Locally linearized particle filter

LOCALLY-LINEARIZED PARTICLE FILTER GAN LUHUI (B.Sc), NTU A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF STATISTICS AND APPLIED PROBABILITY NATIONAL UNIVERSITY OF SINGAPORE 2013 Declaration I hereby declare that this thesis is my original work and it has been written by me in its entirety I have duly acknowdged all the source of information which have been used in the thesis This thesis has also not been submitted for any degree in any university previously i Contents Introduction 1.1 Motivation 1.2 Outline Problem Statement Extended Kalman Filter 3.1 Kalman filter 3.1.1 3.2 Justification of Kalman filter 10 Extended Kalman Filter 12 3.2.1 One-dimensional EKF 13 3.2.2 Derivation of Extended Kalman Filter 14 3.2.3 Multidimensional Extended Kalman Filter 20 Particle Filter 23 4.1 Basic Monte Carlo Method and Importance Sampling 24 4.2 Sequential Importance Sampling 26 ii 4.3 Particle Filter 28 4.4 Mixture Kalman Filter (Rao-Blackwellized PF) 30 Locally-linearized Particle Filter I 32 Locally-linearized Particle Filter II 36 6.1 Motivation 36 6.2 Algorithm 37 6.3 Justification 39 6.4 Comment 40 Numerical Example I 42 7.1 Settings 42 7.2 Results 44 7.3 Remarks 46 Numerical Example II 48 8.1 Settings 48 8.2 Results 49 8.3 Remarks 49 Discussion 53 A Justification of Multidimensional Extended Kalman Filter 57 B Proof of A.0.6 and A.0.7 65 iii C R Code for Example 67 C.1 Initialize 67 C.2 Particle Filter 67 C.3 Extended Kalman Filter 68 C.4 LLPF 69 C.5 LLPF2 70 D R Code for Example 72 D.1 Initialize 72 D.2 Paticle Filter 72 D.3 Extended Kalman Filter 73 D.4 LLPF I 74 D.5 LLPF II 75 iv Summary Nonlinear dynamic systems with Gaussian noise are widely used in applied fields Extended Kalman filter (EKF) and particle filter (PF) are two traditional methods in estimating these systems In the thesis, two algorithms called locally-linearized particle filters are proposed Both of them combine the idea of EKF and PF In the first algorithm, process noises are split into two parts One part is realized in random samples, leaving out a remaining system on which EKF is applied In the second algorithm, same procedure is conducted, but results from EKF are not the final estimates Instead, they are used to construct a sampling scheme that generate samples from the target distribution Results from simulation studies show that both methods gain improvements over EKF and PF v List of Tables 7.1 Errors of Numerical Example I 46 8.1 Errors of Numerical Example II 52 vi List of Figures 7.2.1 Posterior Density Estimation 44 7.2.2 Effective sample sizes for each filter with 1,000 particles and resampling threshould 300 45 8.2.1 Posterior Density Estimation 50 8.2.2 ESS for Example II 51 vii Chapter Introduction 1.1 Motivation Dynamic systems are widely used in applied fields such as signal processing, computer vision and finance Today, many real-world data analysis tasks involve estimating dynamic systems from some inaccurate observations generated by the system In most cases, some prior information about the underlying dynamic system is available These problems are often formulated as Bayesian models, which consist of a prior distribution about the unobserved model parameters and a likelihood that links these unknown parameters with observations Under Bayesian framework, inference about the underlying system is based on the posterior distribution, which can be obtained using Bayes’ theorem Moreover, observations often come in a sequential manner and one would like to make inference on-line, i.e., update