Posterior Cramer-Rao Bounds for Multi-Target Tracking

Size: px
Start display at page:

Download "Posterior Cramer-Rao Bounds for Multi-Target Tracking"

Transcription

1 Poseror Cramer-Rao Bounds for Mul-Targe Trackng C. HUE INRA France J-P. LE CADRE, Member, IEEE IRISA/CNRS France P. PÉREZ IRISA/INRIA France ACRONYMS B1 PCRB compued under he assumpon ha he assocaons are known B PCRB compued under he A1 and A assumpons B3 PCRB compued under he A1 and A3 assumpons CRB Cramér-Rao bounds PCRB Poseror Cramér-Rao bounds IRF Informaon reducon facor EM Expecaon-maxmzaon algorhm EKF Exended Kalman fler KF Kalman fler PDAF Probablsc daa assocaon fler JPDAF Jon probabls daa assocaon fler MHT Mulple hypoheses racker PMHT Probablsc mulple hypoheses racker MOPF Mulple objecs parcle fler RMSE Roo mean square error. Ths sudy s concerned wh mul-arge rackng (MTT. The Cramér-Rao lower bound (CRB s he basc ool for nvesgang esmaon performance. Though bascally defned for esmaon of deermnsc parameers, has been exended o sochasc ones n a Bayesan seng. In he arge rackng area, we have hus o deal wh he esmaon of he whole rajecory, self descrbed by a Markovan model. Ths leads up o he recursve formulaon of he poseror CRB (PCRB. The am of he work presened here s o exend hs calculaon of he PCRB o MTT under varous assumpons. NOTATIONS A º B A B posve sem-defne r X [(@=@ x1,:::,(@=@ xnx ] T Y T X r X r Y E p Expecaon compued w.r.. he densy p J (p E[ log(p] Leer used as an ndex o denoe me varyngbeween0andt Leer used as an exponen o denoe one of he M arges j Leer used as an exponen o denoe one of he m measuremens a me P d Deecon probably Parameer of he Posson law modelng he number of false alarms V observaon volume. I. INTRODUCTION Manuscrp receved June 4, 003; revsed February 14, 004 and Aprl 19, 005; released for publcaon May 1, 005. IEEE Log No. T-AES/4/1/ Refereeng of hs conrbuon was handled by P. K. Wlle. Auhors addresses: C. Hue, INRA, Cenre de Recherches de Toulouse, BP 7, F-3136, Casane, Tolosan Cedex, France, E-mal: (chue@toulouse.nra.fr; J-P. Le Cadre and P. Pérez, IRISA, Campus de Beauleu, 3504 Rennes Cedex, France /06/$17.00 c 006 IEEE Ths sudy s concerned wh mul-arge rackng (MTT,.e., he esmaon of he sae vecor made by concaenang he sae vecors of several arges. As assocaon beween measuremens and arges are unknown, MTT s much more complex han sngle-arge rackng. Exsng MTT algorhms generally presen wo basc ngredens: an esmaon algorhm coupled wh a daa assocaon mehod. Among he mos popular algorhms based on (exended Kalman flers (EKFs are he jon probablsc daa assocaon fler (JPDAF, he mulple hypohess racker (MHT or, more recenly, he probablsc MHT (PMHT. They vary on he assocaon mehod n use. Wh he developmen of he sequenal Mone Carlo (SMC mehods, new opporunes for MTT have appeared. The sae IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 4, NO. 1 JANUARY

2 dsrbuon s hen esmaed wh a fne weghed sum of Drac mass cenered around parcles. The Cramér-Rao lower bound (CRB [1] s wdely used for assessng esmaon performance. Though a grea deal of aenon has been pad o measures of performance such as rack 1 pury and correc assgnmen rao [] hese mehods are based on dscree assgnmens of measuremens o racks and are hus no unversally applcable. Ther neres s, o a large exen, due o he fac ha numerous MTT algorhms rely on hard assocaon. Whn hs framework hs ype of analyss s que pernen; bu here s a need for a smple and versale formulaon of a performance measure n he MTT conex; whch leads us o focus on CRB. These bounds are developed here n a general framework whch employs a probablsc srucure on he measuremen-o-arge assocaon. Agan, he dffculy of obanng CRB for MTT s due o a need for an assocaon beween measuremens and racks, and o ncorporae hs basc sep n he CRB calculaon. Thus, esmaon of he arge saes on he one hand, and of he measuremen-o-rack assocaon probables on he oher, are ghly relaed. On anoher hand, whle he CRB s an essenal ool for analyzng performance of deermnsc sysems, he poseror CRB (PCRB s a measure of he maxmum nformaon whch can be exraced from a dynamc sysem when boh measuremens and sae are assumed o be random, hus evaluang performance of he bes unbased fler. Thus, performance analyss s now consdered n a Bayesan seup. Naurally, hs analyss deals wh racks and dmenson grows lnearly wh me. Que remarkably, has been shown ha a recursve Rcca-lke formulaon of he PCRB could be derved under reasonable assumpons. Here, we show ha hs framework s sll vald n he MTT seup and allows us o derve convenen bounds. Ths paper s organzed as follows. The MTT problem s nroduced n Secon II, followed by a bref background on PCRB for nonlnear flerng (Secon III. Secon IV s he core of hs manuscrp snce deals wh he dervaon of he PCRB for MTT, under varous assocaon modelngs. These bounds are llusraed by compuaonal resuls. II. THE MULTI-TARGET TRACKING PROBLEM A. General Framework Le M be he number of arges o rack, assumed o be known and fxed here. The ndex desgnaes one among he M arges and s always used as 1 By rack, we consder here a sequence of saes assocaed wh a Markovan model. superscrp. MTT consss n esmang he sae vecor made by concaenang he sae vecors of all arges. I s generally assumed ha he arges are movng accordng o ndependen Markovan dynamcs, even hough can be crczed lke n [3]. A me, X =(X 1,:::,XM follows he sae equaon decomposed n M paral equaons: X = F (X 1,V 8 =1,:::,M: (1 The noses (V and(v0 are supposed only o be whe boh emporally and spaally, and ndependen for 6= 0. The observaon vecor colleced a me s denoed by y =(y 1,:::,ym. The ndex j s used as frs superscrp o refer o one of he m measuremens. The vecor y s composed of deecon measuremens and cluer measuremens. The false alarms are assumed o be unformly dsrbued n he observaon area. Ther number s assumed o arse from a Posson densy ¹ f of parameer V where V s he volume of he observaon area and he average number of false alarms per un volume. As we do no know he orgn of each measuremen, one has o nroduce he vecor K o descrbe he assocaons beween he measuremens and he arges. Each componen K j s a random varable ha akes s values among f0,:::,mg. Thus, K j = ndcaes ha y j s assocaed wh he h arge f =1,:::,M and ha s a false alarm f.inhe frs case, y j s a realzaon of he sochasc process: Y j = H (X,Wj f K j = : ( Agan, he noses (W j and(w j0 are supposed only o be whe noses, ndependen for j 6= j 0.Wedo no assocae any knemac model o false alarms. A measuremen recepon, he ndexng of he measuremens s arbrary and all he measuremens have he same pror probably o be assocaed wh a gven model. Thevarables(K j,:::,m are hen supposed dencally dsrbued. Ther common law s defned wh he probably (¼ =1,:::,M : ¼ =P(K j = 8 j =1,:::,m : (3 The probably ¼ s hen he pror probably ha an arbrary measuremen s assocaed wh model. The erm model denoes he arge f =1,:::,M and he model of false alarms f = 0. Inuvely, hs probably represens he observably of arge for =1,:::,M. The¼ vecor s consdered as a realzaon of he sochasc vecor =( 0, 1,:::, M wh he followng pror dsrbuon on : p( =p( 0 p( 1,:::, M j 0 (4 where p( 1,:::, M j 0 s unform on he hyperplane defned by P M =1 = IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 4, NO. 1 JANUARY 006

3 To solve he daa assocaon some assumpons are commonly made [4]: A1. One measuremen can orgnae from one arge or from he cluer. A. One arge can produce zero or one measuremen a one me. A3. One arge can produce zero or several measuremens a one me. Assumpon A1 expresses ha he assocaon s exclusve and exhausve. Unresolved observaons are hen excluded. From a mahemacal pon of vew, P he oal probably heorem can be used and M ¼ = 1 for every. Assumpon A mples for j =1,:::,m are dependen. Assumpon A3 s ofen crczed because may no mach he physcal realy. However, allows o suppose he sochasc ndependence of he varables ha he assocaon varables K j and drascally reduces he complexy of he ¼ vecor esmaon. K j B. Revew of Man MTT Algorhms Le us now brefly revew he reamen of he daa assocaon problem. The followng algorhms essenally dffer accordng o her esmaon srucure (deermnsc or sochasc and her assocaon assumpons. Frs, he daa assocaon problem occurs as soon as here s uncerany n measuremen orgn and no only n he case of mulple arges. In he case of one sngle-arge rackng, he negraon of false alarms n he model hen mples daa assocaon. The probablsc daa assocaon fler (PDAF [5] akes no accoun hs uncerany under he classcal hypoheses A1 and A. The JPDAF s an exenson of he PDAF for mulple arges [6]. Boh hese algorhms are based on Kalman fler (KF and consequenly assume lnear models and addve Gaussan noses n (1 and (. The man approxmaon consss of assumng ha he predced law s sll Gaussan whereas s n realy a sum of Gaussan assocaed wh he dfferen assocaons. The MHT sll uses A1 and A bu allows he deecon of a new arge a each me sep [7]. To cope wh he exploson of he assocaon number, some of hem mus be gnored n he esmaon. For hese hree algorhms ((JPDAF, MHT, a pror sascal valdaon of he measuremens decreases he nal assocaon number. Ths valdaon s based on he fundamenal hypohess ha he law p(y j Y 1: 1 s Gaussan, cenered around he predced measuremen and wh he nnovaon covarance. The valdaon gae s hen usually defned as he measuremen se for whch he Mahalanobs dsance o he predced measuremen s lower han a ceran hreshold. Some deals can be found n [4] TABLE I Classfcaon of Man MTT Algorhms Accordng o Ther Assocaon Assumpon and Esmaon Srucure Assocaon Assumpon Esmaon srucure A1 A A1 A3 Kalman fler (JPDAF MHT EM PMHT parcle fler SIR-JPDAF MOPF for nsance. Ths valdaon gae procedure wll no be consdered hroughou, whch means ha all he measuremens wll be aken no accoun. Unlke he above algorhms, he PMHT s based on he assumpons A1 and A3. I proposes he bach esmaon of mulple arges n cluer va an expecaon-maxmzaon (EM algorhm. Radcally dfferen from a deermnsc approach lke KF-based rackers or EM-based rackers, he sochasc approach developed quckly hese las years. SMC mehods [8] esmae he enre a poseror law of he saes and no only he frs momens of hs law lke KF-based rackers do. In he conex of MTT, parcle flers are parcularly appealng: as he assocaon needs only o be consdered a a gven me eraon, he complexy of daa assocaon s reduced. For a sae of ar of he proposed algorhms he reader can refer o [9]. Agan, we can dsngush algorhms usng A for solvng daa assocaon lke he sequenal mporance resamplng (JPDAF, SIR-JPDAF [10] or usng A3 lke he mulple objecs parcle fler (MOPF [11]. Classfcaon of he above algorhms accordng o her assocaon assumpon and esmaon srucure are summarzed n Table I. III. BACKGROUND ON POSTERIOR CRAMÉR-RAO BOUNDS FOR NONLINEAR FILTERING I s of grea neres o derve mnmum varance bounds on esmaon errors o have an dea of he maxmum knowledge on he saes ha can be expeced and o assess he qualy of he resuls of he proposed algorhms compared wh he bounds. Frs defned and used n he conex of consan parameer esmaon, he nverse of he Fsher nformaon marx, commonly called he Cramér-Rao (CR bound, has been exended o he case of random parameer esmaon n [1], hen called he PCRB. Le X R n x be a sochasc vecor and Y R n y asochasc observaon vecor. The mean-square error of any esmae ˆX(Y sasfes he nequaly E( ˆX(Y X( ˆX(Y X T º J 1 (5 The nequaly means ha he dfference E( ˆX(Y X( ˆX(Y X T J 1 s a posve sem-defne marx. HUE ET AL.: POSTERIOR CRAMER-RAO BOUNDS FOR MULTI-TARGET TRACKING 39

