Progressive Correction for Deterministic Dirac Mixture Approximations

Transcription

1 4th Internationa Conference on Information Fusion Chicago, Iinois, USA, Juy 5-8, Progressive Correction for Deterministic Dirac Miture Approimations Patrick Ruoff, Peter Krauthausen, and Uwe D. Hanebeck Inteigent Sensor-Actuator-Systems Laboratory (ISAS), Institute for Anthropomatics, Karsruhe Institute of Technoogy (KIT), Germany. Emai: Abstract Since the advent of Monte-Caro partice fitering, partice representations of densities have become increasingy popuar due to their feibiity and impicit adaptive resoution. In this paper, an agorithm for the mutipication of a systematic Dirac miture (DM) approimation with a continuous ikeihood function is presented, which appies a progressive correction scheme, in order to avoid the partice degeneration probem. The preservation of sampe reguarity and therefore, representation quaity of the underying smooth density, is ensured by incuding a new measure of smoothness for Dirac mitures, the DM energy, into the distance measure. A comparison to common correction schemes in Monte-Caro methods reveas arge improvements especiay in cases of sma overap between the ikeihood and prior density, as we as for muti-moda ikeihoods. Keywords: state estimation, noninear fitering. I. INTRODUCTION State estimation as the inference of a hidden system state based on noisy observations is a fundamenta probem in engineering. Due to the uncertainty inherent in the probem, Bayesian estimation methods are commony empoyed. The soution to the state estimation probem is then obtained by recursive prediction and correction steps. Even though anaytic soutions eist for some specia type of systems, e.g., inear systems [], in genera no anaytic soutions to the prediction and correction equation are obtainabe. Therefore, suitabe approimations of the systems and/or the densities are required. The rest of this paper is concerned with deriving an approimation of the correction step as needed in a Bayesian state estimator for the case of systematic Dirac miture approimations of the underying densities. For noninear systems, the eisting approaches may be categorized according to the empoyed density approimation schemes, as the specific type of approimation governs the treatment of the correction step. For eampe, the etended Kaman fiter (EKF) [] uses a Gaussian density approimation and inearization of the ikeihood around the current state estimate. Gaussian miture fiters [3] offer a more feibe density representation, but in genera suffer from an eponentia increase in components. The conceptuay simpest density representation is an approimation by a Dirac miture (DM) of equay weighted partices, as used in canonica partice fiters. The partice representation is not ony very advantageous for simuations, but aso offers an impicit adaptive resoution of the density approimation. As pure Monte- Caro partice representations suffer from sow convergence rates to the true density in the number of partices used, different approimation schemes were deveoped, such as quasi-monte-caro approimations [4]. However, as partice representations do not capture the smoothness of the origina density, naïve agorithms for the correction step in a state estimator ead to the typica partice degeneration probem, where eventuay ony a few different partices contribute to the posterior distribution. This paper introduces a reguarized progressive correction step in the case that a certain distance measure to the underying density sha be minimized by deterministicay pacing partices. In contrast to a progressive correction for Monte- Caro methods, this probem is especiay chaenging as the benefit of deterministic approimation schemes, i.e., a representation with fewer partices, is even more ikey to degrade in the correction step. Many different schemes have been proposed to hande the correction step for Monte-Caro partice fiters, cf. [5] for an overview. The approach resembing the present work most is [6], which is a progressive correction step for Monte Caro methods. We etend this scheme to deterministic Dirac miture approimations, spitting the ikeihood adaptivey into severa factors, each of which is easier to process. In the course of the progression, each factor of the ikeihood is mutipied with the current deterministic DM approimation and subsequenty reapproimated with equay weighted Diracs. As wi be shown, this effectivey reocates the Dirac components to regions with higher posterior density and thereby avoids the typica degeneration of the