A Probabilistic Nearest Neighbor Filter for m Validated Measurements.

Transcription

1 A Probabilisic Neares Neighbor iler for m Validaed Measuremes. ae Lyul Sog ad Sag Ji Shi ep. of Corol ad Isrumeaio Egieerig, Hayag Uiversiy, Sa-og 7, Asa, Kyuggi-do, 45-79, Korea Absrac - he simples approach for racig a arge i a cluered evirome is o selec he validaed measureme ha is closes o he prediced measureme ad use i i racig as if i were he rue oe. his mehod is he so-called "eares eighbor filer"(nn) widely used for racig i cluer. O he oher had, he probabilisic eares eighbor filer(pnn) is desiged o o igore he fac ha he NN would be he false measureme ad o calculae he probabiliy of he eve ha he NN is argeorigiaed. he PNN algorihm does o uilize he curre umber of validaed measuremes ha may be helpful for calculaig more reliable esimaes i he realisic siuaio where he spaial desiy of false measuremes is uow. Icorporaig he umber of validaed measuremes io desig of he PNN produces ew daa associaio proposed i his paper. his filer has similar performace o he PNN if he spaial cluer desiy is ow ad i is less sesiive o he uow spaial desiy of false measuremes. Keyword: aa associaio, Neares eighbor, Validaed measuremes. Iroducio Cluer is defied as ay uwaed radar echo ha occurs aroud he arge of ieres i he validaio regio. his leads o he occurrece of several measuremes i he validaio gae for racig a sigle arge. arge racig i cluer requires accurae daa associaio for rac maieace. he simples approach for daa associaio is his wor was suppored by he Brai Korea Projec i 3. o selec he validaed measureme ha is closes o he prediced measureme ad use i as if i were he correc oe i he filer updae sep. he filer wih his daa associaio mehod is he so-called eares eighbor filer(nn) ad i is widely used i arge racig for compuaioal simpliciy. Aoher approach cosiderig he possibiliy ha he eares eighbor(nn) would be origiaed form he false oe is he probabilisic eares eighbor filer(pnn) [] amog a variey of varias of he PNN. [] uilizes he codiioal probabiliy desiy fucios associaed wih he NN which play impora roles i he developme of daa associaio. Approach of [] is suggesed i a effor o refie he resuls of [] ad [3]. aa associaio wih he NN measureme ca be divided io he eves: () here is o validaed measureme( M ); () he NN measureme is origiaed from he arge( M ); (3) he NN measureme is from a false arge( M ). I [], i is assumed ha he hree eves cocerig he eares eighbor have correlaio bewee samplig iervals, ad he eves are assumed as hidde Marov models wih proper rasiio probabiliies. his paper alog wih a compaio paper of [4] cosiders he radomized discree eves o be he oes wihou ay correlaio wih he previous eves. he PNN suggesed i [4] uilizes he fac ha he posiio of he NN measureme is available i he derivaio of he codiioal probabiliy desiy fucio ad he updaed covariace uder he assumpio of M. his algorihm will be discussed i Secio briefly. his paper preses a ew form of he PNN, called he PNN- m, which aes io accou of he eves relaed o he NN measureme seleced amog he m measuremes i he curre validaio gae. he PNN- m may produce more reliable esimaes ha he PNN i real applicaios where he