4 where J = logp X,Y (X,Y=@X s he Fsher nformaon marx and where he expecaons are w.r.. he jon densy p X,Y (X,Y under he followng condons. X,Y (X,Y=@X p X,Y (X,Y=@X exs and are absoluely negrable w.r.. X and Y. The esmaor bas Z B(X= ( ˆX(Y Xp YjX (Y j XdY R ny sasfes: lm B(Xp(X, 8 l =1,:::,n x : X l! 1 (6 Le us consder he nonlnear dscree sysem for a unque objec: ½ X = F (X 1,V (7 Y = H (X,W and he assocaed flerng problem,.e., he esmaon of X gven Y 0: =(Y0,:::,Y. A frs approach consss of usng a lnear Gaussan sysem equvalen o (7 lke n [1] and [13]. The error covarance of he nal sysem s hen lower bounded by he error covarance of he Gaussan sysem. Neverheless, wo major remarks can be made [14]. Frs, he equvalen noon s no precsely defned n [1] and [13]. Second, seems no lkely ha here always exss such a lnear Gaussan sysem for nsance f he probably densy funcon (pdf s mulmodal. A revew of hs approach can be found n [14]. The approach recenly developed by Tchavsky, e al. n [15] orgnally consders he Fsher nformaon marx for he esmaon of X gven Y 0: as a submarx of he Fsher nformaon marx assocaed wh he esmaon of X 0: gven Y 0:. Usng he noaons of [15], J(X 0: denoes he (( +1n x ( +1n x nformaon marx of X 0: and J X denoes he n x n x nformaon submarx of X whch s he nverse of he n x n x rgh lower block of [J(X 0: ] 1. To avod nverson of oo large marces, a recursve expresson of he bound J X has been presenedrecenlyn[15]and[16]andsummarzed by he followng formula: J X+1 = DX DX 1 (J X + DX 11 1 DX 1 (8 where DX 11 = E[ X X logp(x +1 j X ] D 1 X = E[ X +1 X logp(x +1 j X ] DX 1 = E[ X X +1 logp(x +1 j X ] = [DX 1 ] T D X = E[ X +1 X +1 logp(x +1 j X ] + E[ X +1 X +1 logp(y +1 j X +1 ] (9 and where he r and operaors denoe he frs and second paral dervaves, respecvely: r X =,:::,, Y = r X r Y T : (10 xnx The marx J 1 X +1 provdes a lower bound on he mean-square error of esmang X +1. I can be shown n [17] ha hs bound s overopmsc bu has he grea advanage o be recursvely compuable. Le us see now some exensons recenly proposed for he PCRB. A. Inegraon of Deecon Probably In [18], he auhors propose o negrae he deecon probably n he prevous bound. For a scenaro of gven lengh, he bound s compued as a weghed sum on every possble deecon/nondeecon sequence. As he number of erms of hs sum grows exponenally he less sgnfcan are no aken no accoun. B. Exenson o Measuremen Orgn Unceranes Several works have suded CRBs for models wh measuremen orgn unceranes, bu for a sngle-arge. The assocaon of each measuremen o he arge or o he false alarm model can be done under he classcal hypoheses A1 and A or under A1 and A3. As CRB was frs defned for parameer esmaon, models wh deermnsc rajecores have frs been suded. If he nose s Gaussan, has been shown n [19] and [0] ha, under A1 and A, he nverse of he nformaon marx can hen be wren as he produc of he nverse of he nformaon marx whou false alarms by an nformaon reducon facor, noed IRF and lower han uny. In [1], he auhors show ha here s also an IRF for he PMHT measuremen model,.e., under he hypoheses A1 and A3. In he case of dynamc models, he exenson of he bound (8 o he case of lnear and nonlnear flerng wh measuremen orgn uncerany due o cluer has been recenly suded n [] and [3]. The exenson manly consss of replacng he classc pdf of he measuremen gven he sae by he pdf of he measuremen vecor akng no accoun he measuremen uncerany. The conclusons are he followng. 1 Under he assumpon of a Gaussan observaon nose wh a dagonal covarance marx, an IRF dagonal marx appears n he PCRB. The PCRB does no show nsably whereas rackng algorhms can relavely easly be pu no wrong. 40 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 4, NO. 1 JANUARY 006

5 3 The PCRB would be more affeced by a low P d han by a hgh sae or nose covarance or by a hgh cluer densy. 4 For low deecon probables, he PCRB s really overopmsc (versus PDAF RMSE. IV. POSTERIOR CRAMÉR-RAO BOUNDS FOR MULTI-TARGET TRACKING Now, le us see how he PCRB proposed n [15] can be exended and used n he case of mulple arges flerng defned by (1 and (. Noe ha n hs case, he measuremen vecor s composed of deecon measuremens ssued from he dfferen arges and of false alarms. The followng exenson hen akes no accoun smulaneously he measuremen uncerany and he exenson of one o mulple arges. Frs, he recursve equaon (8 can be obaned as well for mulple arges usng he srucure of he jon law: p(x 0:+1,Y 0:+1 =p(x 0:,Y 0: p(x +1 j X p(y +1 j X +1 : Ths srucure s sll rue for mulple arges, whch leads o he same recursve formula for he nformaon marx. As he arges are supposed o move accordng o ndependen dynamcs, we have logp(x 1:M +1 j X1:M = MX =1 (11 logp(x+1 j X : (1 Consequenly, he marces DX 11, DX 1 and he frs erm of DX are smply block-dagonal marces where he h block s compued w.r.. X and X+1.Iremanshe second erm of DX,.e.,E[ X1:M +1 logp(y X 1:M +1 j X+1 1:M]. +1 As n [], we can decompose hs erm accordng o he observaon number usng he oal probably heorem: E[ X1:M +1 logp(y X 1:M +1 j X+1 1:M ] +1 1X = P(m +1 E[ X1:M +1 logp(y m +1 X 1:M +1 j X+1 1:M ] : +1 m +1 =1 B(m +1 The probables P(m +1 aregvenby P(m +1 = ¹= d (13 ¹X ( V d exp V P ¹ d d : (14 d! To compue B(m +1, we have o face agan he assocaon problem: some addonnal hypoheses mus be formulaed o gve explc expressons of he lkelhood p(y m j X +1. The problem s ha hese hypoheses condon he esmaon algorhm, whle hey should no nfluence he heorecal bound. We propose here o derve hree bounds: B1, he PCRB compued under he assumpon ha he assocaons are known. B, he PCRB compued under he A1 and A assumpons. B3, he PCRB compued under he A1 and A3 assumpons. The followng lemma s used hroughou he sequel. LEMMA 1 Le X =(X 1,:::,X M R n x and Y R n y wo sochasc varables and 1, wo negers [1,:::, M], hen he followng expecaon equaly holds rue: E X E YjX [ X logp(y j X] X 1 = E X E YjX [r logp(y j X(r X X logp(y j 1 XT ]: Le us defne he followng noaon: for wo vecors, and p a probably law, (15 J (p =E[ log(p]: (16 In he nex hree paragraphs we descrbe J X1:M +1 (p(y m X 1:M +1 j X1:M +1 accordng o he assocaon +1 assumpons. A. PCRB B1 The assocaon vecor s supposed o be known. We hen have Xm logp(y = y m j X = x,k = k = logp(y j j x kj : (17 The graden of he log-lkelhood w.r.. X s no zero only f here exss j such ha k j =. Inhs case, r X logp(y m j x,k = r X p(yj j x : (18 p(y j j x We fnally oban for all =1,:::,M: J X (p(y m X and r Xp(y j j x j x,k = E X E j (r X p(y j j x T Y jx p(y j j x (19 J X (p(y X 1 j x,k = 0 f 1 6= : (0 HUE ET AL.: POSTERIOR CRAMER-RAO BOUNDS FOR MULTI-TARGET TRACKING 41

