Cramer-Rao Lower Bound for a Nonlinear Filtering Problem with Multiplicative Measurement Errors and Forcing Noise

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1 Preprnts of the 9th World Congress he Interntonl Federton of Automtc Control Crmer-Ro Lower Bound for Nonlner Flterng Problem wth Multplctve Mesurement Errors Forcng Nose Stepnov О.А. Vslyev V.А. Concern CSRI Elektroprbor JSC Ntonl Reserch Unversty of Informton echnology Mechncs Optcs 49 Kronverksky pr. St. Petersburg 97 Russ (e-ml: Concern CSRI Elektroprbor JSC Mly Posdsky str. St. Petersburg 9746 Russ (e-ml: Abstrct: he recurrence lgorthms for the Crmer-Ro lower bound for dscrete-tme nonlner flterng problem n the condtons when forcng nose mesurement errors ntl covrnce mtr depend on the stte vector to be estmted re derved. It s ssumed tht the stte vector beng estmted ncludes subvector of tme-nvrnt unknown prmeters. Some emples re gven to llustrte the pplcblty of the lgorthms obtned. Keywords: nonlner flterng multplctve mesurement errors multplctve forcng nose unknown prmeters ccurcy Crmer lower bound.. INRODUCION When effcent lgorthms for processng of mesurement dt re developed n the contet of the Byesn flterng theory t s common prctce for reserchers to solve two problems: the problem of the nlyss of the potentl ccurcy obtned usng the lgorthm optml n the sense of the chosen crteron the problem of desgn of computtonlly economcl lgorthm tht provdes ccurcy close to potentl. Such n pproch s wdely used n prtculr for the development of lgorthms for nvgton dt processng trckng problems (Dmtrev Stepnov 998 Bergmn Rstc et. l. 4). he covrnce mtr of estmton errors of the optml lgorthm s conventonlly used s chrcterstc of the potentl ccurcy. hs mtr s determned by smulton whch nvolves the procedure for clcultng the optml estmte. It s well known tht generlly t s mpossble to desgn unversl computtonlly convenent optml lgorthm for the problems of nonlner flterng. Despte the fct tht reserchers hve dvnced n desgnng such lgorthms recently due to n prtculr the pplcton of vrous modfctons of the Monte Crlo method (Doucet Gustfsson et l. Rstc 4) the clculton of optml estmtes by these methods s computtonlly ntensve (Snyder 8 Stepnov Berkovsky ). In ths regrd the development of ppromte procedures for the nlyss of potentl ccurcy of estmton s vtlly mportnt for the soluton of ppled problems. One of such procedures s bsed on the clculton of the Crmer-Ro lower bound (CRLB) (Vn rees 968). he methods of obtnng lgorthms for CRLB clculton ther pplcton n nonlner flterng problems hve been the subject mtter of mny publctons (Gldos 98 Vn rees Bell 7). For emple n (Koshev Stepnov 997 chvsky et l. 998 Sml et l. ) the uthors obtned convenent recurrence lgorthms for CRLB clculton for dscrete-tme nonlner flterng problems wth ddtve mesurement errors forcng (process) nose n the eutons for the stte vector. hese lgorthms hve been successfully used to solve wde rnge of problems relted n prtculr to the processng of nvgton dt (Dmtrev Stepnov 998 Bergmn 999 Btst et l ). However n prctce there s often need to solve problems n whch the propertes of forcng nose mesurement errors depend on the unknown stte vector to be estmted thus endowng them multplctve nture. It s to ths problem tht the pper s devoted. Actully we contnue the reserch reported n (Stepnov et l. ). Here we suppose tht not only propertes of forcng nose depend on the unknown stte vector but such dependency s lso vld for mesurement nose the ntl covrnce mtr. hese generlztons re very mportnt n estmtng the prmeters of Mrkov rom processes wdely used n the problems of nvgton trckng dt flterng.. PROBLEM SAEMEN Let us ssume tht we hve composte n r -dmensonl whch ncludes n -dmensonl vector n Mrkov seuence... r -dmensonl vector =(... r ) of unknown tme-nvrnt prmeters descrbed by the followng eutons: - - w we lso hve m -dmensonl mesurements () y s () v. () Copyrght 4 IFAC 9557

