Advanced Scence and Technology Letters, pp.83-87 http://dx.do.org/10.14257/astl.2014.53.20 A Partcle Flter Algorthm based on Mxng of Pror probablty densty and UKF as Generate Importance Functon Lu Lu 1,1, Meng Yang 2, Shu Geng 1,Yong-hu Wang 1, Juan Wang 1, 1 Harbn Insttute of Petroleum,150027,Harbn, Chna 2 Haerbn Engneerng Unversty, 150001,Haerbn,Chna {Lu Lu, Meng Yang, Shu Geng,Yong-hu Wang, Juan Wang, lulu_hsy}@163.com Abstract. As an mportant nonlnear flter theory, partcle flter s a heated ssue n domestc and foregn researches. The opton of mportance densty s one of the ey steps of partcle flter algorthm. The proper opton of mportance densty can mnsh the negatve nfluence of flter algorthm caused by degeneracy problem. Ths paper ntroduces several dely-used optons of mportance densty systemcally, and analyzes ther features and appled perspectves respectvely. The paper also advances a comprehensve method of mportance densty, analyzes ts techncal features, explores the adjudgement and mprovement of ths method based on varous performance, and fnally puts forard the necessary further study accordng to the engneer requrements. Keyords: partcle flter; mportance functon; SIS; proposal dstrbuton Partcle flter s a dely used nonlnear flter algorthm recently. The man dea of partcle flter s descrbng the posteror probablty densty of the random varable usng eghted random samplng ponts. These samplng ponts are called partcles. The major problem of the partcle flter s the partcle degeneraton,.e. most partcles eghts become tremendously less than before after several teraton steps th only a fe partcles have relatvely hgh eghts. So that a lot of calculaton ll be asted on these lo-eghted partcles [1]. Plenty of research results sho that, the best ay to solve ths problem schoosng a proper mportance densty and add the resamplng step nto the algorthm. In order to choose a good mportance densty, one have to consder several factors: frst, the defnton doman of probablty densty should cover all of the posteror probablty dstrbuton,.e. the mportance functon should have a de dstrbuton, second, t should be sampled easly, furthermore, t should consder both pror probablty densty of the status and the neest observaton data so as to get the smallest varance and mae t close to the true posteror probablty densty. In practcal applcaton, there s no common ay to desgn the mportance functon. The mportance functon s usually desgned by choosng a method to meet the performance requrement based on the specfc case. Ths paper descrbes several mostly used methods that are used to desgn the mportance functon. The advantages and dsadvantages of these methods have been analyzed. ISS: 2287-1233 ASTL Copyrght 2014 SERSC
Advanced Scence and Technology Letters 1 The optmum mportance densty functon Theoretcally, after choosng the mportance densty functon, 1, 1,hen the reference dstrbuton equal to the actual dstrbuton, q x x y p x x y the mportance eght s mnmzed. can be teratvely calculated to = p y x p x x d x. But there are to obvous dsadvantages. Frst, the - 1 1 actual dstrbuton p x x, y 1 usually cannot be calculated. Second, the ntegraton of cannot be solved [2]. 2 Desgn the mportance functon through UKF EKF s a partal lnear method, so t s suboptmum for estmaton of mean value and varance of mportance dstrbuton. Smlar to EKF, Unscented Kalman Flter (UKF) can also be used to approxmate the proposal dstrbuton of partcle flter [3]. UKF drectly use the nonlnear system model and observaton model, va several determned Sgma pont to get the statstc property after nonlnear transformaton. It can mae posteror probablty dstrbuton s mean value and varance be exacted to second order or even hgher. So UKF s better than EKF for ts consderaton of the neest observed mportance densty. Ths s so-called Unscented Partcle Flter (UPF). 3 Mxng of Pror probablty densty and UKF to generate mportance functon UPF frst use the last partcle and ts varance to determne a set of sgma pont. The poston and eghts of ths set of pont can only be determned by the expectaton and varance of partcle. It can get the exactly character of partcle probablty dstrbuton, then substtute t nto status equaton to get a ne set of pont. Use the eghted sum of ths set of pont as expectaton, and use the eghted sum of the varance as varance, then use measurement equaton to correct the obtaned expectaton and varance. The corrected value s used as the expectaton and varance of the Gaussan dstrbuton to generate a partcle of currently pont[4]. Because the suffcently consderaton of the nfluence of the current observaton value on the posteror probablty functon, ths algorthm mproves the effcency of partcle. Hoever, the calculaton cost of each partcle s generaton s hgh. A report [5] gave the mprovement of the UPF algorthm: use UKF that based on the estmaton of the former status to get the mportance functon and generate a porton of partcles. The remanng partcles can be generated by pror probablty. In ths ay, the nfluence of both the currently observaton value and the pror probablty on the posteror probablty s under consderaton. Also, the calculaton cost s reduced hle eepng 84 Copyrght 2014 SERSC
