PARTICLE METHODS FOR MULTIMODAL FILTERING

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1 PARTICLE METHODS FOR MULTIMODAL FILTERIG Chrsan Musso ada Oudjane OERA DTIM. BP France. Absrac : We presen a quck mehod of parcle fler (or boosrap fler) wh local rejecon whch s an adapaon of he kernel fler. Ths fler generalzes he regularzed fler. The condonal densy of he sae s recursvely esmaed. The proposed fler allows a precse correcon sep n a gven compuaonal me. In he conex of he 2D rackng problem wh angle and/or range measuremens smulaons show a beer behavor of hs fler compared wh he Kalman fler and wh classcal boosrap fler. We presen also some resuls of a mulple model parcle fler whch can rack maneuverng arges. Keywords : parcle fler boosrap rackng non-lnear flerng Mone-Carlo rejecon. ITRODUCTIO We consder a arge followng a nosy dynamcal equaon whch s parally observed (noaons are he same as n []) X = F( X ) + V () Y = H( X ) + W (2) d d where F: R R and H R q : R are gven funcons ( V ) ( W ) are d varables wh denses V ~ p( v) dv W ~ q( v) dv (3) X he nal sae of densy p s assumed ndependen of ( V ) ( W ). ( X ) s a Markov chan X /( X = x ) ~ p( x F( x )) dx (4) ( Y ) are ndependen condonally o ( X ) and each ( Y ) s condonally o X ndependen from X ( s s ) Y /( X = x ) ~ q( y H( x )) dy (5) The Exended Kalman Fler (EKF) s wdely used o esmae recursvely he mean and he varance of he sae X gven he pas measuremen Y = ( Y... Y ). The EKF assumes ha he condonal densy s Gaussan. Bu when F or H s hghly nonlnear or n case of mulmodaly he EKF s neffcen. The goal of he non-lnear flerng (LF) s o esmae he whole law of he sae X gven he measuremens Y. For example n he rackng conex we wll be able o esmae precsely he probably of he presence of a arge n any poron of he sae space and consenquenly o esmae he poson of he arge. For hs fler here s no hypohess concernng he lneary of F and H and no condons abou he naure of he nose V and W. We wan o esmae recursvely he condonal densy denoed by f/ ( x/ y ). Suppose we know f / and a new measuremen y s avalable. Bayes rules gve easly he formulaon of f/ ( x/ y ) n wo seps f/ ( x/ y ) = p( x/ x ) f / ( x / y ) dx (6) f ( x/ y ) q ( y / x) f ( x/ y ) (7) / / The frs sep (6) s he predcon sep usng he dynamcal law. The predced densy f/ ( x/ y ) s he expecaon of p( x/ X ) where X follows f /. The second sep (7) s he correcon sep. q( y/ x)= qy ( Hx ( ))s he lkelhood a he pon x. and p( x/ x ) = p( x F( x )) When he nose W s Gaussan q can be expressed as q( y/ x) = d / 2 ( 2π ) de( Ω) exp ( y H( x ))' ( ( ))' ) Ω y H x (8) 2

