Monte Carlo Filter Particle Filter

2015 European Control Conference (ECC) July 15-17, 2015. Linz, Austria

Masaya Murata†, Hidehisa Nagano† and Kunio Kashino†‡

Abstract— We propose a new realization method of the B. Sequential Importance Sampling (SIS) sequential importance sampling (SIS) algorithm to derive a new particle filter. The new filter constructs the importance To tackle the problem of interest, the sequential impor- distribution by the Monte Carlo filter (MCF) using sub- tance sampling (SIS) algorithm[4][5] is often used to obtain particles, therefore, its non-Gaussianity nature can be ade- the filtered state estimate. The SIS algorithm approximates quately considered while the other type of particle filter such the p(Xt|{Yt}), that is, the posterior PDF of state at t given as unscented Kalman filter particle filter (UKF-PF) assumes observations up to time t, as follows: a Gaussianity on the importance distribution. Since the state estimation accuracy of the SIS algorithm theoretically improves Sequential Importance Sampling (SIS) as the estimated importance distribution becomes closer to the N true posterior probability density function of state, the new filter 0(i) (i) is expected to outperform the existing, state-of-the-art particle p(Xt|{Yt}) ≈ wt δ(Xt − Xt ) filters. We call the new filter Monte Carlo filter particle filter Xi=1 (MCF-PF) and confirm its effectiveness through the numerical where simulations. (i) (i) Xt ∼ p(Xt|Xt−1 = Xt−1, Yt), i = 1, 2, · · · , N I. INTRODUCTION (i) (i) (i) 0 p(Yt|Xt = X )p(Xt = X |Xt−1 = X − ) w(i) = w (i) t t t 1 A. Problem Definition t t−1 (i) (i) p(Xt = Xt |Xt−1 = Xt−1, Yt) In this paper, we consider the sequential state estimation (i) problem of the following nonlinear non-Gaussian state space 0(i) wt wt = N (j) model. j=1 wt P(i) 0(i) Xt = f(Xt−1) + Φt, Φt ∼ p(Φt) (1) Here, {Xt , wt }, i = 1, 2, · · · , N are weighted set of Yt = h(Xt) + Ωt, Ωt ∼ p(Ωt) (2) particles (particle values and their corresponding normalized 0 importance weights) at time t and N w (i) = 1 holds. Here, X is the L-dimensional random vector called state i=1 t t δ(·) is the Dirac’s delta function and this PDF approximation at time t and this is the estimation target using the P is called the Dirac’s point-mass estimate. p(Xt|Xt−1 = past series of M-dimensional observations {Yt−1} = (i) X − , Yt) is the optimal importance distribution specified by {Yt−1, Yt−2, · · · , Y1} or {Yt} = {Yt, Yt−1, · · · , Y1}. The t 1 ¯ eqs. (1) and (2) and it proves to minimizes the variance of the former is called the predicted state estimate Xt := (i) (i) importance weights. X ∼ p(Xt|Xt−1 = X − , Yt), i = E[Xt|{Yt−1}] and the latter is called the filtered state t t 1 ˆ 1, 2, · · · , N denotes the random (i.i.d.) sampling from the estimate Xt := E[Xt|{Yt}]. Their approximation errors (i) T optimal importance distribution. and are calculated by E[(Xt − X¯t)(Xt − X¯t) |{Yt− }] and p(Yt|Xt = Xt ) 1 (i) (i) T − E[(Xt−Xˆt)(Xt−Xˆt) |{Yt}], respectively. f(·) and h(·) are p(Xt = Xt |Xt 1 = Xt−1) are likelihood and transition nonlinear functions and called state transition and state obser- PDFs specified by eq. (2) and (1), respectively. The above vation, respectively. Φt and Ωt are independent white noises importance weight equation also indicates that the current following arbitrary probability density functions (PDFs) and weight is calculated from the previous