Comparison of Augmented State Track Fusion Methods for Non-full-rate Communication

Transcription

1 8th Internationa Conference on Information Fuion Wahington, DC - Juy 6-9, Comparion of Augmented tate Trac Fuion Method for Non-fu-rate Communication Feix Govaer enor Data and Information Fuion Fraunhofer FIE Wachtberg, Germany feix.govaer@fie.fraunho fer.de Chee-Yee Chong Independent reearcher and conutant Lo Ato, CA 9, UA cychong@ieee.org hozo Mori ytem & Technoogy Reearch, unnyvae, CA, UA hozo.mori@treea rch.com Wofgang och enor Data and Information Fuion Fraunhofer FIE Wachtberg, Germany wofgang.och@fie.fr aunhofer.de Abtract - For inear-gauian non-determinitic dynamic, that i, ytem with non-zero proce noie, it i we nown that tracet fuion baed on equivaent meaurement i optima ony for fu communication rate, i.e., if the oca poterior probabiitie or etimate are communicated and fued after each obervation and update time. Depite thi contraint, tracet fuion ha become very popuar becaue it perform we in many rea word probem even when communication i not at fu rate. By incuding oca tate etimate at mutipe time, augmented tate (A) tracet fuion compute the optima goba etimate depite thi communication contraint. A imiar method with thi property i ditributed accumuated tate denity () fuion, which compute decorreated oca peudo etimate by mean of a reaxed evoution mode. Thi paper compare thee two method by examining their underying principe. Numerica reut compare their performance and ao with that of a centraized aman fiter. The reut how that they have many propertie uch a the etimation accuracy in common depite their different derivation. eyword: Trac fuion, tracet fuion, augmented tate, accumuated tate denity, centraized aman fiter, ditributed aman fiter. Introduction Mutipe enor can provide better etimation performance becaue each additiona enor contribute more information. Centraized fuion of the meaurement from a enor at a inge node i theoreticay optima becaue the information in the meaurement i not degraded by any intermediate proceing. However, centraized fuion i not away feaibe when communication bandwidth i imited. Thu many ytem ue a ditributed etimation or fuion architecture where the individua enor proce their meaurement to generate oca etimate and error covariance, which are then ent to a fuion node to be combined into goba tate etimate and etimation error covariance. In mot ditributed etimation or fuion ytem, the oca etimate that are fued are optima tate etimate given the oca enor meaurement and computed uing ony oca enor mode information []. ince the oca etimation error are not independent, fuion agorithm have to addre thi cro enor dependence or correation. ome fuion agorithm addre dependence that can be characterized by cro-enor covariance of the oca etimation error. Exampe incude maximum ieihood etimate [] [], bet inear unbiaed etimate (BLUE) [] [], and minimum variance (MV) etimate [6] [7]. ince cro-covariance are ued in fuion, the oca enor mode have to be communicated to the fuion ite aong with the oca etimate. Furthermore, the fued etimate may not be gobay optima becaue it i ony the bet etimate given the oca etimate characterized by oca and cro-enor etimation error covariance. A popuar approach for ditributed fuion i tracet fuion or tracet, equivaent-meaurement, or channe-fiter fuion, [8] []. A tracet ue the current and predicted oca etimate to find the new information received ince the at fuion time. However, tracet fuion doe not generate the optima goba etimate when proce noie i preent and the fuion rate i ower than the enor obervation rate. Optimaity can be regained if the tate i augmented to be the entire tate trajectory for a obervation time ince the mot recent fuion [] []. In particuar, [] how that the centraized aman fiter () etimate can be obtained even when the augmented tate incude ony the tate of the mot recent two or three time intant. Thi augmented tate i equivaent to the accumuated tate denity (AD) ued for exact memorye trac fuion [] [7]. Even though augmented tate (A) tracet fuion and ditributed accumuated tate denity () fuion are both optima in computing the etimate, the oca computation are different. In particuar, the augmented tate etimate i the bet etimate given the oca meaurement. On the other hand, the oca AD etimate i non-optima on a oca perpective becaue the prediction tep ue a reaxed evoution mode. The difference in the equation i due to the different form of fuion equation. The convex combination fuion equation in require a reaxed evoution mode in oca proceing. Thu, the oca etimate i not the bet etimate given the oca data. On the other hand, augmented tracet IIF 86

