Applied Mathematical Modelling

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1 Applied Maemaical Modelling 36 () Conens liss available a SciVerse ScienceDirec Applied Maemaical Modelling journal omepage: Recursive sae esimaion for ybrid sysems Evgenia Suzdaleva a,, Ivan Nagy b a Deparmen of Adapive Sysems, Insiue of Informaion Teory and Auomaion of e ASCR, Pod vodárenskou věží 4, 88 Prague, Czec Republic b Faculy of Transporaion Sciences, Czec Tecnical Universiy, Na Florenci 5, Prague, Czec Republic aricle info absrac Aricle isory: Received January Received in revised form 5 Augus Acceped 6 Augus Available online 6 Augus Keywords: Recursive sae esimaion Hybrid sysems Sae-space model Filering Mixed daa Te paper deals wi recursive sae esimaion for ybrid sysems. An unobservable sae of suc sysems is canged bo in a coninuous and a discree way. Fas and efficien online esimaion of ybrid sysem sae is desired in many applicaion areas. Te presened paper proposes o look a is problem via Bayesian filering in e facorized (decomposed) form. General recursive soluion is proposed as e probabiliy densiy funcion, updaed enry-wise. Te paper summarizes general facorized filer specialized for (i) normal sae-space models; (ii) mulinomial sae-space models wi discree observaions; and (iii) ybrid sysems. Illusraive experimens and comparison wi one of e counerpars are provided. Ó Elsevier Inc. All rigs reserved.. Inroducion Sysems wose sae is canging dynamically coninuously in ime and also swicing among several discree values are undersood as ybrid sysems. A sae of a ybrid sysem is modeled by coninuous variables wiin several discree modes, among em a sysem is swicing. Usually sysem parameers are canging according o a paricular mode. Hybrid sysems are widely used in many fields of signal processing (arge racking, medicine, speec recogniion, raffic conrol ec.). Fas and efficien online esimaion of eir sae is desired in some of ese areas. A lo of works are devoed o sae esimaion of ybrid sysems. One of e well-known approaces dealing wi swicing sysems wi Gaussian linear and discree saes is e ineracive muliple model (IMM) algorim []. I performs classical Kalman filer [] for eac mode under assumpion a is paricular mode is a rig one a curren ime sep. Ten e IMM algorim compues a weiged combinaion of updaed sae esimaes produced by all e filers yielding a final Gaussian mean and covariance. Tis mixed sae esimae is aken as e iniial one for e nex ime sep. Te weigs are cosen according o e probabiliies of e models, wic are compued in filering sep of e algorim. Te paper [3] proposes e exac filer for a specialized ybrid sysem sae. Te reference probabiliy meod for idden Markov models (HMM) is employed. Te soluion is presened as Gaussian sum wi explicily compued specific weigs, means and variances. However, a number of saisics grows geomerically in ime, and provided resuls are resriced only by 5 ime seps. Te approac [4] considers anoer special case of a dynamic linear sae-space model wi measuremen marices swicing according o ime-varying independen random process. Te updaing of probabiliies is derived as an applicaion of Bayes rule o e weiged observaion model. Te esimaion of e normal sae is sown as exension of e classical Kalman filer wi involved weiged combinaions of e gain-adjused innovaions. Ieraive ecniques for jump Markov linear sysems are nicely presened in [5]. Te algorims are derived o obain e marginal maximum a poseriori sequence esimae of e finie sae Markov cain. Te paper [6] is concerned wi opimal Corresponding auor. addresses: (E. Suzdaleva), (I. Nagy) X/$ - see fron maer Ó Elsevier Inc. All rigs reserved. doi:.6/j.apm..8.4