6 B. PCRB B We can wre logp(y = y m j X = x A1 A = log X k p(y =(y 1,:::,ym j x,k p(k =log X k Ym p(y j j x,k p(k : (1 The probably p(k = k can be compued from he deecon probably P d, he number of false alarms ª k, her dsrbuon law ¹ f and he bnary varable D K ( equal o one f he objec s deeced, zero else: p(k = k = ª k! m! ¹ f (ª k MY =1 P DK ( d MY (1 P d (: D1 K ( The graden of he log-lkelhood w.r.. X s Q Pk r X logp(y j x = r m X p(yj j x,k p(k : p(y j x (3 Le us denoe by k ¾ he assocaons ha assocae one measuremen o he h arge. Under A, here exss a mos one such measuremen, denoed j. Then, P k ¾ Qj6=j p(y j r X logp(y j x = j x,k p(k r X p(y j j x : p(y j x (4 Usng Lemma 1, we oban for all 1, =1,:::,M: =1 PMHT, he maxmzaon sep for ¼ depends on he preceden esmaes for X and vce versa. The esmaon qualy of one hen srongly affecs he esmaon qualy of he oher. Smlarly for he MOPF, he smulaed values for ¼ are used for smulaed X values and vce versa. In hs conex, seems o us naural o consder he PCRB for he esmaon of he jon vecor (,X. For all ha, he PCRB on he esmaon of X can be deduced from he global one by an nverson formula as we see laer. From he equaly P M ¼ =1andas¼0 s fxed a each nsan, we only consder he M 1 componens 1:M 1 =( 1,:::, M 1. Le us defne =( 1:M 1 ; he jon law s,x 1:M p +1 =p( 0:+1,Y 0:+1 =p p(y +1 j +1 p(x +1 j X p( +1 : (6 Le J( 0: be he nformaon marx of 0: assocaed wh p ; we are neresed n a recursve expresson on of he nformaon submarx J for esmang. Le us recall ha J s he nformaon submarx of whch s he nverse of he rgh lower block of [J( 0: ] 1. Usng he srucure of he jon law p +1 and he same argumen as n [15], he followng recursve formula can be shown (see he proof n he appendx: J +1 = D D 1 (J + D 11 1 D 1 (7 where 0 D 11 = J 0 (p(x +1 j X = 0 DX 11 0 D 1 = J +1 0 (p(x +1 j X = 0 DX 1 E[ X logp(y X 1 j X ] 6 = E X E Y jx 4 Pk ¾ 1 Q j6=j 1 p(yj j x,k p(k r X p(y 1 j1 j x 1 p(y j x X Y k ¾ p(y j j x,k p(k (r X j6=j p(y j 3 7 j x T 5 (5 where E X and E Y jx denoe, respecvely, he expecaon w.r.. he densy p(x andp(y j X. Le us noce ha he negrals w.r.. y are m n y -dmensonal. C. PCRB B3 To our knowledge, algorhms usng A3 need a jon esmaon of X and ¼.Inhsway,forhe D = J (p(y +1 j +1 p(x +1 j X p( +1 (8 0 0 = 0 J X +1 X +1 (p(x +1 j X J (p( J (p(y +1 j +1 : Once J s recursvely compued, a lower bound on he mean-square error of esmang X sgvenbyhe 4 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 4, NO. 1 JANUARY 006

7 nverson formula appled o he rgh lower block J X of J J X # J = : J X E( ˆX(Y X( ˆX(Y X T º [J X J X J 1 J X ] 1 : (9 As a unform pror s assumed for he law, J (p( +1 s zero. To evaluae he hrd erm of D,wecanwre J X and he same expressons for 1 = M by replacng ¼ M by 1 P M 1 ¼. Noce ha under hese assocaon assumpons, all he negrals w.r.. y j are n y -dmensonal. D. Mone Carlo Evaluaon for a Bearngs-Only Applcaon Leusbegnwhhecasewhereheevoluon model s lnear and Gaussan. As n [15], we logp(y = y j = Á A1 A3 Ym = log p(y j j Á # Xm ¼ 0 M 1 = log V X ¼0 p(yj j x M + (p(y j j x p(yj j x M ¼ + p(yj j x M : (30 =1 For 6= M, he graden w.r.. X s Xm r r X logp(y j Á =¼ X p(y j j x p(y j : (31 j Á A smlar expresson for = M s obaned by replacng ¼ M by 1 P M 1 ¼.For =1,:::,M 1: Xm p(y j j x r logp(y j Á = p(yj j x M p(y j : (3 j Á Usng Lemma 1, we oban for 1, 6= M J X (p(y X 1 j =E[r X 1 (r X = E 4 ¼ 1 ¼ logp(y j T ] m X r p(y j j x 1 E X1 j Y j (r X p(y j j x p(y j j Á T 3 5 analycally oban he followng equales: DX 11 = dagff T V 1 F g, 3 DX 1 =dagf F T V 1 g and J X +1 X +1 (p(x +1 j X = dagf V 1 g. In he general case of an observaon model wh an addve Gaussan nose defned as follows: p(y j j x we have =(¼ ny de 1= exp f 1 (yj H(x T 1 (y j H(x g r X p(y 1 j j x 1 =p(yj j x 1 I reads for he PCRB B1: r X 1 (36 H T (x 1 1 (y j H(x 1 : J X (p(y X j X = E X r X H T (x 1 (r X H T (x T (37 (38 (33 and he same expressons for 1 or = M by replacng ¼ M by 1 P M 1 ¼. For 1, 6= M: 3.e., he block-dagonal marx whose h block s equal o F T 1 V F. J 1 Xm (p(y j = E 4 E Y j j (p(y j j x 1 p(yj j x M # 3 (p(y j j x p(yj j x M 5: p(y j (34 j Á For 1, 6= M: J (p(y X 1 j = E 4¼ 1 Xm E Y j j p(y j j x p(yj j x M p(y j r j Á X 1 p(y j j x 1 #3 5 (35 HUE ET AL.: POSTERIOR CRAMER-RAO BOUNDS FOR MULTI-TARGET TRACKING 43

8 for he PCRB B: J X X 1 P (p(y j X = E X r X H T (x k E ¾ p(y 1 j x,k p(k (yj1 H(x 1 Y jx p(y j x X k ¾ p(y j x,k p(k (y j H(x # T 1 (r X H T (x T # (39 and for he PCRB B3: J X (p(y X 1 j = E ¼ 1 ¼ J (p(y X 1 j = E 4¼ 1 r X 1 H T (x 1 1 Xm E Y j j p(y j j x 1 p(yj j x p(y j j Á (y j H(x 1 (yj H(x T # 1 (r X H T (x #: T (40 r X 1 H T (x 1 1 Xm E Y j j p(y j j x p(yj j x M p(y j j Á p(y j j x 1 (yj H(x 1 #3 5: (41 In he bearngs-only applcaon, we have n y = 1andhenH T = H ha leads o some wrng smplfcaons. We deal wh classcal bearngs-only expermens wh hree arges. In he conex of a slowly maneuverng arge, we have chosen a nearly-consan-velocy model. 1 The Scenaro: The sae vecor X represens he coordnaes and he veloces n he x-y plane: X =(x,y,vx,vy for = 1,,3. For each arge, he dscrezed sae equaon assocaed wh me perod s µ X+ = I I X 0 I + I 0 1 AV 0 I (4 where I s he deny marx n dmenson and V s a Gaussan zero-mean vecor wh covarance marx V =dag[¾x,¾ y,¾ x,¾ y ]. A se of m measuremens s avalable a dscree mes and can be dvded no wo subses. 1 One subse s of rue measuremens whch follow (43. A measuremen produced by he h arge s generaed accordng o Y j =arcan µ y y obs x + W j x obs (43 where W j s a zero-mean Gaussan nose wh covarance ¾w :05 rad ndependen of V,andx obs and y obs are he Caresan coordnaes of he observer, whch are known. We assume ha he measuremen produced by one arge s avalable wh a deecon probably P d. The oher subse s of false measuremens whose number follows a Posson dsrbuon wh mean V where s he mean number of false alarms per un volume. We assume hese false alarms are ndependen and unformly dsrbued whn he observaon volume V. The nal coordnaes of he arges and of he observer are he followng (n meer and meer/second, respecvely: X 1 0 = (00,1500,1, 0:5T, X 3 0 =( 00, 1500,1,0:5T X 0 = (0,0,1,0T X0 obs = (00, 3000,1:,0:5 T : (44 The observer s followng a leg-by-leg rajecory. Is velocy vecor s consan on each leg and modfed a he followng nsans, so ha: Ã! vx obs µ 00,600,900 0:6 = vy00,600,900 obs à vx obs 400,800 vy obs 400,800 0:3! µ :0 = 0:3 : The rajecores of he hree objecs and of he observer are ploed n Fg. 1(a. E. The Assocaed PCRB The hree bounds are frs nalzed o J X0 = P 1 X 0 for B1 andb andj 0 = P 1 0 for B3 wherep X0 = (45 44 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 4, NO. 1 JANUARY 006

9 Fg. 1. (a Trajecores of he hree arges and of he observer. (b Measuremens smulaed wh P d :9 and V =3. dagfxcov g wh X cov =dagf150,150,0:1,0:1g and P 0 =dagfdagf0:05, =1,:::,M 1g;P X0 g. Then, o esmae he marces needed n he recurson formulas (8 or (7, we perform Mone Carlo negraon by carryng ou P1 ndependen sae rajecores and for each of hem P ndependen measuremen realzaons, and addonally P3 ndependen realzaons of he ¼ vecor for he PCRB B3 (P1, P, and P3 have been fxed o 100 n he followng compuaons. For nsance, he esmae Ĵ X of J X s compued as X 1 X 1 where J(x p1 Ĵ X X 1 = 1 P1P XP1 XP p1=1 p=1 J(x p1,y p1,p (46,y p1,p s he quany whose expecaon s o be compued n (39. We hen obaned he marx nequales: E( 1:M 1:M ˆX +1 (Y X( ˆX +1 (Y XT º B for =1,,3: (47 In he scenaro descrbed above, he marces B dmenson s equal o dm = 3 4 = 1. To nerpre he nequales (47, we have derved he scalar mean-square error gven by he race of (47: 1:M E( ˆX +1 (Y XT 1:M ( ˆX +1 (Y X rb (48 and he nequaly on he volume of he marces defned as he deermnan a he power 1=dm: [dee( 1:M 1:M ˆX +1 (Y X( ˆX +1 (Y XT ] 1=dm º [deb ] 1=dm : (49 We have compued he race and he volume of he hree bounds for dfferen values of he parameers ¾ x, ¾ y, P d, V. Frs, for a dynamc nose sandard ¾ x = ¾ y :0005 ms 1, a deecon probably P d :9 and V = 1,,3, he race and he volume are ploed agans me on he hree frs rows of Fg.. The resuls on he fourh row have been obaned for a hgher dynamc nose sandard ¾ x = ¾ y :001 ms 1, P d :9 and V =1.Theffh and las row corresponds o a scenaro where a deecon hole s smulaed for he frs objec durng a hundred consecuve nsans, beween mes 600 and 700. Whaever he parameers values, he nsan or he funcon f of he bounds consdered (race or volume, we always have f(b f(b3 f(b1 wh a greaer gap beween f(b3 and f(b1 han beween f(b and f(b3. More precsely, frs means ha he opmal performance whch can be obaned wh an algorhm usng assumpons A1 and A are below he opmal performance whch can be obaned wh an algorhm usng assumpons A1 and A3. Second, he opmal performance obaned wh an algorhm assumng he assocaon s known s far beer han for he wo precedng cases. For all ha, nohng can be concluded on he relave performance of he SIR-JPDA and of he MOPF for nsance. Such sudy needs he esmaon of he RMSE of boh algorhms over a hgh number of realzaons of he process and measuremen nose. For each couple of realzaon of boh noses, several runs of he algorhms are needed. To go back over he analyss of Fg., he plos presen wo peaks around mes 150 and 400. They correspond o nsans where bearngs from he hree arges are very close as shown n Fg. 1(b for one parcular realzaon of he rajecores and of he measuremens. Durng he second peak, he gap beween B and B3 on he one hand and B1 on he oher hand s wdenng. A slgh peak s also observed when he frs arge s no deeced (see las row of Fg.. Fnally, by comparng he hree frs rows, we observe ha he gap beween f(b and f(b3 s wdenng wh he cluer densy V. In all hese scenaros, as he deecon probably P d s srcly nferor o uny, may happen a one nsan ha no arge s deeced. If moreover no cluer measuremen s smulaed a ha nsan, he measuremen vecor Y s empy. In hs case, we smply se he expecaons J X +1 X +1 (p(y +1 j X +1 and J (p( +1 j X +1 o zero and he recursve formula (8 and (7 are reduced. V. CONCLUSION In hs manuscrp, an exenson of he PCRB from a sngle-arge o mul-arge flerng problem HUE ET AL.: POSTERIOR CRAMER-RAO BOUNDS FOR MULTI-TARGET TRACKING 45

