Outlier Detection in 3D Coordinate Transformation with Fuzzy Logic

Transcription

1 Acta Montanstca Slovaca Ročník 7 (22), číslo, -8 Outle Detecton n 3D Coodnate ansfomaton wth Fuzzy Logc Yasemn Ssman, Aslan Dlave 2 and Sebahattn Bektas Coodnate measuements nevtably contan outles that affect the esults of tansfomaton Conventonal methods based on the leasts squaed method detemne the outles by mnmzng the selected objectve functon Fuzzy Logc can be used to analyze the outles In ths study, seveal outle detecton methods ae descbed and appled to a eal case consstng of a tangulaton netwok Results show that fo outle detecton methods ae not as effcent as Fuzzy povdes a non-teatve soluton n contast to conventonal methods Key wods: Coodnate tansfomaton, outles detecton, conventonal methods, fuzzy logc Intoducton he use of satellte technologes n poston navgatng, the establshment of geodetc netwoks and detemnaton of pont coodnates has bought a new appoach n whch both the hozontal and the vetcal coodnates ae nvestgated togethe It s known that the 3D coodnates of the ponts n vaous global datum can be detemned faste and moe accuately usng the Global Postonng System (GPS) method Fo example n the Intenatonal eestal Refeence Fame (IRF96) datum the poston of ponts s detemned by the GPS method, (URL) he cuent applcaton of new technologes n efeence fame defnton have made t necessay to fom 3D pont coodnates fom defned dffeent datum (coodnate systems) and to use the 3D datum (coodnate) tansfomaton fo effcency he known common coodnate ponts n both coodnate systems ae needed to calculate the tansfomaton paametes n coodnate tansfomaton Fst the tansfomaton paametes ae calculated by takng the coodnates of the common ponts as a measuement value and then the coodnates of othe ponts ae tansfomed usng these tansfomaton paametes So, the selecton of common pont selecton s vey mpotant and dectly affects the esults, (Kutoglu and Ayan 26) It s vtually mpossble to avod goss, systematc and andom eos wthn a data set Whle goss and systematc eos can be detected fom the data set wthout calculaton beng equed, outles that ae vey close to andom eos n tems of sze can only be detemned though the applcaton of outle tests he lteatue epots dffeent appoaches have been used to detemne the outle measuements he man appoaches to detemne outles can be gouped as fallows; Conventonal methods, Robust Estmaton and Atfcal Intellgence technques Conventonal outle detecton pocedues, ntoduced by Baada (968), use teatve appoaches to fnd outles n the data set, (Koch 999) Lately, the Robust Estmaton method was poposed by Hube (98), and Hampel at al (986) as an altenatve to conventonal methods A new appoach s usng Atfcal Intellgence technques fo outle detecton fo example Neual Netwoks and Fuzzy Logc technque he latte was suggested by Zadeh (965), can also be used to geodetc netwoks fo outle detecton, (Bebean 995 Sun 994, Alosmanoglu and Akylmaz 22) In the coodnate tansfomaton, the use of pont coodnates as a measuement and n some tansfomaton models, the acceptance of these measues as uneng ae esult n a negatve stuaton In ode to obtan the sgnfcant esults, both analyses of the models that measuements ae eoneous and outle detecton has to be caed out on the esults he measuements ae taken as eoneous n the condtonal adjustment wth unknown models and the total least squaes models In ths study, fstly the 3D coodnate tansfomaton models, one of the man felds of mathematcs and geodesy, ae explaned he soluton s calculated by condtonal adjustment wth unknown models Afte the outle detecton by Conventonal Methods and Fuzzy Logc ae theoetcally explaned, the solutons can be obtaned fo these methods usng a eal case consstng of a tangulaton netwok he advantages and dsadvantages of the outle detecton methods ae assessed 2 Methods 2 Coodnate (Datum) ansfomaton Models When a geodetc netwok s set up, the datum has to be detemned n ode to oent the netwok space Geodetc datum defnes both the efeence ellpsod and the coodnate system heefoe, Yasemn Ssman, Sebahattn Bektas, Ondokuz Mays Unvesty Samsun,ukey, yssman@omuedut, sbektas@omuedut 2 Aslan Dlave, Kaadenz ecncal Unvesty,abzon, ukey, dlave@ktuedut