2 The correcon sep confrons he new measuremen wh he predced densy. The recurson begns wh he assumed known nal densy f/ ( x) = p( x). A naural way o solve (6) and (7) s o dscreze he sae space. Ths can be done when he dmenson (d) of hs space s low (d<4). Oherwse he compung cos s hgh. [2]. Anoher way s o use Mone-Carlo mehods. 2. CLASSICAL PARTICLE FILTER ALGORITHM The am of he parcle fler called also boosrap fler or Mone-Carlo fler s o generae recursvely a sample (parcles) whch follows approxmaely he condonal densy f/ ( x/ y ). Suppose we have a me (-) a -sample ( ) ( ) ( x /... x / ) accordng o f /. The negral (6) can be approxmaed by he emprcal expecaon. In oher words he densy f / s approxmaed by he emprcal measure ( / ) δ ( x) whch x / j = pus unformly he mass on he parcles. (6) and (7) become ˆ f/ ( x/ y ) = p( x/ x / ) j = ˆ f ( x/ y ) q ( y / x) p( x/ x ) / / j = (9) () The error of hs approxmaon s ndependen of he dmenson (d) and s of order / (law of large numbers). Therefore unlke dscrezaon mehods Mone-Carlo mehods can heorcally deal wh large (d) wh a reasonable compung cos. There s wo classcal ways o generae sample from (9) and (). 2. The weghed resamplng mehod (SIR) These flers called SIR fler (Samplng Imporance Resamplng) ([3][4]) or IPF (Ineracng Pacle Fler) ([5]) or CODESATIO (Condonal Densy Propagaon) ([6]) frs generae a sample from (9). Ths can be done by Generae I unform on {...} () 2 Generae X accordng o p( x/ x / ) (2) Sep s a boosrap algorhm and sep 2 for fxed x / gves a predced parcle x / accordng he dynamcal law (). Ths algorhm produces an d sample ( ) ( ) ( x /... x / ). Then n (7) f/ ( x/ y ) s approxmaed by he emprcal densy ( / ) δ ( x) j = x/ fˆ / ( x) = wjδ ( x) j = x/ ( j) ( j) j = / / j = (3) where w q ( y / x )/ q ( y / x ) s he wegh of each parcle proporonal o he lkelhood. Generang a sample from (3) can be done by (correcon sep) Generae I PI ( = ) = w (mulnomal)(4) 2 Pu X = x / (5) The mos lkely predced parcle are he mos duplcaed. () (2) (4) (5) produce ( ) ( ) quckly a new d sample ( x... x ). 2.2 The rejecon mehod (RM) / / The RM [][4] generae he predced ( ) ( ) parcle ( x/... x/ ) wh () and (2) lke n SIR. Bu approxmaon (3) s avoded. The RM produces a exac sample accordng o (). I s easy o check ha he followng algorhm generaes hs sample. Generae I unform on {...} 2 Generae X ~ p( x/ x / ) and U unform on [] (6) 3. If q( y / X) cu accep X x / =X and j=j+ (7) where c q y x x sup ( / ). Seps 23 are repeaed o ge he desred sze of he sample. For he rejecon mehod he correcon sep s exac for fxed unlke he weghed sample mehod. However n hs form he compung cos s hgh. Indeed he probably ha (6)+(7) produce a sample X (accepance probably) s proporonal o

3 c. The mum of q(y/.) can be hgh (see for example (8) c / 2 = ( 2π ) d de( Ω ) ). SIR and RM have a serous drawback. In case of low dynamcal nose we observe ha n mulplyng he hgh weghed parcles he predcon sep wll explore poorly he sae space. There s wh me a degeneracy phenomenon. The parcle clouds wll concenrae on a few pons of he sae space. The dscree naure of he (weak) approxmaons reduce he explorng capacy. Therefore s useful o generae a sample from a smooh dsrbuon whch approxmaes he underlyng dsrbuon f / whch s assumed o be smooh. 2.3 The kernel and he regularzed mehods Huerzeler and Kunsch [] have nroduced he kernel fler (KF) whch uses local rejecon and densy kernel esmaon. The new algorhm proposed n hs paper (L2RPF) s an adapaon of he KF. Regularzed parcle mehods (RPF) ha have been proposed n ([6]...[9]) deal wh weghed sample mehods. The RPF (verson where regularzaon s made afer correcon) esmaes f / by a nonparamerc densy esmaon usng a kernel (K). (3) becomes ˆ f ( x) w K[ h A( x x )] / j / j = (8) where h s he bandwh K he kernel whch s self a densy A he roo of S he covarance marx of he parcles x / ' ( AA = S ). The algorhm. Generae I accordng PI ( = ) = w (9) 2. Generae Z ~ K( x) dx (2) 3. Pu X = x + ha Z (2) / produces a sample accordng o (8). The regularzaon (2) mprove he explorng capacy. oe ha h= gves he SIR (4) (5). K and h are chosen n order o mnmze he L 2 error MSE(Kh) 2 = (ˆ f/ ( x) f/ ( x)) dx [] [] among he even kernels of L 2 norm equal o ) 2 Kx ( ) = cd ( d+ 2)( x ) f x (22) 2 oherwse. c d s he volume of he un sphere. The opmal h s h = AK d + ( ) /( 4 ) / 2 wh /( d + 4) [ d ] d AK ( ) = c 8 ( d+ 4)( 2 π ) (23) I s mporan o when he parcle before he regularsaon because h s he same n all drecons. oe ha he MSE depends now on he dmenson (d) wh he opmal h. 3.THE L2RPF FILTER 3. Descrpon of he fler The Local Rejecon Regularsed Parcle Fler allows a precse correcon sep n a gven compuaonal me. Gven ( x / ) j =... and a scalar α we generae a correced sample wh he followng algorhm (24) (25) (26).Generae I PI ( = ) c ( α ) 2.Generae Z ~ K( x) dx U unform on [] 3. Pu X = x/ + ha Z 4 If q( y / X) α c I( α )U we accep X =X and j=j+ (27) x / The coeffcens c ( α ) (compued below) sasfy c ( α ) sup Σ q ( y / x) (28) j x j ( α ) ( j) ( j) 2 2 Σ j = { x/( x x/ )' S ( x x/ ) α h } s a local ellpsod cenered on he parcle ( j) x. α / s a conrol parameer beween and. Proposon 3. : he L2RPF algorhm produces a sample accordng o ˆ α ( / ) f q y x ( ) [ ( )mn( ( ) )] / x c α = αc α ( ) Kh [ A( x x )] (29) / Indeed wh n R d n he «coordnae by coordnae» sense and pung gx ( ) = q( y / x) he 3 ndependen varables beng I U and Z we have

