Outlier Detection in GPS Networks with Fuzzy Logic and Conventional Methods

Transcription

1 Outler Detecton n GPS Networks wth Fuzzy Logc and Conventonal ethods Ertan GÖALP and Yüksel BOZ, urkey ey Words: Outler, Conventonal ethod, Fuzzy Logc, Statstcal est, GPS SUARY It s assumed that the geodetc observatons always have random errors and these errors are normally dstrbuted. he measurements have random errors that devate from the normal dstrbuton are called as outlers. Outler nvestgaton s one of the frst nterest areas of the statstcans and dfferent approaches have been developed for ths purpose so far. In the lterature, there sn t a method as the best over all. In the conventonal methods lke Data Snoopng (DS), au, and t tests, outlers are determned teratvely by the statstcal test theory and removed from the observaton set. In the fuzzy logc approach, fuzzy set relatons and the statstcal tests are used together to detect the outlers. Fuzzy membershp relatons are used among the resduals and observaton errors. In ths study, commonly used conventonal methods (statstcal tests) and fuzzy logc method have been appled to several GPS networks wth dfferent characterstcs. It has been amed to fnd whch algorthm s convenent among the methods used n ths study for outler detecton n GPS observatons. he baselne components X, Y, and Z of the GPS baselnes have been taken as the measurements. he conventonal methods have been appled to the networks at dfferent sgnfcance levels. As the sgnfcance level gets greater, the senstvty of the tests for outler ncrease and more outlers appear. It s approprate to determne the sgnfcance level regardng the number of the observatons n the network. he capabltes of the fuzzy logc approach to be an alternatve method have been examned. Also, t has been seen that the results of ths method are completely dependent on the statstcal tests. Unlke conventonal methods, no measurement s removed from the observaton set, thus the shape of the network sn t defected. 1/1 FIG Workng Week 5 and GSDI-8 Caro, Egypt Aprl 16-1, 5

2 Outler Detecton n GPS Networks wth Fuzzy Logc and Conventonal ethods Ertan GÖALP and Yüksel BOZ, urkey 1. INRODUCION Outler s an observaton, whch has a random error n a dfferent dstrbuton apart from the observaton set. he random errors are small and nconspcuous. Snce they do not have large magntudes, they can be determned by statstcal tests such as Data Snoopng (DS), au, and t tests. hese methods are conventonal approaches to detect outlers and deal wth the resduals. In Least Squares (LS) adjustment, an observaton that fals the statstcal test s removed from the observaton set. In the adjustment that uses GPS-derved baselne components as observatons, the baselne component dentfed as outler and the other two baselne components are taken away from the observaton set as well after each teraton. In order to consder more accurately the observaton that has a test statstc close to the crtcal value of the statstcal test, a fuzzy logc based method can be used wth statstcal tests. In the fuzzy logc approach, the fnal decson about outlers s gven by means of the observaton errors. Defnton of the membershp functons plays an mportant role n outler detecton. he outlers are obtaned altogether at the end of the method; no observaton s removed from the observaton set.. CONVENIONAL EHODS.1 Data Snoopng DS was suggested by Baarda and s applcable when the theoretcal varance ( ) known, the a pror varance ( ) σ of the observaton of unt weght s known. In ths study, snce the theoretcal varance s not s obtaned from loop closures usng Ferrero equaton s used nstead of t. In the DS method, t s assumed that only one outler s present n the observaton set. In addton usng ths method teratvely, more than one outler can be detected and ther locatons can be estmated (Berberan 1995). Ferrero equaton s as follows: s = w w 9 n t (1) /1 FIG Workng Week 5 and GSDI-8 Caro, Egypt Aprl 16-1, 5

3 where w s the closure of a trangle related to each baselne component, 9 s the number of the observatons belongs to a trangle, n t s the number of the trangles of GPS network (Gökalp and Boz 4). DS s realzed usng normalzed resduals. herefore, the test statstcs are compared wth the crtcal values from normal dstrbuton table. he test statstc s = s ( Pv) ( PQvvP) () where v s the vector of resduals, P s the weght matrx of the observatons, cofactor matrx of the resduals. Q vv s the he crtcal value s q = N F 1 α / = 1,,1 α = χ 1,, 1 α (3) Here α s the sgnfcance level, N represents the normal dstrbuton, F represents the Fscher table, χ represents the Ch-Square table. he sgnfcance level for a sngle observaton α s computed from 1/n ( 1 α) α / n α = 1 (4) where α s the total sgnfcance level and usually chosen as 5%, n s the number of observatons (Bacs et al. 199).. au est If the a pror varance s not known or a value cannot be assgned to t before adjustment, the a posteror varance ( m ) calculated at the end of adjustment s used for outler detecton. he resduals normalzed usng the a posteror varance are not normally dstrbuted. hey are n au ( τ ) dstrbuton (Schwarz and ok 1993). m v Pv v Pv = = f n u + d 3/1 FIG Workng Week 5 and GSDI-8 Caro, Egypt Aprl 16-1, 5