2 s re the known nonlner vector- Here - functons of n m dmensons; () - re the known mtrces of n p m m dmensons the elements of whch re nonlner functons of ther rguments; w v re whte-nose zero-men Gussn seuences of p m dmensons for whch the reltons E{ wl wk } lkql E{ vl vk } lk Rl hold; Q l R l re covrnce mtrces; lk s the Kronecker opertor; re rom vectors wth the known probblty densty functon (PDF) f ( ) f ( /) f () where f( /) s Gussn.е. f ( /) N ; P wth E { } E { } P. Along wth ntroduce composte vectors X... ( we X X Y y... y ) of ( ) n (( ) n r) m dmensons. We cn wrte the followng the Crmer- Ro neulty for the vector X X where (Gldos 98): ˆ ( ) ˆ ( ) EX Y X X Y X X Y J G () ln ln ( ) d f X Y d f X Y J EX Y (4) dx dx f ~ ( X Y ) s the PDF for vectors X seprte the lower ( n r) ( n r) dgonl block n Y. Let us J n( nr) ( ) nr n Ι nr J (5) Ι nr where ( nr) n ( n r) n s zero mtr Ι ( nr ) s unt n r mtr. he mtrces for vectors. determne CRLB he purpose of ths work s to obtn recurrence lgorthm for.. ALGORIHM FOR CRLB Dong mthemtcl opertons n the wy smlr to tht of (Stepnov et l. ) we cn show tht for the followng reltons hold good: F L L (6) L L. (7) Here Q L Q F Φ F s Q R L Q Φ (8) F F F j Φ L s R L Q Q L Φ Q L s R Q Q () Here: E P (). d d F E ln f ln f d d d P d P F E tr P P dl dμ E dln f d ln det P d d d ln det P dln f d d Φ - Φ E Q ( - -) s E R s - s () s E R s - s () sj (9) s - () s E R Φ Φ Φ E Q ( -) Φ Φ Φ ( -) E Q Φ Φ Φ E Q ( -) Q ( ) ( ) Q ( ) Q E Q ( ) R R () () () R () R () R () E tr R () () R l μ l μ=.r 9558

3 ( ) ( ) R = R R R ( lμ) E tr R ( ) ( ) R l μ lμ=. n ( ) ( ) R R R R ( lμ) E tr R ( ) ( ) R l μ Q ( -) Q( ) l Q lμ E tr Q ( -) Q( ) μ Q ( -) Q( ) l Q lμ E tr Q ( j-) Q( ) μ Q ( -) Q( ) l Q lμ E tr Q ( -) Q( ) μ μ=. nl.r lμ=. n lμ=. r μ=. nl.r. In these eutons the followng nottons re used: s( ) s( )... n..... ds( ) ds ( ) ds( )..... d. d..... d sm( ) sm( )... n Note tht when f () N ; P P - wrte F P. Let us consder some specfc cses. P we cn Cse. he subvector s bsent ( r ) the mesurement errors forcng nose re ddtve; moreover ( ) ()= I s unt mtr f ( /) f () P P f () N P ;. For these ssumptons Q s j Q Q j therefore R Q Q ГQ Г s Φ Lj () Φ s Q P.() Snce Q does not depend on the stte vector we cn wrte: Φ - Φ E Q - Q s Q Q ( ) Q P () d( ) where E d ( ) ( ) E Q s ( ) s ( ) s E R. d d d d It s cler tht () concde wth the eutons n (Koshev et l. 997 chvsky 998 Sml et l. ). Let us rewrte () n the form: s Q Q ( ) Q Q ( ) where Q ( ). (4) Applyng the mtr nverson lemm to (4) we cn wrte one more vrnt of the recurrence relton for Q s ( ) ( ).(5) Remrk. It should be noted tht by ncludng the subvector n the stte vector we cn lso obtn the CRLB for usng (5) for some cses (Koshev Stenov 997) whch for emple s true when the euton for the sttevector s lner. It s cler becuse usng (5) we do not need to nvert mtr Q. However we should keep t n mnd tht n generl cse the CRLB for obtned usng the lgorthm wth the vector of unknown prmeters ncluded n the stte vector s hgher or eul to the CRLB for clculted usng the lgorthm obtned n ths pper (Koshev 998). In other words the CRLB for obtned usng the lgorthm consdered n ths pper s more ect. For more detls see vrnt for emple. Cse. he stte vector ncludes subvector ( r ) ll ddtonl ssumptons re the sme s for cse then - F P R Q Q Q Q Q. Eutons (6) (7) re the sme but the eutons for the mtrces ncluded n them re dfferent.е. F L L L L F F s L Φ d d L F E f f Φ ln ln d d L j Φ L s L L Φ s Q Φ P. 9559