representation. In order to ensure that the DM representation maintains the smoothness/reguarity properties of the underying density throughout the progression, a nove energy term is added to the distance measure used for the reapproimation. This reguarization of the DM representation resuts in a reduced approimation error as a comparison to Monte-Caro fiters reveas for a ikeihood and prior density with sma overap as we as for mutimoda ikeihoods. The rest of this paper is structured as foows: After giving the mathematica probem formuation in Sec. II, the required distance measure for Dirac mitures is introduced in Sec. III and the progressive correction agorithm is stated in Sec. IV ISIF 35

2 The benefits of the proposed approach are discussed in an eperimenta vaidation in Sec. V. II. PROBLEM FORMULATION The Bayesian estimation framework consists of two recursivey aternating steps: a prediction and a correction step. The prediction step corresponds to cacuating the net state estimate k+ based on the current estimate k and the system dynamics a k (.). The correction step integrates a measurement ŷ according to the measurement function h k (.) into the current state estimate and thereby corrects the predicted state. This paper is concerned with the correction step ony, i.e., p( k+ y :k+ ) p( k+ y :k ) p(y k+ k+ ), where y :k denotes the set of measurements from timestep to k and focuses on a prior density approimated by a DM with n N equay weighted components minimizing the distance measure described in the Sec. III, p( k+ y :k ) n δ( n i ), () i= where δ( i ) denotes the Dirac measure at position i R N. The goa is to find the optima DM resuting from the mutipication of the prior density with a continuous ikeihood function (.) ( k+ ) p(ŷ k+ k+ ), () where ŷ k+ denotes the current observation and renormaization was omitted. As we focus on a singe correction step, we wi omit the time inde k from now on. The correction step () may be triviay soved by directy mutipying the () with the ikeihood and subsequent renormaization of the new weights w i = ( i ) n j= ( j ). (3) However, this approach wi ead to weight decay over time and thus, reduces the effective resoution per Dirac component. Eventuay, a but very few components wi have neary zero weights and coud be negected. In order to avoid this degeneracy probem, the basic SIR agorithm [7] draws new sampes from the reweighted Diracs in each correction step. This is probematic for systems with ow noise, where the propagated components wi mainy concentrate around the od partice positions. A possibe remedy that is used in post reguarized partice fiters, e.g., [8], is to reguarize the posterior measure by convoving the partice distribution - after samping from the reweighted DM - with a Gaussian kerne. Samping from this reguarized density then effectivey broadens the distribution, but erodes representation detais especiay for mutimoda densities. The origin of these probems is iustrated in Fig. : Let R n ( f) denote the optima representation in the set of DM with n components and equa weights for a given density f. The goa is to find the optima approimation of the underying true density f mutipied with the ikeihood as a DM, R n ( f ), given the optima approimation of the true density f = R n ( f). This probem is obviousy i-posed, as Figure. f R n M f R n f = R n ( f) M (f) R n ( f ) Diagram of the probem to be soved, see tet for epanation. the reduction R n (drawing sampes for Monte-Caro-based methods or minimization of a certain distance measure here) is not invertibe. In order to sove this probem, additiona prior information about the smoothness properties of and f has to be empoyed. Roughy speaking, what one assumes is that the true density has its probabiity mass smeared out uniformy between Dirac components. The goa is then to use an initia DM approimation and the smoothness assumptions in order to construct an agorithm M producing resuts as cose to the optima soution R n ( f ) as possibe. There are two chaenges when trying to incorporate the smoothness assumption of the underying densities. First, the ikeihood evauation at the origina Dirac positions wi yied ony a high resoution in regions where the prior density is high. What is sought, however, is a resoution that is adapted to the posterior density instead. Fig. iustrates this probem. Based on the insight that this probem is ess probematic for ikeihoods with ow variation, the idea of the proposed approach is to spit the ikeihood into factors with ower variation and perform the mutipication