New Applied Methods For Optimum GPS Satellite Selection

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1 New Appled Methods For Optmum GPS Satellte Selecton Hamed Azam, Student Member, IEEE Department of Electrcal Engneerng Iran Unversty of Scence &echnology ehran, Iran Mlad Azarbad Department of Electrcal Engneerng Unversty of Mazandaran Mazandaran, Iran Saed Sane, Senor Member, IEEE Faculty of Engneerng and Physcal Scences, Unversty of Surrey, Unted Kngdom Abstract Geometrc dluton of precson (GDOP) s a powerful, smple and wdely used measure for assessng the effectveness of potental measurements to specfy the precson and accuracy of the data receved from global postonng system (GPS) satelltes. he most correct method to classfy or approxmate the GPS GDOP s to use nverse matrx on all the combnatons and choosng the lowest one, but nversng a matrx puts a lot of computatonal burden on the navgator s processor. hs approach however s a tme-consumng task. o overcome the problem, basc back propagaton neural network (BPNN) was used. Snce the BPNN s too slow for practcal problems, ncludng the GPS GDOP classfcaton, n ths paper several methods, namely, reslent back propagaton (RBP) to tran a NN, nave Bayes classfer, Fsher s lnear dscrmnant (FLD) and k-nearest neghbor (KNN) for classfcaton of the GPS GDOP are proposed. Smulaton results show that these methods are much more effcent to classfy the GPS GDOP data than prevous methods. Keywords- GPS GDOP; reslent back propagaton, nave Bayes classfer; Fsher s lnear dscrmnant; k-nearest neghbor I. INRODUCION Global postonng system (GPS) s a navgaton system based on a network of 4 satelltes n 6 nclned orbts that provdes three-dmenson poston and tme by at least four satelltes that are orbted around the earth [1,]. Dluton of precson (DOP) or geometrc DOP (GDOP) refers the effect of geometry on the relatonshp between measurement errors and poston determnaton errors obtaned by the GPS satelltes [,4]. A GPS recever generally uses GDOP optmal crtera or GDOP sub-optmal crtera as the satellte selecton algorthm [4]. here are bascally two approaches employed to justfy the satellte geometry based on the GPS GDOP, namely, approxmaton and classfcaton. Unlke the GPS GDOP estmator whch computes the values n order to select the optmal subsets of the satelltes, the GPS GDOP classfer s employed to choose one of the acceptable subsets (four optmum satelltes from 4 exstng satelltes) of satelltes for navgaton use [5]. A subset of satelltes wth a GPS GDOP value of less than s deal,.e. the attaned measure of locaton by these satelltes s very relable, whle a hgher GPS GDOP value ndcates poor satelltes postonng and an nferor measurement confguraton. able 1 demonstrates the GPS GDOP ratngs [1]. able 1. GPS GDOP ratngs Class number GDOP value Ratngs Class 1 1 Ideal Class - Excellent Class 4-6 Good Class Moderate Class Far Class Poor he most common approach to obtan the GPS GDOP s to calculate the nverse matrx for all combnatons of satelltes and choose the mnmum one whch t s very tme consumng approach. In [6] Hsu has proposed a method based on tetrahedron volume formed by four user-tosatellte vectors lke Fg. 1. Fg. 1 shows the geometry of the satelltes and ts affect on the GPS GDOP values. (a) (b) Fgure 1. Satellte s dagram and ts relaton wth DOP: (a) Bad DOP and (b) Good DOP However, t s not unversally acceptable because t does not guarantee optmum selecton of satelltes [5]. In order to estmate and classfy the GPS GDOP, frst, Smon and El-Sheref extracted a set of features nclude traces of the measurement matrx and ts second and thrd powers, /1/$ IEEE