2 spaial cluer desiy i he validaio regio is uow. hrough a series of Moe Carlo simulaio rus, i urs ou ha he PNN- m is less sesiive o he uow spaial cluer desiy ad i has similar racig performace o PNN of [4] for he cases of ow spaial cluer desiy.. Probabilisic Neares Neighbor iler he NN is defied as he measureme correspodig o he smalles ormalized disace squared (NS) amog he validaed measuremes. A sadard Kalma filer called he NN is applied o use he NN measureme i he updae sep[5] as if i were arge-origiaed. he drawbac of he NN sems from he fac ha he NN measureme may be origiaed from a cluer. his problem could be solved by employig a probabilisic approach o daa associaio wih he NN measureme as i he probabilisic daa associaio filer(pa)[5], ad he PNN[,4]. I is ow ha he PA has superior performace a a cos of high compuaioal complexiy. he mai differece bewee he PNN of [] ad he PNN of [4] is i he developme of he updaed covariace uder he assumpio of M, besides he aforemeioed hidde Marov model assumpio for probabiliy weighig facor calculaio. I [4], he updaed covariace is derived from he fac ha he posiio of he NN measureme correspodig o a cluer is available uder he assumpio of M. he arge is assumed o be perceivable so ha i exiss ad ca be deeced [6]. I his secio, he PNN of [4] is briefly iroduced. Le z ad ν deoe he -dimesioal measureme vecor ad he residual represeig he differece bewee z ad he ceer of he validaio gae z, he arge prediced measureme, respecively. he validaio gae used is he ellipsoid : { } ( ) = ν : ν S ν () where ν is a zero-mea aussia residual wih covariace of S for he rue measureme, ad is called he gae size. he ormalized disace squared (NS) of z is defied as = ν S ν. he volume of he -dimesioal gae saisfies V = C S 4 where C =, C = π, C 3 = 3 π ec. he validaed measuremes cosis of dimesioal posiio iformaio of he plos of which sigal iesiies exceed a predeermied hreshold value. If we le z be he NN measureme ad ν be he correspodig residual, he volume of he ellipsoid wih he gae size of becomes V = C S. he followig assumpios are used i his paper. A) he probabiliy ha he arge is deeced ad iside he validaio gae is PP where P is he probabiliy of arge deecio idicaig ha he arge sigal ampliude exceeds a hreshold τ ; ad P is he probabiliy ha he arge falls iside he validaio gae. A) he umber of validaed false measuremes i he validaio gae, deoed by m, is Poisso disribued wih a spaial desiy λ such ha ( λv ) λv µ ( m) = P( m = m) = e. (3) m! A3) he sae predicio error e = x x for ay give ime is a zero-mea aussia process wih a covariace ( ;, P ) N e A4) he validaed false measuremes a ay ime are i.i.d. uiformly disribued over he gae. A5) he locaio of a validaed false measureme is idepede of he rue measureme a ay ime ad oher validaed false measuremes a ay oher ime. A6) he arge is exisig ad ca be deecable, i.e., i is m () P. Hece e is said o saisfy

3 perceivable [6]. Appedix A shows a summary of he PNN algorihm. he derivaio of his algorihm is omied i his paper. Ieresed reader should refer o [4] for deailed derivaio. 3. PNN- m he PNN [4] is derived wihou cosiderig he umber of validaed measuremes i he curre validaio gae. Comparig he PNN o he PA of [5], he PA cosiders he umber of measuremes implicily sice he probabiliies ha each validaed measureme would be origiaed from he arge of ieres are calculaed. I a cluer evirome, he umber of validaed measuremes is varyig a every samplig isace hough he spaial desiy of cluer is cosa. I he updae sep of he PNN[,4], he probabiliy β, ha he NN measureme is arge-origiaed, is calculaed wihou aig accou of he umber of validaed measuremes such ha if he NSs of he NN measuremes are equal for differe ime seps, he resulig values of β are ideical regardless of he volume of he validaio gae ad he umber of validaed measuremes. However, i is reasoable o cosider ha β for he same NS of he NN measureme would be saisically differe for he cases of differe m, he umber of he validaed measuremes. Icorporaig his idea io desig of he PNN produces a ew ehaced filer called he PNN- m. Basically, he differece bewee he PNN ad he PNN- m exiss i he calculaio of probabiliy desiy fucios of he NN associaed wih he eves of M ad M. his differece leads boh algorihms o calculae differe probabiliy weighigs ad esimaio error covariace marices, ad hus resuls i differe esimaio performace. A. Probabiliy desiy fucios Le z be he NN measureme ad ν be he correspodig residual, he he NS of z is defied as =ν S ν, he he volume of he ellipse wih he gae size is V C S. he NS of he NN measureme saisfies he followig probabiliy desiy fucio (pdf) uder he codiios ha he umber of validaed measureme is m ad he NN is argeorieed. heorem : wih Assumpio A) ~ A5), f( M is give by f( M = f(, M (4) PM ( m P V = N( ) µ ( m)( ) PM ( where he validaed measureme se sequece Z = { z, z,..., z } hrough is subsumed i he codiioig, arge residual uder replacig N( ) ν o such ha is he aussia pdf of he rue M, ν ~ N( ν ;, S) obaied by N( ) e =, ad π S ( x ) is he ui sep fucio defied as if x, ad for elsewhere. PM ( of (4) saisfies (5) PM ( = f( M,, md ).