progressivey. The proposed approach resembes [6] and can be understood as an etension of this approach to systematic DM approimations. The factors of the ikeihood function are each mutipied pointwise with the current intermediate density and subsequenty reapproimated with an equay weighted DM. This effectivey moves the Dirac components to regions with higher posterior probabiity, thereby samping the ikeihood with a higher resoution in these regions. p() Prior Likeihood Posterior 4 Figure. Iustration of the resoution probem: The bue Dirac components approimate the prior Gaussian we, but offer ow resoution in regions with high posterior density. The optima evauation points for the ikeihood woud correspond to the points that optimay approimate the posterior density (dashed Dirac components). 36

3 A second chaenge is to guarantee consistency with the origina continuous density, in the sense that the processed sampes shoud sti represent a smooth function and therefore, must not form oca custers throughout the progression. Instead of ony working with a genera purpose distance measures on the ower path of Fig., one has to incorporate another term that connects to the underying true density f by enforcing inter-dirac distance preservation in the reapproimation step, which wi be described in Sec. IV-B. First, in Sec. III, the empoyed distance measure is introduced and in Sec. IV, the progressive correction step is epained. III. LOCALIZED CUMULATIVE DISTRIBUTION AND MODIFIED CRAMÉR-VON-MISES DISTANCE In order to compare two given Dirac miture distributions, a suitabe distance measure has to be defined. As pointwise evauation and integra distance measures are not defined for Dirac distributions, this distance measure has to be based on probabiity mass differences under a subset of possibe kerne functions, which motivates the foowing concepts, as originay defined in [9]. Definition (Locaized Cumuative Distribution, [9]) Let f : R N R + be a probabiity density function. The corresponding Locaized Cumuative Distribution (LCD) is defined as F f (m, b) f() K( m, b) d = (f K(., b))(m), R N where denotes the convoution product, b R N + is a width parameter, and K : R N R N [, ] a kerne function. In this paper, ony Gaussian kernes of the ) form K b () K(, b) = ep ( b, with a singe width parameter b R and the Eucidean distance. are used. In anaogy to the Cramér-von Mises distance, a distance between two generic probabiity distributions is defined in [9] via the squared integra distance of their corresponding LCDs. Definition (Modified Cramér-von Mises distance, [9]) Let f : R N R + and g : R N R + be two probabiity distributions and F f / F g their corresponding LCDs. The modified Cramér-von Mises distance (mcvmd) is defined as D(f, g) w(b) (F f (m, b) F g (m, b)) dm db R R N = w(b) K b f K b g L db, R with a suitabe weighting function w : R + R +. As defined above, the mcvmd compares two densities by their differences in probabiity mass under smooth ocaized kernes. As a weighted mean over a possibe kerne positions and kerne widths is performed, the resuting measure is transation-invariant and does not favor the equivaence of the given densities at a particuar scae. The use of isotropic Gaussian kernes and the squared integra distance measure aows for an anaytic soution of a integras in the mcvmd of the LCDs of two DMs. Theorem (mcvmd for Dirac Miture Densities) Let f() = n i= w i δ( i ) and g() = m i= w i δ( i ) denote two N-dimensiona Dirac miture densities. If f and g possess equa means, the mcvmd can be epressed as D(f, g) = πn/ ( w T Γ w w T Γ w + w T Γ w ), 8 where (w) i w i denote the weight vectors and the matrices Γ y are defined via Γ y i,j γ( i y j ) with the function γ(s) s n s. A proof of this theorem is given in []. IV. PROGRESSIVE CORRECTION In this section, the progressive correction step is derived. The ikeihood decomposition and the reguarization term are introduced, as we as the overa agorithm is stated. A. Likeihood Decomposition As motivated, the ikeihood decomposition addresses the probem of misaigned ikeihood evauation points by the introduction of sub-steps. To this end, the ikeihood function is dynamicay decomposed into a product of the form () = () λi, (4) i with the constraints λ i >, λ i =. (5) At a given intermediate progression step k, the DM from the previous progression step f k () n δ( n i ) (6) i= approimates the prior density mutipied with the pseudoikeihoods from a preceding steps, i.e., f k f λ... λ k. Note, that the DM in (6) has equa weights. The net eponent λ k is determined