4 P( X x) K( z) c ( α ) dz du = g( x + ha z) αc( α ) u x + ha z x g( x + ha z) [ c ( α)mn( )] Kz ( ) dz α c ( α ) = x + ha z x (Pung z' = x + ha z becomes) gz [ c ( α (') )mn( α c ( α ) )] = z' x Kh ( Az ( ' x )) dz' We oban (29) afer dervang he las expresson w.r. «x». Proposon 3.2 : he accepance probably P a of he L2RPF s wh c = ( c ( α )) (3) = g( x/ + ha z) Pa = c [ c ( α)mn( )] = αc ( α) Kzdz () (3) ( ) gx ( / ) Pa( α) c c ( α)mn( ) α c ( α ) (32) = ( ) (3) s compued lke n (29). (32) s obaned usng an expanson of ( ) g( x/ + ha z) around h =. Ths approxmaon s n general precse. If we pu α = n (29) he «mn» s c ( α ) q( y/ x) because c ( α ) s a local mum (28) ( ) ˆ α = f ( x) [ q ( y / x) K[ h A( x x )] / / = whch s he KF wh he exac correcon. In hs case P a s mnmal (32) he compuaonal cos s mal. If we pu α = n (29) he «mn» s and c / c = w (Σ j reduces o a parcle). = We oban he RPF (8). In hs case P a = he compuaonal cos s low. oe ha P a ( α ) decreases when α ncreases A each me he choce of α s done by he followng manner : we keep he mal mn value of α such as Pa( α) Pa (wh a coarse dscrezaon of []). P mn a s gven by he compung capably. The hgher α s beer he correcon. When α s chosen we pu e = / Pa( α ) he number of essamples whch ener n he loop (24-27). In pracce α s close o for he frs measuremens hen ncreases o when he parcles concenrae on lkely regons of he sae-space. ow we presen a fas mehod o compue c ( α ) 3.2 compung he c ( α ) coeffcen By Lagrangan mehods we can see ha he coordnaes ( x... x d ) of a pon n Σ j verfes( d) (33) mn j j x = x α h S x x + α h S = x where S s on he dagonal of S. Le C j beng he hyper-cube { / mn x x x x d} (Σ j C j.). c ( α ) wll be he mum of g on C j. Assume ha he measuremen funcon ( Hk (.) k =... q) (2) s locally decreasng for one coordnae or ncreasng for an oher. For example f we measure an angle H ( k x )=arcg(x /x ) H ncreases when x 2 k ncreases and H k decreases when x 2 ncreases (f x x 2 >). The exreme values of H k are (34) exr exr k k m k k M k mn H = H ( x ) H ( x) H ( x ) = H where x exr mn (.) equals x or x. Suppose ha he q componens of he measuremen nose W are ndependen and ha q(.) (5) decreases around he orgn can be seen ha he mum of he lkelhood on C j ( c j = sup(g(x)) s (35) q exr q ( y H ( x)) = q ( y H ) x Cj Wk k k Wk k k k = k = q where H exr k = H mn k f yk H mn k H exr k = y k f mn exr Hk yk Hk and Hk = Hk f y H k k 4. SIMULATIOS We presen hree 2D-rackng problems. L2RPF s appled n each problem excep he las one. The compung cos s abou 3