4 Proof: omied. Uder he eve M, he NN measureme is o origiaed from he arge of ieres. he arge may be exisig i he validaio gae bu o deeced, or he arge may be locaed he ouside of he validaio gae, or he arge may be deeced bu he NS of he arge-origiaed measureme is bewee he NS of he NN measureme ad. Accordigly, he pdf of he NS of he NN measureme associaed wih a false arge uder M ad m ca be obaied from he followig heorem. heorem : Wih Assumpios A)-A6), f ( M, m ) is give by f( M = f(, M PM ( = {( PP) f% C ( m) µ ( ) m PM ( + P ( P P ( )) f% ( m) µ ( m) ( ) R C where f% C ( m) is he codiioal pdf of he NS of he NN measureme origiaed from a cluer uder he assumpio ha he umber of he validaed false measuremes m is equal o m. f % C ( m ) is expressed as m fc ( m) = m } (6) %, (7) f% C ( m) ca be obaied by replacig m i (7) o m. P R () i (6) is he probabiliy ha he arge exiss i he ellipsoid wih gae size such as q C P ( ) ( ;, ) R = N v S dv = q e dq + V π (8) ad P( M saisfies PM ( = f( M,, md ). (9) Proof: omied. Proposiio : P( M of (5) ad P( M of (9) are relaed o PM ( + PM ( = Pm ( ) = ( PP) µ ( m) + PPµ ( m). () Proof: omied. Uder he eve M, here is o measureme i he validaio gae ad he pdf for he arge-origiaed measureme is ideical o he oe for he PNN[4]. he pdf is expressed as V N ( ) P ( ) f( M ) = PP ( ) () his pdf is used i he updae sep for calculaig he esimaio error covariace uder M. I order o calculae he updaed covariace for he eve M which implies ha he available NN measureme is origiaed from a cluer, he pdf of, he NS of he arge-origiaed measureme, codiioed o M ad m is required. Le us deoe he NS of he NN measureme as, he f(, M is obaied from he followig heorem. Noe ha cluers of ieres are locaed iside he validaio gae such ha. heorem 3: Wih Assumpios A)-A6), he pdf of he codiioed o M, m, ad he NS of he NN measureme is give by f(,, M, ) (,, ) m f M m = f(, M

5 % ( ) ( ) V ( ) C ( ) ( ) P f m µ m + N( ) ( ) P ( ) ( ) f% C ( m) µ ( ) m = ( PP) f% C ( m) µ ( ) m + P( P PR( ) ) f% C ( m) µ ( ) m () Proof: omied B. Updaed Covariace available NS ad m, is expressed by M P ( ) = E e,, e M m = P KSK + α KSK where K is he filer gai ad α saisfies (5) If oe ows ha here is o measureme i righ afer he samplig ime, he updaed arge sae esimae x ˆ remais he same as he prediced esimae x however, he updaed esimaio error covariace is o merely he prediced covariace P due o he perceivabiliy assumpio of A6). he updaed error covariace codiioed o M [7] is derived from () ad give by PP( Cτ ) PM, = P + KSK PP where K is he filer gai ad g C τ saisfies g (3) λ( PPCτg )( V V ) + P( PCτg PR( ) Cτ( ))( m) α = λ( PP)( V V) + P( P PR( ))( m) (6) of which V represes he volume of he ellipsoid of gae size such ha give by Proof: omied. V = C S ad Cτ ( ) is q qe dq Cτ ( ) =. (7) q q e dq C τ g q q e dq. (4) q q e dq = or =, Cτ ( ) = e e + PR ( ) = e. ad Hece, C τ g= e + e Noe also ha P = e, for =. M for =. By usig f(, M, P ( ), he updaed esimaio error covariace codiioed o he umber of validaed measuremes m ad he available NS he NN measureme assumed o be origiaed form a cluer for M ca be obaied. heorem 4: Wih assumpio A)-A6), P ( ), he updaed esimaio error covariace for M wih he M of he updaed covariace codiioed o M is equivale o he oe i he sadard Kalma filer such as P = P KSK. ˆ he probabiliy weighig facor for he updae of he NN measureme is derived from a poseriori pdfs. he probabiliy β ha he NN measureme is argeorigiaed is expressed as β = f(, M (8) c where he codiioig o Z is suppressed as before,

6 ad c is he ormalizaio cosa comprised of c= f( M, + f( M,. he probabiliy ha he NN measureme is o arge-origiaed becomes β = β. he pdfs ivolved i (8) are available from (4) ad (6). he PNN-m algorihm is summarized below. xˆ Pˆ = x M PP( Cτ ) = PM, = P + KSK PP xˆ = x + he PNN-m algorihm Predicio sep Ideical o he sadard Kalma filer Updae sep () or he case of () or he case of M Kβ ν g M P ( ) = P + ( α ) KSK ˆ M P = βp ( ) + β( P KSK ) + ββkν ν K A = I ad he process oise I τ s, I w= ( w, w ) B = I τ s () is a whie aussia oise vecor wih zero-mea ad power specral desiy of s A I τσ., he variace of he arge acceleraio, is se o be σ A X Y. ( m / sec ) ad τ s, he ime-cosa of arge maeuver, is 5sec. he measureme equaio a ime = is expressed as z = ( I,, ) x + v () where v is a zero-mea whie aussia oise vecor sequece wih he variace R = diag( m, m ). he arge used i his sudy has a speed of 38 m / sec wih he headig agle of 3 degrees from he X-axis, ad he iiial ierial posiio is se o be (7 m, 4 m ) as depiced i ig.. he algorihms are operaig a Hz. Y 4m A V = 38m/sec = 3 4. Simulaio Resuls he NN, he PNN of [4] summarized i Appedix A, ad he PNN- m are applied o a plaar racig problem i a cluer evirome o verify he performaces hrough a series of simulaio rus. I his sudy, filer saes are composed of arge posiio, velociy ad acceleraio. he coiuous dyamic model is represeed by where x = ( X, Y, X&, Y&, A, A ), x& = Ax + Bw (9) X Y ig.. 7m Iiial codiios he associaio probabiliies for M of he PNN ad he PNN- m are depiced i ig.. Comparig β of he PNN o ha of he PNN- m, i ca be see ha he PNN- m is more adapable o he curre evirome. able Ⅰ represes rac loss perceages obaied from 5 rus of Moe Carlo simulaio for he case of ρ = 5, = 9, A =.5g a = 5sec ad varyig X P