2 fuion i a natura generaization of the uua tracet fuion by uing augmented tate to retore conditiona independence of the tracet meaurement given the tate. Thu, the oca etimate are equivaent to the optima etimate given the oca data. Thi paper compare the two fuion method that ue augmented tate etimate. We dicu the difference in the derivation that reut in different oca proceing and different goba fuion agorithm. We ao compare their performance by mean of imuation experiment. Our goa i to undertand the imiaritie and difference in the two method. Thi may aow u to deveop a method that ha the bet feature of both approache. The ret of thi paper i tructured a foow. ection II preent the trac fuion probem. ection III review the ditributed accumuated tate denity () fiter fuion. ection IV provide a imiar review of augmented tate fuion. ection V decribe the imuation experiment, and ection VI preent the reut. ection VII contain the concuion. Trac Fuion Probem Thi ection preent the target and enor mode for the tracing probem and a trac fuion architecture. Even though we ca the probem trac fuion, the reut are appicabe to genera ditributed etimation probem when the tate i not that of a moving target.. tate and Meaurement Mode The tate to be etimated i modeed by the inear ytem x = Fx + + Gw () n where x R i the tate at time t with =,,,..., F and G are matrice repreenting the ytem dynamic, and w i a zero-mean Gauian white random proce with covariance Q. We aume that the tate i oberved by enor with z = H x + v () for =,...,, and =,,..., where z i the meaurement of the -th enor at time t, H i the meaurement matrix, and v i a zero-mean white noie proce with covariance R. The meaurement noie are aumed to be independent with each other and the proce noie. The initia tate x i independent of the noie with mean x and covariance P. For impicity, we aume ynchronou obervation by a enor but the reut can be generaized to non-ynchronou meaurement with appropriate modification of the agorithm.. Fuion Architecture Define the cumuative meaurement of enor to be ( ) Z = z j j=. Let x and P be the optima (oca) etimate of x and it error covariance given Z. Loca aman fitering at enor conit of the foowing prediction and update tep. Prediction Update x = F x () P = F P F + G Q G () T T ( P ) x = ( P ) x + i () ( P ) = ( P ) + I (6) with initia condition x and P, i (H ) T (R ) z, and I (H ) T (R ) H. Each oca proceor ony now it own enor mode. At the fuion time t, each oca proceor communicate ome oca etimate and it error covariance to the fuion ite. When fuion tae pace after each enor obervation, i.e., = +, where t i the at fuion time, then communicating the oca etimate x and error covariance P i ufficient for recontruction of the optima etimate after fuion. Thi i the tandard tracet fuion method. When >, the oca proceor have to communicate augmented tate etimate for the fuion ite to recontruct the etimate. The foowing two ection dicu two different method of computing the oca augmented tate etimate and the fuion agorithm. In dicuing augmented tate etimation, it i convenient to define Z : (z j ) j= t, Z : (Z : ) = Z (Z ) = a the enor meaurement from t to a a meaurement from t to t, and a a cumuative meaurement. Ditributed Accumuated tate Denity () The fiter i a ditributed, memorye fiter [7], which mean that the fuion center doe not fue the received data to update a centra trac but combine them without uing the centra trac. The reut i the goba etimate, which i not required for future fuion tep. imiar to the Ditributed aman Fiter [8], the idea of the i to obtain a product repreentation of the fued poterior denity. In contrat to the DF, the require arger communication bandwidth a the tranmitted parameter are rather high dimeniona. However, in contrat to the DF, the 86