2 Yasemn Ssman, Aslan Dlave and Sebahattn Bektas: Outle Detecton n 3D Coodnate ansfomaton wth Fuzzy Logc the tansfomatons between two datum and between two coodnate systems ae n fact the same opeaton Coodnate tansfomaton s a common pactce n geodetc studes Fo 3D coodnate tansfomaton, many models have been developed ncludng Busa-Wolf, Molodensky-Badekas and Ves, (hompson, 976) hese methods defne 3D datum wth 7 paametes (3 tanslatons (X,Y,Z ), 3 otates (ε x,ε y,ε z ) and scale paamete (κ)) Busa-Wolf and Molodensky-Badekas models ae the most popula hese models defne and solve the elatonshp between two othogonal coodnate systems wth tansfomaton paametes smlaty tansfomaton as gven below X Y Z X Y Z + ( + κ) ε ε y z ε z ε x ε ε x U V W y () Hee, (U,V,W ) and (X,Y,Z ) show the postons n the fst and seconday coodnate systems, espectvely In the Molodensky-Badekas model, the shfted and educed values of the pont coodnates accodng to a local statng pont s used Equaton () s lneazed accodng to the datum paametes used as unknown and shfted common pont coodnates paametezed as below: Ax + Bv + w (2) Equaton (2) s the man equaton of condtonal adjustment wth unknown models he measuements ae eoneous n ths adjustment method In Equaton (2); A and B ae the desgn matces, x s an estmate of the tansfomaton paametes, v s esdual of measuements and w s the msclosue vecto In the soluton of Equaton (2) by Least Squaed method (LSM), Q ll, the nvese weghts matx of the coodnates used as measuements, Q xx, the cofacto matx of datum tansfomaton paametes, and Q vv, the cofacto matx of the esduals, can be calculated (Vancek, 972) N BQ v Q ll ll B B ; x k; Q vv Q -( A N A) A N w; Qxx ( A N A) ll B N N A( A N A) A N BQll he valdty of the esults of the soluton poduced by LSM depends on the completeness and accuacy of the mathematcal model that has been bult In the test of the model hypothess, the equalty of a po 2 vaance σ and a posteo vaance s 2 should be statstcally explcated In the applcaton, both null H and altenatve H S hypotheses ae fomed and test sze s computed he valdty of the hypothess whch has been statstcally explcated In the case whee H s vald, t s assumed that the dffeences ae deved fom andom eos, and the model may be accepted accodng to a cetan pobablty level Othewse, H S wll be vald In that case, the functonal model s extended and tested Aftewads, fo the test of the stochastc model, the outle detecton pocess s stated, (hompson, 976) 22 Outle Detecton Methods It s unavodable that thee ae goss o outle measuements n the data set hese goss eos can be detemned and elmnated whle the lneazaton equatons of the adjustment model ae fomed Accodng to eo theoy, othe eos such as outles that ae vey close to andom measuement eos can be detemned only by explcatng the solutons found by the LSM Not all the outles ae due to bad data havng goss eos In some cases, these measuements may contan mpotant nfomaton fo the data set If the model s bult well fo example, consdeng the dstbuton of the data, the outle measuements may be emoved fom the goup wthout an addtonal evaluaton Futhemoe, thee may be a falue n the shape of the mathematcal model because of the excessve amount of outle measuements, (Hampel, 986) 22 Conventonal Outle Detecton Methods Conventonal Outle Detecton Methods ae the Data-Snoopng Method (W-est), au est and t- est In these methods the hypothess test s pefomed to detemne the l eo n any l measuement n a l measuement vecto hen, the test value s calculated fo each measuement and the test value s compaed wth the ctcal value of the dstbuton table If the test value s lage than ctcal value, then (3) 2

