Gaussian proposal density using moment matching in SMC methods

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1 Stat Comput (009 19: DOI.07/s Gaussian poposal density using moment matching in SMC methods S. Saha P.K. Mandal Y. Boes H. Diessen A. Bagchi Received: 3 Apil 007 / Accepted: 30 June 008 / Published online: 31 July 008 The Autho(s 008. This aticle is published with open access at Spingelin.com Abstact In this aticle we intoduce a new Gaussian poposal distibution to be used in conjunction with the sequential Monte Calo (SMC method fo solving non-linea filteing poblems. The poposal, in line with the ecent tend, incopoates the cuent obsevation. The intoduced poposal is chaacteized by the exact moments obtained fom the dynamical system. This is in contast with ecent wos whee the moments ae appoximated eithe numeically o by lineaizing the obsevation model. We show futhe that the newly intoduced poposal pefoms bette than othe simila poposal functions which also incopoate both state and obsevations. Keywods Bayesian filteing Nonlinea dynamic system Sequential Monte Calo methods Paticle filteing Impotance sampling Moment matching This wo was suppoted by a eseach gant fom THALES Nedeland BV. S. Saha ( P.K. Mandal A. Bagchi Depatment of Applied Mathematics, Univesity of Twente, 7500 AE, Enschede, The Nethelands s.saha@math.utwente.nl P.K. Mandal p..mandal@math.utwente.nl A. Bagchi a.bagchi@math.utwente.nl Y. Boes H. Diessen THALES Nedeland BV, Haasbegestaat 49, 7554 PA, Hengelo, The Nethelands Y. Boes yvo.boes@nl.thalesgoup.com H. Diessen hans.diessen@nl.thalesgoup.com 1 Intoduction Conside a nonlinea dynamic system given by x f(x 1,w, (1 y h(x,v, 1,,... ( whee (x ae the unobsevable system values (the state with (nown initial pio density p(x 0 p(x 0 x 1 and (y ae the obseved values (the measuements. The pocess noises (w ae assumed to be independent of the measuement noises (v. The poblem is to estimate the unobseved system value x n in some optimal manne fom all the obsevations y 1:n (y 1,y,...,y n,uptotimen, o equivalently, estimate the conditional density (also nown as filteed density p(x n y 1:n. An analytical solution can be found only fo a few special cases such as when both the system and obsevation equations (1 ( ae linea and the noise pocesses ae Gaussian (Kalman filte. Fo geneal models, analytical appoximations such as Extended Kalman filte and Gaussian sum filte (Andeson and Mooe 1979; Jazwinsi 1970; Bagchi 1993 and othe appoximate methods using numeical integation (Kitagawa 1987, the unscented Kalman filte (Julie and Uhlmann 1997; Wan and van de Mewe 000 and the Gaussian quadatue Kalman filte (Ito and Xiong 000 ae poposed in the liteatue. Simulation based sequential Monte Calo (SMC methods, also nown as Paticle Filtes (PF s, povide the taget filte density p(x n y 1:n in the fom of a cloud of paticles (Handschin and Mayne 1969; Aashi and Kumamoto 1975; Godon et al. 1993; West1993; Pitt and Shephad 1999; Doucet et al. 001; Aulampalam et al. 00. The biggest advantage of the SMC method is that it can easily adapt to

2 04 Stat Comput (009 19: nonlineaity in the model and/o non-gaussian noises. The efficiency of the PF algoithm depends on the so-called impotance function π(, also often efeed to as the poposal distibution, used to geneate the paticles. It has been shown in (Doucet et al. 000 that the impotance function of the fom π p(x x 1,y is optimal in a cetain sense. Thee ae, howeve, two majo, in pactice pohibitive, dawbacs fo using this impotance function. Fistly, dawing samples accoding to p(x x 1,y is, in geneal, difficult. Secondly, it is also difficult to get an analytical expession which