2 348 E. Suzdaleva, I. Nagy / Applied Maemaical Modelling 36 () filering for ybrid sysems wi non-gaussian noises. Te derived filer is opimal in e sense of e mos probable rajecory (MPT) esimae. Te sae and e observaion are considered as a pair of deerminisic processes wi swicing coefficien as a random process. Despie e claimed generaliy of soluion, is can resric applicaion in pracice. Te paper [7] proposes mixure Kalman filer based on a special sequenial Mone Carlo meod using a random mixure of Gaussian disribuions for approximaion of arge poserior disribuion. Te approac deals wi condiional dynamic linear models (CDLM) wi mixed Gaussian noises defined via known indicaor process. Te weiged sample of e indicaors is used wiin e proposed effecive filer. A series of oer researc in e field of nonlinear ybrid sysems [8] and online realime sae esimaion [9] can be also found. Te presened paper is focused on modeling of sysem saes as condiionally dependen enries of e sae vecor. Teir enry-wise recursive esimaion is subsequenly reaced via facorizaion of e sae-space model and prior disribuions for Bayesian filering []. A par of e work dealing wi esimaion of discree sae is also closely relaed o algorims based on idden Markov models (HMM) eory []. However, ese algorims run in offline mode suppored by Mone Carlo compuaions. Imporan feaures of e proposed eory are a: e algorims used run in online mode, numerical procedures are applied only in a pars, wic canno be compued analyically. In is way e amoun of compuaions as well as e risk of collapsing is minimized, general probabilisic approac is universal for e disribuions used, i opens a way o recursive esimaion of discree sysem modes dependen on evoluion of coninuous saes. Tis is planned for fuure researc. Te paper is srucured as follows. Necessary preliminaries are provided in Secion. Secion 3 presens general probabilisic soluion of e facorized form of Bayesian filering. Te paper summarizes general facorized filer specialized for (i) normal sae-space models in Secion 4; (ii) mulinomial sae-space models wi discree-valued observaions in Secion 5 and (iii) ybrid sysems in Secion 6. Secion 7 demonsraes examples wi real daa and comparison wi e IMM filer. Remarks in Secion 8 close e paper. Derivaions of e proposed formulas are provided in Appendix A.. Preliminaries.. Sae-space model Te sysem is described by e sae-space model in e form of e following condiional probabiliy (densiy) funcions (p(d)fs) for simpliciy denoed as pdfs wiin is paper observaion model f ðy jx ; u Þ; ðþ sae evoluion model f ðx þ jx ; u Þ; ðþ were e sysem oupu y and e conrol inpu u are measured a discree ime momens = {,...,T}. In general, e variables are column vecors suc a y =[y ;,...,y Y; ],u =[u ;,...,u U; ]. Te sysem sae x =[x ;,...,x X; ] is no direcly observed and as o be esimaed in an online (recursive) mode... Bayesian filering Bayesian filering, esimaing e sysem sae, includes e following coupled formulas. Daa updaing f ðx jdðþþ ¼ f ðy jx ; u Þf ðx jdð ÞÞ R / f ðy f ðy jx ; u Þf ðx jdð ÞÞdx jx ; u Þf ðx jdð ÞÞ; ð3þ incorporaes informaion conained in observaions D()=(d,...,d ), were d (y,u ). Relaion (3) also comprises e naural condiions of conrol [], according o ose f ðx ju ; Dð ÞÞ ¼ f ðx jdð ÞÞ: Time updaing Z f ðx þ jdðþþ ¼ f ðx þ jx ; u Þf ðx jdðþþdx ; ð4þ fulfills sae predicion. Te prior pdf f(x jd()) wic expresses e subjecive prior knowledge on e sysem iniial sae sars e recursions. Applicaion of (3,4) o linear Gaussian sae-space model provides Kalman filer [].