3 Acta Montanstca Slovaca Ročník 7 (22), číslo, -8 the lagest value s consdeed to be the outle and t should be emoved fom the data set he teaton contnues untl thee s no outle left n the data set he test value, ctcal value and dstbuton nfomaton ae gven n able ; α s the sgnfcance level, f s the feedom degee, s 2 s the posteo vaance elmnated fom the model eos, and N, τ, t epesent espectvely nomal, tau and student dstbuton, (Koch 999) ab Conventonal outle detecton methods Method Data-Snoopng au-est t-est est Value W σ v Q v v s est Dstbuton N(, ) τ f, ( -α / 2) t f-, ( -α/ 2) Ctcal Value N ( / 2) τ f, ( -α 2 ) / t f-, ( -α/ 2) v Q v v t s v Q v v 223 Fuzzy Logc and Fuzzy Set heoy Recently, many appoaches have been tested on decson makng theoes Some of the Atfcal Intellgence technques that ae used n outle analyss ae, Neual Netwoks, Suppot Vecto Machne, Fuzzy Logc, (Caten at al 28) Fuzzy logc s a logcal model povdng a geneal dea about the decson pocess n the analyss of the data set he fuzzy logc suggested by Zadeh (965) s, essentally, an appoach that allows tanston values to make a defnton between the conventonal values, such as ght/wong, yes/no, hgh/low he man pupose of the method s to bng a cetanty to assgnng a membeshp degee to the concepts, whch ae had to expess o have dffcult meanng, (Sh at al 999) Fuzzy set theoy was suggested as an altenatve to classcal set theoy In a fuzzy set, fo any component to have a membeshp degee between and, the lmt of the classcal set s extended and the degees of a fuzzy event to occu o to exst ae measued In classcal set theoy, an object s ethe a component of a set o not, wheeas n fuzzy logc an object can be gven membeshps n many sets A fuzzy system conssts of thee man pats, whch ae fuzzfcaton, ule base and defuzzfcaton Fstly, fuzzfcaton can be defned as a tansfe between a defnte system and a fuzzy system and t descbes a popety of an object n a cetan fuzzy set he objects can belong to low, mddle, hgh popety classes wth membeshp functons and each object s assgned a membeshp degee between and Secondly, the ule base combnes the membeshp functons fom the fuzzfcato wth the ule handlng data such as f, and, although, f not whch s based on the database and stoed thee hdly, n the defuzzfcaton unt, the ule esults that ae obtaned fom the ule handlng unt ae evaluated n the defuzzfcato and tuned nto defnte esults, (Alosmanoglu and Akylmaz, 22) he membeshp functons defne the degee to whch an nput belongs to a fuzzy set hese membeshp functons ae chosen empcally and optmzed usng a sample nput/output data he If-then ules defne a connectng the antecedent to the consequent (e nput to output) hese ules ae gven weghts based on the ctcalty, (Syed and Cannon, 24) Wth ths appoach, measuements can be classfed accodng to the membeshp degees by adequate membeshp functons In geneal, the membeshp degees between and ae eal numbes, shows that thee s no membeshp and shows full membeshp he dffeent foms of membeshp functons can be chosen whle fuzzy sets ae fomed he most common membeshp functons nclude the tangle, tapezod, Gauss cuve and sgmod As the membeshp functons epesent the fuzzy set, the selecton of the shape and fom dectly affects the decson pocess In classcal set theoy, the values of measuement ae gouped nto two pats In fuzzy set theoy, moe goups ae fomed up he classfcaton of goup membes s done effectvely by usng the membeshp functon Many geodetc poblems, defomaton analyss, paamete estmaton, geod detemnaton, data analyss n a geogaphcal nfomaton system, outle detecton of levelng netwoks and GPS netwoks, can be solved wth the help of fuzzy model, (Hampel at al, 986; Konak at al, 25; Kuttee, 22) he most commonly used methods n fuzzy set theoy ae Mamdan and akag Sugeno methods, (Xue-Gong, 2) hese methods defne a membeshp functon, then measuements ae gouped accodng to the membeshp degee Consequently, the decson can be made as to whethe a measuement s an outle o not, thus an outle data set can be fomed Also, an outle data set can be fomed usng the membeshp values obtaned fom the fuzzy set theoy n the LSM soluton, (Konak at al, 25) 3