4 Here f s degree of freedom, n s the number of the observatons, u s the number of unknown parameters, and d s the datum defect. he test statstc s (5) = m ( Pv) ( PQvvP) (6) he crtcal value of τ table can be obtaned as follows (Heck 1981): q = τf,1 α/ = f t f 1+ f 1,1 α / t f 1,1 α / (7) where t represents the student (t) table, and τ represents the au table..3 he t est If an observaton ( l ) ncludes an error ( ), outler detecton usng the a posteror varance obtaned from the nvald adjustment model s not approprate. In ths stuaton t s a more accurate approach to compute the m value from the resduals that are free from the model errors. m 1 = f m f 1 v ( ) Qvv (8) Here m s the a posteror varance calculated from the resduals free from the model errors. he test statstcs of the observatons as follows: he crtcal value of t table s = m ( Pv) ( PQvvP) (9) 4/1 FIG Workng Week 5 and GSDI-8 Caro, Egypt Aprl 16-1, 5

5 .4 Fuzzy Logc ethod q = t f 1,1 α / (1) he conventonal methods are nsuffcent n case of exstng more than one outler n the observaton set. As seen from relaton between resduals and errors v = QvvP (11) the magntudes of resduals depend on observaton errors. In equaton (11), the multplcaton Qvv P equals to redundancy matrx R. he expanded form of R and equaton (11) s as follows: v1 r 11 v r 1 = vn r n1 r1 r r n r r r 1n n nn 1 n (1) A normal observaton can be seen as an outler due to other observaton errors. In order to overcome ths problem, the fnal decson about outlyng observatons s made usng the observaton errors. Fuzzy membershp values are assgned to them and judgment about an observaton to be an outler or not are gven accordng to the magntudes of these membershp values. he procedures to ths stage are presented as follows (Sun 1994): At ntal stage, the resduals are separated nto two man classes va a statstcal test: abnormal resduals C{v }, whch fal the test and normal resduals D{v }, whch pass the test. hen, membershp functon values of the resduals are defned n one of these classes. q 1. µ ( ) = > q C v e 1. + q (13) Here, e s the standardzaton component denotng the sgnfcant devaton of the test statstc from the crtcal value (onak and Dlaver 1998). After determnng the membershp values of resduals n set C, correspondng values n set D are obtaned utlzng from the complementaton property of fuzzy sets. ( v ) = 1 µ ( v ) µ D C (14) 5/1 FIG Workng Week 5 and GSDI-8 Caro, Egypt Aprl 16-1, 5

6 In order to get the membershp values of the observaton errors, the redundancy matrx s used n addton to the membershp values of resduals. Each row element n R s dvded to the greatest value n the correspondng row. r j rj = max j j ( r ) (, j = 1,,, n) (15) hs new generaton matrx s called relatve redundancy matrx. v v v 1 n = r 11 r 1 r n1 + 1 r r1n 1+ r + + r n 1+ rn + + rnn n n n (16) he rows of the relatve redundancy matrx ndcate the relatve contrbutons of all observaton errors to an ndvdual resdual and the columns ndcate the relatve contrbuton of an ndvdual error to all resduals. It s assumed that observatons, the most lkely affected by gross errors, are that have the greatest contrbuton to the most lkely abnormal resduals and also have the least contrbuton to the most lkely normal resduals. Let G be the set of the observaton errors have greatest effect on the most lkely abnormal resduals and L be the set of the observaton errors have least effect on the most lkely normal resduals. he ntersecton set H ncludes the observaton errors so called gross errors. he related observaton to the error n set H s treated as outler. But some procedures are necessary to make the fnal decson. In order to obtan the membershp values of the observaton errors n the set G, the maxmum relatve contrbuton of the th observaton error to the resduals that have membershp values equal or greater than.5 n the set C s looked for. hen, ths relatve value s multpled by the membershp value of the correspondng resdual as follows: ( ) r j = max r k k = u,v,..., w ( ) = r µ ( v ) µ G j C j (17) (18) 6/1 FIG Workng Week 5 and GSDI-8 Caro, Egypt Aprl 16-1, 5