weight and together they are called system noise and observation noise, respec- with the fact that the current particle value depends on the tively. Equations (1) and (2) are referred to as system model previous value, this algorithm can be sequentially performed and observation model, and these two equations constitute after the appropriate importance distribution selection. Then ˆ the state space model of interest. For this model, since both the filtered state estimate at time t (denoted as Xt) can be state and noises do not always follow Gaussian distributions, obtained as follows: the well-studied, state-of-the-art Gaussian filters[1] such as N ˆ 0(j) (i) extended Kalman filter (EKF)[2] and unscented Kalman Xt = wt Xt (3) filter (UKF)[3] can not be adequately employed. Therefore, Xi=1 designing the sequential state estimator for this model still Note that the SIS algorithm is not designed to obtain remains as a challenging problem. the predicted state estimate at time t (denoted as X¯t). †Masaya Murata, Hidehisa Nagano and Kunio Kashino are research This algorithm proves to easily encounter the degeneracy scientists at NTT Communication Science Laboratories, NTT Corpora- problem that almost all the particles have zero or nearly tion, 3-1, Morinosato Wakamiya, Atsugi-Shi, Kanagawa 243-0198, Japan. zero weights as the time proceeds and therefore, the particle murata.masaya at lab.ntt.co.jp ‡Kunio Kashino is also a visiting professor at National Institute of resampling procedure is often performed after obtaining (i) 0(i) Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430 Japan. {Xt , wt }, i = 1, 2, · · · , N. The resampling makes the

978-3-9524269-3-7 ©2015 EUCA 2836 importance weights for all of the new particles become 1/N. filter is expected to be superior over the existing filters. We The new particles are randomly drawn from the discrete call the new filter Monte Carlo filter particle filter (MCF-PF) (i) 0(i) set {Xt }, i = 1, 2, · · · , N with probabilities {wt }, i = and confirm its effectiveness using the numerical examples. 1, 2, · · · , N. Note that after the resampling procedure, the The rest of this paper is organized as follow. Section I sequential weight equation becomes I describes the MCF algorithm and its relationship to the (i) (i) (i) BF algorithm. In section III, we explain the proposed filter (i) p(Yt|Xt = Xt )p(Xt = Xt |Xt−1 = Xt−1) formulation and summarize the sequential state estimation wt ∝ (i) (i) p(Xt = Xt |Xt−1 = Xt−1, Yt) algorithm. Numerical examples are shown in section IV and (4) section V summarizes the basic findings and conclusion of 0(i) this paper. since the resampling makes {wt }, i = 1, 2, · · · , N become 1/N. II. MONTE CARLO FILTER (MCF) A. Algorithm C. Importance Distribution Selection As well as the SIS algorithm, the Monte Carlo filter The selection of the importance distribution affects the (MCF) is also a promising solution for the aforementioned entire state estimation performance of the SIS algorithm and sequential state estimation problem. Unlike the SIS, the MCF generally speaking, the smaller the difference between the approximates the Bayesian filter algorithm using randomly assumed importance distribution and true posterior PDF of drawn samples called particles. And, the MCF not only state, the better the state estimation accuracy becomes. The (i) provides the filtered state estimate, but also provides