4 If n ddton j s Φ then ll Lj j ; therefore or F (6) F F s Φ s Q (7) P (8) s Q P ( ) ( ).(9) Below we gve some smplest emples to llustrte the pplcton of the reltons obtned. 4. EXAMPLE Assume tht we need to estmte n unknown prmeter of rom wlk (Wener process) zt () by ts dscrete mesurements wth ddtve mesurement errors. Let us consder dfferent vrnts of the problem soluton. Model. We use the followng model for dscrete tme: where tw () y z v s ( ) v v. () - - s ( ) t w t s the smplng ntervl; v w re zero-men Gussn whte nose wth vrnces v w respectvely; re ndependent rom vlues wth PDF N f ( ) s PDF for whch the f ( ) ( ; ) E( ) E( ) re known. A feture of ths model s tht the shpng flter for z does not depend on nonlnerty s only due to nonlnerty n mesurements. Usng the bove reltons we cn wrte: Q ( ) - w tw s E v v Φ Φ Q Q ( wt) t Φ j w t s E v v s Φ w t R Q Q L j.. kng nto consderton the fct tht ths emple corresponds to cse n ddton ll Lj j. we cn use (6) (7) (9). hus: tw v s ( ) Q F F s F F F j Fnlly w t v ( ) /..5( ) tw F () v tw v It s lso esy to see tht for. () f ( ) N( ; ) then F. hus we cn stte the fct tht n the cse under consderton the type of f ( ) t fed vlues of E( ) E( ) does not prctclly ffect the fnl result. It should be noted tht the CRLB for the model () () s euvlent to the covrnce n the lner estmton problem of vector () by mesurements ln ln y v ln ln y t w v where v ln re ndependent of zero-men Gussn whte-nose seuences wth vrnces v wheres re ndependent of ech other. Gussn rom vlues wth vrnces F. In other words the vlue tht determnes the CRLB for corresponds to the cse of estmton from mesurements of the form () under the ssumpton tht s replced by the known coeffcent t w. In turn the CRLB for corresponds to the cse of estmton from the sme mesurements but under nother ssumpton nmely tht s replced by the known coeffcent. Model. Let us nclude the unknown prmeters n the stte vector ( ) use the sme model () () but n so dong our m s to fnd the recurrence relton for the CRLB for vector. In ths cse tkng nto consderton the fct tht s ( ) s ( ) - t w we cn use (5) whch does not reure nonsngulrty of 956

5 mtr then Q. Snce s ds ( ) s ( ) s ( ) ( ) d( ) E. t w By vrtue of the fct tht the form: s E (9) tkes tw t w where P. F (4) It s esy to see tht the result genertng by (4) concde wth () (). kng nto consderton the bove remrk we note tht n ths emple the CRLBs correspondng to dfferent lgorthms re dentcl. Model. We cn use nother shpng flter for z : tw (5) y v (6) where v w re the sme s n the prevous cse. As n the frst two models we ssume tht for f ( ) the frst two moments E( ) E( ) re known besdes the vlue of E f ( ) d s lso determned. It should be noted tht Gussn PDF does not stsfy the ltter reurement becuse such ntegrl dverges. he feture of ths sttement s tht the model for the mesurements re lner wheres euton for the stte vector s nonlner snce the coeffcent of the forcng nose depends on the unknown prmeter. From (5) (6) t follows tht - - s -. In ths cse: Q ( ) - w t Φ E wt s v s Φ s Φ E R Q Q w t Φ Q ( ) ( ) Q ( ) t w Q E wt d d j. F E ln f ln f d d Q E wt Lj. Snce here too ll Lj j. usng (9) we cn wrte F F d d ln ln d d F E f f v v v ( ) P. ( ) herefore we hve F (7) v P. v (8) In ths emple we clculte the CRLB for whch determnes the propertes of the Wener process. It s nterestng to compre () (7) for the sme vlues of. Let ssume tht f.5 e (Webull PDF). In ths cse π π 4 F where s the Gmm functon. Fg. presents the results of the CRLB clcultons obtned usng () (7) wth.9. v.. w t. Here we lso gve the vlues of the root-men-sure (RMS) error for the optml estmte computed usng Monte-Crlo L smulton s MC ( j j ) ˆ L j where L-number of j smples n Monte-Crlo smulton; ˆ j re the smples optml estmtes clculted usng the lgorthm descrbed for emple n (Ivnov et l. ). 956