2 and the determnant of the matrx. hen, n order to advantages of computatonal effcency, they use the basc back propagaton for neural networks (BPNNs) to classfy and approxmate the GPS GDOP [5]. However, n many applcatons, ncludng GPS GDOP classfcaton, the BPNN has many defcences such as too slow convergence speed, easy to fall nto local mnma, and easly affected by sudden peaks n the sgnal trend durng the learnng process. o overcome these problems Jwo and La n [5] have proposed to use the basc BP wth momentum, the optmal nterpolatve (OI) network, probablstc NN (PNN) and general regresson NN (GRNN) to classfy the GPS GDOP. In order to overcome these problems and ncreasng the accuracy of the PNN, reslent back propagaton (RBP) to tran an NN, nave Bayes classfer, Fsher s lnear dscrmnant (FLD) and k-nearest neghbor (KNN) are suggested. RBP s an enhanced statc NN and t s known to provde faster local adaptaton of weghts and bases wthout sacrfcng accuracy [7,8]. he nave Bayes classfer that s a knd of the Bayesan classf`er s smple, fast, and an acceptable method for data classfcaton. hs classfer when the dmensonalty of the nputs s large s partcularly used [9]. he KNN s one of the well establshed and smple classfer methods. When there s a lttle or no pror nformaton about the dstrbuton of data, the same as our applcaton, ths method s one of the best methods for classfyng the data [10]. he FLD s a method manly used n pattern recognton and machne learnng to dscover a lnear combnaton of features whch characterze or dvde several classes of objects or events [11]. he paper s organzed as follows. he background knowledge for the proposed methods ncludng the concept of the GPS GDOP and the classfers used here are brefly expressed n Secton. Secton ntroduces the proposed method for the GPS GDOP classfcaton. hen, the computer smulaton results are dscussed n Secton 4. Fnally, conclusons are gven n secton 5. II. BACKGROUND KNOWLEDGE FOR HE PROPOSED MEHODS In ths secton, the concept of the GPS GDOP, and four proposed classfers have brefly been explaned. A. Geometrc Dluton of Precson Bascally, the GPS accuracy s reled to the GPS GDOP or GDOP. GDOP s a relevant factor for postonng accuracy whch changes wth the satellte geometrc locaton. herefore, GDOP factor s a measure of postonng error. he recevers generally employ the GDOP as the satellte selecton crteron [1]. GPS recevers often report the qualty of satelltes geometry assessment based on the horzontal dluton of precson (HDOP), vertcal dluton of precson (VDOP) and tme dluton of precson (DOP) as follows: GDOP = HDOP + VDOP + DOP (1) We resume the defntons of GDOP computaton, useful for the sequel of the document and help understandng our contrbutons. he absolute dstance between a user and a satellte s defned as follows [1]: ono, trop, R = ρ +Δ ρ +Δ ρ () where: ρ = ( X X ) + ( Y Y ) + ( Z Z ) cδ t () u u u u Δ ρ ono, trop, and Δ ρ whch are the errors nduced by the onospherc and the tropospherc propagaton, are calculated from a model, ( X, Y, Z, δ t ) are the four u u u u system unknowns and δ t s the correcton the recever has u to apply to ts own clock. Also, c s the speed of the lght. In order to resolve ths system we need four equatons whch mean four pseudo-ranges from four dfferent satelltes. he pseudo-ranges can be approxmated by a aylor expanson. We obtan [1]: ^ ^ ^ ^ ^ ρ = ( X X ) + ( Y Y ) + ( Z Z ) cδ t (4) u u u u he aylor expanson at the frst order s: ^ Δ ρ = ρ ρ = a Δ x + a Δ y + a Δz cδ t (5) x u y u z u u where: ^ ^ ^ X X Y Y Z Z a = u ; a = u ; a = u ; x ^ y ^ z ^ r r r (6) ^ ^ ^ ^ r = ( X X ) + ( Y Y ) + ( Z Z ) u u u Let assume N be the number of vsble satelltes. sat he matrx H s as follows: a a a 1 x1 y1 z1 a a a 1 x y z H = (7) a a a 1 xn yn zn sat sat sat Let defne the G matrx: G = ( H 1 (8) he GDOP s [11]: GDOP = trace[ G] (9) B. Implemented Classfers RBP s a powerful method for supervsed learnng n feedforward NNs that s known as a fast convergng algorthm.

