There are many problems in science in which the state of a

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1 Implct samplng for partcle flters Alexandre J. Chorn 1 and Xuemn Tu Department of Mathematcs, Unversty of Calforna and Lawrence Berkeley Natonal Laboratory, Berkeley, CA 9470 Contrbuted by Alexandre J. Chorn, August 13, 009 sent for revew June 6, 009 We present a partcle-based nonlnear flterng scheme, related to recent work on chanless Monte Carlo, desgned to focus partcle paths sharply so that fewer partcles are requred. The man features of the scheme are a representaton of each new probablty densty functon by means of a set of functons of Gaussan varables a dstnct functon for each partcle and step and a resamplng based on normalzaton factors and Jacobans. The constructon s demonstrated on a standard, ll-condtoned test problem. pseudo-gaussan Jacoban chanless samplng There are many problems n scence n whch the state of a system must be dentfed from an uncertan equaton supplemented by a stream of nosy data ref. 1. A natural model of ths stuaton conssts of a stochastc dfferental equaton SDE dx = fx, t dt + gx, t dw, [1] where x = x 1, x,..., x m sanm-dmensonal vector, w s an m-dmensonal Brownan moton, f s an m-dmensonal vector functon, and gx, t sanm by m dagonal matrx. The Brownan moton encapsulates all the uncertanty n ths equaton. The ntal state x0 s assumed gven and may be random as well. As the erment unfolds, t s observed, and the values b n of a measurement process are recorded at tmes t n. For smplcty, assume t n = nδ, where δ s a fxed tme nterval and n s an nteger. The measurements are related to the evolvng state xt by b n = hx n + GW n, [] where h s a k-dmensonal, generally nonlnear, vector functon wth k m, G s a dagonal matrx, x n = xnδ, and W n s a vector whose components are ndependent Gaussan varables of mean 0 and varance 1, ndependent also of the Brownan moton n Eq. 1. The task s to estmate x on the bass of Eq. 1 and the observatons n Eq.. If the system n Eq. 1 and Eq. are lnear and the data are Gaussan, the soluton can be found va the Kalman Bucy flter. In the general case, t s natural to try to estmate x va ts evolvng probablty densty. The ntal state x s known and so s ts probablty densty; all one has to do s evaluate sequentally the densty P n+1 of x n+1 gven the probablty denstes P k of x k for k n and the data b n+1. Ths evaluaton can be done by followng partcles replcas of the system whose emprcal dstrbuton approxmates P n. In a Bayesan flter 3 10, one uses the probablty densty functon PDF P n and Eq. 1 to generate a pror densty, and then one uses the new data b n+1 to generate a posteror densty P n+1. In addton, one has to sample backward to take nto account the nformaton each measurement provdes about the past as well as to avod havng too many dentcal partcles. Ths can be very ensve, n partcular because the number of partcles needed can grow catastrophcally 11, 1. In ths paper, we offer an alternatve to the standard approach n whch P n+1 s sampled more drectly and backward samplng s done wthout chans 13. Our drect samplng s based on a pseudo-gaussan representaton of a varable wth densty P n+1,.e. a representaton by a collecton of functons of Gaussan varables wth sample-dependent parameters. The constructon s related to chanless samplng as descrbed n ref. 13. The dea n chanless samplng s to produce a sample of a large set of varables by sequentally samplng a growng sequence of nested, condtonally ndependent subsets, wth dscrepances balanced by samplng weghts. As observed n refs. 14 and 15, chanless samplng for a SDE reduces to nterpolatory samplng, as laned below. Our constructon wll be laned n the followng sectons through an example where the poston of a shp s deduced from the measurements of an azmuth, already used as a test bed n a number of prevous papers 7, 16, 17. We call our samplng mplct by analogy wth mplct schemes for solvng dfferental equatons, where the determnaton