3 E. Suzdaleva, I. Nagy / Applied Maemaical Modelling 36 () Cain rule An operaion inensively used rougou e paper is: Cain rule f ða; bjcþ ¼f ðajb; cþf ðbjcþ; wic decomposes e join pdf f(a,bjc) ino a produc of condiional pdfs for any random variables a, b and c. ð5þ 3. General soluion in a facorized form Bayesian filering (3,4) is proposed o be done in one inegraion sep, i.e., 8 9 Z >< >= f ðx þ jdðþþ / f ðx þ jx ; u Þ f ðy jx ; u Þf ðx jdð ÞÞ >: fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} >; dx ; ð6þ /f ðx jdðþþ wic is obained by a rivial subsiuion of e sae esimae updaed by measuremens (3) in e ime updaing (4). A basic idea of e approac is o apply e cain rule (5) o models (,) and o (6). Aferwards, models (,) are facorized as f ðy jx ; u Þ¼ Y j¼ f ðx þ jx ; u Þ¼ YX f ðy j; jy ðjþþ:y; ; x :X; ; u :U; Þ; f ðx i;þ jx ðiþþ:x;þ ; x :X; ; u :U; Þ; a is a produc of facors a are condiional pdfs of corresponding disribuions. A noaion of e form x (i+):x; denoes a sequence {x i+;,x i+;,...,x X; } for curren ime insan, wic is empy, wen (i +)P X. Subsiuion of (7,8) in (6) and applicaion of e cain rule o e prior pdf f(x jd( )) provide e following facorized form of (6), i.e., Y X Z Y X Y f ðx i;þ jx ðiþþ:x;þ ;DðÞÞ / f ðx i;þ jx ðiþþ:x;þ ;x :X; ;u :U; Þ f ðy j; jy ðjþþ:y; ;x :X; ;u :U; Þ YX f ðx i; jx ðiþþ:x; ;Dð ÞÞdx ; were inegraion is assumed o be done over x =[x ;,...,x X; ]. Formal facorizaion ino e facors elps in designing e resuling algorims as all e facors are scalar pdfs of respecive disribuions. j¼ ð7þ ð8þ ð9þ 4. Facorized filer for linear normal models Le us apply e proposed facorized soluion (9) o linear normal sae-space model. In is field, e sequenial Kalman filer [3] can be found closely relaed o e proposed one. In conras o e sequenial filer, e facorized soluion is no resriced by a diagonal measuremen covariance marix (as well as e process one). Tis is a significan benefi of e approac, since full covariances conribue o a beer qualiy of esimaion of normally disribued sae. Furermore, facorizaion of covariance marices for Kalman filering is ofen aimed a more compuaional sabiliy via a lesser rank of e marix, e.g. e Square-Roo and U-D Kalman filers [3]. Te presened algorim explois marix facorizaion for reacing e enry-wise updaing of sae esimae. Te normal observaion model () as e form z} { f ðy ju ; x ÞN Cx þ Hu fflfflfflfflfflffl{zfflfflfflfflfflffl} ; R v mean covariance 8 9 >< A >= ¼ðpÞ Y jrv j exp ½y Cx Hu Š R v ½y Cx Hu Š >: fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} >; ; ðþ were N ðþ denoes normal disribuion; C and H are parameers supposed o be known or esimaed offline; Rv is a known covariance marix of e measuremen Gaussian noise wi zero mean; Q y denoes a quadraic form inside e exponen. Similarly, e sae evoluion model () is 8 9 >< >= f ðx þ ju ; x ÞN xþ ðax þ Bu ; R w Þ¼ðpÞ X jrw j exp ½x þ Ax Bu Š R w ½x þ Ax Bu Š >: fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} >; ; ðþ Q x Q y

4 35 E. Suzdaleva, I. Nagy / Applied Maemaical Modelling 36 () were A and B are known parameers of appropriae dimensions; R w is a known covariance marix of e process Gaussian noise wi zero mean; and Q x is a quadraic form inside e exponen. Applicaion of recursion (9) o normal models (,) leads o a facorized version of Kalman filer. For normal disribuion, e poserior sae esimae preserves is form Y X f ðx i;þ jx ðiþþ:x;þ ; DðÞÞ; for facors via LDL decomposiion [] of e precision (i.e., inverse covariance) marices. Suc a decomposiion supposes L o be a lower riangular marix wi uni diagonal, D o be a diagonal one and