7 ad λ. he rac loss is declared if he rue ad he esimaed posiios are deviaed more ha imes he sadard deviaio of he measureme oise. ABLE Ⅰ. rac loss perceages P λ NN PNN PNN-m able Ⅱ shows he resuls of he sesiiviy aalysis represeig wheher reliable esimaes are produced i real siuaio where he spaial cluer desiy is uow. able Ⅱ idicaes he rac loss perceages obaied from 5 Moe Carlo simulaio rus. I his sudy, he rue spaial desiy is se o be λ =.5 ad he filers use he guessed spaial desiy λˆ. Oher simulaio codiios are he same as he oes for ableⅠ. Judgig from ables Ⅰad Ⅱ he PNN- m has similar performace o he PNN ad i is less sesiive o he uow spaial cluer desiy, which idicaes ha PNN- m has subsaial advaages i pracice as he spaial desiy is o ow exacly beforehad ABLE Ⅱ Sesiiviy o λ i erms of rac loss perceages λ P ˆλ PNN PNN-m Probabiliies PNN PNN-m ime(sec) ig. β hisories

8 5. Coclusios he PNN-m ha uilizes he umber of he validaed measuremes is proposed here as a useful daa associaio algorihm for arge racig i cluer. he PNN- m is esed by a series of simulaio rus alog wih he PNN of [4] ad he NN o evaluae he esimaio performace. he PNN- m has similar performace o he PNN for rac maieace. However, i is foud ha he PNN- m is less sesiive o he spaial cluer desiy due o he fac ha he umber of validaed measuremes icorporaed i he algorihm maes i more adapable o he curre cluer evirome. herefore i ca be said ha he PNN- m produces more reliable esimaes i real siuaio where he spaial desiy of false measuremes is uow. Appedix A. he PNN algorihm Predicio sep Ideical o he sadard Kalma filer Updae sep () or he case of M xˆ = x P ( ) ˆ P Cτ g P = P, M = P + KSK PP () or he case of M xˆ = x + Kβv ( τ ( )) PP ( ) C M P ( ) = P + KSK, = v S v PP( ) M Pˆ = β P ( ) + β ( P KSK β P P ( ) R τ = P + P PR ( ) R ) + β β Kv v ( C ( ) ) β KSK K + β β Kv v K where β = (, ) f M c c = f(, M ) + f(, M ) CP f(, M ) = e e λ + π V V ( ) R λ C S f(, M) = P P ( ) e λ Noe ha Cτg, C τ ( ) ad PR ( ) are represeed by (4), (7) ad (8), respecively. Refereces [] X. R. Li, "he pdf of eares eighbor measureme ad a probabilisic eares eighbor filer for racig i cluer," Proceedigs of he 3d CC, Sa Aoio, exas, ec [] S. R. Rogers, iffusio aalysis of rac loss i cluer, IEEE ras. o Aerospace ad Elecroic Sysems, Vol. 7, , 99. [3] A. aria ad S. Pardii, rac-while-sca algorihm i a cluer evirome, IEEE ras. o Aerospace ad Elecroic Sysems, Vol. 4, , 978. [4]. L. Sog, J. H. Ryu ad W. C. Oh, A probabilisic eares eighbor filer for uderwaer arge moio aalysis, Proceedigs of he U, Korea, Oc.. [5] Y. Bar-Shalom ad. E. orma, "racig ad aa Associaio," Academic Press, New Yor, 988. [6] N. Li ad X. R. Li, arge perceivabiliy ad is applicaios, IEEE ras. o Sigal Processig, Vol.49, No., Nov.. [7] X. R. Li, racig i cluer wih sroges eighbor measuremes Par Ⅰ: heoreical aalysis, IEEE ras. o Auomaic Corol, Vol. 43 No., Nov [8] K. J. Rhee ad. L. Sog, A probabilisic sroges eighbor filer algorihm based o umber of validaed measuremes, Proceedig of he 6h Ieraioal sessios, JSASS, Yoohama, Oc.