is needed fo the weight update. In this aticle we popose as the impotance function a Gaussian appoximation of p(x x 1,y by matching the exact moments up to second ode of the distibution of (x,y conditional on x 1. Recently, Doucet et al. (000 and Guo et al. (005 also poposed simila impotance functions. Howeve, Doucet et al. (000 use the lineaised appoximation of the obsevation model to calculate the moments, while Guo et al. (005 appoximate the moments by diffeent numeical methods such as the Gaussian-Hemite quadatue ule o the Julie-Uhlmann quadatue ule. Besides the possibility that the use of exact moments may lead to bette appoximation of the optimal poposal distibution, ou method has one distinct advantage ove the one in Guo et al. (005, namely that, it is computationally less demanding. We also note that, the dynamical systems consideed by Doucet et al. (000 and Guo et al. (005 ae with additive Gaussian noise pocesses, wheeas ou method would wo fo moe geneal models, so long as the obsevation model is a polynomial. Fo compaison we have also consideed unscented paticle filte (van de Mewe et al. 000 which, too, wos fo geneal dynamical systems. Ou expeimental esults with additive Gaussian noise pocesses show that the oveall pefomance of ou poposal function is bette than that of the othe poposals, consideing the tade off between the RMSE and the computational load. The est of the aticle is oganized as follows. In Sect. the geneal SMC method is eviewed vey biefly and the ole of the impotance function is discussed. We descibe ou poposed impotance function using the exact moments in Sect. 3. Constuction of othe impotance functions as poposed by Doucet et al. (000, Guo et al. (005 and van de Mewe et al. (000 ae biefly eviewed in Sect. 4. Implementation issue of ou poposed method is discussed in Sect. 5. Section 6 contains the numeical compaison esults of these methods based on two examples one with polynomial (Sect. 6.1 and the othe with non-polynomial (Sect. 6. obsevation equation. Finally, Sect. 7 concludes the aticle. Geneal SMC method and impotance function Suppose that the system dynamics ae given by (1 (. The sequential Monte Calo method is based on impotance sampling and allows one to estimate ecusively in time the distibution function p(x y 1:. The estimate is given in the fom of a weighted paticle cloud {(x (i, w(i, i 1,...,N}. Given the obsevations up to the cuent time (y 1: and the paticles up to time 1(x (i 0: 1, x(i is dawn accoding to a nomalized impotance function π(x x (i 0: 1,y 1: whose suppot includes that of the tue posteio and subsequently the impotance weights ae updated. Fo a full account we efe the eades to Aulampalam et al. (00. Usually in pactice, the impotance function is taen to be the tansition density, i.e., π(x x (i 0: 1,y 1: p(x x (i 1 because it is easily available fom the model. It is nown that this algoithm suffes fom the degeneacy poblem, that is to say, the vaiance of the impotance weights can only incease ove time. It has been shown by Doucet et al. (000 that an impotance function of the fom π(x x (i 0: 1,y 1: p(x x (i 1,y addesses this issue by minimizing the vaiance of the (unnomalized impotance weight w (i conditional upon x (i 0: 1 and y 1:. In geneal, though, this choice of the impotance function is not pactical as it is difficult to geneate samples fom this distibution. Futhemoe, one needs an analytical expession of the impotance function to be used in the weight update equation, which is also geneally difficult with this choice of the impotance function. 3 Impotance function based on exact moment matching (EMM Suppose the system dynamics ae given by (1 (. We futhe assume the following. Assumption A All the moments of (x,y conditional on x 1 up to second ode, i.e., E(x x 1, E(x x T x 1, E(y x 1, E(y y T x 1, and E(x y T x 1 ae nown. To detemine the impotance function, to be used in conjunction with the paticle filteing algoithm, we poceed as follows. We appoximate the joint distibution of (x,y, conditional on x 1, by a Gaussian distibution with matching moments up to the second ode. Let the coesponding mean μ ( and the covaiance ( be given by μ ( ( ( μ 1 μ ( ( ( and ( 11 ( 1 ( ( 1 T (. (3