denoing ransposiion. Tis ype of marix decomposiion is an analogue of facorizaion (7,8) via e cain rule for normal models (,). Te facorizaion of (,) can be clearly demonsraed via exploiaion of e quadraic forms Q y and Q x. Le us firsly facorize e observaion model (). Marix Rv is invered ino a precision marix and decomposed so a R v ¼ L v D v L v : ð3þ Te resuled quadraic form corresponding o normal disribuion () is 3 6 Q y ¼ L v y L v Hu L fflffl{zfflffl} {z} v C 7 4 x 5 D v L v y q Ax ; ð4þ q A wic elps o express e j oupu facor as scalar pdf variance N yj; q j; X Y L k¼jþ v;kjy k; þ X z} { X A B jlx l; ; fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} D v;jj A ; mean value were L v;kj ; A jl and Dv;jj are elemens of marices L v ; A and Dv respecively. Normal model () is facorized quie similarly via e following operaions wi marix R w and quadraic form Q x, i.e., R w ¼ L wd w L w ; 3 ð6þ 6 Q x ¼ L w x þ L w Bu Lw fflffl{zfflffl} {z} A 7 4 x 5 D w L w x þ z Nx ; ð7þ z N resuling ino normal facor of e i sae N xi;þ z i; X k¼iþ L w;ki x k;þ þ X N il x l; ; D w;ii ðþ ð5þ! ; ð8þ were L w;ki, N il and D w;ii are elemens of marices L w, N and D w respecively. Te prior sae disribuion is cosen as e normal one wi mean l and covariance marix P for =. I is ransformed o a similar form as follows. P ¼ L pj D pj L pj ; i ð9þ Dpj i Q pj ¼ L pj x l f L pj x l f ; were l f ¼ L pjl ; ðþ were Q pj is a resuled quadraic form for e normal disribuion of e iniial sae x, wic enables expressing e prior facorized sae esimae as N xi; l f i; X k¼iþ L pj;ki x k; ; D pj;ii! ; ðþ were L pj;ki and D pj;ii are elemens of marices L pj and D pj respecively. Usage of e quadraic forms (4), (7) and () allows o represen an elegan form of soluion (9) for normal models. Subsiuing e facorized disribuions in (9) and afer all rearrangemens, one obains e poserior sae esimae in e preserved form (), or precisely (), for e i facor, i.e.,! X l f i;þ L pjþ;ki x k;þ ; ðþ N xi;þ for a i olds k¼iþ D pjþ;ii

5 Q pjþ ¼ 6 4L uj L w fflffl{zfflffl} L pjþ l f þ ¼ L uj z þ D e 3 x þ l f 7 þ5 D w NC E. Suzdaleva, I. Nagy / Applied Maemaical Modelling 36 () D uj {z} D pjþ Lpjþ x þ l f þ A D v L v y q þ Lpj D pj l f i ; ð3þ ð4þ and D w D w NC N D w ¼ D e ¼ L uj D uj L uj i ; ð5þ X i C ¼ N; A; L pj N; A; L pj ; ð6þ X ¼ diag½d w ; D v ; D pj Š: Deailed derivaions can be found in Appendix A. 4.. Algorim Te obained resuls can now be summarized in e form of an algorim. Iniial par of e algorim. Load daa y, u and parameers A, B, C, H, R w and Rv.. Se prior values l and P. 3. Facorize e observaion model () according o (3) and obain q ¼ L v Hu ; A¼L v C: 4. Facorize e sae evoluion model () according o (6) and compue z ¼ L w Bu ; N ¼ L w A: 5. Facorize e prior disribuion according o (9,) o obain L pj, D pj and l f. Online par of e algorim For ime from o T. Make diagonal marix X = diag[d w,dv,d pj ]. i X i. Compue C ¼ N; A; L pj N; A; L pj : 3. Compue marix D e ¼ D w D w NC N D w. 4. Facorize marix D e ¼ L uj D uj L uj. 5. Compue e facorized sae esimae () according o (3) and (4), i.e., l f ¼ L uj L pj ¼ L uj L w ; D pj ¼ D uj : z þ e D End of e cycle for. D w NC 5. Facorized filer for discree models A D v L v y q þ Lpj D pj l f ; Le us apply e proposed facorized soluion (9) o discree models wi mulinomial disribuion. In is area e HMM approaces are widely used. However, e presened online filer is based on explici soluion and avoids Mone Carlo compuaions. Here facors are obained naurally since mulivariae discree variables are reduced o scalars wi finie number of possible values. Te mulinomial