in the foowing way: In order to contro the oca error in this progression step due to misaigned ikeihood evauation points, we demand that the maimum and minimum Dirac weights after pointwise mutipication with the current pseudo-ikeihood λ k must not differ by a factor greater than α. The maimum eponent λ k meeting this requirement may be cacuated by λ k ma n α λ k = α λ k =, (7) min n ma n min where ma ma i {,..,n} ( i ) and min min i {,..,n} ( i ) denote the maimum and minimum ikeihood coefficients. The subprobem in this progression step is then to find a good DM representation of f k λ k given a DM representation of f k. As the weights of f k mutipied with λ k differ i= 37

4 p().4 p().4 p() (a) Step. (b) Step 6. (c) Step. Figure 3. Progressive correction with α = appied to the eampe in Fig., eading to correction steps. The pots show the current Dirac components before the respective progression step in red, which are reweighted with the current pseudo-ikeihood (dash-dotted curve), a broader version of the origina ikeihood. The resuting Diracs after reapproimation are shown in back. As the progression continues, the Dirac positions move into regions with higher posterior density. ony by a factor of α, the intermediate product f k λ k is cose to f k and, hence, the ikeihood evauation points are cose to optima, cf. Fig.. The resuting reweighted DM is then reapproimated by an equay weighted Dirac miture f k. As a resut, the DM components move into areas with higher posterior probabiity and consequenty increase the ikeihood samping resoution in those areas during the net progression step. This procedure is repeated unti the whoe ikeihood is processed. Fig. 3 shows an iustration of different stages in the progression. B. Reguarization As motivated before, processing the DM progressivey soey with a generic distance measure may produce soutions that are cose to the origina Dirac miture mutipied with the ikeihood f but not necessariy cose to the rea product f that one wants to approimate, cf. Fig.. In order to incorporate the smoothness assumption of the underying density during the progression s reapproimation step, a measure for the reguarity and smoothness of the underying density has to be defined. The energy of a DM provides such a measure. Definition 3 (Energy of a Dirac Miture) Let f n i= w iδ( i.) denote a DM density. The miture s energy is defined as n n w i w j E i j. (8) i= j= j i The definition of the energy in (8) is an anaogy from physics, resembing the energy of an eectric fied of charged partices at positions i carrying the charge w i. This new measure of reguarity or smoothness of the underying density is then empoyed in the reapproimation step as an additiona penaty term: In addition to a ow mcvmd to the reweighted DM, the reapproimating DM shoud aso have simiar smoothness properties and hence simiar energy vaues. The proportion β between approimation quaity and smoothness preservation quantifies the importance of the representation quaity for the probem at hand and may be determined by generic mode seection agorithms []. C. Agorithm In each progression step, the agorithm is composed of two main parts: the ikeihood factorization with the weight computation based on the mutipication of the preceding DM with the pseudo-ikeihood and the penaized reapproimation of the obtained DM. First, the new weights are cacuated based on the current ikeihood eponent as described in Sec. IV-A. As the constraints (5) have to be met, some additiona bookkeeping needs to be performed in order to obtain a vaid decomposition. Second, the subprogram REAPPROX in Ag., reapproimates the reweighted Dirac miture f = n i= w j δ( i.) with an equay weighted DM g[x] = n n i= δ( i.), where X i,j = (j) i denotes the matri of a Dirac positions, by minimizing the distance measure presented in Sec. III penaized by the energy difference as defined in Sec. IV-B. In summary, the foowing optimization probem is soved X = argmin D(f, g[x]) + β (E(f) E(g[X])) (9) s.t. T X/n = w T X. This probem may be soved with standard sovers, e.g., with a imited memory Broyden-Fetcher-Godfarb-Shanno (L- BFGS) agorithm []. The required gradient of the objective function is provided in the appendi. The agorithm summarizing both main parts is given in Ag.. V. EXPERIMENTS In this section, we perform an eperimenta comparison of the proposed agorithm with standard methods in Monte-Caro partice fitering. To this end, we set up two eperiments, which aow for an anaytic soution in order to compute different performance metrics on the resuts. First, the eperimenta setup of both eperiments is epained. Then, the different 38