10 Fg.. Trace and volume of he hree PCRB marces: B (dashed,b3 (sold,b1 (dashdoed. Lef column: race. Rgh column: volume. Frs (op row: ¾ x = ¾ y :0005 ms 1 and V = 1. Second row: ¾ x = ¾ y :0005 ms 1 and V =. Thrd row: ¾ x = ¾ y :0005 ms 1 and V = 3. Fourh row: ¾ x = ¾ y :0001 ms 1 and V = 1. Ffh (boom row: ¾ x = ¾ y :0005 ms 1 and a deecon hole beween mes 600 and 700 for objec IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 4, NO. 1 JANUARY 006

11 has been suded. Three bounds have been derved accordng o he assocaon assumpons beween he measuremens and he arges. Based on Mone Carlo negraon, esmaes of hese hree bounds have fnally been proposed and evaluaed for he bearngs-only applcaon. APPENDIX. RECURSIVE FORMULA OF PCRB B By defnon, he nformaon marx J( 0:+1 of 0:+1 assocaed wh he law p +1 can be expressed as J( 0:+1 = 6 4 J 0: 1 0: 1 (p +1 J 0: 1 (p +1 J : 1 (p J 0: 1 (p +1 J (p +1 J +1 (p +1 J 0: 1 +1 (p +1 J +1 (p +1 J (p +1 where J (p =E[ log(p]. Usng (6, reads J 0: 1 0: 1 (p +1 =J 0: 1 0: 1 (p J 0: 1 (p +1 =J 0: 1 (p J 0: 1 +1 (p +1 =J 0: 1 +1 (p (50 + J 0: 1 0: 1 (p(y +1 j +1 p(x +1 j X p( +1 (51 + J 0: 1 (p(y +1 j +1 p(x +1 j X p( +1 (5 + J 0: 1 (p(y +1 j +1 p(x +1 j X p( +1 J (p +1 =J (p +J (p(x +1 j X + J (p(y +1 j +1 p( +1 J +1 (p +1 =J +1 (p(x +1 j X + J +1 (p J (p +1 =J (p + J +1 (p(y +1 j +1 p( +1 + J (p(y +1 j +1 p(x +1 j X p( +1 : (53 (54 (55 (56 Usng (51 (56 and he noaon: A B J( 0: = we have he recursve formula: A B 0 6 J( 0:+1 = 4B C + D 11 where D 11 = J (p(x +1 j X D 1 = J +1 (p(x +1 j X B T C 0 D 1 T D 1 D D = J (p(y +1 j +1 p(x +1 j X p( +1 : ( (58 (59 Now, J +1 s he nverse of he rgh lower block of J( 0:+1 1. Usng wce a classcal nverson lemma, we oban 1 A J +1 = D [0 D 1 B 0 ] REFERENCES B T C + D 11 = D D 1 [C + D 11 B T A 1 B ] 1 D 1 D 1 = D D 1 [J + D 11 ] 1 D 1 : (60 [1] Van Trees, H. L. Deecon, Esmaon, and Modulaon Theory (Par I. New York: Wley, [] Chang, K. C., Mor, S. and Chong, C. Y. Performance evaluaon of rack naon n dense arge envronmens. IEEE Transacons on Aerospace and Elecronc Sysems, 30, 1 (1994, [3] Mahler, R. Mul-source mul-arge flerng: A unfed approach. SPIE Proceedngs, 3373 (1998, [4] Bar-Shalom, Y., and Formann, T. E. Trackng and daa assocaon. New York: Academc Press, [5] Bar-Shalom, Y., and Tse, E. Trackng n a cluered envronmen wh probablsc daa assocaon. In Proceedngs of he 4h Symposum on Nonlnear Esmaon Theory and s Applcaons, [6] Formann, T. E., Bar-Shalom, Y., and Scheffe, M. Sonar rackng of mulple arges usng jon probablsc daa assocaon. IEEE Journal of Oceanc Engneerng, 8 (July 1983, [7] Red, D. An algorhm for rackng mulple arges. IEEE Transacons on Auomaon and Conrol, 4, 6 (1979, [8] Douce, A., De Freas, N., and Gordon, N. (Eds. Sequenal Mone Carlo Mehods n Pracce. New York: Sprnger, 001. [9] Hue, C., Le Cadre, J-P., and Pérez, P. Sequenal Mone Carlo mehods for mulple arge rackng and daa fuson. IEEE Transacons on Sgnal Processng, 50, (Feb. 00, HUE ET AL.: POSTERIOR CRAMER-RAO BOUNDS FOR MULTI-TARGET TRACKING 47

12 [10] Oron, M., and Fzgerald, W. A Bayesan approach o rackng mulple arges usng sensor arrays and parcle flers. IEEE Transacons on Sgnal Processng, 50, (00, [11] Hue, C., Le Cadre, J-P., and Pérez, P. Trackng mulple objecs wh parcle flerng. IEEE Transacons on Aerospace and Elecronc Sysems, 38, 3 (July 00, [1] Bobrovsky, B. Z., and Zaka, M. A lower bound on he esmaon error for Markov processes. IEEE Transacons on Auomac Conrol, 0, 6 (Dec. 1975, [13] Galdos, J. I. ACramér-Rao bound for muldmensonal dscree-me dynamcal sysems. IEEE Transacons on Auomac Conrol, 5, 1 (1980, [14] Kerr, T. H. Saus of Cramér-Rao-lke lower bounds for nonlnear flerng. IEEE Transacons on Aerospace and Elecronc Sysems, 5, 5 (Sep. 1989, [15] Tchavský, P., Muravchk, C., and Nehora, A. Poseror Cramér-Rao bounds for dscree-me nonlnear flerng. IEEE Transacons on Sgnal Processng, 46, 5(May 1998, [16] Bergman, N. Recursve Bayesan esmaon: Navgaon and rackng applcaons. Ph.D. dsseraon, Lnköpng Unversy, Sweden, [17] Bobrovsky, B. Z., Mayer-Wolf, E., and Zaka, M. Some classes of global Cramér-Rao bounds. The Annals of Sascs, 15, 4 (1987, [18] Farna, A., Rsc, B., and Tmmoner, L. Cramér-Rao bound for non lnear flerng wh P d < 1 and s applcaon o arge rackng. IEEE Transacons on Sgnal Processng, 50, 8 (00, [19] Jauffre, C., and Bar-Shalom, Y. Track formaon wh bearng and frequency measuremens n cluer. IEEE Transacons on Aerospace and Elecroncs, 6, 6 (1990, [0] Krubajan, T., and Bar-Shalom, Y. Low observable arge moon analyss usng amplude nformaon. IEEE Transacons on Aerospace and Elecorncs, 3, 4 (1996, [1] Ruan, Y., Wlle, P., and Sre, R. A comparson of he PMHT and PDAF rackng algorhms based on her model CRLBs. In Proceedngs of SPIE Aerosense Conference on Acquson, Trackng and Ponng, Orlando, FL, Apr [] Zhang, X., and Wlle, P. Cramér-Rao bounds for dscree-me lnear flerng wh measuremen orgn unceranes. In Workshop on Esmaon, Trackng, and Fuson: A Trbue o Yaakov Bar-Shalom, May 001. [3] Hernandez, M., Marrs, A., Gordon, N., Maskell, S., and Reed, C. Cramér-Rao bounds for nonlnear flerng wh measuremen orgn uncerany. In Proceedngs of 5h Inernaonal Conference on Informaon Fuson, July IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 4, NO. 1 JANUARY 006

13 Carne Hue was born n She receved he M.Sc. degree n mahemacs and compuer scence n 1999 and he Ph.D. degree n appled mahemacs n 003, boh from he Unversy of Rennes, France. Snce he end of 003 she has been a full-me researcher a INRA, he French Naonal Insue for Agrculural Research. Her research neress nclude sascal mehods for model calbraon, daa assmlaon, sensvy analyss, and n parcular, he Bayesan approach for agronomc models. Jean-Perre Le Cadre (M 93 receved he M.S. degree n mahemacs n 1977, he Docora de 3 eme cycle n 198, and he Docora d Ea n 1987, boh from INPG, Grenoble. From 1980 o 1989, he worked a he GERDSM (Groupe d Eudes e de Recherche en Deecon Sous-Marne, a laboraory of he DCN (Drecon des Consrucons Navales, manly on array processng. Snce 1989, he s wh IRISA/CNRS, where he s Dreceur de Recherche a CNRS. Hs neress are now opcs lke sysem analyss, deecon, mularge rackng, daa assocaon, and operaons research. Dr. Le Cadre has receved (wh O. Zugmeyer he Eurasp Sgnal Processng bes paper award (1993. Parck Pérez was born n He graduaed from ÉcoleCenralePars,France, n 1990 and receved he Ph.D. degree from he Unversy of Rennes, France, n Afer one year as an Inra pos-docoral researcher n he Deparmen of Appled Mahemacs a Brown Unversy, Provdence, RI, he was apponed a Inra n 1994 as a full me researcher. From 000 o 004, he was wh Mcrosof Research n Cambrdge, U.K. In 004, he became senor researcher a Inra, and he s now wh he Vsa research group a Irsa/Inra-Rennes. Hs research neress nclude probablsc models for undersandng, analysng, and manpulang sll and movng mages. HUE ET AL.: POSTERIOR CRAMER-RAO BOUNDS FOR MULTI-TARGET TRACKING 49