3 Stat Comput (009 19: Note that μ ( and ( can be calculated fom the moments which ae assumed to be nown fom Assumption A. Subsequently, we tae the impotance function to be the conditional distibution of x,given(x 1,y, deived fom the appoximated Gaussian distibution above. Thus, we assume π(x x (i 0: 1,y 0: N (m,, whee m μ ( 1 + ( 1 ( 11 ( 1 [ (] 1 ( y μ (, (4 [ ( ] 1 ( ( 1 T. (5 We note hee that a sufficient condition fo Assumption A to hold is: Condition 1 Both E(y x and E(y y T x ae polynomials in x of degee at most m and all conditional moments of x, given x 1, ae nown up to ode m. In the special case when the noise pocesses in (1 (ae additive Gaussian, suppose the system dynamics ae given by x f(x 1 + w, w N (0,Q, (6 y h(x + v, v N (0,R, 1,,... (7 then the conditional moments of all ode of x given x 1 ae nown. If, in addition, h( in (7 is a polynomial, then Condition 1 would be satisfied and hence Assumption A would hold. The pecise fomulas fo the quantities can be found in Appendix A. 4 Othe Gaussian impotance functions Thee ae othe Gaussian impotance functions poposed in the liteatue. We mention thee of them hee. The fist two ae based on the Gaussian appoximation of the optimal impotance function p(x x 1,y which ae ideologically simila to EMM, but the moments ae appoximated in diffeent ways. The thid one, on the othe hand, uses a ban of unscented Kalman filtes to obtain the poposal density. 4.1 Impotance function by lineaization (LIN In an ealie pape Doucet et al. (000 conside a dynamical system with additive Gaussian noise, given by (6 (7. Obseving that the optimal impotance function p(x x 1,y is Gaussian when h( in the obsevation model (7 is linea, the authos lineaize the obsevation equation (7 to obtain y h(f (x 1 + C (x f(x 1 + v (8 whee C h x (f (x 1. Subsequently, they use the coesponding Gaussian distibution as impotance function. This essentially educes to appoximating the conditional distibution of (x,y,givenx 1, by the Gaussian distibution with mean vecto μ and covaiance matix given by μ ( f(x 1 h(f (x 1 and ( Q QC T C Q C QC T + R. 4. Numeically appoximated moment matching In a moe ecent aticle, Guo et al. (005 conside the same dynamical system given by (6 (7. The impotance function poposed by them is also in effect deived fom a Gaussian appoximation of the joint distibution of (x,y conditional on x 1. Guo et al. (005, howeve, appoximate the moments in (3 by vaious numeical techniques, such as the Gauss-Hemite quadatue (GHQ ule and the Julie-Uhlmann quadatue (JUQ ule. We efe the eade to the oiginal aticle fo the details. 4.3 Unscented paticle filte (UPF Unscented paticle filte algoithm of van de Mewe et al. (000 is suitable fo geneal dynamical systems given by (1 (. In this method, fo each paticle a sepaate unscented Kalman filte is popagated to geneate the Gaussian poposal distibution. Once again we efe to the oiginal aticle fo details. 