3 hs algorthm, whch s a frst order optmzaton algorthm, was suggested by Redmller and Braun n 199 [14]. he method s based on the sgn of the respectve partal dervatve at the current and the prevous epoch [8]. It s clear that the changes of the learnng rate can sgnfcantly nfluence the performance of the tranng. he RBP tres to allevate ths problem by usng adaptvely computed parameters whch change n every teraton [8]. Actually, these parameters are adjusted durng the learnng process based on the drecton of convergence. Second method used n ths paper s nave Bayes classfer. In spte of ts smplcty, Bayesan classfer s able to often outperform more sophstcated classfcaton methods. hs classfer s based on the Bayesan theorem. In ths paper we use a nave Bayes classfer whch s a smple but effectve Bayesan classfer for vector data (.e. data wth several attrbutes) that assumes that attrbutes are ndependent gven the class. In smple terms, nave Bayes classfer supposes that the presence (absence) of a partcular feature of a class s not related to the presence (absence) of any other feature gven the class varable. he Bayesan decson makng wth applcatons to pattern recognton s treated n detals n [9]. he FLD s a generatve classfer that looks for a model, gven tranng data and the optmal decson boundary between the classes. he FLD whch s a knd of lnear dscrmnant analyss (LDA) method tres to express one dependant varable as a lnear combnaton of other measurements. In fact, t s an acceptable and wdely used classfcaton method that s descrbed n chapter of the book [15] or n the book [16]. he KNN algorthm measures the dstance between a query object and a set of objects n the data set. he dstance between two objects usng some dstance functons such as absolute dstance measurment or Eucldean dstance measurement s computed. In ths algorthm an object s classfed by a majorty vote of ts neghbors (k) that s expermentally chosen [15] III. PROPOSED MEHODS In ths research, a large set of experments was carred out usng the followng set-up: a standard GPS recever was nstalled n a fxed pont and was connected to a PC. In order to get the GPS GDOP real collecton, the azmuth (Az) and the elevaton (E) of each observed satellte are measured by usng a developed embedded system. After collectng the GPS nformaton on DRAM, these data were transformed to seral port of PC for processng. Fg. shows the entre GPS data collectng embedded system. Fgure. GPS data collectng embedded system used n our experments. In order to make the nstructonal data, all of the nput and output values are normalzed to between 0 and 1 to reduce the nstructon tme. Snce H H s a 4 4 matrx, t has four λ ( = 1,,,4) egenvalues. We know that the four egenvalues for the matrx ( H 1 are as λ 1 ( = 1,,,4). Based on the fact that the trace of a matrx s the sum of ts egenvalues, equaton (9) would be as below [17,18]: GDOP = λ (10) 1 4 Mappng wth the defnton of four varants would be done as below: x ( λ) = λ = trace( H (11) x ( λ) = λ = trace[( H ] (1) 1 4 x ( λ) = λ = trace[( H ] (1) 1 4 x ( λ) = λ λ λ λ = det( H (14) where det denotes determnant of a matrx. Mappng from Y to the GPS GDOP classes s often hghly non-lnear and cannot be determned analytcally, but t can be determned specfcally usng an NN or other classfers. In ths research, four classfers s appled to do the mappng from Y to the GPS GDOP classes. he entre classfcaton block dagram of the GPS GDOP usng the NN s shown n Fg..

about 4.86% n accuracy by the RBP and usng the GPS GDOP measurement data compared wth that n [5]. Fgure.

SIMULAION RESULS In ths paper the smulatons have been carred out usng a DELL-PC wth Intel (R) Core (M) CPU M50.7 GHz and -GB RAM by MALAB R010a.

Decdng how many neurons to use n the hdden layer s one of the most mportant characterstcs n an NN.

If too many numbers of neurons are used for an NN, t not only ncreases the tranng tme, but also may reduce the performance of the NN.

For all of these classfcaton methods we tran the classfers wth 50% of the GPS GDOP measurement data and then use the rest of the data for testng these algorthms.

4 to 9 show comparatve analyses of dfferent tranngs for varous neurons of the NN hdden layer wth 10, 0, 50, 100, 150, and 00 teratons, respectvely.