the simplest choice is p(Xt|Xt−1 = Xt−1, Yt) ≈ p(Xt|Xt−1 = (i) predicted state estimate. The algorithm of MCF can be Xt−1) and it derives the bootstrap filter (BF)[6]. Here, the summarized as follows: (i) p(X |X − = X ) is the state transition PDF specified by t t 1 t−1 Monte Carlo Filter MCF eq. (1) and the BF is one of the most popular particle filters ( ) ˆ (i) ¯ ¯ (i) when tackling the problem of interest. However, one obvious X0 ∼ N (X0, P0), wˆ0 = 1/N, i = 1, 2, · · · , N problem is that the possible large deviation between the For each t = 1, 2, · · · assumed importance distribution and the true one is likely to Predicted state estimate arise. To alleviate this gap, we may need the large number of (i) particles and it sometimes results in the severe computational Φt ∼ p(Φt), i = 1, 2, · · · , N ¯ (i) ˆ (i) (i) burden. Xt = f(Xt−1) + Φt The other often-used selection is to assume p(Xt|Xt−1 = 0(i) 0(i) (i) w¯t = wˆt−1 − where − is supposed Xt−1, Yt) ≈ p(Xt|X(i),t 1, Yt) X(i),t 1 N 0 to follow eqs. (1) and (2) and E[X(i),t−1|{Yt−1}] is set ¯ (j) ¯ (j) (i) Xt = w¯t Xt to Xt−1. We call X(i),t−1 ith particle state. Moreover the Xj=1 p(Xt|X(i),t−1, Yt) is assumed as Gaussian distributed. Then, Filtered state estimate using N particles which specify the expectations of the N ˆ (i) ¯ (i) Xt = Xt particle states, N different p(Xt|X i ,t− , Yt) are estimated ( ) 1 (i) 0(i) ˆ (i) by using the Gaussian filter such as EKF or UKF. The wˆt = w¯t p(Yt|Xt = Xt ) former filter is called extended Kalman filter particle filter (i) 0(i) wˆt (EKF-PF)[7] and the latter one is called unscented Kalman wˆt = j N wˆ( ) filter particle filter (UKF-PF)[7][8]. Since the latest obser- j=1 t NP vation Y is incorporated when constructing the importance 0 t ˆ (j) ˆ (j) distribution, these filters theoretically are expected to be Xt = wt Xt Xj=1 more accurate than the BF that does not take Yt into account. The UKF-PF is also theoretically better than the ˆ (i) (i) Here, X0 ∼ p(X0) and Φt ∼ p(Φt) are the ith particles EKF-PF since the unscented transformation (UT)[3] in the randomly drawn from the initial state PDF and the system UKF algorithm handles the nonlinear state transformation ¯ (i) noise PDF, respectively. p(Yt|Xt = Xt ) is the likelihood with higher accuracy than the truncated Taylor series based PDF specified by eq.(2). X¯t and Xˆt are the predicted and linearization approach in the EKF algorithm. the filtered state estimates, and they are calculated by the D. Contribution weighted sum of the particle values as shown in the algorithm. As well as the SIS algorithm, the particle resampling In this paper, we propose a method to construct the (i) procedure is performed after the filtered state estimate to p(Xt|Xt−1 = Xt−1, Yt) by using Monte Carlo filter randomly drawn particles from the estimated posterior state (MCF)[9] with sub-particles. Therefore the non-Gaussianity 0(i) PDF. Then {w }, i = 1, 2, · · · , N all become 1/N and the of the importance distribution is taken into account, con- t weight equation in the filtered state estimate can be simply tributing to the improvement in the state estimation accuracy. expressed as follows: Although the computational burden increases due to the MCF (i) ˆ (i) execution for all of the particles under consideration, the new wˆt ∝ p(Yt|Xt = Xt ) (5)