5 Agorithm Progressive Correction t whie t < do i ( i ) i Cac. net eponent λ n α/(n ma n min ) if λ + t > then λ t end if w i λ i /( n j= j λ ) i Cac. new weights REAPPROX(, w) t t + λ end whie Reapproimation performance metrics are epained. Finay, the resuts are shown and discussed. A. Eperimenta Setup Eperiment (Unimoda ikeihood): The first eperiment considers a Gaussian prior distribution f = N ( µ; Σ ) with zero mean µ = [, ] T and covariance matri [ ] Σ =, which is to) be mutipied by a singe Gaussian ikeihood = N (µ ; Σ with mean µ = [3, ] T and covariance matri [ ].4.3 Σ =..3.4 The difficuties for partice agorithms in this eperiment ie in the reativey sma overap between prior density and ikeihood and, as the ikeihood covariance is not ais-aigned, to capture the rotation of the posterior distribution s covariance eipse in the right way. Eperiment (Bimoda ikeihood): The second eperiment consists of a zero-mean Gaussian prior distribution with unit covariance, which is to be mutipied with a bimoda ikeihood of the form N ( [ 3, ] T ) ( ; Σ + N [3, ] T ) ; Σ, with [ ]. Σ = 3, i.e., a Gaussian miture with two components that form two narrow strips in y-direction at = ±3. The chaenges for partice agorithms in this eperiment stem from the bimoda ikeihood that aso yieds a bimoda posterior distribution. Hence, the Gaussian assumption usuay made in additiona Monte-Caro reguarization steps is inappropriate. B. Initia Approimation and Performance Metrics First, the Gaussian prior density has to be approimated by a DM with equa weights. For the proposed deterministic agorithm this is performed by minimizing the mcvmd [3]. For the Monte-Caro agorithms, the initia density is approimated by random samping. In order to compare the resuts of the different agorithms quantitativey, suitabe performance metrics have to be defined. As the processed partice density is sti a DM, integra distance measures cannot be empoyed, so we imit ourseves to compare the first and second moment of the resuting densities as we as the mean modified Cramér-von Mises distance (MMCvMD). In the case of Monte-Caro methods, an average over n run = runs is performed, whereas the proposed deterministic agorithms are run ony once. If we define the residua probabiity in the k th run as r k f M (f) k, the empoyed error metrics are defined via RMSE m RMSE C ( n run n run k= i= n run n run / N ) E[r k ] i, k= i,j= N C[r k ] i,j MMCvMD n run D( n f, M (f) k ), run k= with E / C denoting the mean / covariance and N the dimension of the system. C. Resuts and Discussion Figure 4 shows a comparison of eempary runs of the two eperiments with 56 partices. In the foowing, the resuts of the different agorithms wi be discussed. COND: As epected, due to the ony weaky overapping ikeihood and prior density, the naïve condensation agorithm seects ony a few different partices which then have higher weight, thereby reducing the effective number of partices. PR: The probem of degenerating partice approimations is often addressed by adding a post-reguarization step, i.e., convoving the given partice density with a Gaussian kerne with a scaed version of the DM s covariance. The optima scaing parameter h opt = (4/(N + )) N+4 n N+4 with n the number of partices can be derived in a density estimation contet [4], but is ony optima for Gaussian densities. This epains the bad performance of post-reguarization in the second eampe: As the true density is bimoda, with arger etent in -direction, the smoothing kernes variance in this direction is too arge, which resuts in a arge partice variation in this direction. PPR: The same probem arises in progressive postreguarization [6], which uses h opt / as scaing parameter but yieds comparabe covariances to PR, as the error of mutipe post-reguarization steps accumuate. SRA: The naïve approach for a systematic mutipication in the mcvmd-minimization contet woud be to mutipy the given prior DM pointwise with the ikeihood and then reapproimate the resuting DM by equay weighted Diracs minimizing the mcvmd. This systematic reapproimation does not produce a high representation quaity as not enough information of the ikeihood is taken into account. /, 39