of a next value requres the soluton of algebrac equatons. If the SDE n Eq. 1 and the observaton functon n Eq. are lnear, our constructon becomes a reformulaton of sequental mportance samplng wth an optmal mportance functon, see refs. 5 and 6. Samplng by Interpolaton and Iteraton Frst, we lan how to sample va nterpolaton and teraton n a smple problem, related to the example and the constructon n ref. 14. Consder the scalar SDE dx = f x, tdt + β dw, [3] where β s a constant. We want to fnd sample paths x = xt, 0 t 1, subject to the condtons x0 = 0, x1 = X. Let Na, v denote a Gaussan varable wth mean a and varance v. We frst dscretze Eq. 3 on a regular mesh t 0, t 1,..., t N, where t n = nδ, δ = 1/N, 0 n N, wth x n = xt n, and, followng ref. 14, use a balanced, mplct dscretzaton 18, 19: x n+1 = x n + f x n, t n δ + x n+1 x n f x n, t n δ + W n+1, where f x n, t n = f x n x n, t n and W n+1 s N0, β/n. The jont probablty densty of the varables x 1,..., x N 1 s Z 1 N 1 0 V n, where Z s the normalzaton constant and V n = 1 δf x n+1 x n δf βδ = xn+1 x n δf /1 δf, β n where f, f are functons of the x n, t n, and β n = βδ/1 δf 0. One can obtan sample solutons by samplng ths densty, e.g. by Markov chan Monte Carlo, or one can obtan them by nterpolaton chanless samplng, as follows. Let a n = f x n, t n δ/1 δf x n, t n. Consder frst the specal case f x, t = f t, so that n partcular f = 0; we recover a verson of a Brownan brdge 1. Each ncrement x n+1 x n s now a Na n, β/n varable, wth the a n = f t n δ known lctly. Let N be a power of. Consder the varable x N/. On the one hand, A.J.C. and X.T. desgned research, performed research, analyzed data, and wrote the paper. The authors declare no conflct of nterest. 1 To whom correspondence should be addressed. E-mal: chorn@math.berkeley.edu. APPLIED MATHEMATICS / cg / do / / pnas PNAS October 13, 009 vol. 106 no

2 N/ x N/ = x n x n 1 NA 1, V 1, 1 where A 1 = N/ 1 a n, V 1 = β/. On the other hand, so that wth X = x N/ + N x n x n 1, N/+1 x N/ NA, V, N 1 A = X a n, V = V 1. N/ The PDF of x N/ s the product of the two PDFs; one can check that x A 1 x A x ā φ, V 1 V v where v = V 1 V V 1 +V, ā = V A 1 +V 1 A V 1 +V, and φ = A A 1 V 1 +V ; e φ s the probablty of gettng from the orgn to X, up to a normalzaton constant. Pck a sample ξ 1 from the N0, 1 densty; one obtans a sample of x N/, by settng x N/ = ā + vξ 1. Gven a sample of x N/, one can smlarly sample x N/4, x 3N/4, then x N/8, x 3N/8, etc., untl all the x j have been sampled. If we defne ξ = ξ 1, ξ,..., ξ N 1, then for each choce of ξ we fnd a sample x = x 1,..., x N 1, such that ξ 1 + +ξ N 1 X n a n β x1 x 0 a 0 x x 1 a 1 β/n β/n xn x N 1 a N 1 β/n, [4] where the factor X n an on the left s the probablty β of the fxed end value X up to a normalzaton constant. In ths lnear problem, ths factor s the same for all the samples and therefore harmless. The Jacoban J of the varables x 1,..., x N 1 wth respect to the varables ξ 1,..., ξ N 1 can be seen to be a constant ndependent of the sample and s also mmateral. One can repeat ths samplng process for multple choces of the varables ξ; each sample of the correspondng set of x n s ndependent of any prevous samples of ths set. Now return to the general case. The functons f, f are now functons of the x j. We obtan a sample of the probablty densty we want by teraton. The smplest teraton proceeds as follows. Frst, pck ξ = ξ 1, ξ,..., ξ N 1, where each ξ l, l = 1,..., N 1 s drawn ndependently from the N0,1 densty ths vector remans fxed durng the teraton. Make a frst guess x 0 = x0 1, x 0,..., xn 1 0 for example, f X = 0, pck x 0 = 0. Evaluate the functons f, f at x j note that now f = 0, and therefore the varances of the varous dsplacements are no longer constants. We are back n the prevous case and can fnd values of the ncrements x n+1 j+1 xn j+1 correspondng to the values of f, f we have. Repeat the process startng wth the new terate. If the vectors x j converge to a vector x = x 1,..., x