2837 Such particle resampling is also effective to avoid the particle Substituting eqs. (8) and (9) into eq. (6) yields, impoverishment problem that almost all of the particles have (i) the same values after the time proceeds to some extent. p(Xt|Xt−1 = Xt−1, Yt) M p(Y |X ) B. Relationship to the Bootstrap Filter (BF) ≈ t t δ(X − X¯ (i),(j)) M ¯ (i),(k) t t The difference between the MCF and BF mentioned in k=1 p(Yt|Xt = Xt ) Xj=1 the previous section is that while the MCF is based on PM 0(i),(j) ˆ (i),(j) the Bayesian filter, the BF is formulated from the SIS = wˆt δ(Xt − Xt ) (10) (i) (i) (i) (i) ˆ ¯ ˆ Xj=1 algorithm. The MCF uses Xt = Xt = f(Xt−1) + Φt to approximate the posterior PDF of state, while the BF where Xˆ (i),(j) = X¯ (i),(j) (i) (i) t t uses X ∼ p(X |X − = X ). Then, since their weight (i),(j) (i),(j) t t t 1 t−1 wˆ = p(Y |X = X¯ ) equations are the same (eq. (4) in section I-B becomes eq. t t t t (i),(j) 0 wˆ (5) in section II-A by selecting the transition prior as the wˆ (i),(j) = t importance distribution), the MCF is very similar to the BF. t M (i),(k) k=1 wˆt Therefore the same problem occurring in the BF algorithm P also exists in the MCF algorithm such that the large number Equation (10) means that the importance distribution for the of particles are required to guarantee the state estimation ith particle is approximated by the MCF algorithm using accuracy due to the deviation between the simple importance the M sub-particles. Therefore, random sampling from the distribution and the true posterior PDF of state. To address estimated importance distribution can be replaced with the ˆ (i),(j) this issue, we propose the new particle filter formulation uniform sampling from the discrete set {Xt }, i = 0(i),(j) that employs the MCF algorithm to accurately estimate the 1, 2, · · · , M with probabilities {wˆt }, i = 1, 2, · · · , M. importance distribution. The resulting estimated importance Such sampling corresponds to the particle resampling proce- distribution is expected to be much closer to the true posterior dure mentioned in the previous section. In short, we construct PDF of state than that for the BF. the importance distribution for each particle in conjunction with the MCF algorithm using the sub-particles and obtain III. MONTE CARLO FILTER PARTICLE FILTER (MCF-PF) the new particle by performing the resampling procedure to A. Importance Distribution Estimation using Sub-particles the constructed importance distribution. The new filter called Monte Carlo filter particle filter (MCF-PF) constructs the importance distribution in the SIS B. Importance Weight Calculation algorithm by using MCF algorithm. The importance distri- As explained in the previous section, by using eq. (10), bution becomes as follows: the new particle sampled from the estimated importance (i) distribution can be expressed as follows: p(Xt|Xt−1 = Xt−1, Yt) (i) (i) (i) ˆ − p(Yt|Xt)p(Xt|Xt−1 = Xt−1) Xt ∼ p(Xt|Xt 1 = Xt−1|Yt) = (i) M p(Yt|Xt)p(Xt|Xt−1 = Xt−1)dXt 0(i),(j) ˆ (i),(j) ∼ wˆt δ(Xt − Xt ) R1 (i) j = p(Yt|Xt)p(Xt|Xt−1 = Xt−1) (6) X=1 Ct ˆ (i) ˆ (i),(1) ˆ (i),(2) ˆ (i),(M) (i) Since the Xt is either Xt , Xt , · · · , or Xt , Here, Ct ≡ p(Yt|Xt)p(Xt|Xt−1 = Xt−1)dXt. By random (i) (i),(l) (i) we can also denote Xˆ as Xˆ , where l is either 1, sampling fromR p(X |X − = X ) specified by eq. (1), the t t t t 1 t−1 2, · · · , or M. Therefore, the sequential importance weight following Dirac’s point-mass approximation holds: calculation and the normalized