Filtrage particulaire et suivi multi-pistes Carine Hue Jean-Pierre Le Cadre and Patrick Pérez

Filtrage particulaire et suivi multi-pistes Carine Hue Jean-Pierre Le Cadre and Patrick Pérez Chaînes de Markov cachées e flrage parculare 2-22 anver 2002 Flrage parculare e suv mul-pses Carne Hue Jean-Perre Le Cadre and Parck Pérez Conex Applcaons: Sgnal processng: arge rackng bearngs-onl rackng

More information

Fall 2010 Graduate Course on Dynamic Learning

Fall 2010 Graduate Course on Dynamic Learning Fall 200 Graduae Course on Dynamc Learnng Chaper 4: Parcle Flers Sepember 27, 200 Byoung-Tak Zhang School of Compuer Scence and Engneerng & Cognve Scence and Bran Scence Programs Seoul aonal Unversy hp://b.snu.ac.kr/~bzhang/

More information

John Geweke a and Gianni Amisano b a Departments of Economics and Statistics, University of Iowa, USA b European Central Bank, Frankfurt, Germany

John Geweke a and Gianni Amisano b a Departments of Economics and Statistics, University of Iowa, USA b European Central Bank, Frankfurt, Germany Herarchcal Markov Normal Mxure models wh Applcaons o Fnancal Asse Reurns Appendx: Proofs of Theorems and Condonal Poseror Dsrbuons John Geweke a and Gann Amsano b a Deparmens of Economcs and Sascs, Unversy

More information

V.Abramov - FURTHER ANALYSIS OF CONFIDENCE INTERVALS FOR LARGE CLIENT/SERVER COMPUTER NETWORKS

V.Abramov - FURTHER ANALYSIS OF CONFIDENCE INTERVALS FOR LARGE CLIENT/SERVER COMPUTER NETWORKS R&RATA # Vol.) 8, March FURTHER AALYSIS OF COFIDECE ITERVALS FOR LARGE CLIET/SERVER COMPUTER ETWORKS Vyacheslav Abramov School of Mahemacal Scences, Monash Unversy, Buldng 8, Level 4, Clayon Campus, Wellngon

More information

Robustness Experiments with Two Variance Components

Robustness Experiments with Two Variance Components Naonal Insue of Sandards and Technology (NIST) Informaon Technology Laboraory (ITL) Sascal Engneerng Dvson (SED) Robusness Expermens wh Two Varance Componens by Ana Ivelsse Avlés avles@ns.gov Conference

More information

HEAT CONDUCTION PROBLEM IN A TWO-LAYERED HOLLOW CYLINDER BY USING THE GREEN S FUNCTION METHOD

HEAT CONDUCTION PROBLEM IN A TWO-LAYERED HOLLOW CYLINDER BY USING THE GREEN S FUNCTION METHOD Journal of Appled Mahemacs and Compuaonal Mechancs 3, (), 45-5 HEAT CONDUCTION PROBLEM IN A TWO-LAYERED HOLLOW CYLINDER BY USING THE GREEN S FUNCTION METHOD Sansław Kukla, Urszula Sedlecka Insue of Mahemacs,

More information

Dynamic Team Decision Theory. EECS 558 Project Shrutivandana Sharma and David Shuman December 10, 2005

Dynamic Team Decision Theory. EECS 558 Project Shrutivandana Sharma and David Shuman December 10, 2005 Dynamc Team Decson Theory EECS 558 Proec Shruvandana Sharma and Davd Shuman December 0, 005 Oulne Inroducon o Team Decson Theory Decomposon of he Dynamc Team Decson Problem Equvalence of Sac and Dynamc

More information

( ) () we define the interaction representation by the unitary transformation () = ()

( ) () we define the interaction representation by the unitary transformation () = () Hgher Order Perurbaon Theory Mchael Fowler 3/7/6 The neracon Represenaon Recall ha n he frs par of hs course sequence, we dscussed he chrödnger and Hesenberg represenaons of quanum mechancs here n he chrödnger

More information

Solution in semi infinite diffusion couples (error function analysis)

Solution in semi infinite diffusion couples (error function analysis) Soluon n sem nfne dffuson couples (error funcon analyss) Le us consder now he sem nfne dffuson couple of wo blocks wh concenraon of and I means ha, n a A- bnary sysem, s bondng beween wo blocks made of

More information

CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 4

CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 4 CS434a/54a: Paern Recognon Prof. Olga Veksler Lecure 4 Oulne Normal Random Varable Properes Dscrmnan funcons Why Normal Random Varables? Analycally racable Works well when observaon comes form a corruped

More information

Relative controllability of nonlinear systems with delays in control

Relative controllability of nonlinear systems with delays in control Relave conrollably o nonlnear sysems wh delays n conrol Jerzy Klamka Insue o Conrol Engneerng, Slesan Techncal Unversy, 44- Glwce, Poland. phone/ax : 48 32 37227, {jklamka}@a.polsl.glwce.pl Keywor: Conrollably.

More information

On One Analytic Method of. Constructing Program Controls

On One Analytic Method of. Constructing Program Controls Appled Mahemacal Scences, Vol. 9, 05, no. 8, 409-407 HIKARI Ld, www.m-hkar.com hp://dx.do.org/0.988/ams.05.54349 On One Analyc Mehod of Consrucng Program Conrols A. N. Kvko, S. V. Chsyakov and Yu. E. Balyna

More information

WiH Wei He

WiH Wei He Sysem Idenfcaon of onlnear Sae-Space Space Baery odels WH We He wehe@calce.umd.edu Advsor: Dr. Chaochao Chen Deparmen of echancal Engneerng Unversy of aryland, College Par 1 Unversy of aryland Bacground

More information

Linear Response Theory: The connection between QFT and experiments

Linear Response Theory: The connection between QFT and experiments Phys540.nb 39 3 Lnear Response Theory: The connecon beween QFT and expermens 3.1. Basc conceps and deas Q: ow do we measure he conducvy of a meal? A: we frs nroduce a weak elecrc feld E, and hen measure

More information

( t) Outline of program: BGC1: Survival and event history analysis Oslo, March-May Recapitulation. The additive regression model

( t) Outline of program: BGC1: Survival and event history analysis Oslo, March-May Recapitulation. The additive regression model BGC1: Survval and even hsory analyss Oslo, March-May 212 Monday May 7h and Tuesday May 8h The addve regresson model Ørnulf Borgan Deparmen of Mahemacs Unversy of Oslo Oulne of program: Recapulaon Counng

More information

In the complete model, these slopes are ANALYSIS OF VARIANCE FOR THE COMPLETE TWO-WAY MODEL. (! i+1 -! i ) + [(!") i+1,q - [(!

In the complete model, these slopes are ANALYSIS OF VARIANCE FOR THE COMPLETE TWO-WAY MODEL. (! i+1 -! i ) + [(!) i+1,q - [(! ANALYSIS OF VARIANCE FOR THE COMPLETE TWO-WAY MODEL The frs hng o es n wo-way ANOVA: Is here neracon? "No neracon" means: The man effecs model would f. Ths n urn means: In he neracon plo (wh A on he horzonal

More information

CHAPTER 10: LINEAR DISCRIMINATION

CHAPTER 10: LINEAR DISCRIMINATION CHAPER : LINEAR DISCRIMINAION Dscrmnan-based Classfcaon 3 In classfcaon h K classes (C,C,, C k ) We defned dscrmnan funcon g j (), j=,,,k hen gven an es eample, e chose (predced) s class label as C f g

More information

Outline. Probabilistic Model Learning. Probabilistic Model Learning. Probabilistic Model for Time-series Data: Hidden Markov Model

Outline. Probabilistic Model Learning. Probabilistic Model Learning. Probabilistic Model for Time-series Data: Hidden Markov Model Probablsc Model for Tme-seres Daa: Hdden Markov Model Hrosh Mamsuka Bonformacs Cener Kyoo Unversy Oulne Three Problems for probablsc models n machne learnng. Compung lkelhood 2. Learnng 3. Parsng (predcon

More information

Time-interval analysis of β decay. V. Horvat and J. C. Hardy

Time-interval analysis of β decay. V. Horvat and J. C. Hardy Tme-nerval analyss of β decay V. Horva and J. C. Hardy Work on he even analyss of β decay [1] connued and resuled n he developmen of a novel mehod of bea-decay me-nerval analyss ha produces hghly accurae

More information

Math 128b Project. Jude Yuen

Math 128b Project. Jude Yuen Mah 8b Proec Jude Yuen . Inroducon Le { Z } be a sequence of observed ndependen vecor varables. If he elemens of Z have a on normal dsrbuon hen { Z } has a mean vecor Z and a varancecovarance marx z. Geomercally

More information

Approximate Analytic Solution of (2+1) - Dimensional Zakharov-Kuznetsov(Zk) Equations Using Homotopy

Approximate Analytic Solution of (2+1) - Dimensional Zakharov-Kuznetsov(Zk) Equations Using Homotopy Arcle Inernaonal Journal of Modern Mahemacal Scences, 4, (): - Inernaonal Journal of Modern Mahemacal Scences Journal homepage: www.modernscenfcpress.com/journals/jmms.aspx ISSN: 66-86X Florda, USA Approxmae

More information

A HIERARCHICAL KALMAN FILTER

A HIERARCHICAL KALMAN FILTER A HIERARCHICAL KALMAN FILER Greg aylor aylor Fry Consulng Acuares Level 8, 3 Clarence Sree Sydney NSW Ausrala Professoral Assocae, Cenre for Acuaral Sudes Faculy of Economcs and Commerce Unversy of Melbourne

More information

Part II CONTINUOUS TIME STOCHASTIC PROCESSES

Part II CONTINUOUS TIME STOCHASTIC PROCESSES Par II CONTINUOUS TIME STOCHASTIC PROCESSES 4 Chaper 4 For an advanced analyss of he properes of he Wener process, see: Revus D and Yor M: Connuous marngales and Brownan Moon Karazas I and Shreve S E:

More information

Sampling Procedure of the Sum of two Binary Markov Process Realizations

Sampling Procedure of the Sum of two Binary Markov Process Realizations Samplng Procedure of he Sum of wo Bnary Markov Process Realzaons YURY GORITSKIY Dep. of Mahemacal Modelng of Moscow Power Insue (Techncal Unversy), Moscow, RUSSIA, E-mal: gorsky@yandex.ru VLADIMIR KAZAKOV