6 Number of mesurement Fg.. CRLB for model (5) (6); CRLB for model () () RMS error for the optml estmte for two models L=5. From the curves bove t follows tht frstly the CRLB s unfortuntely sgnfcntly less thn the rel RMS error secondly the CRLB depends on the model used for z. It s cler tht n the ccurcy nlyss t mkes sense to use n upper envelope correspondng to the two CRLBs. 5. CONCLUSIONS Recurrence reltons hve been obtned for the clculton of CRLB n the dscrete-tme nonlner flterng problem n the condtons when the forcng nose mesurement errors ntl covrnce mtr depend on the stte vector to be estmted whch lso ncludes the subvector of unknown tmenvrnt prmeters. Some specfc cses hve been consdered. he relton between the derved recurrence lgorthm for the CRLB clculton the known lgorthms correspondng to the cse of ddtve forcng mesurement nose hs been estblshed. An emple of CRLB clculton n the estmton problem of the prmeters of the rom wlk process hs been consdered. he results obtned llowed the concluson tht there s n obvous dependence of CRLB on the type of the model used to descrbe the process under study for nonlner flterng problem. hs dependence s worth further study. REFERENCES Btst P. Slvestre C. Olver P. (). Prelmnry results on the estmton performnce of sngle rnge source loclzton. Proceedngs of st Medterrnen Conference on Control & Automton. Pltns-Chn Crete Greece June 5-8 pp Bergmn N. (999). Recursve Byesn estmton. Nvgton trckng pplctons. Lnkopng Studes n Scence echnology. Dsserttons No Deprtment of Electrcl Engneerng Lnkopng Unversty SE-58-8 Lnkopng Sweden. Bergmn N. () Poster Crmer-Ro bounds for seuentl estmton. In Doucet A. No de Frets Gordon N (ed.) Seuentl Monte Crlo methods n prctce. Pp. 8. New York: Sprnger-Verlg. Dmtrev S.P. Stepnov O.A. (998). Nonlner flterng аnd nvgton. Proceedngs of 5 th Snt Petersburg Interntonl Conference on Integrted Nvgton Systems. Snt Petersburg. р Doucet A. No de Frets Gordon N (). Seuentl Monte Crlo methods n prctce. New York: Sprnger-Verlg 58 p. Gldos J.I. (98). A Crmer-Ro bound for multdmensonl dscrete-tme dynmcl systems. IEEE rns. Automt. Contr. vol. AC-5 N pp Gustfsson F. et l. () Prtcle Flters for Postonng Nvgton rckng. IEEE rnsctons on Sgnl Processng vol. 5 N. pp Ivnov V.M. Stepnov O.A. Korenevsk M.L. (). Monte Crlo Methods for Specl Nonlner Flterng Problem. Proceedngs of th IFAC Interntonl Workshop Control Applctons of Optmzton. vol.. P Koshev D.A. (998). Comprson of lower bounds of ccurcy n problems of nonlner estmton. Journl of Computer Systems Scences Interntonl vol. 7 N pp Koshev D.A. Stepnov O.A. (997). Applcton of the Ro-Crmer Ineulty n Problems of Nonlner Estmton. Journl of Computer Systems Scences Interntonl vol. 6 N pp. 7. Rstc B. Arulmplm S. Gordon N. (4). Beyond the Klmn flter: Prtcle flters for trckng pplctons. Artech House Publshers. Sml M. Krlovec J. chvsky P. () Flterng predctve smoothng Crmer-Re bounds for dscrete-tme nonlner dynmc systems Automtc vol. 7 pp Snyder C. Bengtsson. Bckel P. Anderson B. (8). Obstcles to hgh-dmensonl prtcle flterng. Monthly Wether Revew vol. 6 N pp Stepnov O.A. Vslyev V.A. Dolnkov A.S. (). Crmer-Ro lower bound for prmeters of rom processes n nvgton dt processng. Proceedngs of st Medterrnen Conference on Control & Automton. Pltns-Chn Crete Greece June 5 8 pp. 4. Stepnov О.А. Berkovsky N.А. (). Investgtng the clculton error of the optml Byesn estmte for nonlner problem wth the use of the Monte-Crlo method. Proceedngs of 8th World Congress Mlno August 8 September. chvsky P. Murvchk C. Nehor A. (998) Posteror Crmer-Ro bounds for dscrete-tme nonlner flterng. IEEE rnsctons on Sgnl Processng Vn rees H. L. (968). Detecton estmton modulton theory. Prt I: Detecton estmton lner modulton theory (Prt ). John Wley& Sons. Vn rees H. L. Bell K. L. (7). Byesn bounds for prmeter estmton nonlner flterng/trckng. Wley-IEEE Press. Reserch supported by the Russn Foundton for Bsc Reserch project no