weight become M (i) 1 (i),(j) (i) (i) (i) − ¯ ˆ ˆ p(Xt|Xt 1 = Xt−1) ≈ δ(Xt − Xt ) (7) (i) 0(i) p(Yt|Xt = Xt )p(Xt = Xt |Xt−1 = Xt−1) M wˆ = wˆ − Xj=1 t t 1 ˆ (i) (i) p(Xt = Xt |Xt−1 = Xt−1, Yt) (i),(j) (i) 0(i) Here, X¯ ∼ p(Xt|Xt−1 = X − ), j = 1, 2, · · · , M t t 1 = wˆt−1 × and the sub-particle number M is not necessarily equal to ˆ (i),(l) ˆ (i),(l) (i) N (particle number in the SIS algorithm). We can then obtain p(Yt|Xt = Xt )p(Xt = Xt |Xt−1 = Xt−1) 0(i),(l) the following equations: wˆt δ(0) (i) (11) p(Yt|Xt)p(Xt|Xt−1 = Xt−1) (i) 0 M (i) wˆt 1 wˆt = k (12) ≈ p(Y |X ) δ(X − X¯ (i),(j)) (8) N wˆ( ) M t t t t k=1 t Xj=1 P Although δ(0) appears in eq. (11), it will be omitted when M 0 1 (i),(k) (i) C = p(Y |X = X¯ ) (9) calculating the normalized weight wˆt in eq. (12). That is, t M t t t Xk=1 the weight calculated without the δ(0) (right side of the

2838 following equation) yields the same normalized weight as Filtered state estimate follows: Importance distribution estimation (i) 0(i) (i),(j) (i) X¯ ∼ p(X |X − = X − ), wˆt ∝ wˆt−1 × t t t 1 t 1 ˆ (i),(l) ˆ (i),(l) (i) (j = 1, 2, · · · , M) p(Yt|Xt = Xt )p(Xt = Xt |Xt−1 = Xt−1) 0 (i),(j) ¯ (i),(j) (i),(l) wˆt = p(Yt|Xt = Xt ) wˆt (i),(j) (13) 0(i),(j) wˆt wˆt = M wˆ(i),(k) Therefore, when executing the MCF-PF, eq. (13) can be used k=1 t to obtain the one-step-ahead importance weight. Sequential impPortance sampling M However, although an approximation error is included, we 0 Xˆ (i) ∼ wˆ (i),(j)δ(X − Xˆ (i),(j)), can also substitute eq. (7) into eq. (11) to cancel the δ(0) as t t t t Xj=1 follows: (i = 1, 2, · · · , N) (i),(l) ˆ 1 i , l i (i) 0(i) p(Yt|Xt = Xt ) δ(0) ˆ ( ) ( ) ˆ ( ) M Xt ≡ Xt wˆt ≈ wˆt−1 0 i , l wˆ ( ) ( )δ(0) (i) t wˆt ∝ i , l ˆ ( ) ( ) (i),(l) (i),(l) (i) 0(i) p(Yt|Xt = Xt ) ˆ ˆ (14) p(Yt|Xt = Xt )p(Xt = Xt |Xt−1 = Xt−1) ≈ wˆt−1 0(i),(l) 0 i , l Mwˆt ( ) ( ) wˆt This formulation may be more appropriate that the eq. (13) or ˆ (i),(l) since the eq (11) is equal to the division by infinity. (i) p(Yt|Xt = Xt ) wˆt ∝ 0(i),(l) Since the importance weights all become 1/N after the wˆt particle resampling procedure and the sub-particle number (i) 0 wˆ in eq. (14) do not change the resulting normalized (i) t M wˆt = N (k) weight, above sequential weight equations can be expressed k=1 wˆt N as follows: P 0 Xˆ = wˆ (k)Xˆ (k) ˆ (i),(l) ˆ (i),(l) (i) t t t (i) p(Yt|Xt = Xt )p(Xt = Xt |Xt−1 = Xt−1) Xk=1 wˆt ∝ 0(i),(l) wˆt The algorithm for obtaining the predicted state estimate (15) is the same as that for the MCF. We call the MCF-PF p(Y |X = Xˆ (i),(l)) using the weight equation in eq. (15) and that using the wˆ(i) ∝ t t t (16) t 0(i),(l) weight equation in eq. (16) as MCF-PF-1 and MCF-PF- wˆ t 2, respectively. Their performance difference is numerically In either case, the computational cost for performing the SIS investigated in the next section in which we compare the algorithm is increased compared to the other SIS realization performance of the MCF-PF with those of the other state- such as EKF-PF or UKF-PF due to the sub-particle calcula- of-the-art filters. tion. However, since no assumptions are imposed when constructing the importance distribution such as the Gaussianity, IV. NUMERICAL SIMULATIONS the MCF-PF algorithm is expected to outperform these PFs. We first investigate the effectiveness of the MCF-PF on We summarize the MCF-PF algorithm for obtaining the the scalar nonlinear state-space