N 1, we obtan, n the lmt, Eq. 4, where now on the rght sde β depends on n so that β = β n, and both a n, β n are functons of the fnal x. The left hand sde of Eq. 4 becomes ξ 1 + +ξ N 1 X n a n n β. n The factor F X n an s now dfferent from sample to n βn sample and changes the relatve weghts of the dfferent samples. The Jacoban J of the x varables wth respect to the ξ varables, s now also a functon of the sample. It can be evaluated step by step the last tme the teraton s carred out, ether by an mplct dfferentaton or by repeatng the teraton for a slghtly dfferent value of the relevant ξ and dfferencng. In averagng, one should take the product F J as weght or resample as descrbed at the end of the followng secton. In order to obtan more unform weghts, one also can use the strateges n refs. 13 and 14. One can readly see that ths teraton converges f KL < 1, [5] where K s the Lpshtz constant of f, and L s the length of the nterval on whch one works here L = 1. If ths teraton fals to converge, more sophstcated teratons are avalable. One should of course choose N large enough so that the results are converged n N. We do not provde more detals here because they are extraneous to our purpose, whch s to lan chanless/nterpolatory samplng and the use of reference varables n a smple context. Fnally, we chose the reference densty to be a product of ndependent N0, 1 varables, whch s a convenent but not mandatory choce. In applcatons, one may well want to choose other varances or make the varables be dependent. The Shp Azmuth Problem The problem we focus on s dscussed n refs. 7, 16 and 17, where t s used to demonstrate the capabltes of partcular flters. A shp sets out from a pont x 0, y 0 n the plane and undergoes a random walk, x n+1 = x n + u n+1, y n+1 = y n + v n+1, [6] for n 0, u n+1 = Nu n, β, v n+1 = Nv n, β,.e., each dsplacement s a sample of a Gaussan random varable whose varance β does not change from step to step and whose mean s the value of the prevous dsplacement. An observer makes nosy measurements of the azmuth arctany n /x n for the sake of defnteness, we take the branch n [π/, π/, recordng b n = arctan yn + N0, s. [7] xn where the varance s s also fxed; here, the observed quantty b n s scalar and s not denoted by a boldfaced letter. The problem s to reconstruct the postons x n = x n, y n from Eqs. 6 and 7. We take the same parameters as ref. 7: x 0 = 0.01, y 0 = 0, u 1 = 0.00, v 1 = 0.06, β = , s = We follow numercally M partcles, all startng from X 0 = x 0, Y 0 = y 0, as descrbed n the followng sectons, and we estmate the shp s poston at tme nδ as the mean of the locatons X n = X n, Y n, = 1,..., M of the partcles at that tme. The authors of ref. 7 also show numercal results for runs wth varyng data and constants; we dscuss those refnements n the numercal results secton below. Forward Step Assume that we have a collecton of M partcles X n at tme t n = nδ whose emprcal densty approxmates P n ; now we fnd dsplacements U n+1 = U n+1, V n+1 such that the emprcal densty of X n+1 = X n + U n+1 approxmates P n+1. P n+1 s known mplctly: It s the product of the densty that can be deduced from / cg / do / / pnas Chorn and Tu

3 the dynamcs and the one that comes from the observatons, wth the approprate normalzaton. If one s gven sample dsplacements, ther probabltes p the denstes P n+1 evaluated at the resultng postons X n+1 can be evaluated, so p s a functon of U n+1, p = pu n+1. For each partcle, we are gong to sample a reference densty, obtan a reference sample of probablty ρ, and then attempt to solve by teraton the equaton ρ = pu n+1 [8] to obtan U n+1. Defne f x, y = arctany/x and f n = f X n, Y n. We are workng on one partcle at a tme, so the ndex can be temporarly suppressed. Pck two ndependent samples ξ x, ξ y from a N0, 1 densty the reference densty n the present calculaton, and set ρ = 1 π ξ x ξ y ; the varables ξ x, ξ y reman unchanged untl the end of the teraton. We are lookng for dsplacements U n+1, V