3 oca AD computation doe not require nowedge of a meaurement mode except for the number of enor.. Underying Principe of Approach Let : = [ xt,..., xt ]T be the augmented tate from t to t with >. We preent the approach for the genera probem of computing p( : Z ), the etimate and their error covariance ince from (), the fiter ony compute peudo etimate at the enor patform. Then oca proceing conit of the foowing prediction and update tep. Prediction " F x # :! = $!!! % &$!:! '% conditiona probabiity denity of : given Z in term of ome peudo " F P F T + G!Q!GT! F!P!:! # P:! = $!!!! % () P!:!FT! P!:! %' $& conditiona probabiity denitie p!( : Z ). From Baye rue, we have p( : Z ) = C! p( Z : : ) p( : Z! ) (7) where C i a normaizing contant. ince the enor meaurement Z : for the enor are conditionay independent given the augmented tate :, (7) become " # p( : Z ) = C! $ ( p( Z : : ) % p( : Z! ) & = ' where F! = #% F! n"n (!! ) $& and the initia condition are = x and P = P. Update For oca information parameter repreenting meaurement z, the update formua are given by (8) Define p! ( : Z! ) o that where ( p( : Z! ) = p! ( : Z! ) ) (9) Then (8) become () the (P: )! : = (P:! )! :! + J i () (P: )! = (P:! )! + J I J T () J = [ I n,n!n( " ") ]T i a n(! ) " n matrix that eect the x in to generate the meaurement z... Communication p( : Z ) = C! " p! ( : Z ) etimate : fuion node. with p! ( : Z ) = C! p(z : : ) p! ( : Z! ) At the fuion time t, each oca node end the peudo () = ().. Fuion proceing The goba etimate : and error covariance P : are The probabiity p! ( : Z ) i not neceariy the true conditiona probabiity of : given the oca meaurement obtained by the foowing fuion equation Z becaue () ue p! ( : Z! ) intead of p( : Z! ). When the probabiity denitie are Gauian, () become a convex combination of the oca etimate and information matrice, and () become the oca proceing equation for AD. Equation (9) ead to the reaxed evoution mode.. and peudo error covariance P : to the P!: : = " (P : )! : = (6) P!: = " (P : )! (7) = Loca Proceing and Fuion Equation The derivation for the foowing equation can be found in [7]. Augmented tate Trac Fuion When the communication (and fuion) rate i ower than the obervation rate, the new meaurement Z : + coected by.. Loca proceing Let : and P: be the mean and covariance of p! ( : Z ), and :! and P:! be the mean and ). Thee are not the oca optima covariance of p! ( : Z! the enor ince the at fuion time are no onger conditionay independent given the tate at a inge time becaue of the common proce noie. Thu equivaent meaurement or tracet fuion no onger produce the optima goba etimate. However, the meaurement Z : + 86

4 for the enor are conditionay independent given the augmented tate : +. Thi conditiona independence motivate the augmented tate fuion agorithm, which i imiar but not the ame a the fuion agorithm preented in ection III.. Underying Principe of Approach Let t and t be the current and at fuion time. Then p ( Z ) = C pz ( ) p ( Z ) (8) : + : + : + : + where C i a normaizing contant. ince the enor meaurement Z + for the enor are conditionay : independent give the augmented tate : +, (8) become p ( Z ) ( ) ( ) : = C pz p Z + : + : + : + = (9) The ieihood pz ( ) : + : + repreent the new information on the augmented tate received by enor ince t and can be computed from the oca conditiona probabiitie by pz ( ) = Cp ( Z )/ p ( Z ) () : + : + : + : + Equation (9) i the fuion equation that combine the conditiona probabiity denity at the at fuion time with the oca ieihood of the individua enor. Thee ieihood repreent the new information in the tracet of meaurement received ince the at fuion time and can be computed from true oca conditiona probabiitie denitie. When the probabiity denitie are Gauian, (9) and () become the augmented tate (A) tracet fuion equation. The difference between