models[10] corrupted by predicted and the filtered state estimates in the next section. Gaussian system and observation noises [Prob.1]. We then C. Algorithm examine the performance of the MCF-PF on non-Gaussian system and observation noises such as the Laplace system The algorithm of MCF-PF is shown as follows: noise and Cauchy observation noise [Prob.2]. Monte Carlo Filter Particle Filter (MCF − PF) The comparison filters are UKF (unscented Kalman filter), MCF (Monte Carlo filter), GPF (Gaussian particle filter[11]), ˆ (i) 0(i) X0 ∼ p(X0), wˆ0 = 1/N, i = 1, 2, · · · , N and UKF-PF (unscented Kalman filter particle filter). Here, For each t = 1, 2, · · · the GPF assumes the Gaussian-distributed posterior PDF of Predicted state estimate state in the MCF algorithm and at every time step, new (i) particles are randomly drawn from that PDF. Therefore the Φ ∼ p(Φ ), i = 1, 2, · · · , N t t particle resampling procedure required in the MCF algorithm ¯ (i) ˆ (i) (i) Xt = f(Xt−1) + Φt is not necessary for the GPF and the so-called particle 0(i) 0(i) impoverishment problem that the almost all of the particles w¯t = wˆt−1 N have the same values can be circumvented. The GPF is ¯ 0(j) ¯ (j) expected to work with relatively small number of particles, Xt = w¯t Xt kX=1 leading to the less computational burden compared to the

2839 MCF. Therefore, we also compare the performance of the N = 10 for all of the PFs and M = 10 in the MCF- MCF-PF with that of the GPF in this section. PF algorithm. The results are shown in Table II below and The [Prob.1] is on the sequential state estimation problem again, the bold number indicates the smallest state estimation for the following state-space model. error. From Table II, we can see that when the number of TABLE II [Prob.1] AVERAGE ABSOLUTE STATE ESTIMATION ERRORS OVER 100 MONTE 1 25xt−1 N = 10 M = 10 − CARLO SIMULATIONS WITH AND SETTING. xt = xt 1 + 2 + 8cos(1.2t) + φt, 2 1 + (xt−1) MCF GPF UKF-PF MCF-PF-1 MCF-PF-2 2 1 φt 3.41 3.24 3.04 3.43 3.12 where φt ∼ N(φt; 0, 1) = exp − (2π) 2 (x )2 p y = t + ω , particles became small, the GPF became superior over the t 20 t 2 MCF. This was due to the random sampling effect from 1 ωt where ωt ∼ N(ωt; 0, 1) = exp − the Gaussian-assumed posterior state PDF and the GPF (2π) 2 seemed to work for the particle impoverishment problem that We generated the observation datapfrom t = 1 to t = 1000 was inherent in particle resampling procedure for the MCF based on the initial state x0 = 0. The initial conditions for algorithm. We also found that the UKF-PF scored the lowest the ﬁlters were all set to N(0, 102). Table I shows average estimation error and this observation leads to the conclusion 1 1000 that when the system and observation noises follow Gaussian absolute state estimation errors ( 1000 t=1 |xt − xˆt|) of UKF, MCF, GPF, UKF-PF, MCF-PF-1 Pand MCF-PF-2 over distributions, the UKF-PF is the best choice for addressing 100 Monte Calro simulations. Here, the xt is the true state the ﬁltering problem. at time t. Particle numbers for the MCF, GPF, UKF-PF and The state-space model for the [Prob.2] is described as MCF-PF were all N = 100 and the number of sub-particles follows. At this time, the system noise follows a Laplace used in the MCF-PF algorithms was M = 100. In Table I, distribution and the observation noise follows a Cauchy the bold number indicates the smallest state estimation error. distribution.