5 Implementation of the EMM Clealy, the EMM as descibed in Sect. 3 can be implemented if the Assumption A holds. A pope classification of models fo which Assumption A holds is not vey easy. Howeve, as mentioned in Sect. 3, if the dynamical system is given by (6 (7 with h( in (7 a polynomial function, then EMM can be implemented. When the exact values of the quantities in (3 cannot be calculated, we popose to appoximate the obsevation equation by one of polynomial fom and implement the EMM to deive the impotance function. Fo instance, conside a eal-valued dynamical system given by (6 (7. We assume futhe that the function h( is n times diffeentiable. We appoximate h( locally by its n-th degee Taylo polynomial aound x f(x 1 to get the following obsevation equation. y a m (x x m + v with a m 1 m! ( m h(x x m x x (9. (

4 06 Stat Comput (009 19: Then the quantities in (3 can be appoximated by the coesponding quantities fo the dynamical system govened by (6 and (. See Appendix B fo details. Note that the appoach poposed above extends the methodology used by Doucet et al. (000 whee the obsevation equation is appoximated by the fist degee Taylo polynomial, wheeas we conside highe degee polynomials. It is also wothwhile to note the diffeence between the appoach followed by Guo et al. (005 and the one poposed above. Guo et al. (005 wo with the given nonlinea model and duing setting up of the Gaussian impotance density, they appoximate the moments. We, on the othe hand, fist appoximate the obsevation equation with a n-th degee polynomial and futhe deive the Gaussian impotance density using the exact moments (based on the appoximated polynomial model. In the following section we pesent two illustative numeical examples. We notice that EMM pefoms bette than the othe methods in the sense that its computational load is consideably less, while the qualities of the filtes (in tems of RMSE ae compaable. 6 Numeical simulation esults In this section we conside two examples one with a polynomial obsevation model and the othe with a nonpolynomial model and compae the filteed estimates obtained by diffeent methods. In both examples we conside additive Gaussian noise pocesses. 6.1 Polynomial obsevation model As in Doucet et al. (000 we conside the system dynamics to be given by (6 (7 with f(x 1 x 1 + 5x x cos(1., (11 h(x x 0. (1 In ou simulations, we set Q, R 1 and geneate a time seies data of length 0 stating with x 0 N (0, 1. Given only the noisy obsevations y, paticle filte algoithm is pefomed with the impotance functions descibed in Sect. 4 (LIN, GHQ, JUQ, UPF and the new one (EMM poposed in Sect. 3. We estimate the state sequence x, 1,,...,0, with all the diffeent methods mentioned above. Note that, in this case, the diffeences in the moments used in EMM and LIN can be clealy seen. The moments used in EMM, as given in (3, ae ( f(x 1 μ ( f and (x 1 ( ( 0 + Q 0 Q f(x 1 Q f(x 1 Q f (x 1 Q 0 + Q 00 + R, while the moments used in LIN, as given in (9, ae ( f(x 1 μ and ( f (x 1 0 Q f(x 1 Q f(x 1 Q f (x 1 Q 0 + R. Fo GHQ, we use the five point quadatue ule and fo JUQ, the thee (n 1 sigma points wee calculated using κ. Fo UPF, the paametes ae taen to be the same as that of van de Mewe et al. (000 with α 1,β 0,κ and P 0 1. Howeve, while esampling we use systematic esampling scheme wheeas van de Mewe et al. (000use esidual esampling scheme. Fo all methods, the initial distibution p(x 0 is taen to be N (0, 5 and esampling was done when the effective sample size became less than one-thid of the oiginal sample size N. Fo each method, we fist calculate the oot mean squaed eo (RMSE ove M 0 uns fo each time point and then the aveage (ove time RMSE, given by ( M 1 Mj1 ( ˆx j xj 1.Heex j is the tue (simulated state fo time in the j-th un and ˆx j is the coesponding (point estimate using a PF method. Each of these methods is implemented with diffeent Monte Calo sample sizes