n+1, and parameters a x, a y, v x, v y, φ, such that: πρ U n+1 U n β f n+1 b n+1 s φ V n+1 V n β U n+1 a x v x V n+1 a y v y. [9] The frst equalty states what we wsh to accomplsh. Indeed, dvde ths frst equalty by φ. The equalty now defnes mplctly new dsplacements U n+1, V n+1, functons of ξ x, ξ y, wth the probablty of these dsplacements wth respect to P n+1 gven up to an unknown normalzaton constant. The second equalty n Eq. 9 defnes parameters a x, a y, v x, v y, all functons of X n and ξ x, ξ y that wll be used to actually fnd the dsplacements U n+1, V n+1. One should remember that n our example, the mean of U n+1 before the observaton s taken nto account s U n, wth a smlar statement for V n+1. We use the second equalty n Eq. 9 to set up an teraton for vectors U n+1, j = U j for brevty that converges to U n+1. Start wth U 0 = 0. We now lan how to compute U j+1 gven U j. Approxmate the observaton n Eq. 7 by f X j + f x U j+1 U j + f y V j+1 V j = b n+1 + N0, s, [10] where the dervatves f x, f y are, lke f, evaluated at X j = X n + U j,.e., approxmate the observaton equaton by ts Taylor seres anson around the prevous terate. Defne a varable η j+1 = f x U j+1 + f y V j+1 / fx + f y. The approxmate observaton equaton says that η j+1 s a Na 1, v 1 varable, wth a 1 = f f x U j f y V j b n+1, fx + f y s v 1 = fx + f. y On the other hand, from the equatons of moton one fnds that η j+1 s Na, v, wth a = f x U n + f y V n / fx + f y and v = β. Hence the PDF of η j+1 s, up to normalzaton factors, x a 1 x a x ā φ, v 1 v v where v = v 1 v v 1 +v, ā = a 1 v +a 1 v v 1 +v, φ = a 1 a v 1 +v = φ j+1. We can also defne a varable η j+1 + that s a lnear combnaton of U j+1, V j+1 and s uncorrelated wth η j+1 : + = f y U j+1 + f x V j+1. fx + f y η j+1 The observatons do not affect η j+1 +, so ts mean and varance are known. Gven the means and varances of η j+1, η j+1 + one can easly nvert the orthogonal matrx that connects them to U j+1, V j+1 and fnd the means and varances a x, v x of U j+1 and a y, v y of V j+1 after ther modfcaton by the observaton the subscrpts on a, v are labels, not dfferentatons. Now one can produce values for U j+1, V j+1 : U j+1 = a x + v x ξ x, V j+1 = a y + v y ξ y, where ξ x, ξ y are the samples from N0, 1, chosen at the begnnng of the teraton. Ths completes the teraton. Ths teraton converges to X n+1, such that f X n+1 = b n+1 + N0, s, and the phases φ j converge to a lmt φ = φ, where the partcle ndex has been restored. The tme nterval over whch the soluton s updated n each step s short, and there are no problems wth convergence, ether here or n the next secton see Eq. 5; n all cases, the teraton converges n a small number of steps. We now calculate the Jacoban J of the U n+1 varables wth respect to ξ x, ξ y. The relaton between these varables s lad out n the frst equalty of Eq. 9. Take the log of ths equaton, partton t nto a part parallel to the drecton n whch the observaton s made.e., parallel to the vector f x, f y and a part orthogonal to that drecton. Because the ncrement U n+1, V n+1 s now known, the evaluaton of J s merely an exercse n mplct dfferentaton. J can also be evaluated numercally by fndng the ncrement U n+1, V n+1 that corresponds to nearby values of ξ x, ξ y, and dfferencng. Do ths for all the partcles and obtan new postons wth weghts W j φ j J j, where J j s the Jacoban for the jth partcle. One can get rd of the weghts by resamplng,.e., for each of M random numbers θ k, k = 1,..., M drawn from the unform dstrbuton on [0, 1], choose a new X n+1 k = X n+1, such that A 1 1 j=1 W j < θ k A 1 j=1 W j where A = M j=1 W j, and then suppress the hat. We have traded the usual Bayesan resamplng based on the posteror probabltes of the samples for a resamplng based on the normalzng factors of the several Gaussan denstes; ths s a worthwhle trade because n a Bayesan flter one gets a set of samples, many of whch may have low probablty wth respect to P n+1, and here we have a set of samples, each