4 Yasemn Ssman, Aslan Dlave and Sebahattn Bektas: Outle Detecton n 3D Coodnate ansfomaton wth Fuzzy Logc 223 Outle Detecton By Fuzzy Logc Accodng to the LSM, when the gven functonal model ncludes the l measuement eo vecto, the mathematcal elatonshp between the esduals and the measuement eos s establshed wth R edundancy matx R can be wtten followng equaton v v2 vn n n n 2n nn 2 n (4) Fom Equaton (4), t can be seen that each measuement s affected by all measuement eos dependng on the ato of the sze of the component coespondng to ts edundancy matx Both n coelaton and uncoelaton adjustment, the ank of the R s equal to the excess measuement numbe; hence, ths matx s not postvely defned In othe wods, eos cannot be calculated wth a sngle meanng In ths case, nstead of measuement eos l, the sze of esduals s dealt wth n both conventonal outle tests based on statstc decson-makng pocess Howeve, n fuzzy logc, membeshp elatonshps detemnng the effects of the eos on esduals may be fomed usng the edundancy matx Accodng to Equaton (4), ow elements of edundancy matx show the total effect on one esdual of eos, whch can possbly occu n all measuement eos, the column elements of edundancy matx show the total of effect on the othe esduals of each measuement eo hs povdes a possblty of examnng the eos, deved fom the dstbuton of eos and also to absob them he steps of the fuzzy set theoy ae gven below: Fuzzfcaton: Afte the ntal adjustment, takng the elatonshp gven n Equaton (4) nto consdeaton, the test values fo each elaton ae calculated Afte compang the statstcal lmts wth the test value of each esdual, the esduals n the fuzzy set ae dvded nto two measuement goups: one, the nomal esduals (the test values unde the statstcal lmts) N{V }, and the othe, the abnomal esduals (the test values above the statstcal lmts) M{V } Afte the hypothess tests, the membeshp functon shows the esduals of the membes he components of the N{V } subgoup consstng of the test values below the statstcal lmt ae gven zeo membeshp value he components of the M{V } subgoup consstng of the test values above the statstcal lmts ae gven a membeshp value between and, dependng on the statstcal lmt devatons, degee of feedom and the eos pobablty of the test q ( ) V > q cj + d M 2 (5) If each esdual has test value of measuements, q ctcal value, c j elaton between the test value and the ctcal value, and d the scale facto, membeshp functon s calculated by the above equaton (Kutoglu 26) Usng the Equaton (5), esduals, whch ae not affected fom the outles, can easly be calculated wth fuzzy set theoy V ( ) N V M ( ) (6) Rule Base: o detemne the elatons between measuement eos and fuzzy elatonshps, j, the components of the edundancy matx nomalzed between and, ae used Measuement eos can be sepaated nto two goups, smla to the esduals Fo example, the A subgoup may consst of the measuement eos, whch have maxmum affect on the abnomal esduals and the B subgoup, may consst of the measuement eos whch have mnmum effect on the nomal esduals If R nomalzed edundancy matx s used to defne the membeshp values of A and B sets and these membeshp values ae taken as A ( ) and B ( ), n the calculaton of the membeshp values he membeshp functon A ( ) 4