observaion model (), i.e., f ðy jx ; u Þ is provided by e oupu ransiion able and a known (or esimaed offline) probabiliy a qjl,n wi muli-index qjl,n. Tis muli-index denoes realizaions q {,...,Q} of random discree variable y a ime insan according o a se of is possible values {,...,Q}, were Q is a finie number. Realizaion q in e muli-index qjl,n is condiioned by realizaions l {,...,L}of discree sae x and n {,...,N} of discree inpu u from eir ses of possible values wi finie numbers L and N. Noaion a qjl,n reflecs probabiliy of ransiion of oupu y o e discree value q, i.e., y = q condiioned by x = l and u = n. I olds ð7þ ð8þ

6 35 E. Suzdaleva, I. Nagy / Applied Maemaical Modelling 36 () X Q q¼ a qjl;n ¼ ; and a qjl;n P 8q; l; n: Similarly, e sae evoluion model () f ðx þ jx ; u Þ; is e mulinomial disribuion presened by e sae ransiion able conaining known probabiliy b ljm,n wi a muli-index ljm,n. Here e muli-index is evolved in a similar way as for e observaion model bu e condiion m {,...,L}, wic relaes o value of e discree sae x a ime insan, wile l ere belongs o x +. I olds X L b ljm;n ¼ ; and b ljm;n P 8l; m; n: Te prior disribuion of e discree sae is cosen as e mulinomial one f ðx jdð ÞÞ ¼ p x ; a as e form of a vecor conaining e iniial probabiliies p l "l {,...,L} a ime insan, and i as o be recursively esimaed for ime +. I olds X L p l ¼ ; and p l P 8l: Subsiuing models (8,9) in (9) (ere precisely (6)) wi incorporaion of e prior disribuion (3), one obains e following expression, wic simulaneously updaes e esimae by acual measuremens and predics e sae, i.e., f ðx þ jdðþþ / X x f ðx þ jx ; u Þf ðy jx ; u Þf ðx jdð ÞÞ; ð3þ ð9þ ð3þ were inegraion is replaced by regular summaion. For eac value l {,...,L} ofx + and wi discree observaions y = q {,...,Q} and u = n {,...,N} available a ime insan e prediced probabiliy p l for ime insan + is explicily compued as p l ¼ b ljn a qjn p þ b ljn a qjn p þþb ljln a qjln p L ; and en normalized, i.e., p l p l ¼ P L p ; l ð3þ resuling in e mulinomial disribuion f ðx þ jdðþþ ¼ p xþ ; ð33þ wic preserves e original form (3) and can be used for e nex sep of recursive esimaion. 6. Facorized filer for ybrid sysems Le us consider a ybrid sysem wi e observed oupu y ¼ y c ; yd i wi y c ¼ yc ; ;...; yc Y ; and y d ¼ y Y;, were superscrip c denoes a coninuous ype of a variable, wile superscrip d belongs o a discree variable. Here e case bo wi normally disribued and mulinomial variables is considered. Te conrol inpu is similarly u ¼ u c ; ud ¼ i, u c ; ;...; uc U ; ; ud and e unobserved sae o be esimaed is x ¼ x c ; i xd ¼ x c ; ; ;...; xc X ; ; x X; were xx; ¼ x d : Facorizaion of pdfs sown in (9) allows o represen i in e following way YX f x c i;þ jx ðiþþ:x;þ; DðÞ f x d þ jdðþ Z X Y X / x c xc i;þ jx ðiþþ:x;þ; x c :X ; ; uc f x d þ jxd ; ud x d fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} f ðx þ jx ;u Þ Y Y j¼ yc j; jy ðjþþ:y;; x c :X ; ; uc f y d jxd ; ud fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} f ðy jx ;u Þ Y X f xc i; jx ðiþþ:x;; Dð Þ f x d jdð Þ ; dx c fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} prior pdf ð34þ