More information

CHAPTER 5: MULTIVARIATE METHODS

CHAPTER 5: MULTIVARIATE METHODS CHAPER 5: MULIVARIAE MEHODS Mulvarae Daa 3 Mulple measuremens (sensors) npus/feaures/arbues: -varae N nsances/observaons/eamples Each row s an eample Each column represens a feaure X a b correspons o he

More information

Variants of Pegasos. December 11, 2009

Variants of Pegasos. December 11, 2009 Inroducon Varans of Pegasos SooWoong Ryu bshboy@sanford.edu December, 009 Youngsoo Cho yc344@sanford.edu Developng a new SVM algorhm s ongong research opc. Among many exng SVM algorhms, we wll focus on

More information

Appendix H: Rarefaction and extrapolation of Hill numbers for incidence data

Appendix H: Rarefaction and extrapolation of Hill numbers for incidence data Anne Chao Ncholas J Goell C seh lzabeh L ander K Ma Rober K Colwell and Aaron M llson 03 Rarefacon and erapolaon wh ll numbers: a framewor for samplng and esmaon n speces dversy sudes cology Monographs

More information

Online Appendix for. Strategic safety stocks in supply chains with evolving forecasts

Online Appendix for. Strategic safety stocks in supply chains with evolving forecasts Onlne Appendx for Sraegc safey socs n supply chans wh evolvng forecass Tor Schoenmeyr Sephen C. Graves Opsolar, Inc. 332 Hunwood Avenue Hayward, CA 94544 A. P. Sloan School of Managemen Massachuses Insue

More information

CH.3. COMPATIBILITY EQUATIONS. Continuum Mechanics Course (MMC) - ETSECCPB - UPC

CH.3. COMPATIBILITY EQUATIONS. Continuum Mechanics Course (MMC) - ETSECCPB - UPC CH.3. COMPATIBILITY EQUATIONS Connuum Mechancs Course (MMC) - ETSECCPB - UPC Overvew Compably Condons Compably Equaons of a Poenal Vecor Feld Compably Condons for Infnesmal Srans Inegraon of he Infnesmal

More information

Existence and Uniqueness Results for Random Impulsive Integro-Differential Equation

Existence and Uniqueness Results for Random Impulsive Integro-Differential Equation Global Journal of Pure and Appled Mahemacs. ISSN 973-768 Volume 4, Number 6 (8), pp. 89-87 Research Inda Publcaons hp://www.rpublcaon.com Exsence and Unqueness Resuls for Random Impulsve Inegro-Dfferenal

More information

CS286.2 Lecture 14: Quantum de Finetti Theorems II

CS286.2 Lecture 14: Quantum de Finetti Theorems II CS286.2 Lecure 14: Quanum de Fne Theorems II Scrbe: Mara Okounkova 1 Saemen of he heorem Recall he las saemen of he quanum de Fne heorem from he prevous lecure. Theorem 1 Quanum de Fne). Le ρ Dens C 2

More information

Robust and Accurate Cancer Classification with Gene Expression Profiling

Robust and Accurate Cancer Classification with Gene Expression Profiling Robus and Accurae Cancer Classfcaon wh Gene Expresson Proflng (Compuaonal ysems Bology, 2005) Auhor: Hafeng L, Keshu Zhang, ao Jang Oulne Background LDA (lnear dscrmnan analyss) and small sample sze problem

More information

DEEP UNFOLDING FOR MULTICHANNEL SOURCE SEPARATION SUPPLEMENTARY MATERIAL

DEEP UNFOLDING FOR MULTICHANNEL SOURCE SEPARATION SUPPLEMENTARY MATERIAL DEEP UNFOLDING FOR MULTICHANNEL SOURCE SEPARATION SUPPLEMENTARY MATERIAL Sco Wsdom, John Hershey 2, Jonahan Le Roux 2, and Shnj Waanabe 2 Deparmen o Elecrcal Engneerng, Unversy o Washngon, Seale, WA, USA

More information

FI 3103 Quantum Physics

FI 3103 Quantum Physics /9/4 FI 33 Quanum Physcs Aleander A. Iskandar Physcs of Magnesm and Phooncs Research Grou Insu Teknolog Bandung Basc Conces n Quanum Physcs Probably and Eecaon Value Hesenberg Uncerany Prncle Wave Funcon

More information

GENERATING CERTAIN QUINTIC IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS. Youngwoo Ahn and Kitae Kim

GENERATING CERTAIN QUINTIC IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS. Youngwoo Ahn and Kitae Kim Korean J. Mah. 19 (2011), No. 3, pp. 263 272 GENERATING CERTAIN QUINTIC IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS Youngwoo Ahn and Kae Km Absrac. In he paper [1], an explc correspondence beween ceran

More information

Dual Approximate Dynamic Programming for Large Scale Hydro Valleys

Dual Approximate Dynamic Programming for Large Scale Hydro Valleys Dual Approxmae Dynamc Programmng for Large Scale Hydro Valleys Perre Carpener and Jean-Phlppe Chanceler 1 ENSTA ParsTech and ENPC ParsTech CMM Workshop, January 2016 1 Jon work wh J.-C. Alas, suppored

More information

ON THE WEAK LIMITS OF SMOOTH MAPS FOR THE DIRICHLET ENERGY BETWEEN MANIFOLDS

ON THE WEAK LIMITS OF SMOOTH MAPS FOR THE DIRICHLET ENERGY BETWEEN MANIFOLDS ON THE WEA LIMITS OF SMOOTH MAPS FOR THE DIRICHLET ENERGY BETWEEN MANIFOLDS FENGBO HANG Absrac. We denfy all he weak sequenal lms of smooh maps n W (M N). In parcular, hs mples a necessary su cen opologcal

More information

New M-Estimator Objective Function. in Simultaneous Equations Model. (A Comparative Study)

New M-Estimator Objective Function. in Simultaneous Equations Model. (A Comparative Study) Inernaonal Mahemacal Forum, Vol. 8, 3, no., 7 - HIKARI Ld, www.m-hkar.com hp://dx.do.org/.988/mf.3.3488 New M-Esmaor Objecve Funcon n Smulaneous Equaons Model (A Comparave Sudy) Ahmed H. Youssef Professor

More information

Should Exact Index Numbers have Standard Errors? Theory and Application to Asian Growth

Should Exact Index Numbers have Standard Errors? Theory and Application to Asian Growth Should Exac Index umbers have Sandard Errors? Theory and Applcaon o Asan Growh Rober C. Feensra Marshall B. Rensdorf ovember 003 Proof of Proposon APPEDIX () Frs, we wll derve he convenonal Sao-Vara prce

More information

Tools for Analysis of Accelerated Life and Degradation Test Data

Tools for Analysis of Accelerated Life and Degradation Test Data Acceleraed Sress Tesng and Relably Tools for Analyss of Acceleraed Lfe and Degradaon Tes Daa Presened by: Reuel Smh Unversy of Maryland College Park smhrc@umd.edu Sepember-5-6 Sepember 28-30 206, Pensacola

More information

Cubic Bezier Homotopy Function for Solving Exponential Equations

Cubic Bezier Homotopy Function for Solving Exponential Equations Penerb Journal of Advanced Research n Compung and Applcaons ISSN (onlne: 46-97 Vol. 4, No.. Pages -8, 6 omoopy Funcon for Solvng Eponenal Equaons S. S. Raml *,,. Mohamad Nor,a, N. S. Saharzan,b and M.

More information

Bayes rule for a classification problem INF Discriminant functions for the normal density. Euclidean distance. Mahalanobis distance

Bayes rule for a classification problem INF Discriminant functions for the normal density. Euclidean distance. Mahalanobis distance INF 43 3.. Repeon Anne Solberg (anne@f.uo.no Bayes rule for a classfcaon problem Suppose we have J, =,...J classes. s he class label for a pxel, and x s he observed feaure vecor. We can use Bayes rule

More information

Lecture 6: Learning for Control (Generalised Linear Regression)

Lecture 6: Learning for Control (Generalised Linear Regression) Lecure 6: Learnng for Conrol (Generalsed Lnear Regresson) Conens: Lnear Mehods for Regresson Leas Squares, Gauss Markov heorem Recursve Leas Squares Lecure 6: RLSC - Prof. Sehu Vjayakumar Lnear Regresson

More information

Mechanics Physics 151

Mechanics Physics 151 Mechancs Physcs 5 Lecure 9 Hamlonan Equaons of Moon (Chaper 8) Wha We Dd Las Tme Consruced Hamlonan formalsm H ( q, p, ) = q p L( q, q, ) H p = q H q = p H = L Equvalen o Lagrangan formalsm Smpler, bu

More information

Volatility Interpolation

Volatility Interpolation Volaly Inerpolaon Prelmnary Verson March 00 Jesper Andreasen and Bran Huge Danse Mares, Copenhagen wan.daddy@danseban.com brno@danseban.com Elecronc copy avalable a: hp://ssrn.com/absrac=69497 Inro Local

More information

Department of Economics University of Toronto

Department of Economics University of Toronto Deparmen of Economcs Unversy of Torono ECO408F M.A. Economercs Lecure Noes on Heeroskedascy Heeroskedascy o Ths lecure nvolves lookng a modfcaons we need o make o deal wh he regresson model when some of

More information

[ ] 2. [ ]3 + (Δx i + Δx i 1 ) / 2. Δx i-1 Δx i Δx i+1. TPG4160 Reservoir Simulation 2018 Lecture note 3. page 1 of 5

[ ] 2. [ ]3 + (Δx i + Δx i 1 ) / 2. Δx i-1 Δx i Δx i+1. TPG4160 Reservoir Simulation 2018 Lecture note 3. page 1 of 5 TPG460 Reservor Smulaon 08 page of 5 DISCRETIZATIO OF THE FOW EQUATIOS As we already have seen, fne dfference appromaons of he paral dervaves appearng n he flow equaons may be obaned from Taylor seres

More information

PARTICLE METHODS FOR MULTIMODAL FILTERING

PARTICLE METHODS FOR MULTIMODAL FILTERING PARTICLE METHODS FOR MULTIMODAL FILTERIG Chrsan Musso ada Oudjane OERA DTIM. BP 72 92322 France. {mussooudjane}@onera.fr Absrac : We presen a quck mehod of parcle fler (or boosrap fler) wh local rejecon