one of whch has hgh probablty wth respect to a PDF close to P n+1 see Numercal Results and Concluson. Note also that the resamplng does not have to be done at every step for example, one can add up the phases for a gven partcle and resample only when the rato of the largest cumulatve weght φ log J to the smallest such weght exceeds some lmt L the summaton s over the weghts accrued to a partcular partcle snce the last resamplng. If one s worred by too many partcles beng close to each other depleton n the Bayesan termnology, one can dvde the set of partcles nto subsets of small sze and resample only nsde those subsets, creatng a greater dversty. As wll be seen n the numercal results secton, none of these strateges wll be used here, and we wll resample fully at every step. Fnally, note that f the SDE n Eq. 1 and the observaton n Eq. are lnear, and f at tme nδ one s gven the means and the covarance matrx of a Gaussan x, then our algorthm produces, n one teraton, the means and the covarance matrx of a standard Kalman flter. APPLIED MATHEMATICS Chorn and Tu PNAS October 13, 009 vol. 106 no

4 Backward Samplng The algorthm of the prevous secton s suffcent to create a flter, but accuracy, when the problem s not Gaussan, may requre an addtonal step. Every observaton provdes nformaton not only about the future but also about the past t may, for example, tag as mprobable earler states that had seemed probable before the observaton was made; n general, one has to go back and correct the past after every observaton ths backward samplng s often msleadngly motvated solely by the need to create greater dversty among the partcles n a Bayesan flter. As wll be seen below, ths backward samplng does not provde a sgnfcant boost to accuracy n the present problem, but t must be descrbed for the flter to be of general use as well as be generalzable to problems nvolvng smoothng. Gven a set of partcles at tme n+1δ, after a forward step and maybe a subsequent resamplng, one can fgure out where each partcle was n the prevous two steps and have a partal hstory for each partcle : X n 1, X n, Xn+1 f resamples had occurred, some parts of that hstory may be shared among several current partcles. Knowng the frst and the last members of ths sequence, one can nterpolate for the mddle term as n the frst example above, thus projectng nformaton backward. Ths requres that one recompute U n. Let U tot = U n + U n+1 ; n the present secton, ths quantty s assumed known and remans fxed. In the azmuth problem dscussed here, one has to deal wth the slght complcaton due to the fact that the mean of each dsplacement s the value of the prevous one, so that two successve dsplacements are related n a slghtly more complcated way than usual. The dsplacement U n s a NU n 1, β varable, and U n+1 s a NU n, β varable, so that one goes from X n 1 to X n+1 by samplng frst a U n 1,4β varable that takes us from X n 1 to an ntermedate pont P, wth a correcton by the observaton half-way up ths frst leg, and then one samples a NU tot, β varable to reach X n+1, and ths s done smlarly for Y. Let the varable that connects X n 1 to P be U new, so that what replaces U n s U new /. Accordngly, we are lookng for a new dsplacement U new = U new, V new and for parameters a new x, a new y, v new x, v new y, such that ξ x + ξ y U new U n 1 V new V n 1 8β 8β f new b n s U new U tot V new V tot β β U new ā x V new ā y, v new x v new y φ where f new = f X n 1 +U new /, Y n 1 +V new / and ξ x, ξ y are ndependent N0, 1 Gaussan varables. As n Eq. 9, the frst equalty embodes what we wsh to accomplsh fnd dsplacements, whch are functons of the reference varables that sample the new PDF at tme nδt. The new PDF s defned by the forward moton, by the constrant mposed by the observaton, and by knowledge of the poston at tme n + 1δt. The second equalty states that ths s done by fndng partcle-dependent parameters for a Gaussan densty. We agan fnd these parameters as well as the dsplacements by