DAD and A tracet fuion i that force the fued conditiona probabiity denity into a product of oca conditiona probabiity denitie () when conditiona dependence doe not upport the factorization. On the other hand, A fuion decompoe the fued conditiona probabiity into product of oca ieihood and the conditiona probabiity after the at fuion.. Loca Proceing and Fuion Equation.. Loca proceing The oca proceor perform the prediction and update function to etimate the oca tate and it error covariance. T T T In addition, it ao compute = [( x ),..,( x ) ] +, the etimate of the augmented tate vector : + given the meaurement foowing equation. Prediction F P Z, and P where n n( ) Update, it error covariance by the F x = F P = P T P F P () () = F and the initia condition are = x and P = P. ( P ) = ( P ) + J i () ( P ) = ( P ) + JIJ () T where J [, ] T = I i a n ( ) n matrix that n n n( ) eect the x in to generate the meaurement z. Note that oca proceing ony ue the oca enor mode. Thi i different from oca proceing that ue a reaxed evoution mode with expicit dependence on the number of enor. Thee oca etimate and error covariance are optima or exact given the oca meaurement... Communication At the fuion time t, each oca node end it augmented tate etimate ( n ( ) vector) and error covariance P ( n ( ) n ( ) matrix) to the fuion node... Fuion proceing The fuion node compute recurivey the prediction and P for enor uing the foowing equation P F x = F P F + G Q G F P = T T T P F P () (6) with initia condition and P received at the at communication time. It ao compute and P from the at fued etimate imiar equation. and error covariance P uing 86

5 T T T The goba etimate = [ x,.., x ] + of the augmented tate and it error covariance P are given by ( ( ) ( ) ) P = P + P P = ( ( ) ( ) ) = (7) P = P + P P (8) The goba etimate x and error covariance P can be extracted from the augmented etimate and error covariance. The augmented tate fuion agorithm compute the optima goba etimate when the number of tate equa the number of obervation for a enor between fuion time (note the difficuty for non-ynchronou obervation). Reducing the ength or dimenion of the augmented tate produce a uboptima goba etimate but require e communication bandwidth. It i hown in [] that augmented tate with a very hort ength uch a ha performance imiar to fu augmented tate under ome condition. imuation Experiment We ue the foowing imuation to compare the performance of the fuion agorithm.. Target Mode The target move according to the dimeniona Orntein- Uhenbec mode ued in [] with the proce noie intenity q = βσ and VEL F exp( A t ); A I (9) βi The initia condition at t i t G Q G T e Aτ e ATτ dτ () qi. Meaurement Mode P = diag[ σ I, σ I ]. PO VEL Five enor oberve the poition of the target, i.e., H = [ I, ] for =,,..., at time t with t t = + t. Communication and fuion tae pace at time t < t <!, with +, when each enor end it oca etimate and error covariance for proceing. The nomina imuation parameter are: t =, σ =, σ VEL =, β =, q=., and the covariance of the meaurement noie of a enor i given by R =. The tota number of can i. PO 6 imuation Reut We evauate the performance of the A tracet fuion, the ditributed AD, and a centraized aman fiter (). The cope of the evauation i on communication iue. Thu, we conider ix different cenario in which the communication or other parameter differ. The figure pot the root mean quare poition error (RME) of Monte Caro imuation a a function of time. 6. cenario. Perfect Communication. The term perfect communication refer to a high bandwidth etup where a enor are abe to tranmit their oca data at each intant of time. Thu in cenario, a enor tranmit their meaurement at each time tep to the fuion center. The fuion center then compute the goba etimate by the three fuion method RME, Perfect Communication A Tracet time [] Fig. : Performance for perfect communication A expected, Fig. how that,, and A tracet fuion produce the ame reut. 6. cenario. Batch Communication. In cenario, a enor eep their oca data unti the very at time tep at. Then, they tranmit the batch containing meaurement, AD etimate, or augmented tate etimate to the fuion center. 866