7 E. Suzdaleva, I. Nagy / Applied Maemaical Modelling 36 () wi assumpions a coninuous enries can be omied from e condiion for discree sae and oupu, and e pas discree sae and a discree inpu from e condiion for y c, as well as ud for xc þ. Using relaions (3,4), (6) and (3), one can see a e prescribed original form of e poserior pdf is desroyed in (34): i is a sum of disribuions X x d f x d þ jdðþ Y X f x c i;þ jx ðiþþ:x;þ; DðÞ : ð35þ I is necessary o resore e original form o use i for e nex sep of esimaion. An approximaion based on Kerridge inaccuracy [4] is an explici soluion, wic resores e original form of e pdf via compuaion of a specific weiged combinaion of e pdfs involved in (35). Kerridge inaccuracy is a par of Kullback Leibler divergence [5] adoped as a eoreically jusified proximiy measure. Tis divergence is known o be an opimal ool wiin e Bayesian approac []. For any random variable a, Kerridge inaccuracy is used o measure e proximiy of pdfs f(a) and ^f ðaþ Z K a ðf ðaþk^f ðaþþ ¼ a f ðaþ ln ^f ðaþ da; and is minimizaion allows o find e approximaed pdf ^f ðaþ. According o is approximaion [], e original form of pdf is resored and e produc YX ^f x c i;þ jx ðiþþ:x;þ; DðÞ f x d þ jdðþ ; ð37þ is used as e prior pdf for e nex sep of recursive esimaion (34). Le us apply e presened soluion for e sysem wi normal facors provided by (5), (8) and () and discree facors from (8) (3). Soluion (34) relaed o normal facors coincides wi a proposed in Secion 4 running for eac value l of discree sae. A par of (34) ouside e inegral corresponds o discree facors and is explained in Secion 5. Relaion (35) in is case is e mixure disribuion X L p l Y X N xi;þ l f i;þ X k¼iþ L pjþ;ki x c k;þ ; D pjþ;ii! : ð36þ 5 Car queues on arm 5 Car queues on arm Queue leng [cars] 5 Queue leng [cars] ime [periods] 5 3 ime [periods] Car queues on arm 3 Car queues on arm 4 Queue leng [cars] real esimaed Queue leng [cars] ime [periods] 3 ime [periods] Fig.. Queue leng esimaion wi e proposed filer.

8 354 E. Suzdaleva, I. Nagy / Applied Maemaical Modelling 36 () Resoring e original normal form needs o use e approximaion based on Kerrigde inaccuracy [4]. According o [], for e case of normal pdfs e Kerridge inaccuracy (36) is minimized wi e following mean and covariance marix of e approximaed disribuion ^l þ ¼ XL p l l l;þ ; were l l;þ ¼ L lðpjþþ l f lðþþ ; ð38þ bp þ ¼ XL p l P l;þ þ XL p l ð^l þ l l;þ Þ ; were ; P l;þ ¼ L lðpjþþ D lðpjþþ L lðpjþþ ð39þ were subscrip l denoes resuls obained for eac value l of discree sae. Te approximaion (38) is en facorized according o (9,) and used as e prior normal disribuion for e nex sep of e recursion. To summarize e obained soluion, one can srucure i as follows.. Compue e sae esimae for discree facors, see Secion 5.. Compue e normal sae esimaes, see Secion 4., running e algorim for eac discree sae value. 3. Resore e original form via (38) and facorize i. Noe a e above specializaion is sown for vecor i ; x ¼ x c ; ;...; xc X ; ; x X; i were x X; ¼ x d. For anoer case, for insance, xc ; ;...; x X ;; x c X; wi xx ; ¼ x d and xc X; one sould use disribuions modeling discree variables dependen on coninuous ones. Te proposed facorizaion enables o consider is ask a will be presened soon. Car queues on arm 5 Car queues on arm Queue leng [cars] ime [periods] Queue leng [cars] ime [periods] Queue leng [cars] Car queues on arm 3 real esimaed Queue leng [cars] Car queues on arm 4 3 ime [periods] 3 ime [periods] Fig.. Queue leng esimaion wi e IMM filer.

9 E. Suzdaleva, I. Nagy / Applied Maemaical Modelling 36 () Experimens To es e proposed approac, a real daa sample conaining inensiy (number of cars per ime uni) of e raffic flow in a cosen poin of raffic communicaions in Prague as been aken. In pracice in e field of raffic-flow conrol, fas online Discree sae esimaion wi e facorized filer real esimaed 3 Level of service ime [periods] Fig. 3. LoS esimaion wi e proposed filer. Discree sae esimaion wi e IMM filer real esimaed 3 Level of service ime [periods] Fig. 4. LoS esimaion wi e IMM filer.