N 0, 50, 500 and 00. In Table 1 the aveage RMSE s ae pesented. Also epoted ae the aveage (ove the 0 uns CPU time, in seconds, to complete a un and the aveage numbe of esampling steps (NRS out of the 0 time steps. Fist of all, we see fom the table that, as expected, the pefomances (as measued by RMSE of all the methods become simila as sample size N inceases. This is in confomity with the fact that fo any poposal distibution the paticle filte conveges to the tue posteio as N. The UPF seems to have a consideably highe computational load than the othe methods. This can be explained by the fact that one needs to un the unscented Kalman filte fo each paticle (at each step to calculate the poposal. Pefomances of GHQ, JUQ and EMM ae moe o less simila (which is bette than LIN, but the time taen to aive at the estimate is less in EMM than that by GHQ and JUQ. It appeas that the numbes of esampling steps ae almost the same fo GHQ, JUQ and EMM, which is slightly bette than LIN. So, the exta computational load fo GHQ and

5 Stat Comput (009 19: Table 1 Compaison of the pefomance of diffeent poposal distibutions with a polynomial obsevation model Table Compaison of the pefomance of diffeent poposal distibutions with a nonpolynomial obsevation model N LIN GHQ JUQ UPF EMM RMSE CPU NRS RMSE CPU NRS RMSE CPU NRS RMSE CPU NRS N LIN GHQ JUQ UPF EMM RMSE CPU NRS RMSE CPU NRS RMSE CPU NRS RMSE CPU NRS JUQ elative to ou EMM method can be constued as a esult of computing the moments numeically. Thus one can conclude that the EMM method is moe efficient compaed to the othe methods as it is computationally less demanding in aiving at the compaable level of efficiency. 6. Non-polynomial obsevation model Let us conside the following model x x 1 + 5x x cos(1. + w, w N (0,, (13 y tan 1 (x + v, v N (0, 1, 1,,... (14 Once again, a time seies data of length 0 was simulated stating with x 0 N (0, 1 and the diffeent paticle filtes wee applied on the obsevation y. Hee the exact moments given in (3 ae unnown. Fo EMM we have consideed a nd degee Taylo polynomial, as descibed in Sect. 5. The othe setup ae as in the pevious example in Sect The pefomances of diffeent methods ae pesented in Table. Again, compaing the RMSE s we obseve that the pefomances of GHQ, JUQ, UPF and EMM ae faily simila, and they ae all bette than LIN. But when CPU times ae compaed, UPF is the wost pefome. A vey close loo eveals that GHQ and JUQ may poduce slightly lowe RMSE compaed to EMM. Howeve, this elative gain is achieved at the expense of high computational load. Thus, consideing the tade of between the RMSE and the computational cost, EMM appeas to povide a pactical and efficient poposal density. 7 Conclusion In this aticle a new impotance function has been poposed which is based on the Gaussian appoximation of the conditional distibution of (x,y,givenx 1, with the fist two moments matched exactly to those of the tue conditional distibution. To use the poposed method one needs to now the moments of the system dynamics up to the second ode. A specific case in which this is satisfied is when the noise pocesses ae additive Gaussian and the obsevation equation is polynomial.when the exact moments ae not nown but the noise pocesses ae additive Gaussian and the obsevation model is smooth, we use a polynomial appoximation of the obsevation model to deive the impotance function. With the help of two examples it has been shown that the poposed EMM method povides a moe pactical and efficient poposal density consideing the tade off between the pefomance (RMSE and the computational load. Open Access This aticle is distibuted unde the tems of the Ceative Commons Attibution Noncommecial License which pemits any noncommecial use, distibution, and epoduction in any medium, povided the oiginal autho(s and souce ae cedited. Appendix A Conside the system dynamics as given in (6 (7. Then, the conditional moments of x given x 1 can be deived as follows. m ( m E(x m x 1 [f(x 1 ] E(w m 0 m ( m [f(x 1 ] μ m, (15 0