4 about 4.86% n accuracy by the RBP and usng the GPS GDOP measurement data compared wth that n [5]. Fgure. Classfcaton block dagram of the GPS GDOP In ths research, n addton to the BPNN, GRNN and PNN, four dfferent classfers have been appled. IV. SIMULAION RESULS In ths paper the smulatons have been carred out usng a DELL-PC wth Intel (R) Core (M) CPU M50.7 GHz and -GB RAM by MALAB R010a. For basc BP we expermentally use a feed-forward NN wth three layers. he momentum and ntal learnng rate for basc BP are set to 0.85 and 0.05, respectvely. Decdng how many neurons to use n the hdden layer s one of the most mportant characterstcs n an NN. When the number of neurons s too low, the NN cannot model complex data and the resultng may be unacceptable. If too many numbers of neurons are used for an NN, t not only ncreases the tranng tme, but also may reduce the performance of the NN. herefore, we change the number of neurons of the hdden layer from 10 to 100 and then test ther performances. For all of these classfcaton methods we tran the classfers wth 50% of the GPS GDOP measurement data and then use the rest of the data for testng these algorthms. Because of uncertan behavor of several of these algorthms, we smulate all these algorthms 0 tmes, and the averages of the results are represented. Fg. 4 to 9 show comparatve analyses of dfferent tranngs for varous neurons of the NN hdden layer wth 10, 0, 50, 100, 150, and 00 teratons, respectvely. As llustrated n these fgures, the RBP can classfy the GPS GDOP data better than the basc BP for dfferent teratons. Also, the best numbers of neurons of the feed-forward NN, traned usng basc BP and RBP, are 90 and 80 neurons for 00 teratons, respectvely. Fnally, t has been found that the RBP can mprove the classfcaton accuracy from 9.16% [5] to 98.0% accuracy by usng the GPS GDOP measurement data and 00 teratons. When we ncrease the number of teratons to more than 00, the accuracy for the RBP doesn t change sgnfcantly, but the tranng tme ncreases consderably. hus, we propose to use 00 teratons for tranng a feed-forward NN by usng the RBP tranng algorthms. Jwo and La have also used NN for the GPS GDOP classfcaton wth BPNN learnng and PNN [5]. her methods have been used as a benchmark for comparson wth many other methods. However, the advantages of the proposed methods are hgh accuracy and low CPU tme. In ths paper the GPS GDOP classfcaton has been mproved Fgure 4. he performance of usng the RBP for varous number of neurons n the NN hdden layer for 10 teratons. Fgure 5. he performance of usng the RBP for varous number of neurons n the NN hdden layer for 0 teratons. Fgure 6. he performance of usng the RBP for varous number of neurons n the NN hdden layer for 50 teratons.

Fgure 7. he performance of usng the RBP for varous number of neurons n the NN hdden layer for 100 teratons. Fgure 8.

he performance of usng the RBP for varous number of neurons n the NN hdden layer for 00 teratons.

It should be noted that k s very mportant parameter n the KNN method. hus, n ths paper we change k from 1 to 0, and deduce that k=4 s the best selecton for ths applcaton.

5 Fgure 7. he performance of usng the RBP for varous number of neurons n the NN hdden layer for 100 teratons. Fgure 8. he performance of usng the RBP for varous number of neurons n the NN hdden layer for 150 teratons. Fgure 9. he performance of usng the RBP for varous number of neurons n the NN hdden layer for 00 teratons. able llustrates the average correct classfcaton methods usng BPNN, GRNN, and PNN (exstng methods), RBP, nave Bayes classfer, FLD, and KNN wth k=4. It should be noted that k s very mportant parameter n the KNN method. hus, n ths paper we change k from 1 to 0, and deduce that k=4 s the best selecton for ths applcaton. Also, for the algorthm the dstance between two objects s computed by absolute dstance measurement. As can be seen n able, although the nave Bayes classfer and FLD method consderably decreased the tranng tme, ther classfcaton accuraces are reduced slghtly. he KNN can mprove slghtly both the tranng tme and the correct classfcaton rate. As mentoned before, frst the dataset has been randomly dvded nto two tranng and test datasets that each dataset contaned 1000 ponts. o test the proposed methods we N have used the true postve (P) defned as P = t N where N t s the number of true classfed ponts and N shows the number of the ponts n the test dataset (1000). able : Comparson of classfcaton rates and tranng tmes for proposed methods and three well-known exstng classfers Classfcaton Methods BPNN [5] GRNN [5] PNN [5] RBP nave Bayes classfer FLD KNN Correct classfcaton rate 9.16% 97.9% 97.9% 98.0% 9.4% 96.5% 98.61% CPU tme about 1.5 s for 00 teratons about 1.5 s about 1.5 s about 1.5 s for 00 teratons s s 0.656