5 Acta Montanstca Slovaca Ročník 7 (22), číslo, -8 whch s the maxmum elatve effect of th measuement eos of esduals havng a M ( v ) 5 membeshp value n the M ( v ) fuzzy functon set, and then; the membeshp functon B ( ) whch s the maxmum elatve effect of th measuement eos of esduals havng a N ( v ) 5 membeshp value, s detemned A V ( ) j M ( ) ; max ( ) B (7) j k ( ) ( ) m N ( v ) ; max( ) (8) m k he measuements that ae mmensely affected by the goss eos have maxmum affect on the abnomal esduals wheeas these have mnmum affect on nomal esduals he mnmum value of the membeshp functon gven n Equaton (7) and Equaton (8) shows the degee of l obsevatons, whee the mnmum value s detemned to be out of the lmts Accodng to fuzzy set theoy, the ntesecton of fuzzy sets A and B ae composed of an H set [ ] l ( ) mn H A( ), l B( ) (9) Defuzzfcaton: o detemne a sgnfcant lmt ctcal value, the equatons gven below calculated by weghted aveage fuzzfcaton method can be used C H P H ( l ) ; P P H H ( ) A( ) ( ) B( ) P j P Hee P s weght matx of the measuements he obtaned membeshp functon H ( l ) s compaed wth the C H value and f H ( l ) >C H, then t s decded whethe the values ae out of lmt, whch means that they ae n a dffeent set, (Alosmanoglu and Akylmaz 22) 3 Result and dscusson Fo ths study, a geodetc netwok (Fg ) fomed by he Dectoate of Land Regsty and Cadaste, was chosen Both the Euopean Datum-5 and IRF96 (URL) coodnate values of the 45 ponts of the netwok ae known m () Fg he Netwok Selected fo the Study he netwok ponts wee poduced accodng to coodnates and vectos at a 998 efeence epoch of uksh Natonal GPS Netwok ponts All the measuements wee pefomed n June 25 usng GPS 5

6 Yasemn Ssman, Aslan Dlave and Sebahattn Bektas: Outle Detecton n 3D Coodnate ansfomaton wth Fuzzy Logc eceves Data educton and post-pocessng wee caed out usng Leca LGO 2 softwae Adjustment computaton was pefomed acceptng the coodnates of uksh Natonal GPS Netwok ponts as stable n the measuement epoch hen, coodnates of ponts (X, Y, Z) wee calculated usng coodnated of ponts (φ, λ, h) 3 Coodnate ansfomaton Methods Between the Euopean Datum-5 and IRF96 datum, the coodnate tansfomaton paametes wee estmated usng Molodensky-Badekas models, and the sgnfcance of the calculated paametes was examned (able 2) In the esult of these tests, t was found (Z ) paamete was not statstcally sgnfcant Afte the examnaton, t was concluded that thee mght be an outle n the data set and an outle detecton methods needs to be appled to the data set ab 2 Coodnate tansfomaton paametes X Y Z ε x ε y ε z κ Paametes -42 m 228 m -568 m ppm Sgnfcance est Vald Vald Invald Vald Vald Vald Vald 32 Outle Detecton Fstly, Conventonal methods (Data-Snoopng, au and t-test) wee appled to detect outles fom the esults obtaned afte the coodnate tansfomaton was caed on the applcaton netwok he outles wee found and the esults ae shown n able 3 Data-Snoopng and t-test methods found the followng outles 5, 5,, 43, 48, 3, 32 In tau test, 53 pont found as outle addtonal these outles he teatve soluton was estmated untl a consstent measuement goup was obtaned hen the Fuzzy Logc method was also appled to outle detecton usng the esults obtaned afte the coodnate tansfomaton he outles found by the fuzzy logc method ae shown n able 4 Fstly, Fuzzy Logc method was detemned the ponts of membeshp value owned n outle goup and the test value was calculated hen, a compason was made between the ponts of membeshp value and the test value Hee, eght ponts wee detemned to have membeshp values n outle goups, thee ponts wee taken out of the outles goup due to havng smalle membeshp values than the test value he eght ponts yelded equvalent esult to conventonal methods As a esult, only 5 ponts wee consdeed to be outle n Fuzzy Logc methods ab 3 Outles detemned by conventonal methods Data-Snoopng au est t-est Iteaton PN est Value PN est Value PN est Value Consstent Consstent Consstent ab 4 Outle measuement detemned by Fuzzy Logc PN Membeshp Value est Value Result Outle Outle Outle Outle Outle Consstent Consstent Consstent 6