7 In order to obtan the membershp values of the observaton errors n set L, the maxmum relatve contrbuton of the th observaton error to the resduals that have membershp values equal or greater than.5 n the set D s looked for. hen, the complementary value of ths relatve value s multpled by the membershp value of the correspondng resdual as follows: ( ) r m = max r k k= x,y,...,z (19) ( ) ( = 1. r ) µ ( v ) µ L m D m () he membershp values of the observaton errors n the ntersecton set H are acqured by means of ntersecton property of fuzzy sets. µh( )=mn (µg( ), µl( )) ( = 1,,..., n) (1) Now there s a queston to be answered that whch observaton errors wll be consdered as gross errors? A crtcal value must be defned that the membershp values wll be compared. he crtcal value ( Cµ H ) can be calculated takng nto account the relatve effect values used durng the calculaton of the membershp values of observaton errors and correspondng membershp values n the set H (onak and Dlaver 1998). p p = rj p = 1 rm f f µ H µ H ( ) µ G ( ) ( ) µ ( ) L () Cµ H p ( ) = µ H p (3) he observaton errors, whch have membershp values greater than the crtcal value, are assumed as gross errors. But, ths assumpton must be verfed. At ths stage, the magntudes of the errors exceeded the crtcal value are estmated and the sgnfcance of them are tested. he followng procedure s proposed by Sun (1994): 7/1 FIG Workng Week 5 and GSDI-8 Caro, Egypt Aprl 16-1, 5

8 n m = (4) Here, n s the number of observatons, m s the number of observaton errors that exceeded the crtcal value, and s the locaton matrx of gross errors. For each gross error, 1 s wrtten at correspondng row. he weght matrx and the magntudes of gross errors are calculated by means of ths locaton matrx. Pss = P s = Pss 1 ( PA) A P PA A 1 P v (5) (6) In equatons (5) and (6), Pss m m s the weght matrx of gross errors, P n n s the weght matrx of observatons, A n u s the desgn matrx, v n 1 s the vector of resduals, and s m 1 s the vector of gross errors. As stated above, the magntudes of gross errors ( s ) are tested usng a statstcal test and the decson about ther sgnfcance s made. 3. APPLICAION AND PROGRAING he conventonal methods and fuzzy logc approach have been appled to two GPS networks named as Ordu-1 and Ordu- that were establshed n Ordu, urkey. he propertes of networks are presented below. In these networks, free adjustment has been performed. In order to adjust the measurements and detect outlers, two programs, one for conventonal methods and one for the fuzzy logc method, have been wrtten n ALAB techncal computng language. In applcaton of conventonal methods, the sgnfcance level of statstcal tests has been obtaned as.3 from equaton (4) for each network. Snce, such a sgnfcance level may cause nsenstvty for outlers, t has been consdered as.1 n statstcal tests. eanwhle, the sgnfcance level has also been chosen as.1 and t has been seen that when the sgnfcance level gets greater, more outler appears. So, the results at sgnfcance level.1 are presented n ths study. 8/1 FIG Workng Week 5 and GSDI-8 Caro, Egypt Aprl 16-1, 5

9 4. GPS NEWORS Ordu-1 GPS network conssts of 15 ponts, 5 baselnes, and 84 trangles (Fgure 1). he a pror standard devaton has been calculated as mm from equaton (1). Ordu- GPS network conssts of 14 ponts, 43 baselnes, and 5 trangles (Fgure ). he a pror standard devaton of ths network s mm Fgure 1 Ordu-1 GPS Network 8 Fgure : Ordu- GPS Network 5. RESULS OF PROGRAS able 1: Results of conventonal methods for Ordu-1 GPS network İteraton est Statstc Crtcal Value Number ( ) (q) Outler au Y Y passed the test - DS Y passed the test - t Y passed the test - Statstcal est Observaton Number 9/1 FIG Workng Week 5 and GSDI-8 Caro, Egypt Aprl 16-1, 5

10 As seen from able 1, the results of three statstcal tests are dentcal except for the observaton Y 8-4 n au test. But, t must be pad attenton that the test statstc of ths observaton s very close to the crtcal value. In applcaton of fuzzy logc method, at the begnnng (frst test) and at the end (last test) of the method, the same statstcal test has been used at sgnfcance level.1. hree dfferent values have been assgned to the standardzaton component of membershp functon. If the membershp value of an observaton error n ntersecton set H t equals to the crtcal value, t s treated as gross error. he condtons mentoned here are appled to both GPS networks. he 68 th observaton seems as outler after every applcaton of fuzzy logc method n Ordu-1 GPS network (able ). able : Results of fuzzy logc method for Ordu-1 GPS network Frst Standardzaton Observaton Crtcal Last est Crtcal Outler µh( ) est Component Number Value est Statstc Value au.1 Y au au.5 Y au au.1 Y au DS.1 Y DS DS.5 Y DS DS.1 Y DS he results related to Ordu- GPS network are as follows: able 3: Results of conventonal methods for Ordu- GPS network Statstcal est İteraton est Statstc Crtcal Value Number ( ) (q) Outler au Z X Y Z passed the test - DS Y passed the test - t Z X Y Z passed the test - Observaton Number In able 3, the observatons close to the crtcal value are dentfed as outlers. Effects of other observaton errors may cause such a result. Some of them are most lkely normal observatons. 1/1 FIG Workng Week 5 and GSDI-8 Caro, Egypt Aprl 16-1, 5

