aer. As see i Table, the krigig aroach was suerior to the ANN aroach i all cases with 25% testig/traiig set artitioig. As stated earlier, the ANN estimatios imroved with more traiig data available. As a result, it rovided higher R 2 values tha the krigig aroach did i all cases with 75% testig/traiig set artitioig, excludig 45T-75%-40. For 50% testig/traiig set artitioig, the R 2 values of both aroaches were very close. I summary, the krigig aroach teds to rovide better estimatio i the absece of a adequate umber of survey samles. A third observatio is that for a give grid size the erformace of both aroaches deteriorated with the icreasig umber of trasmitters i the target area. For examle, the R 2 values sigificatly deteriorated from 4T-75%-20 to 27T-75%-20 ad similarly from 27T-75%-20 to 45T-75%-20. The test roblems have the same size target area, ad as the umber of trasmitters icreases for a give area, the SNR coverage ma becomes much more comlex. This exlais the oor erformace for the 45-tower roblem with grids. To aalyse regios i the target area with high estimatio errors, absolute estimatio error was lotted agaist the x-y coordiates of test oits. Figure 4 illustrates the distributio of absolute estimatio error over the target area for 4T-75%-20 usig the krigig aroach with triagulatio. I the figure, it is clear that high estimatio errors occur at the regios with iadequate samles of kow oits. For examle, estimatio error is high for the test oits o the borders ad at the corers of the target area. This ca be exlaied by the lack of surveyed oits beyod the borders of the target area. I other words, the estimatio error is exected to be high for test oits which are ot eclosed by traiig oits [e.g., similar to the case give i Figure 2(b)]. If the border oits are excluded, R 2 for this case becomes higher tha To icrease the estimatio accuracy, therefore, more samles should be take beyod or o the boudaries of a target area. As see i Table, betwee the two selectio methods of kow oits i the eighbourhood of a ukow oit u, the -earest eighbour aroach yielded slightly better R 2 tha the triagulatio aroach did. O the other had, this aroach requires determiig a good value of arameter. Sice triagulatio automatically determies atural eighbours of a oit, a arameter is ot eeded. I the exerimets, all values of betwee 2 ad 20 were tried ad the results reorted i Table are for = 8. Figure 5 illustrates the observed R 2 values with differet levels of for the 4-tower test roblems. Iitially, R 2 quickly icreased with the iclusio of the first a few earest eighbours, ad the it remaied steady as additioal, but farther oits were icluded. The other roblems also showed similar atters as show i Figure 5. This result demostrates a advatage of krigig; it determies a otimal set of oits to iterolate by adjustig their weights. Icreasig the eighbourhood size i the -earest eighbour aroach may cause clusterig of traiig oits. However, krigig successfully comesates for the egative effects of this clusterig by assigig idividual oits withi a cluster less weight tha isolated oits. Figure 4 Figure 5 R Coclusios Distributio of absolute estimatio error over the target area for 4T-75%-20 usig the krigig aroach with triagulatio The relatioshi betwee R 2 ad the size of the eighbourhood i the -earest eighbour aroach for the 4-tower roblems 4T-25%-20 4T-50%-20 4T-75% earest eighbor This aer rooses krigig to redict sigal coverage i wireless etworks based o available samles take i a active site survey. To the best of our kowledge, this aer is the first imlemetatio of krigig as a tool i wireless etwork survey ad desig. The objective of the roosed aroach is to reduce the cost of time cosumed ad labour itesive site surveys. I additio, the roosed aroach ca be used to validate samles take durig a site survey. The roosed aroach was comared to a radial basis ANN aroach ad show to be cometitive with this aroach. The roosed aroach outerformed the ANN aroach articularly i cases with limited traiig oits

7 A krigig aroach to redictig coverage i wireless etworks 7 available. As further research, the krigig ad ANN aroach ca be itegrated to imrove estimatio quality. I this aer ordiary krigig was used to estimate SNR. It would be iterestig further research to fid ways to icororate toology of the target area or objects such as buildigs ito satial correlatios. The roosed aroach ca be easily embedded withi site survey software to iterolate sigal coverage data for oits which are ot surveyed i the target area. Zvaovec, S., Pechac, P. ad Kleal, M. (2003) Wireless LAN etworks desig: site survey or roagatio modelig?, Radioegieerig, Vol. 2, No. 4, Refereces Badma, L. (2006) Visualize your WLAN [Ekahau s site survey 2.2], Network Comutig, Vol. 7, No. 6, Bailey, T. ad Gatrell, T. (996) Iteractive Satial Data Aalysis, Pretice Hall. Bartolacci, M.R., Koak, A. ad Whitaker, R.M. (2004) A coectivity decisio suort system (CDSS) for wireless etwork, Asia Joural of Iformatio Techology, Vol. 3, No. 9, Buhma, M.D. (2003) Radial Basis Fuctios: Theory ad Imlemetatios, Cambridge Uiversity Press. Che, Y. ad Kobayashi, H. (2002) Sigal stregth based idoor geolocatio, , New York, NY, USA. Cheg, B. ad Titterigto, D.M. (994) Neural etworks: a review from a statistical ersective, Statistical Sciece, Vol. 9, Hessami, M., Actil, F. ad Viau, A.A. (200) Delauay imlemetatio to imrove krigig comutig efficiecy, Comuters & Geoscieces, Vol. 27, No. 2, Hills, A. (200) Large-scale wireless LAN desig, IEEE Commuicatios Magazie, Vol. 39, No., Hills, A. ad Schlegel, J. (2004) Rollabout: a wireless desig tool, IEEE Commuicatios Magazie, Vol. 42, No. 2, Hills, A., Schlegel, J. ad Jekis, B. (2004) Estimatig sigal stregths i the desig of a idoor wireless etwork, IEEE Trasactios o Wireless Commuicatios, Vol. 3, No.,.7 9. Issaks, E.H. ad Srivastava, R.M. (989) A Itroductio to Alied Geostatistics, Oxford Uiversity Press. Krige, D.G. (95) A statistical aroach to some basic mie valuatio roblems o the Witwatersrad, Joural of the Chemical, Metallurgical ad Miig Society of South Africa, Vol. 52, No. 6, Mathero, G. (963) Priciles of geostatistics, Ecoomic Geology, Vol. 58, No. 8, Nasereddi, M., Koak, A. ad Bartolacci, M.R. (2005) A eural etwork-based aroach for redictig coectivity i wireless etworks, Iteratioal Joural of Mobile Network Desig ad Iovatio, Vol., No., Neskovic, A., Neskovic, N. ad Pauovic, D. (2000) Idoor electric field level redictio model based o the artificial eural etworks, IEEE Commuicatios Letters, Vol. 4, No. 6, Sarkar, T.K., Zhog, J., Kyugjug, K., Medouri, A. ad Salazar-Palma, M. (2003) A survey of various roagatio models for mobile commuicatio, IEEE Ateas ad Proagatio Magazie, Vol. 45, No. 3,.5 82.