6 08 Stat Comput (009 19: whee μ j is the j-th (aw moment of the N (0,Q distibution, given by a m E[(x x m x 1 ] + x μ +1 0 and μ (!! Q, a m μ m+1 + x a m μ m, (1 fo 0, 1,,... (16 When h(x is a polynomial of the fom, h(x n 0 a x, E(y y T x 1 a m a l E[(x x m+l x 1 ] both E(y x and E(y y T x ae polynomials in x, thus l0 satisfying Condition 1. Subsequently, we have a m a l μ m+l, ( E(y x 1 a m E(x m x l0 1 whee μ j s ae given by (16. m ( m a m [f(x 1 ] μ m, 0 (17 Refeences E(x y T x 1 a m E(x m+1 x 1 m+1 ( m + 1 a m 0 [f(x 1 ] μ m+1, (18 E(y y T x 1 a m a l E(x m+l x 1 l0 m+l l0 0 ( m + l a m a l [f(x 1 ] μ m+l. (19 Then (15 (19 ensue the validity of Assumption A. Appendix B Conside the dynamical system descibed by (6 and (. Obseving that the conditional distibution of (x x given x 1 is N(0,Q, the following moments can be calculated as E(y x 1 E(x y T x 1 a m E[(x x m x 1 ] a m μ m, (0 a m E[x (x x m x 1 ] a m E[(x x m+1 x 1 ] Aashi, H., Kumamoto, H.: Constuction of discete-time nonlinea filte by Monte Calo methods with vaiance-educing techniques. Syst. Contol 19(4, 11 1 (1975. (In Japanese Andeson, B., Mooe, J.: Optimal Filteing. Pentice-Hall, Englewood Cliffs (1979 Aulampalam, S., Masell, S., Godon, N., Clapp, T.: A tutoial on paticle filtes fo online nonlinea/non-gaussian Bayesian tacing. IEEE Tans. Signal Pocess. 50(, (00 Bagchi, A.: Optimal Contol of Stochastic Systems. Seies in Systems and Contol Engineeing. Pentice-Hall Intenational, Englewood Cliffs (1993 Doucet, A., de Feitas, J., Godon, N. (eds.: Sequential Monte Calo Methods in Pactice. Spinge, New Yo (001 Doucet, A., Godsill, S., Andieu, C.: On sequential Monte Calo sampling methods fo Bayesian filteing. Stat. Comput., (000 Godon, N., Salmond, D.J., Smith, A.: Novel appoach to nonlinea/ non-gaussian Bayesian state estimation. IEE Poc. F, Rada Signal Pocess. 140(, (1993 Guo, D., Wang, X., Chen, R.: New sequential Monte Calo methods fo nonlinea dynamic systems. Stat. Comput. 15(, (005 Handschin, J.E., Mayne, D.Q.: Monte Calo techniques to estimate the conditional expectation in multistage nonlinea filteing. Int. J. Contol 9(5, (1969 Ito, K., Xiong, K.: Gaussian filtes fo nonlinea filteing poblems. IEEE Tans. Automat. Cont. 45(5, 9 97 (000 Jazwinsi, A.: Stochastic Pocesses and Filteing Theoy. Academic Pess, New Yo (1970 Julie, S., Uhlmann, J.: A new extensions of the Kalman filte to nonlinea systems. In: Poceedings of AeoSense: The 11th Intenational Symposium on Aeospace/Defense Sensing, Simulation and Contols, Multi Senso Fusion, Tacing and Resouce Management, vol. II, pp SPIE, Bellingham (1997 Kitagawa, G.: Non-Gaussian state-space modeling of nonstationay time seies. J. Am. Stat. Assoc. 8(400, 3 63 (1987 Pitt, M.K., Shephad, N.: Filteing via simulation: Auxiliay paticle filtes. J. Am. Stat. Assoc. 94(446, (1999 van de Mewe, R., de Feitas, N., Doucet, A., Wan, E.: The unscented paticle filte. Technical Repot CUED/F-INFENG/TR 380, Cambidge Univesity Engineeing Depatment, Cambidge (000 Wan, E., van de Mewe, R.: The unscented Kalman filte fo nonlinea estimation. In: Poceedings of the IEEE 000 Adaptive Systems fo Signal Pocessing, Communications and Contol Symposium (AS-SPCC, Lae Louise, Albeta, Canada. IEEE Pess, New Yo (000 West, M.: Mixtue models, Monte Calo, Bayesian updating and dynamic models. Comput. Sci. Stat. 4, (1993