7 Acta Montanstca Slovaca Ročník 7 (22), číslo, -8 4 Concluson he coodnate tansfomaton models Molodensky-Badekas models have been dscussed theoetcally he method was statstcally tested usng a eal case consstng of a tangulaton netwok data set 3D coodnaton tansfomaton esults wee then used fo outle detecton usng two appoaches, whch wee analyzed both theoetcally and pactcally he man obsevatons fo the thee methods ae gven below; In conventonal methods; an teatve appoach s used to each a soluton, fo each teaton only one outle appeaed and emoved fom data set, the esults of method ae dectly affected by the outles, outle detecton s caed out usng esduals, a functon of measuement eos and s not completely clea In the Fuzzy Logc method; esults can be ntepeted wthout any teaton opeatons, outle detecton s caed out usng a edundancy matx whch matx s ceated n the soluton estmated wth LSM and the elatonshp between the esduals and the measuement eos hus the soluton s cleae than the othe methods, a classfcaton of outles can be ealzed by usng the membeshp value of measuements As a esult, f measuements goup nclude eos as a coodnate tansfomaton methods, these eos ae taken nto consdeaton In the soluton the method can be caed out by usng as condtonal adjustment wth unknowns o total least squae methods to detemned measuements goup eos he outle detecton usng esduals nstead of measuements eos s uncetan heefoe, n the soluton the measuement eos must be used fo the outle detecton In ths study, t can be seen that the soluton of the condtonal adjustment wth unknowns s sgnfcant fo 3D coodnate tansfomaton and the Fuzzy Logc method s moe applcable than the conventonal methods he futhe nvestgatons should be focused on total least squaes models fo coodnate tansfomaton and Atfcal Intellgence technques fo outle detecton 5 Refeences Alosmanoglu S and Akylmaz, O: A compason between statstcal & fuzzy technques n outle detecton, Intenatonal Assocaton of Geodesy Symposa, EdsAdam J, Schwaz, K P, Spnge, , 22 Baada, W A: estng pocedue fo use n geodetc netwoks, Publcatons on Geodesy Nethelands Geodetc Commsson, Delft, 968 Bebean A: Multple outle detecton: a eal case study, Suvey Revew, 33, 255, 4-49, 995 Caten, S, Colla, V, Vannucc, M: Advances n obotcs, Automaton and Contol, www-technolnecom, Eds Aámbuo J and evño, A R, , 28 Hampel, F, Ronchett, E M, Rousseeuw P J, Stahel, W A: Robust statstcs: the appoach based on nfluence functons, A Wley-Intescence Publcaton John Wley & Sons, New Yok, USA; 8-34, -6, 986 Hube, PJ: Robust statstcs, John Wley and Sons Inc, New Yok,, USA; -7, 98 Koch, KR: Paamete estmaton and hypothess testng n lnea models, Spnge, Beln-Hedelbeg, NewYok, USA, , 32-37, 999 Konak, H, Dlave A and Oztuk, E: he effects of obsevaton plan and pecson on the duaton of outle detecton and fuzzy logc: a eal netwok applcaton, Suvey Revew, 38, 298, 33-34, 25 Kutoglu, H S and Ayan: he ole of common pont dstbuton n obtanng elable paametes fo coodnate tansfomaton, Appled Mathematcs and Computaton, 76, , 26 Kutoglu, H S, Ayan and Mekk, C: Integatng GPS wth natonal netwoks by collocaton method, Appled Mathematcs and Computaton, 7, 58-54, 26 Kuttee, H: Jont teatment of andom vaablty and mpecson n GPS data analyss, Jounal of Global Postonng Systems, (2), 96-5, 22 Sh, Y, Ebenhat R and Chen, Y: Implementaon of evolutonay fuzzy systems, IEEE ansactons on Fuzzy Systems, 7(2), 9-9, 999 Sun, W: A new method fo localsaton of goss eos, Suvey Revew, 32, 252, , 994 7

8 Yasemn Ssman, Aslan Dlave and Sebahattn Bektas: Outle Detecton n 3D Coodnate ansfomaton wth Fuzzy Logc Syed, S, Cannon ME: Fuzzy logc based-map matchng algothm fo vehcle navgaton system n uban canyons, ION Natonal echncal Meetng, San Dego, CA, 26-28, 24 homson, D B: Combnaton of geodetc netwoks, Depatment of Suveyng Engneeng, Unvesty of New Bunswck, Canada, (accessed 9 Septembe 2) Vancek, P, Wells DE: he least squaes appoxmaton, Depatment of Geodesy and Geomatcs Engneeng Unvesty of New Bunswck, Canada, 972 Xue-Gong, Z: akag-sugeno Fuzzy logc vesus mamdan fuzzy logc, MSc hess, Advanced echnologes n Electoncs, Unvesty of he West of England, 2 Zadeh, L A: Fuzzy sets, Jounal of Infomaton and Contol, 8(3), , 965 8