More information

Mechanics Physics 151

Mechanics Physics 151 Mechancs Physcs 5 Lecure 9 Hamlonan Equaons of Moon (Chaper 8) Wha We Dd Las Tme Consruced Hamlonan formalsm Hqp (,,) = qp Lqq (,,) H p = q H q = p H L = Equvalen o Lagrangan formalsm Smpler, bu wce as

More information

e-journal Reliability: Theory& Applications No 2 (Vol.2) Vyacheslav Abramov

e-journal Reliability: Theory& Applications No 2 (Vol.2) Vyacheslav Abramov June 7 e-ournal Relably: Theory& Applcaons No (Vol. CONFIDENCE INTERVALS ASSOCIATED WITH PERFORMANCE ANALYSIS OF SYMMETRIC LARGE CLOSED CLIENT/SERVER COMPUTER NETWORKS Absrac Vyacheslav Abramov School

More information

( ) [ ] MAP Decision Rule

( ) [ ] MAP Decision Rule Announcemens Bayes Decson Theory wh Normal Dsrbuons HW0 due oday HW o be assgned soon Proec descrpon posed Bomercs CSE 90 Lecure 4 CSE90, Sprng 04 CSE90, Sprng 04 Key Probables 4 ω class label X feaure

More information

P R = P 0. The system is shown on the next figure:

P R = P 0. The system is shown on the next figure: TPG460 Reservor Smulaon 08 page of INTRODUCTION TO RESERVOIR SIMULATION Analycal and numercal soluons of smple one-dmensonal, one-phase flow equaons As an nroducon o reservor smulaon, we wll revew he smples

More information

Advanced Machine Learning & Perception

Advanced Machine Learning & Perception Advanced Machne Learnng & Percepon Insrucor: Tony Jebara SVM Feaure & Kernel Selecon SVM Eensons Feaure Selecon (Flerng and Wrappng) SVM Feaure Selecon SVM Kernel Selecon SVM Eensons Classfcaon Feaure/Kernel

More information

SOME NOISELESS CODING THEOREMS OF INACCURACY MEASURE OF ORDER α AND TYPE β

SOME NOISELESS CODING THEOREMS OF INACCURACY MEASURE OF ORDER α AND TYPE β SARAJEVO JOURNAL OF MATHEMATICS Vol.3 (15) (2007), 137 143 SOME NOISELESS CODING THEOREMS OF INACCURACY MEASURE OF ORDER α AND TYPE β M. A. K. BAIG AND RAYEES AHMAD DAR Absrac. In hs paper, we propose

More information

FTCS Solution to the Heat Equation

FTCS Solution to the Heat Equation FTCS Soluon o he Hea Equaon ME 448/548 Noes Gerald Reckenwald Porland Sae Unversy Deparmen of Mechancal Engneerng gerry@pdxedu ME 448/548: FTCS Soluon o he Hea Equaon Overvew Use he forward fne d erence

More information

THE PREDICTION OF COMPETITIVE ENVIRONMENT IN BUSINESS

THE PREDICTION OF COMPETITIVE ENVIRONMENT IN BUSINESS THE PREICTION OF COMPETITIVE ENVIRONMENT IN BUSINESS INTROUCTION The wo dmensonal paral dfferenal equaons of second order can be used for he smulaon of compeve envronmen n busness The arcle presens he

More information

Comparison of Differences between Power Means 1

Comparison of Differences between Power Means 1 In. Journal of Mah. Analyss, Vol. 7, 203, no., 5-55 Comparson of Dfferences beween Power Means Chang-An Tan, Guanghua Sh and Fe Zuo College of Mahemacs and Informaon Scence Henan Normal Unversy, 453007,

More information

doi: info:doi/ /

doi: info:doi/ / do: nfo:do/0.063/.322393 nernaonal Conference on Power Conrol and Opmzaon, Bal, ndonesa, -3, June 2009 A COLOR FEATURES-BASED METHOD FOR OBJECT TRACKNG EMPLOYNG A PARTCLE FLTER ALGORTHM Bud Sugand, Hyoungseop

More information

This document is downloaded from DR-NTU, Nanyang Technological University Library, Singapore.

This document is downloaded from DR-NTU, Nanyang Technological University Library, Singapore. Ths documen s downloaded from DR-NTU, Nanyang Technologcal Unversy Lbrary, Sngapore. Tle A smplfed verb machng algorhm for word paron n vsual speech processng( Acceped verson ) Auhor(s) Foo, Say We; Yong,

More information

Chapter 6 DETECTION AND ESTIMATION: Model of digital communication system. Fundamental issues in digital communications are

Chapter 6 DETECTION AND ESTIMATION: Model of digital communication system. Fundamental issues in digital communications are Chaper 6 DEECIO AD EIMAIO: Fundamenal ssues n dgal communcaons are. Deecon and. Esmaon Deecon heory: I deals wh he desgn and evaluaon of decson makng processor ha observes he receved sgnal and guesses

More information

Hidden Markov Models Following a lecture by Andrew W. Moore Carnegie Mellon University

Hidden Markov Models Following a lecture by Andrew W. Moore Carnegie Mellon University Hdden Markov Models Followng a lecure by Andrew W. Moore Carnege Mellon Unversy www.cs.cmu.edu/~awm/uorals A Markov Sysem Has N saes, called s, s 2.. s N s 2 There are dscree meseps, 0,, s s 3 N 3 0 Hdden

More information

Econ107 Applied Econometrics Topic 5: Specification: Choosing Independent Variables (Studenmund, Chapter 6)

Econ107 Applied Econometrics Topic 5: Specification: Choosing Independent Variables (Studenmund, Chapter 6) Econ7 Appled Economercs Topc 5: Specfcaon: Choosng Independen Varables (Sudenmund, Chaper 6 Specfcaon errors ha we wll deal wh: wrong ndependen varable; wrong funconal form. Ths lecure deals wh wrong ndependen

More information

F-Tests and Analysis of Variance (ANOVA) in the Simple Linear Regression Model. 1. Introduction

F-Tests and Analysis of Variance (ANOVA) in the Simple Linear Regression Model. 1. Introduction ECOOMICS 35* -- OTE 9 ECO 35* -- OTE 9 F-Tess and Analyss of Varance (AOVA n he Smple Lnear Regresson Model Inroducon The smple lnear regresson model s gven by he followng populaon regresson equaon, or

More information

Mechanics Physics 151

Mechanics Physics 151 Mechancs Physcs 5 Lecure 0 Canoncal Transformaons (Chaper 9) Wha We Dd Las Tme Hamlon s Prncple n he Hamlonan formalsm Dervaon was smple δi δ Addonal end-pon consrans pq H( q, p, ) d 0 δ q ( ) δq ( ) δ

More information

Analysis And Evaluation of Econometric Time Series Models: Dynamic Transfer Function Approach

Analysis And Evaluation of Econometric Time Series Models: Dynamic Transfer Function Approach 1 Appeared n Proceedng of he 62 h Annual Sesson of he SLAAS (2006) pp 96. Analyss And Evaluaon of Economerc Tme Seres Models: Dynamc Transfer Funcon Approach T.M.J.A.COORAY Deparmen of Mahemacs Unversy

More information

Performance Analysis for a Network having Standby Redundant Unit with Waiting in Repair

Performance Analysis for a Network having Standby Redundant Unit with Waiting in Repair TECHNI Inernaonal Journal of Compung Scence Communcaon Technologes VOL.5 NO. July 22 (ISSN 974-3375 erformance nalyss for a Nework havng Sby edundan Un wh ang n epar Jendra Sngh 2 abns orwal 2 Deparmen

More information

Chapter 6: AC Circuits

Chapter 6: AC Circuits Chaper 6: AC Crcus Chaper 6: Oulne Phasors and he AC Seady Sae AC Crcus A sable, lnear crcu operang n he seady sae wh snusodal excaon (.e., snusodal seady sae. Complee response forced response naural response.

More information

A NEW TECHNIQUE FOR SOLVING THE 1-D BURGERS EQUATION

A NEW TECHNIQUE FOR SOLVING THE 1-D BURGERS EQUATION S19 A NEW TECHNIQUE FOR SOLVING THE 1-D BURGERS EQUATION by Xaojun YANG a,b, Yugu YANG a*, Carlo CATTANI c, and Mngzheng ZHU b a Sae Key Laboraory for Geomechancs and Deep Underground Engneerng, Chna Unversy

More information

Machine Learning Linear Regression

Machine Learning Linear Regression Machne Learnng Lnear Regresson Lesson 3 Lnear Regresson Bascs of Regresson Leas Squares esmaon Polynomal Regresson Bass funcons Regresson model Regularzed Regresson Sascal Regresson Mamum Lkelhood (ML)

More information

. The geometric multiplicity is dim[ker( λi. number of linearly independent eigenvectors associated with this eigenvalue.

. The geometric multiplicity is dim[ker( λi. number of linearly independent eigenvectors associated with this eigenvalue. Lnear Algebra Lecure # Noes We connue wh he dscusson of egenvalues, egenvecors, and dagonalzably of marces We wan o know, n parcular wha condons wll assure ha a marx can be dagonalzed and wha he obsrucons

More information

Lecture VI Regression

Lecture VI Regression Lecure VI Regresson (Lnear Mehods for Regresson) Conens: Lnear Mehods for Regresson Leas Squares, Gauss Markov heorem Recursve Leas Squares Lecure VI: MLSC - Dr. Sehu Vjayakumar Lnear Regresson Model M

More information

Effect of Resampling Steepness on Particle Filtering Performance in Visual Tracking

Effect of Resampling Steepness on Particle Filtering Performance in Visual Tracking 102 The Inernaonal Arab Journal of Informaon Technology, Vol. 10, No. 1, January 2013 Effec of Resamplng Seepness on Parcle Flerng Performance n Vsual Trackng Zahdul Islam, Ch-Mn Oh, and Chl-Woo Lee School

More information

Lecture 18: The Laplace Transform (See Sections and 14.7 in Boas)

Lecture 18: The Laplace Transform (See Sections and 14.7 in Boas) Lecure 8: The Lalace Transform (See Secons 88- and 47 n Boas) Recall ha our bg-cure goal s he analyss of he dfferenal equaon, ax bx cx F, where we emloy varous exansons for he drvng funcon F deendng on

More information

J i-1 i. J i i+1. Numerical integration of the diffusion equation (I) Finite difference method. Spatial Discretization. Internal nodes.