teraton. Much of the work s separate for the X and Y components of the equatons of moton, so we wrte some of the equatons for the X component only. Agan set up an teraton for varables U new, j = U j, whch converge to U new. Start wth U 0 = 0. To fnd U j+1 gven U j, approxmate the observaton n Eq. 7, as before, by Eq. 10; defne agan varables η j+1, η j+1 +, one n the drecton of the approxmate constrant and one orthogonal to t. In the drecton of the constrant, multply the PDFs as n the prevous secton; construct new means a 1 x, a1 y and new varances v1 x, v1 y at tme n, takng nto account the observaton at tme n as before. Ths also produces a phase φ = φ 0. Now take nto account that the locaton of the shp at tme n+1 s known; ths creates a new mean ā x, a new varance v x, and a new phase φ x,by v = v 1 v, ā v 1 +v x = a 1 v +a v 1, φ v 1 +v x = a 1 a, where v 1 +v a 1 = a 1, v 1 = 4v 1 x, a = X tot, v = β. Fnally, fnd a new nterpolated poston U j+1 = anew x v + new x 4 ξ x the calculaton for V j+1 s smlar, wth a phase φ y, and we are done. The total phase for ths teraton s φ = φ 0 + φ x + φ y. As the terates U j converge to U new, the phases converge to a lmt φ = φ. One also has to compute Jacobans and set up a resamplng. After one has values for X new, a forward step gves corrected values of X n+1 ; one can use ths nterpolaton process to correct estmates of X k by subsequent observatons for k = n 1, k = n,..., or as many as are useful. Numercal Results Before presentng examples of numercal results for the azmuth problem, we dscuss the accuracy one can ect. We run the shp once and record synthetc observatons wth the approprate nose, whch wll reman fxed. Then we fnd other shp trajectores compatble wth these fxed observatons as follows. We have 160 observatons. We note that the maxmum lkelhood estmate of s gven 160 observatons s a random varable wth mean zero and varance.11s. Then we make other runs of the shp, record the azmuths along the path and calculate the dfferences between these azmuths and the fxed observatons. If the set of these dfferences n any run s a lkely set of 160 samples of a N0, s varable whch s what the nose s supposed to be, then we declare that the new run s compatble wth the fxed observatons. We vew the set of dfferences as lkely f ther emprcal varance s wthn one standard varaton.11s of the nomnal varance s of the observaton nose. One can mpose further requrements for example, one may demand that the emprcal covarance of two successve noses be wthn a standard devaton of zero, but these turn out to be weaker requrements. To lghten the burden of computaton, we make the new shp runs have fxed dsplacements n the observed drecton equal to those that the frst shp erenced and sample new dsplacements only n the drecton orthogonal to the observed drecton. We use the varablty of the compatble runs as an estmate of the lower bound on the possble accuracy of the reconstructon. In Table 1, we dsplay the standard devatons of the dfferences between the resultng paths and the orgnal path that produced the observatons after the number of steps ndcated there the means of these dfferences are statstcally ndstngushable from zero. Ths table provdes an estmate of the accuracy we can ect. It s far to assume that these standard devatons are underestmates of the uncertanty a maxmum varaton of a sngle standard devaton n s s a strct requrement, and we allowed no varablty n β. In partcular, our constructon, together wth the partcular set of drectons n the lnearzed observaton equatons that arses wth our data, conspre to make the error estmate n the x component unrealstcally small. Table 1. Intrnsc uncertanty n the azmuth problem Step x component y component / cg / do / / pnas Chorn and Tu