11 able 4: Results of fuzzy logc method for Ordu- GPS network Frst Standardzaton Observaton Crtcal Last est Crtcal Outler µh( ) est Component Number Value est Statstc Value au.1 Z au au.5 Z au au.1 Z au DS.1 Z DS DS.5 Z DS DS.1 Z DS In fuzzy logc method, the 39 th observaton Z 8-4 seems as outler. hs observaton also has been conspcuous at the frst teraton n conventonal methods. 6. CONCLUSIONS AND RECOENDAIONS In GPS networks wth redundant observatons, choosng the sgnfcance level as.1 s suffcent to realze outler detecton procedure. Workng wth great sgnfcance levels produces unrelable results. In conventonal methods, a normal observaton may seem as outler at the end of teratons and may be removed from observaton set. hus, the shape of the network s defected. On the other hand, more relable results are obtaned wth fuzzy logc method. In contrast to the conventonal methods, the observatons that have test statstcs close to the crtcal value are examned more accurately. It has not any teraton and removal about observatons. here s no dsadvantage of shape defecton of network. he outlers are determned altogether at the end of the method. ACNOWLEDGEEN We would lke to thank aradenz echncal Unversty Research Fund for ther support to our study. REFERENCES Berberan, A., 1995, ultple Outler Detecton. A Real Case Study, Survey Revew, 33, Bacs, Z. F., rakwsky, E. J., and Lapucha, D., 199, Relablty Analyss of Phase Observatons n GPS Baselne Estmaton, Journal of Surveyng Engneerng, 116, 4, 4-4. Gökalp, E. and Boz, Y., 4, Uyuşumuz Ölçülern Belrlenmesnde Bulanık antık ve Geleneksel Yaklaşımların İrdelenmes, UJ 4 Çalıştayı ühendslk Ölçmelernde Jeodezk Ağlar Çalıştayı, October, Zonguldak. Heck, B., 1981, Der Enfluss enzelner Beobachtungen auf das Ergebns ener Ausglechung und de Suche nach Ausressern n den beobachtungen, AVN, 88, onak, H. and Dlaver, A., 1998, Jeodezk Ağlarda Uyuşumsuz Ölçülern Yerelleştrlmesnde ullanılan Yöntemlern Davranışları-II Fuzzy Logc (Bulanık antık) Yaklaşımı, Harta ve adastro ühendslğ, 85, /1 FIG Workng Week 5 and GSDI-8 Caro, Egypt Aprl 16-1, 5

12 Schwarz, C. R. and ok, 1993, J. J., Blunder Detecton and Data Snoopng n LS and Robust Adjustments, Journal of Surveyng Engneerng, 119, 4, Sun, W., 1994, A New ethod for Localsaton of Gross Errors, Survey Revew, 3, 5, BIOGRAPHICAL NOES Ertan Gökalp s an assocate professor at aradenz echncal Unversty (U), urkey. He graduated from the Department of Geodesy and Photogrammetry Engneerng at U n He got hs.eng. degree from the Department of Surveyng Engneerng at Unversty of New Brunswck (UNB), Fredercton, Canada n He got hs Ph.D. degree from the Department of Geodesy and Photogrammetry Engneerng at U n He s currently workng at the Department of Geodesy and Photogrammetry Engneerng at U. Hs nterest areas are GPS (Global Postonng System), Engneerng Surveyng, and Satellte Geodesy. He s a member of Chamber of Surveyng Engneers. Yüksel Boz s a.sc. student at aradenz echncal Unversty (U), urkey. He graduated from the Department of Geodesy and Photogrammetry Engneerng at aradenz echncal Unversty (U) n. Hs nterest areas are Satellte Geodesy and GPS. He s a member of Chamber of Surveyng Engneers. CONACS Ertan Gökalp aradenz echncal Unversty Department of Geodesy and Photogrammetry Engneerng 618 rabzon UREY. el Fax Emal: ertan@ktu.edu.tr Yüksel Boz aradenz echncal Unversty Department of Geodesy and Photogrammetry Engneerng 618 rabzon UREY. el Fax Emal: yboz@ktu.edu.tr 1/1 FIG Workng Week 5 and GSDI-8 Caro, Egypt Aprl 16-1, 5