10 356 E. Suzdaleva, I. Nagy / Applied Maemaical Modelling 36 () Table Esimaion error and correc poin esimaes. EE CPE Facorized filer.96 4 IMM filer sae esimaion is imporan: i can influence e adapive conrol of e microregion via adequae green lig ime. A normally disribued sae x c of e considered ybrid sysem is a four-dimensional queue leng of cars waiing for passing roug a raffic microregion. A full dimension of e aken normal sae is eig, since occupancy of a measured deecor is added o e vecor o ensure observabiliy of e model. A discree sae x d is a level of service (LoS) of e microregion. I expresses a degree of raffic sauraion in a sense ow easy cars can pass roug e microregion wi 4 possible values: from (e bes) o 4 (e wors). Te measured daa used were: y c car ougoing inensiy along wi occupancy of a measured deecor; yd a ime mode of a workday (morning peak-our ime, lunc, lae afernoon peak-our ime, evening); u c a relaive ime of e green lig; u d a discree variable, reflecing weer e sauraed sraegy of e adapive conrol is used or no. A duraion of e online filering was workday, wic corresponds o 88 ime periods. Te filering sared a midnig a simplifies a coice of prior disribuions (i.e., zero queue leng and LoS = ). Te sae esimaion was performed via e presened approac and, o compare, wi e elp of e IMM filer implemened in oolbox [6]. Comparison of ese filers provided e following resuls. Significan difference beween ese meods is a e proposed one considers e probabilisic sae-space model in a general form bo for e normal and e discree saes and akes ino accoun ybrid observaions and conrol inpus. Te discree sae esimaion in e IMM filer is based mosly on e sae ransiion able, i.e., oer discree variables bringing some informaion are no aken ino accoun. Tis caused a worsened sabiliy of e IMM filer during e esing. Resuls of e online queue leng esimaion for four arms of e considered raffic microregion (ere an inersecion) are sown in Figs.,, were e firs figure corresponds o e proposed ybrid filer, and e second figure o e IMM filer. Te LoS esimaion for bo e filers is demonsraed in Figs. 3, 4. Te esimaion error (EE) compued as EE ¼ X x c T l þ x c l þ ; were T = 88 is e duraion of e esimaion and x c is e sae idenified wi e real one, is provided in Table for bo e meods. A number of correcly poin-esimaed saes (CPE) from e oal 88-daa sample was evaluaed for bo e filers and sown in Table. I is assumed a for beer qualiy of esimaion, EE sould ave a minimal value, and CPE on e conrary a maximal (from 88) value. An advanage of e facorized ybrid filer is raer significan. I sould be also noed a bo e oupu and e sae ransiion ables used for e facorized filer were given as raer uncerain models. However, usage of more deerminisic ransiion able for e IMM filer does no improve is resuls. 8. Conclusion Te paper is devoed o recursive sae esimaion a can be applied o ybrid sysems. Te proposed soluion is based on e facorized form of Bayesian filering. Te paper summarizes applicaion of general facorized filer o normal, discree and ybrid models. Te presened algorims run in online mode avoiding numerical procedures as far possible. A number of saisics does no grow in ime and e risk of collapsing is minimized. An imporan conribuion is a e presened approac can be evolved for recursive esimaion of discree modes dependen on evoluion of coninuous saes. Te proposed meod demonsraes beer sabiliy and qualiy of esimaion in comparison wi e IMM filer. Acknowledgemens Te researc was suppored by projecs MŠMT M57 and TAČR TA33. Appendix A. Derivaions for normal models Subsiuing e facorized normal disribuions wi quadraic forms (4), (7) and () ino (9), one obains e following funcion o be inegraed 8 Z >< 9 >= exp ½Q x þ Q y þ Q pj Š >: fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} >; dx ; ða:þ Q e