5 Table. Mean and standard varaton of the dscrepancy between synthetc data and ther reconstructon,,000 runs, no back step, 100 partcles x component y component Number of steps Mean SD Mean SD If one wants relable nformaton about the performance of the flter, t s not suffcent to run the shp once, record observatons, and then use the flter to reconstruct the shp s path, because the dfference between the true path and the reconstructon s a random varable that may be accdentally small or large. We have therefore run a large number of such reconstructons and computed the means and standard devatons of the dscrepances between path and reconstructon as a functon of the number of steps and of other parameters. In Tables and 3, we dsplay the means and standard devatons of these dscrepances not of ther mean! n the the x and y components of the paths wth,000 runs, at the steps and numbers of partcles ndcated, wth no backward samplng. ref. 7 used 100 partcles. On average, the error s zero so that the flter s unbased and the standard devaton of the dscrepances cannot be ected to be better than the lower bound of Table 1, and n fact t s compatble to that lower bound. The standard devaton of the dscrepancy s not catastrophcally larger wth one partcle and no resamplng at all! than wth 100 the man source of the dscrepancy s the nsuffcency of the data for accurate estmaton of the trajectores. The more-sophstcated resamplng strateges dscussed above make no dscernble dfference here because they are unable to remedy the lmtatons of the dataset. One can check to see that backward samplng also does not make much dfference for ths problem, where the underlyng moton s Gaussan and the varance of the observaton nose s much larger than the varance n the model. In Fg. 1 we plot a sample shp path, ts reconstructon, and the reconstructons obtaned when the ntal data for the reconstructon are strongly perturbed here, the ntal data for x, y were perturbed ntally by, respectvely, 0.1 and 0.4, and when the value of β assumed n the reconstructon s random: β = Nβ 0, ɛβ 0, where β 0 s the constant value used untl now and ɛ = 0.4 but the calculaton s otherwse dentcal. Ths produces varatons n β of the order of 40%; any larger varance n the perturbatons produced here a negatve value of β. The dfferences between the reconstructons and the true path reman wthn the acceptable range of errors. These graphs show that the flter has lttle senstvty to perturbatons we dd not calculate statstcs here because the nsenstvty holds for each ndvdual run. We now show that the parameter β can be estmated from the data. The flter needs an estmate of β to functon; call ths estmate β assumed.ifβ assumed = β, the other assumptons used to produce the Fg. 1. Some shp trajectores as laned n the text. dataset e.g. ndependence of the dsplacements and of the observatons are also false, and all one has to do s detect the fallacy. We do t by pckng a trajectory of a partcle and computng the quantty D = K U j+1 U j + K V j+1 V j K U j+1 U j + K V. j+1 V j If the dsplacements are ndependent, then on the average D = 1; we wll try to fnd the real β by fndng a value of β assumed for whch ths happens. We chose K = 40 the early part of a trajectory s less nosy than the later parts. As we already know, a sngle run cannot provde an accurate estmate of β, and accuracy n the reconstructon depends on how many runs are used. In Table 4 we dsplay some values of D averaged over 00 and over 3,000 runs as a functon of the rato of β assumed to the value of β used to generate the data. From the longer computaton, one can fnd the correct value of β wth an error of about 3%, whereas wth 00 runs the uncertanty s about 10%. The lmted accuracy reported n prevous work can of course be acheved wth a sngle run. A detaled dscusson of parameter estmaton usng our algorthm wll be presented elsewhere. Conclusons The numercal results for the test problem are comparable wth those produced by other flters. What should be noted s that our flter behaves well as the number of partcles decreases down to a sngle partcle n the test problem. There s no lnearzaton or other uncontrollable approxmaton. Ths good behavor perssts as the number of varables ncreases. The dffculty encountered by Bayesan partcle flters when the number of varables ncrease Table 4. The mean of the dscrmnant D as a functon of σ assumed /σ, 30 partcles σ assumed /σ 3,000 runs 00 runs APPLIED MATHEMATICS Table 3. Mean and standard varaton of the dscrepancy between synthetc data and ther reconstructon,,000 runs, no back step, one partcle x component y component Number of steps Mean SD Mean SD ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0.04 Chorn and Tu PNAS October 13, 009 vol. 106 no