5 mes bgger han he EKF wh P a mn =.2. The number of parcles () s 5. The number of Mone-Carlo (MC) s 5. In hese problems he dynamcal nose level s equal o zero. So we can easly compue he Cramer-Rao Lower Bound. 4. Bearng only The arge has a unform sragh moon 2 ( x x x x 2 ) =(km -m/s km m/s) (fgure ()). The observer ( xo yo) s on he orgn a me. I has a 2 legs moon: (5m/s m/s) speed for he frs (durng s) and (-5m/s 5m/s) for he second. Durng 2s he observer measures every second a nosy angle wh sandard devaon (sd)=.5 H( X ) = arcg(( x xo )/ ( x 2 yo )). The nal esmae X(/) (cener of he cloud) of he arge s a Gaussan varable cenered on he rue poson wh covarance marx P(/)=dag(5km 3m/s 5km 3m/s). Fgure () shows he esmaed rajecory (cener of he cloud) of he L2RPF. Fgure (3) shows he evoluon of he accepance probably wh he correspondng conrol parameer α (Fgure (2)). For each MC ral we compue he rajecory esmaon error n order o oban he sd (for he 5 ral) of he arge poson. As you can see on fgure (5) he sd of he horzonal poson of he arge s very close o he Cramer-Rao Lower Bound (CRLB) (whou bas (fgure (4)). In hs conex (observably problem) he EKF has dverged 5 mes over parcle cloud (=9s) > < parcle cloud (=2s) y (m) 8 6 <--- arge poson esmaon (L2RPF) 4 2 X(/) <--- observer x (m) Fgure () : rue and esmaed rajecores emprcal heorecal.8.7 alpha accepance probably me (s) me (s)

6 Fgure (2) : conrol parameer evoluon fgure (3) : accepance probably L2RPF 35 CRLB L2RPF 2 3 bas (m) 5 sandard devaon (m) me (s) Fgure (4) : bas for x-poson me (s) Fgure( 5) : sandard devaon for x-poson 4.2 Range and Bearng The arge has a unform sragh moon 2 ( x x x x 2 ) =(5km -2m/s 5km - 2m/s). The observer s on he orgn. Durng 2s he observer measures every second a nosy angle wh sd= and a range wh sd=m (very precse) Y = [ arcg( x / x 2 ) ( ) 2 x ( 2 x ) 2 ] +. The nal esmae X(/) of he arge s a Gaussan varable cenered on he rue poson wh covarance marx P(/)=dag(.5km 5m/s.5km 5m/s). Resuls wh he 5 MC are shown below L2RPF and EKF are compared. Fgure (6) shows he x-poson esmaor bas. And we observe on Fgure (7) ha unlke he EKF he L2RPF converge rapdly o he CRLB. RPF has been performed. The resuls are comparable wh he L2RPF for he sd. Bu he varance of he clouds are bgger wh he RPF. Error esmaon s more precse wh he L2RPF. 5 4 L2RPF EKF 2 CRLB L2RPF EKF 5 bas (m) -5 - sandard devaon (m) me (s) Fgure (6) : x-poson bas me (s) Fgure (7) : x-poson sd 4.3 Mulple Model Parcle Fler (MMPF) The MMPF s presened n [2]. By means of he formalsm of Ineracng Mulple Model [3] where he dynamcal model θ of he arge has o be esmaed among some fxed models. Ths s a case of mulmodaly. We suppose ha { θ } s a dscree Markov chan wh a gven ranson marx. Therefore we can apply he heory of he parcle fler wh he new augmened sae E