Advanced Scence and Technology Letters the accuracy of flterng. The mproved UPF algorthm s shon belo: (1) Intalzaton 0 Get the samplng ponts x ~ p x, set 1/ 0 (2) Importance samplng 0 0, 1,..., For each samplng pont x, usng UKF to get the mean value x 1 and varance P of the set of partcles. (3) Generaton of partcles Generate 0.5 partcles from the result of UKF x, P. Generate 0.5 partcles from the pror probablty dstrbuton p x x 1. (4) Update of eght value p y x p x x 1 1 x x, y 1 1: The mportance probablty densty functon s: x x, y x, P. It ntroduces the neest observaton value, so the 1 1: performance of the flter s mproved. (5) Get the normalzed eght value j 1 j (6) Resamplng Defne for evaluaton of the number of effectve partcle e ff If e ff th r, resample for x, 1,...,, generate ne set x 1,..., readjust the eghts of partcle as: (7) Update the status x x 1 1 /.,, The number rato of the partcles that generated by UKF to the partcles that generated by pror probablty densty s not alays fxed. A parameter c ( 0 c 1) can be ntroduced to control the rato. The evaluaton of c can be based on the accuracy requrement and the speed requrement of the flterng. For smaller c, the calculaton tme s shorter hle the flterng accuracy s loer; For bgger number c, the calculaton tme s longer and the flterng accuracy s hgher. Table 1. The calculaton tme th dfferent c c 0.1 0.2 0.3 0.4 0.5 Calculaton tme(s) 0.6832 0.8455 0.9559 1.1522 1.3117 c 0.6 0.7 0.8 0.9 1 Copyrght 2014 SERSC 85
Advanced Scence and Technology Letters Calculaton tme(s) 1.4237 1.6365 1.8387 1.9771 2.1644 Fg. 1. The status estmaton for dfferent partcle flterng algorthm (n hch, c s 0.35 n UPF algorthm) Performance analyss and comparson of several partcle algorthm has been done usng smulaton. Example smulaton uses the nonlnear non-gaussan model. The status equatons and observaton equatons are shon belo: x e x v 2 0.2 x n 3 0 y 0.5 x 2 n 3 0 1 s n (( 4 2 ) ( 1)) 0.5 1 1 Durng the smulaton, the algorthms used are PF, EKPF and the UPF hch s suggested n ths report. The number of partcles s set to 200 for each algorthm. The observaton tme s T=60. The smulaton conducts 100 tmes ndependent experments. From the smulaton results, the flter accuracy that suggested n ths report s obvously better than classcal PF algorthm and EKPF algorthm. 4 Concluson As a concluson, n the desgn of flter algorthm, the frst thng s analyzng the detals of problem to be solved, hle systematcally consderng the statstcal property of nose and observng the nfluence of the relatonshp beteen the statstcal propertes on the partcle flter s performance, mang the best usage of pror dstrbuton functon, lelhood functon and the neest observaton; second, the relatonshp beteen the flter s performance and the calculaton cost, complexty of calculaton can also affect the flter s performance. Only f the above factors are consdered, a good mportance functon can be desgned hen usng proper method. 86 Copyrght 2014 SERSC
Advanced Scence and Technology Letters Acnoledgements. Ths study as supported by Helongjang Provncal Department of Educaton Scence and Technology Research Projet(12543037). Reference 1. Janmn Yao, Study on Partcle Flter Based Vsual Tracng Method[D]. Ph.D Dssertaton of CAS, 2004 2. Sh-qang Hu, Zhong-lang Jng, Overve of partcle flter algorthm[j], Control and Decson, vol.20, o. 4, 361-365(2005) 3. Mere R V, Doucet A, Fretas De. The Unscented Partcle Flter[R]. Techncal Report CUED/F-IPEG/TR 380, Cambrdge Unversty Engneerng Department,2000 4. Quan Pan, Feng Yang, Lang Ye, Yan Lang and Yong-me Cheng, Survey of a nd of nonlnear flters-ukf[j], Control and Decson, vol. 20, o.5, 481-489(2005) 5. Juler S J, Uhlmann J K. Reduced sgma pont flters for the propagaton of means and covarances though nonlnear transformatons[a]. Proceedng of the Amercan Control Conference[C].Anchorage AK,2002 Copyrght 2014 SERSC 87