6 RME, Batch Communication. RME, Frequent Communication A Tracet time []. A Tracet time [] Fig. : Performance for batch communication In Fig., the RME reut are computed from the moothed etimate at time. Thu they are maer than thoe in Fig..,, and A tracet fuion a have the ame performance. 6. cenario. Frequent Communication. In cenario, a enor are abe to tranmit their oca data at every n-th time tep, where n =. The data tranmitted refer to the compete ag. Fig. : Performance for frequent communication with moothing by. A expected, Figure how that the moothed etimate of have maer RME than the fued augmented tate etimate. 6. cenario. Random Loe. In thi cenario, the enor try to end the data of a inge time tep after each update. However, three out of the five enor fai to tranmit uccefuy, o that ony the data of two enor wi be received by the fuion center. The enor indice of the tranmiion oe are permutated randomy. 6 RME, Frequent Communication 8 RME, Random Loe 7 6 A Tracet time [] Fig. : Performance for frequent communication The reut in Fig. are the fiter etimate and the and A fued etimate are moothed after each fuion time. Thu ha arger RME than and A tracet fuion etimate between fuion time. A Tracet time [] Fig. : Performance for communication with random o ince the enor index of a communication faiure i random, the augmented tate etimate may contain more information than the meaurement of a inge time tep. If ome previou tranmiion of enor have faied, the augmented tate ti contain information from meaurement that are communicated. Thu and A tracet fuion perform better than in Fig.. 867

7 6. cenario. Mimatched Number of enor. ince oca proceing in the agorithm depend on the number of enor, thi cenario evauate the effect of a mode mimatch in the number of enor. The rea number of enor i two but both oca AD proceing and goba fuion aume the number of enor to be. Fig. 6 how the reut when the both oca AD proceing and fuion have the ame prior. More pecificay, the prior are: enor and enor AD for =,: x = (,,,) T, P = I I Fuion Center AD for =,,: x = (,,,) T, P = I I Fuion i performed ony at time. It turn out that the fuion center can compenate thi mode mimatch when it aume the ame number of enor a oca AD proceing and ha the ame prior. Thi can be expained a foow. When the fuion center of the doe not receive the AD of a enor, it predict the AD from the previou tranmiion. If there wa never any tranmiion, a when the number of aumed enor i arger than the true number, the prior i ued to mae the prediction. When both the fuion center and the oca AD proceing have the ame prior, the enor number mimatch ha no effect on the fuion equation. Thi can be een from (8) to () of ection III A. The fuion center of the compute the etimate under the aumption that the fictitiou enor never had any detection. Fig. 6 how that the optima etimate can be recovered depite the mimatch. proceing and fuion aume the ame incorrect number of enor but the fuion center i not aware of an initia prior. A a conequence the fuion center cannot predict the fictitiou etimate and ony reie on the tranmiion. Thi i the cae of a mode match between oca AD proceing and fuion proceing. Figure 7 how that fuion ha degraded performance a compared to or augmented tate tracet fuion RME, Mimatched No of enor A Tracet time [] Fig. 7: RME for mimatched number of enor. The fuion center and oca proceing have different prior. 6.6 cenario 6. Communication Outage Communication in in rea appication often cannot be aumed to be tabe and reiabe during the compete tracing proce. In thi cenario, the communication brea down for ten time tep after and after again. RME, Communication Outage 6 RME, Mimatched No of enor A Tracet A Tracet time [] Fig. 6: RME for mimatched number of enor. The fuion center can compenate the mimatch by mean of a common prior. time [] Fig 8: Performance for communication outage. Fig. 8 how that a fuion method have the ame performance. ince the aumption of a common prior i not away atified, we conider a different cenario when oca AD 868