11 E. Suzdaleva, I. Nagy / Applied Maemaical Modelling 36 () were o faciliae algebraic rearrangemens, e following addiional noaions in e quadraic form Q e appear, i.e., Q e ¼ L w x 7 4 þ z Nx 5 D w L w fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} x 6 þ z Nx þ L 7 4 v y q fflfflfflfflffl{zfflfflfflfflffl} Ax 5 D v L v y q Ax b b 3 6 þ 4 l f {z} b 3 L pj x 7 5 D pj l f L pj x i ; ða:þ were b, b, b 3 and b =[b ;b ; b 3 ] are column vecors. To ave e variable x in (A.) inegraed ou, one as o fulfill e compleion of squares [] in (A.) for x. Afer a and subsequen inegraion of non-normalized Gaussian pdf [7], e variable x is being inegraed ou. Te compuaional resul of e filering (A.) is proporional o exp k wi e following remainder k obained afer inegraion i i X k ¼ b X X N; A; L pj C N; A; L pj b; ða:3þ were X ¼ diag½d w ; D v ; D pj Š; ða:4þ i X i C ¼ N; A; L pj N; A; L pj : ða:5þ Wi e elp of algebraic rearrangemen of e remainder (A.3) using compleion of squares for x +, one obains e following quadraic form L w x þ z D e D w NC A D v L v y i q þ Lpj D pj l f e D L w x þ z D e D w NC A D v L v y i q þ Lpj D pj l f ; ða:6þ were ed ¼ D w D w NC N D w : ða:7þ Te marix e D, obained in (A.7) is decomposed so a ed ¼ L uj D uj L uj : ða:8þ Te decomposiion (A.8) and facorizaion of e quadraic form (A.6) (i.e., is muliplicaion by riangular marix L uj ) enable o preserve e prior form () and obain e following resul 3 i Q pjþ ¼ 6 4L uj L w x þ l f 7 þ fflffl{zfflffl} 5 D uj Lpjþ {z} x þ l f þ ; ða:9þ L D pjþ pjþ were l f þ ¼ L uj z þ D e D w NC A D v L v y q þ Lpj D pj l f : ða:þ References [] Y. Bar-Salom, T. Kirubarajan, X.-R. Li, Esimaion wi Applicaions o Tracking and Navigaion, Wiley, New York, NY, USA,. [] M. Grewal, A. Andrews, Kalman Filering: Teory and Pracice Using MATLAB, Second ed., Wiley,. [3] F.D.R.J. Ellio, D. Sworder, Exac ybrid filers in discree ime, IEEE Transacions on Auomaic Conrol 4 () (996) [4] R. Sumway, D. Soffer, Dynamic linear models wi swicing, Journal of e American Saisical Associaion 86 (45) (99) [5] A. Douce, C. Andrieu, Ieraive algorims for sae esimaion of jump markov linear sysems, IEEE Transacions on Signal Processing 49 (6) () 6 7. [6] Q. Zang, Opimal filering of discree-ime ybrid sysems, Journal of Opimizaion Teory and Applicaions (). [7] R. Cen, J.S. Liu, Mixure kalman filers, Journal of e Royal Saisical Sociey Series B 6 () [8] J.S.-H. Tsai, C.-T. Wang, C.-C. Kuang, S.-M. Guo, L.-S. Sie, C.-W. Cen, A narmax model-based sae-space self-uning conrol for nonlinear socasic ybrid sysems, Applied Maemaical Modelling 34 () () [9] J.-B. Seu, Y.-H. Cou, A. Cen, Socasic modeling and real-ime predicion of inciden effecs on surface sree raffic congesion, Applied Maemaical Modelling 8 (5) (4) [] M. Kárný, J. Böm, T.V. Guy, L. Jirsa, I. Nagy, P. Nedoma, L. Tesař, Opimized Bayesian Dynamic Advising: Teory and Algorims, Springer, London, 5. [] M.J. Beal, Z. Garamani, C.E. Rasmussen, Te infinie idden markov model, in: Advances in Neural Informaion Processing Sysems, Vol. 4, (). [] V. Peerka, Bayesian sysem idenificaion, in: P. Eykoff (Ed.), Trends and Progress in Sysem Idenificaion, Pergamon Press, Oxford, 98, pp [3] D. Simon, Opimal Sae Esimaion: Kalman, H Infiniy, and Nonlinear Approaces, Wiley, 6. [4] D. Kerridge, Inaccuracy and inference, Journal of Royal Saisical Sociey B 3 (96)

12 358 E. Suzdaleva, I. Nagy / Applied Maemaical Modelling 36 () [5] S. Kullback, R. Leibler, On informaion and sufficiency, Annals of Maemaical Saisics (95) [6] J. Harikainen, S. Särkkä, Opimal filering wi kalman filers and smooers a manual for malab oolbox ekf/ukf, <p://www.lce.u.fi/researc/ mm/ekfukf/> (February 8). [7] E. Suzdaleva, M. Kárný, Facorized Filering, Tec. Rep. 6, ÚTIA AV ČR, Praa, <p://www.library.uia.cas.cz/prace/69.pdf> (6).

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