6 s due to the fact that the relatve sze of that part of space that the data desgnate as probable decreases, so that t s harder for a Bayesan flter to produce probable samples. Ths stuaton can be modeled as the lmt of a problem where both the varance β of the model and the varance s of the observaton nose tend to zero; t s easy to see that n ths lmt, our teraton produces the correct trajectores wthout dffculty. In refs. 11 and 1 Snyder, Bckel, et al. produced a smple, many-dmensonal problem where a Bayesan flter collapses because a sngle partcle hogs all the probablty; one can see that n that problem our flter produces the same weghts for all the partcles wth any number of varables. ACKNOWLEDGMENTS. We thank Prof. G.I. Barenblatt, Prof. R. Kupferman, Prof. R. Mller, and Dr. J. Weare for askng searchng questons and provdng good advce, and most partcularly, Prof. J. Goodman, who read the manuscrpt carefully and ponted out areas that needed work. Ths work was supported n part by U.S. Department of Energy Contract No. DE-AC0-05CH1131 and by the Natonal Scence Foundaton Grant DMS Doucet A., de Fretas N, Gordon, N, eds 001 Sequental Monte Carlo Methods n Practce Sprnger, New York.. Bozc S 1994 Dgtal and Kalman Flterng Butterworth-Henemann, Oxford. 3. Maceachern S, Clyde M, Lu J 1999 Sequental mportance samplng for nonparametrc Bayes models: the next generaton. Can J Stat 7: Lu J, Sabatt C 000 Generalzed Gbbs sampler and multgrd Monte Carlo for Bayesan computaton. Bometrka 87: Doucet A, Godsll S, Andreu C 000 On sequental Monte Carlo samplng methods for Bayesan flterng. Stat Comp 10: Arulampalam M, Maskell S, Gordon N, Clapp T 00 A tutoral on partcle flters for onlne nonlnear/nongaussan Bayesan trackng. IEEE Trans Sg Proc 50: Glks W, Berzun C 001 Followng a movng target Monte Carlo nference for dynamc Bayesan models. J R Stat Soc B 63: Chorn AJ, Krause P 004 Dmensonal reducton for a Bayesan flter. Proc Natl Acad Sc USA 101: Dowd M 006 A sequental Monte Carlo approach for marne ecologcal predcton. Envronmetrcs 17: Doucet A, Johansen A 009 Partcle flterng and smoothng: Ffteen years later. Handbook of Nonlnear Flterng, eds Crsan D, Rozovsky B Oxford Unv Press, Oxford, preprnt. 11. Snyder C, Bengtsson T, Bckel P, Anderson J 008 Obstacles to hgh-dmensonal partcle flterng. Monthly Weather Rev 136: Bckel P, L B, Bengtsson T 008 Sharp falure rates for the bootstrap partcle flter n hgh dmensons. IMS Collectons: Pushng the Lmts of Contemporary Statstcs: Contrbutons n Honor of Jayanta K Ghosh Insttute of Math Stat, Beachwood, OH 3: Chorn AJ 008 Monte Carlo wthout chans. Commun Appl Math Comput Sc 3: Weare J 007 Effcent Monte Carlo samplng by parallel margnalzaton. Proc Natl Acad Sc USA 104: Weare J 009 Partcle flterng wth path samplng and an applcaton to a bmodal ocean current model. J Comput Phys 8: Gordon N, Salmond D, Smth A 1993 Novel approach to nonlnear/non-gaussan Bayesan state estmaton. IEEE Proc F 140: Carpenter J, Clfford P, Fearnhead P 1999 An mproved partcle fter for nonlnear problems. IEEE Proc Radar Sonar Navg 146: Mlsten G, Platen E, Schurz H 1998 Balanced mplct methods for stff stochastc systems. SIAM J Num Anal 35: Kloeden P, Platen E 199 Numercal Soluton of Stochastc Dfferental Equatons Sprnger, Berln. 0. Stuart A, Voss J, Wlberg P 004 Condtonal path samplng of SDEs and the Langevn MCMC method. Commun Math Sc 4: Levy P 1954 Le Mouvement Brownen Gauthers-Vllars, Pars n French / cg / do / / pnas Chorn and Tu