TABLE I [Prob.2] 100 1 25xt−1 AVERAGE ABSOLUTE STATE ESTIMATION ERRORS OVER MONTE − xt = xt 1 + 2 + 8cos(1.2t) + φt, CARLO SIMULATIONS WITH N = 100 AND M = 100 SETTING. 2 1 + (xt−1) 1 UKF MCF GPF where φ ∼ Laplace(φ ; 0, 1) = exp(−|φ |) t t 2 t 4.27 1.74 1.77 2 (xt) UKF-PF MCF-PF-1 MCF-PF-2 yt = + ωt, 1.71 1.81 1.70 20 1 where ωt ∼ Cauchy(ωt; 0, 1) = 2 π (1 + (ωt) ) From the result in Table I, we first confirmed that the PFs are much better than the Gaussian filter represented by As well as the previous problem, we again generated the the UKF in terms of state estimation accuracy. However, observation data from t = 1 to t = 1000 based on 2 since the UKF was much faster than the other PFs, whether x0 = 0. The initial conditions for the filters were N(0, 10 ). using the UKF or PFs depends on the problem we solve. Table III shows the average absolute state estimation errors 1 N(=1000) For example, when the case that the filtering (processing) ( N−10 t=11 |xt − xˆt|) of MCF, GPF, UKF-PF, MCF- time is rather important, the UKF still remains as one of the PF-1 andP MCF-PF-2 over 100 Monte Calro simulations. candidate filtering algorithms. Although the GPF was also Here, when employing UKF and UKF-PF, we assumed superior in terms of calculation cost over the other PFs, its that the Laplace system noise was approximated by N(0, 2) state estimation accuracy was very slightly worse than that and also assumed that the Cauchy observation noise was for the MCF. This was due to the Gaussian assumption on approximated by N(0, 10), respectively. These Gaussian- the posterior PDF of state. assumed PDF settings were required for the UKF and the The UKF-PF scored the second best estimation accuracy. UKF-PF algorithms. As well as in the previous simulations, The MCF-PF-2 was better than the MCF-PF-1. This result we set N = 100 for all of the PFs and M = 100 for the two indicated that the delta function in the weight equation MCF-PF algorithms. (11) was better to be eliminated. Although the statistical From Table III, we can see that the UKF was not capa- significance difference was not observed between the results ble of filtering observations corrupted by the non-Gaussian of UKF-PF and MCF-PF-2, the MCF-PF-2 scored the best noises. MCF was better than the GPF while sacrificing the state estimation accuracy among the comparison filters. filtering speed. The UKF-PF sometimes suffered from the We also executed the same 100 Monte Carlo simulations problem that all of the importance weights became zero, with less number of particles for all of the PFs to investigate which implied that sampled particles from the importance the performance dependency on the particle number. We set distributions were not located within the support domain of

2840 the likelihood PDF or the state transition PDF. Because of MCF-PF-2 and it was the fastest filtering algorithm among the large noise realization of the Cauchy observation noise, the five comparison filters. The MCF-PF-2 scored the second the observation update for each particle state sometimes pro- best estimation accuracy, showing the effectiveness of the duced the deviated importance distributions for each particle, proposed filter. and that made the sampled particles being located out of We found that the low state estimation accuracy of the the aforementioned support domains. This result indicated UKF were implying that the prior state PDF was not better that for filtering problems of noisy observations, the UKF- to be assumed as Gaussian. On the other hand, since the GPF PF might be not suitable. The setting of large observation was successful, the posterior state PDF could be assumed as noise variance in the UKF calculation is one remedy for Gaussian for resampling new particles. this problem, however, it leads to degrade the entire state To summarize the overall results, when the relatively large estimation accuracy. number of particles was allowed in the filter execution, TABLE III we confirmed that the MCF-PF-2 was the best among the AVERAGE ABSOLUTE STATE ESTIMATION ERRORS OVER 100 MONTE other PFs. However, for a situation that the small number of CARLO SIMULATIONS WITH N = 100 AND M = 100 SETTING. particles was desired due to the computational concern, the UKF MCF GPF GPF provided good state estimation results. To the contrary, 9.34 2.69 2.86 when the sufficiently large number of particles is allowed, UKF-PF MCF-PF-1 MCF-PF-2 the accuracy difference between MCF and MCF-PF becomes 4.21 2.85 2.66 negligible. In that case, since the MCF is much faster than the MCF-PF, the choice of the MCF will be preferred.