7 E = ( X θ ) (36) Assumng ha gven θ θ s ndependen of X predcon sep (6) s gven by PE ( = e / y ) = PX ( / θ X ) P( θ / θ ) PE ( = e / y ) de (37) { } If we have a sample E from E y / he followng algorhm produces a predced sample { E / } accordng o (37) for each parcle E ( X θ ) = Generae θ accordng o p( θ / θ = θ ) 2 Generae X / accordng o px ( / X ) θ Correcon sep s done wh he RPF verson ((8)...(2)). The updaed sample sae { X / } s gven by he margnal drbuon of { E / }. and a urn wh consan velocy (2 sae models). Fgure (8) shows he geomery. The observer placed on he orgn measures bearng (sd= ) and range (sd=2m) every s. The duraon of he frs USM s 6s he duraon of he urn s 8s (urn rae=.6 rd/s) and he duraon of he las USM s 6s. The nal USM mode probably s.99 and he ranson Markov marx p( θ / θ ) s The nal esmae X(/) of he arge s a Gaussan varable cenered on he rue poson wh covarance marx P(/)=dag(45m 63m/s 42m 6m/s). The number of Mone-Carlo s. Classcal IMM fler and he MMPF are compared. MMPF esmaes he angular urn rae (dmenson of he sae=5). The IMM knows hs rae (oherwse for hs conex IMM s no sable). everheless he behavor of he 2 flers are comparable. Probables of he USM mode are shown n Fgure (9) hey follow he change of he dynamc. On Fgure () we can see a good angular urn rae esmaon for he MMPF. In our smulaon he arge can have 2 moons : unform sragh moon (USM) M.M.P.F. Sae I.M.M Y POSITIO (km) parcle cloud X POSITIO (km) Fgure (8) : rue and esmaed rajecores for he 2 flers

8 MODE PROBABILITY M.M.P.F. IMM AGULAR TUR RATE STD (rad/s).8 x TIME (s) Fgure( 9) : unform moon mode probably TIME (s) Fgure () : sd of he angular urn rae REFERECES [] M Huerzeler H.R. Kunsch. «Mone Carlo approxmaons for general sae space models». Journal of Compuaonal and Graphcal Sascs (7) : pp [2] C. Zhqang F. Le Gland Z. Hulong «An adapave local grd refnemen mehod for nonlnear flerng» rappor de recherche 2679 IRIA Oc. 995 [3].Gordon D. Salmond A.Smh. «ovel approach o nonlnear/ non-gaussan Bayesan sae esmaon». IEE Proceedngs Par F vol 4 pp. 7-3 Aprl. 993 [4] M. K. P. Shephard. «Flerng va smulaon : auxlary parcle fler» Oc [5] P. Del Moral. «onlnear flerng : neracng parcle soluon» Markov Processes and relaed Felds vol. 2 no4 pp [6] M. Isard A. Blake. «Conour rackng by sochasc propagaon of condonal densy». Proc European Conf. Compuer Vson. pp 343 Cambrdge UK 996 [6] C. Musso. Oudjane. «regularsaon schemes for branchng parcle sysems as a numercal solvng mehod of he nonlnear flerng problem». ISSC 998. Dubln (Ireland) June [7]. Oudjane C. Musso. «regularsed parcle schemes appled o he rackng problem». IRS 98. Munchen (Germany) Sepember [8] F. Le Gland C. Musso. Oudjane. «An analyss of regularzed neracng parcle mehods for nonlnear flerng». Preprns of he 3rd IEEE European Workshop on Compuer--Inensve Mehods n Conrol and Daa Processng Prague 998. pp Sepember 998 [9] C. Musso. Oudjane. «mehodes sasques en flrage parculare appqué au psage». Inernaonal Conference on Radar Sysems. Bres. France Ma 999 [] B. W. Slverman «Densy Esmaon for Sascs and Daa Analyss». Chapman & Hall 986 [] L. Devroye L. Gyorf. «onparamerc Densy Esmaon». John Wley & Sons 984. [2]. Oudjane C. Musso. «Mulple Model Parcle Fler». 7ème colloque Gres sur le raemen du sgnale e des mages. Vannes 3-7 Sepembre 999 [3] A.P. Henk Blom and Y. Bar-Shalom. «The neracng mulple model algorhm for sysems wh Markovan swchng coeffcens. IEEE Transacons On Auomac Conrol 78:

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