6 PSRA w/o energy: Appying the naïve systematic reapproimation procedure progressivey without the additiona energy distance term surprisingy does not improve the situation much. As the progression continues, partices tend to form oca custers. The probem here is that the reduction steps minimizing the mcvmd aone introduce sma errors that accumuate. Minimization of the mcvmd as a genera concept is not specificay taiored to preserve the smoothness properties of a DM, i.e., to assure that the approimated DM is sti the reduction of an underying smooth function. PSRA: In order to incorporate the assumption that the underying densities possess strong smoothness properties, another distance term based on the DM energy has to be introduced. The proposed progressive systematic reapproimation then produces high-quaity resuts as the resuting partices cover the posterior density more uniformy. Fig. 5 shows a quantitative evauation of the different agorithms as a function of the number of partices. One can ceary see that the proposed agorithm outperforms standard Monte- Caro techniques and the naïve systematic approimation in amost a cases. However, the main advantage of the proposed agorithm is the reguar coverage of the posterior distribution with partices, which increases the samping resoution for further processing and cannot be reay quantified by the empoyed error metrics. VI. CONCLUSION In this paper, we have proposed a progressive fiter step for deterministic Dirac miture approimations, which avoids partice degeneration by preserving an additiona energy term throughout the progression. The eperimenta comparison with current (progressive) Monte-Caro methods for the correction step, such as the method proposed by [6], reveas a much higher representation quaity per partice, as the partices are paced deterministicay to cover the posterior density homogeneousy. In contrast, Monte-Caro methods treat each sampe individuay, which on the one hand reduces the computationa compeity to grow ony ineary with the number of partices. On the other hand, this independence of partices eads to wasted sampes in terms of the posterior samping resoution and therefore to a worse representation quaity per sampe. The higher computationa costs invoved with the noninear optimization in this approach can become negigibe if compe system and measurement modes are empoyed. In such cases, the number of sampes and thereby evauations of the system mode, needed for Monte-Caro methods to reach the same representation quaity as the proposed method, becomes computationay unfeasibe. VII. APPENDIX Theorem (Gradient of Modified Distance Measure) Let f = n i= w j δ( i.) and g[] = n i= w iδ( i.) denote two Dirac miture densities. The gradient of the modified distance measure in (9) with respect to the k th component of the th Dirac ( position is then given as D(f, g[]) + β (E(f) E(g[])) ) = with (k) (k) (k) D(f, g[]) + β (E(f) E(g[])) (k) E(g[]), ( D(f, g[]) = w T (k) Γ ) ( w w T (k) Γ ) w n = 4w (w j γ( j )( (k) (k) j ) j= w j γ( j )( (k) where γ(s) = n s + and (k) E(g[]) = w n j= j REFERENCES (k) j w j (k) j )), (k) j 3. [] R. Kaman et a., A new approach to inear fitering and prediction probems, Journa of basic Engineering, vo. 8, no., pp , 96. [] S. Juier and J. Uhmann, A new etension of the Kaman fiter to noninear systems, in Int. Symp. Aerospace/Defense Sensing, Simu. and Contros, vo. 3. Citeseer, 997, p. 6. [3] D. Aspach and H. Sorenson, Noninear Bayesian estimation using Gaussian sum approimations, Automatic Contro, IEEE Transactions on, vo. 7, no. 4, pp , 97. [4] D. Guo and X. Wang, Quasi-Monte Caro fitering in noninear dynamic systems, Signa Processing, IEEE Transactions on, vo. 54, no. 6, pp , 6. 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7 Eperiment Eperiment COND PR PPR SRA PSRA w.o. energy PSRA Figure 4. Eempary runs of typica partice fiter methods and the proposed deterministic agorithm for 56 partices. (COND) denotes the simpe condensation agorithm, (PR) adds a post-reguarization step, (PPR) is the progressive post-reguarization method as in Oudjane et a. [6] with α = 64, (SRA) is the naive systematic reapproimation, (PSRA w.o. energy) the proposed agorithm with α = 64 but β =, and (PSRA) the proposed agorithm with α = 64 and β =, cf. Sec. V-C for further detais. 33

8 Eperiment Eperiment.5 Mean Error og (RMSEm).5 og (RMSEm) og (n) og (n) Covariance Error og (RMSEC).5 og (RMSEC) og (n) og (n) MMCvMD og (MMCvMD).5.5 og (MMCvMD) og (n) og (n) PRSA SRA PPR PR Figure 5. Performance of the proposed method with α = 64 and β = (PRSA) versus the naïve systematic reapproimation (SRA), condensation and post-reguarization (PR) and progressive post-reguarization [6] with α = 64 (PPR) in different error metrics: RMSE of the mean and covariance matri, as we as the (shifted) mean mcvmd. More epanation can be found in the tet. 33