6 V. CONCLUSIONS here s no doubt that the GPS GDOP plays the man role n the desgn of many navgaton systems. he GPS GDOP concept s employed for an ndvdual user to select measurements whch best work n ther partcular crcumstances. In ths paper, n order to reduce the computatonal burden and tranng tme, four approaches for the GPS GDOP classfcaton, namely, RBP, nave Bayes classfer, FLD, KNN have been proposed. Although the nave Bayes classfer and FLD method sgnfcantly have mproved the tranng tme, ther correct classfcaton rates have only slghtly reduced comparng wth prevous methods. RBP could only ncrease the accuracy of the GPS GDOP classfcaton. KNN could mprove both the tranng tme and the correct classfcaton rate. Not only the tranng tme for KNN s 5.84 tmes less than those of PNN, GRNN and BPNN, ths method can classfy the GPS GDOP data wth an accuracy of 1.% more than those of the best prevous methods. Also, because of the speed of KNN, t can be used n onlne applcatons. REFERENCES [1] M. R. Mosav and H. Azam, Applyng neural network ensembles for clusterng of GPS satelltes, Journal of Geonformatcs, vol. 7, no., pp. 7-14, 011. [] H. Azam, S. Sane and H. Alzadeh, GPS GDOP Classfcaton va Advanced Neural Network ranng Internatonal Conference on Contemporary Issues n Computer and Informaton Scences, Brown Walker Press, USA, pp. 15-0, 01. [] N. Levanon, Lowest GDOP n -D scenaros, IEE Proceedngs Radar, Sonar and Navgaton, vol. 147, no., pp , 000. [4] M. Zhang and J. Zhang, A fast satellte selecton algorthm: beyond four satelltes, IEEE Journal of Selected opcs n Sgnal Processng, vol., no. 5, pp , 009. [5] D. J. Jwo and C. C. La, Neural network-based GPS GDOP approxmaton and classfcaton, Journal of GPS Solutons, vol. 11, no. 1, pp , 007. [6] H. DY. Relatons between dlutons of performance and volume of the tetrahedron formed by four satelltes, IEEE Poston Locaton and Navgaton Symposum, pp , [7] C. Igel and M. Husken, Emprcal evaluaton of the mproved RPROP learnng algorthms, Journal of Neurocomputng, vol. 50, pp , 00. [8] H. Azam, S. Sane and K. Mohammad, Improvng the neural network tranng for face recognton usng adaptve learnng rate, reslent back propagaton and conjugate gradent algorthm, Journal of Computer Applcatons, vol. 4, no., pp. -6, 011. [9] M. I. Schlesnger and V. Hlavac, en lectures on statstcal and structural pattern recognton, Sprnger, Kluwer Academc Publshers, 00. [10] H. Parvn, H. Alzadeh and B. Mnat, A modfcaton on K- nearest neghbor classfer, Global Journal of Computer Scence and echnology, vol. 10, no. 14, pp. 7-41, 010. [11] D. M. Wtten and R. bshran, Penalzed classfcaton usng Fsher s lnear dscrmnant Journal of the Royal Statstcal Socety, vol. 7, no. 5, pp , 011. [1] X. Bo and B. Shao, Satellte selecton algorthm for combned GPS-Galleo navgaton recever, Internatonal Conference on Autonomous Robots and Agents, pp , 009. [1] M. R. Mosav and M. Shroe, Effcent evolutonary algorthms for GPS satelltes classfcaton, he Araban Journal for Scence and Engneerng, Sprnger-Verlog, 01 (onlne publshed) [14] P. A. Mastorocostas, Reslent back propagaton learnng algorthm for recurrent fuzzy neural networks, Journal of Electroncs Letters, vol. 40, no. 1, 004. [15] R. O. Duda, P. E. Hart and D. G. Stork, Pattern classfcaton, John Wley & Sons, nd edton, 001. [16] S. heodords and K. Koutroumbas, Pattern Recognton Academc Press, 006. [17] S. H. Doong, A closed-form formula for GPS GDOP computaton, Journal of GPS Solutons, vol. 1, no., pp , 009 [18] H. Azam, M. R. Mosav and S. Sane, Classfcaton of GPS satelltes usng mproved back propagaton tranng algorthms, Wreless Personal Communcatons, Sprnger- Verlog, DOI /s , 01.

ANNUAL OF NAVIGATION 11/2006

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