J i-1 i. J i i+1. Numerical integration of the diffusion equation (I) Finite difference method. Spatial Discretization. Internal nodes. umercal negraon of he dffuson equaon (I) Fne dfference mehod. Spaal screaon. Inernal nodes. R L V For hermal conducon le s dscree he spaal doman no small fne spans, =,,: Balance of parcles for an nernal

More information

Computing Relevance, Similarity: The Vector Space Model

Computing Relevance, Similarity: The Vector Space Model Compung Relevance, Smlary: The Vecor Space Model Based on Larson and Hears s sldes a UC-Bereley hp://.sms.bereley.edu/courses/s0/f00/ aabase Managemen Sysems, R. Ramarshnan ocumen Vecors v ocumens are

More information

2.1 Constitutive Theory

2.1 Constitutive Theory Secon.. Consuve Theory.. Consuve Equaons Governng Equaons The equaons governng he behavour of maerals are (n he spaal form) dρ v & ρ + ρdv v = + ρ = Conservaon of Mass (..a) d x σ j dv dvσ + b = ρ v& +

More information

TSS = SST + SSE An orthogonal partition of the total SS

TSS = SST + SSE An orthogonal partition of the total SS ANOVA: Topc 4. Orhogonal conrass [ST&D p. 183] H 0 : µ 1 = µ =... = µ H 1 : The mean of a leas one reamen group s dfferen To es hs hypohess, a basc ANOVA allocaes he varaon among reamen means (SST) equally

More information

Advanced time-series analysis (University of Lund, Economic History Department)

Advanced time-series analysis (University of Lund, Economic History Department) Advanced me-seres analss (Unvers of Lund, Economc Hsor Dearmen) 3 Jan-3 Februar and 6-3 March Lecure 4 Economerc echnues for saonar seres : Unvarae sochasc models wh Box- Jenns mehodolog, smle forecasng

More information

. The geometric multiplicity is dim[ker( λi. A )], i.e. the number of linearly independent eigenvectors associated with this eigenvalue.

. The geometric multiplicity is dim[ker( λi. A )], i.e. the number of linearly independent eigenvectors associated with this eigenvalue. Mah E-b Lecure #0 Noes We connue wh he dscusson of egenvalues, egenvecors, and dagonalzably of marces We wan o know, n parcular wha condons wll assure ha a marx can be dagonalzed and wha he obsrucons are

More information

Single-loop System Reliability-Based Design & Topology Optimization (SRBDO/SRBTO): A Matrix-based System Reliability (MSR) Method

Single-loop System Reliability-Based Design & Topology Optimization (SRBDO/SRBTO): A Matrix-based System Reliability (MSR) Method 10 h US Naonal Congress on Compuaonal Mechancs Columbus, Oho 16-19, 2009 Sngle-loop Sysem Relably-Based Desgn & Topology Opmzaon (SRBDO/SRBTO): A Marx-based Sysem Relably (MSR) Mehod Tam Nguyen, Junho

More information

Kernel-Based Bayesian Filtering for Object Tracking

Kernel-Based Bayesian Filtering for Object Tracking Kernel-Based Bayesan Flerng for Objec Trackng Bohyung Han Yng Zhu Dorn Comancu Larry Davs Dep. of Compuer Scence Real-Tme Vson and Modelng Inegraed Daa and Sysems Unversy of Maryland Semens Corporae Research

More information

Computer Robot Vision Conference 2010

Computer Robot Vision Conference 2010 School of Compuer Scence McGll Unversy Compuer Robo Vson Conference 2010 Ioanns Rekles Fundamenal Problems In Robocs How o Go From A o B? (Pah Plannng) Wha does he world looks lke? (mappng) sense from

More information

On computing differential transform of nonlinear non-autonomous functions and its applications

On computing differential transform of nonlinear non-autonomous functions and its applications On compung dfferenal ransform of nonlnear non-auonomous funcons and s applcaons Essam. R. El-Zahar, and Abdelhalm Ebad Deparmen of Mahemacs, Faculy of Scences and Humanes, Prnce Saam Bn Abdulazz Unversy,

More information

Modélisation de la détérioration basée sur les données de surveillance conditionnelle et estimation de la durée de vie résiduelle

Modélisation de la détérioration basée sur les données de surveillance conditionnelle et estimation de la durée de vie résiduelle Modélsaon de la dééroraon basée sur les données de survellance condonnelle e esmaon de la durée de ve résduelle T. T. Le, C. Bérenguer, F. Chaelan Unv. Grenoble Alpes, GIPSA-lab, F-38000 Grenoble, France

More information

Notes on the stability of dynamic systems and the use of Eigen Values.

Notes on the stability of dynamic systems and the use of Eigen Values. Noes on he sabl of dnamc ssems and he use of Egen Values. Source: Macro II course noes, Dr. Davd Bessler s Tme Seres course noes, zarads (999) Ineremporal Macroeconomcs chaper 4 & Techncal ppend, and Hamlon

More information

How about the more general "linear" scalar functions of scalars (i.e., a 1st degree polynomial of the following form with a constant term )?

How about the more general linear scalar functions of scalars (i.e., a 1st degree polynomial of the following form with a constant term )? lmcd Lnear ransformaon of a vecor he deas presened here are que general hey go beyond he radonal mar-vecor ype seen n lnear algebra Furhermore, hey do no deal wh bass and are equally vald for any se of

More information

Normal Random Variable and its discriminant functions

Normal Random Variable and its discriminant functions Noral Rando Varable and s dscrnan funcons Oulne Noral Rando Varable Properes Dscrnan funcons Why Noral Rando Varables? Analycally racable Works well when observaon coes for a corruped snle prooype 3 The

More information

Foundations of State Estimation Part II

Foundations of State Estimation Part II Foundaons of Sae Esmaon Par II Tocs: Hdden Markov Models Parcle Flers Addonal readng: L.R. Rabner, A uoral on hdden Markov models," Proceedngs of he IEEE, vol. 77,. 57-86, 989. Sequenal Mone Carlo Mehods

More information

Lecture 2 M/G/1 queues. M/G/1-queue

Lecture 2 M/G/1 queues. M/G/1-queue Lecure M/G/ queues M/G/-queue Posson arrval process Arbrary servce me dsrbuon Sngle server To deermne he sae of he sysem a me, we mus now The number of cusomers n he sysems N() Tme ha he cusomer currenly

More information

Clustering (Bishop ch 9)

Clustering (Bishop ch 9) Cluserng (Bshop ch 9) Reference: Daa Mnng by Margare Dunham (a slde source) 1 Cluserng Cluserng s unsupervsed learnng, here are no class labels Wan o fnd groups of smlar nsances Ofen use a dsance measure

More information

Comb Filters. Comb Filters

Comb Filters. Comb Filters The smple flers dscussed so far are characered eher by a sngle passband and/or a sngle sopband There are applcaons where flers wh mulple passbands and sopbands are requred Thecomb fler s an example of

More information

ABSTRACT KEYWORDS. Bonus-malus systems, frequency component, severity component. 1. INTRODUCTION

ABSTRACT KEYWORDS. Bonus-malus systems, frequency component, severity component. 1. INTRODUCTION EERAIED BU-MAU YTEM ITH A FREQUECY AD A EVERITY CMET A IDIVIDUA BAI I AUTMBIE IURACE* BY RAHIM MAHMUDVAD AD HEI HAAI ABTRACT Frangos and Vronos (2001) proposed an opmal bonus-malus sysems wh a frequency

More information

Chalmers Publication Library

Chalmers Publication Library Chalmers Publcaon Lbrary Exended Objec Tracng usng a Radar Resoluon Model Ths documen has been downloaded from Chalmers Publcaon Lbrary CPL. I s he auhor s verson of a wor ha was aeped for publcaon n:

More information

Epistemic Game Theory: Online Appendix

Epistemic Game Theory: Online Appendix Epsemc Game Theory: Onlne Appendx Edde Dekel Lucano Pomao Marcano Snscalch July 18, 2014 Prelmnares Fx a fne ype srucure T I, S, T, β I and a probably µ S T. Le T µ I, S, T µ, βµ I be a ype srucure ha

More information

Introduction ( Week 1-2) Course introduction A brief introduction to molecular biology A brief introduction to sequence comparison Part I: Algorithms

Introduction ( Week 1-2) Course introduction A brief introduction to molecular biology A brief introduction to sequence comparison Part I: Algorithms Course organzaon Inroducon Wee -2) Course nroducon A bref nroducon o molecular bology A bref nroducon o sequence comparson Par I: Algorhms for Sequence Analyss Wee 3-8) Chaper -3, Models and heores» Probably

More information

Hidden Markov Models

Hidden Markov Models 11-755 Machne Learnng for Sgnal Processng Hdden Markov Models Class 15. 12 Oc 2010 1 Admnsrva HW2 due Tuesday Is everyone on he projecs page? Where are your projec proposals? 2 Recap: Wha s an HMM Probablsc

More information

NEW TRACK-TO-TRACK CORRELATION ALGORITHMS BASED ON BITHRESHOLD IN A DISTRIBUTED MULTISENSOR INFORMATION FUSION SYSTEM

NEW TRACK-TO-TRACK CORRELATION ALGORITHMS BASED ON BITHRESHOLD IN A DISTRIBUTED MULTISENSOR INFORMATION FUSION SYSTEM Journal of Compuer Scence 9 (2): 695-709, 203 ISSN: 549-3636 203 do:0.3844/jcssp.203.695.709 Publshed Onlne 9 (2) 203 (hp://www.hescpub.com/jcs.oc) NEW TRACK-TO-TRACK CORRELATION ALGORITHMS BASED ON BITHRESHOLD

More information

Tight results for Next Fit and Worst Fit with resource augmentation

Tight results for Next Fit and Worst Fit with resource augmentation Tgh resuls for Nex F and Wors F wh resource augmenaon Joan Boyar Leah Epsen Asaf Levn Asrac I s well known ha he wo smple algorhms for he classc n packng prolem, NF and WF oh have an approxmaon rao of

More information

Machine Learning 2nd Edition

Machine Learning 2nd Edition INTRODUCTION TO Lecure Sldes for Machne Learnng nd Edon ETHEM ALPAYDIN, modfed by Leonardo Bobadlla and some pars from hp://www.cs.au.ac.l/~aparzn/machnelearnng/ The MIT Press, 00 alpaydn@boun.edu.r hp://www.cmpe.boun.edu.r/~ehem/mle

More information

5th International Conference on Advanced Design and Manufacturing Engineering (ICADME 2015)

5th International Conference on Advanced Design and Manufacturing Engineering (ICADME 2015) 5h Inernaonal onference on Advanced Desgn and Manufacurng Engneerng (IADME 5 The Falure Rae Expermenal Sudy of Specal N Machne Tool hunshan He, a, *, La Pan,b and Bng Hu 3,c,,3 ollege of Mechancal and

More information