Figure 1 shows the state filtering results for the MCF-PF- V. CONCLUSION 2. We can observe two large estimation errors at the noisy We propose the Monte Carlo filter particle filter (MCF-PF) in which the importance distribution in the SIS algorithm is approximated by the MCF using sub-particles. Therefore, the non-Gaussianity nature of the importance distribution can be incorporated and it leads to the better state estimation accuracy than the other PFs such as GPF and UKF-PF. The numerical simulations show the effectiveness of the MCF-PF and also reveal that when the limited number of particles is allowed, the GPF outputs good state estimation results.

REFERENCES [1] K. Ito and K. Xiong. “Gaussian Filters for Nonlinear Filtering Problems”, IEEE Trans. on Automatic Control, Vol. 45, No. Fig. 1. Plots of observation examples, true states and state estimates of 5, pp. 910-927, 2000. MCF-PF-2. The horizontal axis is the time step from t = 400 and t = 450. [2] A. Jazwinski. “Adaptive Filtering”, Automatica, 5(4), pp. 475- 485, 1969. [3] S. J. Julier, J. K. Uhlmann and H. F. Durrant-Whyte. “A New observations corrupted by the realizations of the additive Approach for Filtering Nonlinear Systems”, In Proc. American Cauchy noise. The MCF-PF-2 estimated the other true states Control Conference, pp. 1628-1632, 1995. well from the nonlinear, bimodal observations. [4] A. Doucet. “On Sequential Simulation-based Methods for The MCF-PF-2 was also better than the MCF-PF-1 for Bayesian Filtering”, Tech. Rep. CUED/FINFENG/TR 310, this problem. The MCF-PF-2 scored the best state estimation Department of Engineering, Cambridge University, 1998. [5] S. Sarkka. “Bayesian Filtering and Smoothing”, Cambridge accuracy among the comparison ﬁlters. As well as in the University Press, Cambridge CB2 8BS, 2013. previous problem, we also executed the same 100 Monte [6] N. Gordon, D. J. Salmond and A. F. M. Smith. “Novel Carlo simulations with N = 10 setting for all of the PFs and Approach to Nonlinear/Non-Gaussian Bayesian State Estima- M = 10 setting in the MCF-PF algorithm. The results are tion”, IEEE Proceedings-F 140(2); pp. 107-113, 1993. shown in Table IV. From Table IV, when the limited number [7] R. Van der Merwe and N. De Freitas, A. Doucet and E. TABLE IV Wan. “The Unscented Particle Filter”, Advances in Neural Information Processing Systems 13, pp. 584-590, 2001. AVERAGE ABSOLUTE STATE ESTIMATION ERRORS OVER 100 MONTE [8] A. Doucet, S. J. Godsill and C. Andrieu. “On Sequential Monte N M CARLO SIMULATIONS WITH = 10 AND = 10 SETTING. Carlo Sampling Methods for Bayesian Filtering”, Statistics and MCF GPF UKF-PF MCF-PF-1 MCF-PF-2 Computing, 10(3), pp. 197-208, 2000. 4.06 3.70 5.08 4.45 3.80 [9] G. Kitagawa. “Monte Carlo Filter and Smoother for Non- Gaussian Nonlinear State Space Models”, Journal of Com- putational and Graphical Statistics, 5(1), pp. 1-25, 1996. [10] T. Katayama. “Nonlinear Kalman Filter”, Asakura-Shoten, of particles was allowed, we found that the GPF became 2011 (in Japanese). superior over the MCF. The UKF-PF again suffered from the [11] J. H. Kotecha and P. M. Djuric. “Gaussian Particle Filtering”, problem of importance weights becoming zero after the time IEEE Trans. Signal Processing, Vol.51, No.10, pp.2592-2601, proceeded. For this problem, the GPF also outperformed the 2003.

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