8 7 Concuion We review two fuion method that ue augmented tate etimate invoving the tate at mutipe time. Athough the oca proceing and fuion equation are different, both method compute the optima etimate when the target dynamic invove non-zero proce noie and the fuion rate i ower than the enor obervation rate. The difference in the equation i due to the different derivation required to obtain the fuion equation. imuation reut how that both method have good performance a compared to. In particuar augmented tate tracet fuion ha the ame performance a at the fuion time. When there i a mimatch in the number of enor, fuion compute the optima etimate when both fuion and oca proceing aume the ame number of enor and ame prior. When they have different prior, performance degradation i oberved. ince and augmented tate tracet fuion ha imiar performance, and [] how that augmented tate with very hort ength are adequate for mot fuion probem, either method can be ued for trac fuion. However, trac aociation performance can be improved ignificanty by uing augmented tate etimate of the trac [9]. ince thee are true etimate computed by the oca augmented tate etimation equation, augmented tate tracet fuion may be a better approach for trac fuion than fuion. REFERENCE [] C. Y. Chong,. Mori,. C. Chang, and W. H. Barer, Architecture and agorithm for trac aociation and fuion, IEEE Aeropace and Eectronic ytem Magazine, vo., no., pp., Jan.. [] Y. Bar-haom, and L. Campo, The effect of the common proce noie on the two-enor fued-trac covariance, IEEE Tran. Aeropace and Eectronic yt., vo., no. 6, pp. 8 8, Nov []. C. Chang, R.. aha, and Y. Bar-haom, On optima trac-to-trac fuion, IEEE Tran. on Aeropace and Eectronic yt., vo., no., pp. 7 76, Oct [] Y. Zhu, and. R. Li, Bet inear unbiaed etimation fuion, Proc. nd Int. Conf. on Information Fuion. unnyvae, CA, 999. []. R. Li, Y. Zhu, J. Wang, and C. Han, Optima inear etimation fuion part I: unified fuion rue, IEEE Tran. on Information Theory, vo. 9, no. 9, pp. 9 8, ep.. [6]. Mori, W. H. Barer, C. Y. Chong, and. C. Chang, Trac aociation and trac fuion with non-determinitic target dynamic, IEEE Tran. on Aeropace and Eectronic yt. vo. 8, no., pp , Apr.. [7]. C. Chang, T. Zhi,. Mori, and C. Y. Chong, Performance evauation for MAP tate etimate fuion, IEEE Tran. on Aeropace and Eectronic yt., vo., no., pp. 76 7, Apr.. [8] C. Y. Chong, Hierarchica etimation, Proc. MIT/ONR Worhop on C, Monterey, CA, 979. [9] O. Drummond, Tracet and a hybrid fuion with proce noie, Proc. PIE, vo. 6, 997. []. Tian and Y. Bar-haom, Exact agorithm for four trac-to-trac fuion configuration: a you wanted to now but were afraid to a, Proc. th Int. Conf. on Information Fuion, eatte, UA, 9. [] C. Y. Chong and. Mori, Graphica mode for noninear ditributed etimation, Proc. 7th Int. Conf. on Information Fuion, tochom, weden,. [] C. Y. Chong,. C. Chang, and. Mori, Fundamenta of ditributed etimation and tracing, Proc. PIE, vo. 89,. [] C. Y. Chong,. C. Chang, and. Mori, Fundamenta of ditributed etimation, in D. Ha, C. Y. Chong, J. Lina, and M. Liggin II, editor, Ditributed Data Fuion for Networ-Centric Operation, CRC Pre,. [] C. Y. Chong,. Mori, F. Govaer, and W. och, Comparion of tracet fuion and ditributed aman fiter for trac fuion, Proc. 7th Int. Conf. on Information Fuion, aamanca, pain,. [] W. och and F. Govaer, On accumuated tate denitie with appication to out-of-equence meaurement proceing, IEEE Tran. Aeropace and Eectronic yt., vo. 7, no., pp , Oct.. [6] W. och, F. Govaer, and A. Charih, An exact oution to trac-totrac fuion uing accumuated tate denitie, Worhop on enor Data Fuion: Trend, oution, Appication (DF),. [7] W. och and F. Govaer, On decorreated trac-to-trac fuion baed on accumuated tate denitie, Proc. 7th Int. Conf. on Information Fuion, aamanca, pain,. [8] F. Govaer and W. och, An exact oution to trac-to-trac-fuion at arbitrary communication rate, IEEE Tran. Aeropace and Eectronic yt., vo. 8, no., pp , Juy. [9] C. Y. Chong and. Mori, Trac aociation uing augmented tate etimate, Proc. 8th Int. Conf. on Information Fuion, Wahington, D. C. UA,. 869