the posterior distribution whenever a new observation is available When the underlying dynamic system is a linear one with Gaussian error, there is an analytic expression for the posterior distribution An algorithm to compute this distribution was provided by Kalman (1960) However, for non-linear Gaussian state-space model, there is no universal effective approach to on-line estimation Several methods have been proposed to surmount these problems The effectiveness of them depends on the nature of individual problems These methods can be classified into two main streams The first stream focus on approximating the system with locally linear (sometimes higher order) functions and derive an analytic solution Some well known algorithms are extended Kalman filter (Gelb, 1974), Gaussian sum filters (Anderson and Moore, 1979) and iterated extended Kalman filter (Jazwinski, 1970) The second stream of methods attempt to draw representative samples directly from the posterior distributions and use these samples for inference It began with the seminal paper introducing sequential Monte Carlo method (also called particle filter) for filtering (Gordon, Salmond and Smith, 1993) Since its origin, modifications and improvements have been suggested, including mixture Kalman filter (Liu and Chen, 2000) and Rao-Blackwellization (Doucet, Godsill and Andrieu, 2000) In the thesis, two algorithms are proposed We call them locally-linearized particle filters (LLPF I and LLPF II) The motivation of them is splitting the process noise into two parts, realizing part of system in a random sample and using approximation algorithms to estimate the remaining part In the first algorithm, approximation is conducted using extended Kalman filter while in the second algorithm, results from extended Kalman filter are not directly used as estimates Rather, they are used to Updating Step Assume that xt+1 ∼ N(xˆt+1|t , Pˆt+1|t ) Suppose a new observation yt+1 is available, we would like to update our estimate of xt+1 We make a further assumption that (xt+1 , yt+1 ) follows a joint normal distribution, i.e.:    xt+1      |y1:t ∼ N     yt+1  xˆ   t+1|t   ,     yˆt+1|t     Pˆt+1|t Cov(xt+1 , yt+1 )         Cov(xt+1 , yt+1 ) Var(yt+1 ) Our optimal estimate will be the posterior distribution of xt+1 given yt+1 , which is a normal distribution by assumption To compute the posterior mean and variance, we would need to compute yˆt+1|t , Cov(xt+1 , yt+1 ) and Var(yt+1 ) The computation of an estimate of yˆt+1|t and Var(yt+1 ) is very similar to that in prediction step We have: yˆt+1|t = g(xˆt+1|t ) + n ∑ eitr(Hgi Pˆk+1|k ), i=1 ˆ t+1 ) = Jg (xˆt+1|t )Pˆt+1|t [Jg (xˆt+1|t )]T + St+1 +W Var(y (A.0.9) (A.0.10) where Vt+1|t is a k × k matrix with the component in i-th row j-th column given by tr[Hgi (xˆt+1|t )Pˆt+1|t Hg j (xˆt+1|t )Pˆt+1|t ] To obtain an estimate of Cov(xt+1 , yt+1 ), we expand yt+1 by Taylor expansion, similar to : 62 yt+1 = g(xˆt+1|t ) + Jg (xˆt+1|t )(xt+1 − xˆt+1|t ) + n ∑ ei(xt+1 − xˆt+1|t )T Hgi (xˆt+1|t )(xt+1 − xˆt+1|t ) + HOT + vk (A.0.11) i=1 Thus: Cov(xt+1 , yt+1 ) = Cov(xt+1 − xˆt+1|t , yt+1 ) = Jg (xˆt+1|t )Var(xt+1 − xˆt+1|t )+ n Cov[xt+1 − xˆt+1|t , ∑ ei (xt+1 − xˆt+1|t )T Hgi (xˆt+1|t )(xt+1 − xˆt+1|t )] + HOT i=1 The second term is essentially 0, by the argument in Appendix Therefore, ignoring higher order terms, we have: ˆ t+1 , yt+1 ) = Jg (xˆt+1|t )Var(xt+1 − xˆt+1|t ) = Jg (xˆt+1|t )Pˆt+1|t Cov(x The joint distribution of (xt+1 , yt+1 ) is: 63 (A.0.12)   xt+1      |y1:t ∼ N     yt+1   xˆ   t+1|t   ,     yˆt+1|t    ˆ t+1 , yt+1 ) Pˆt+1|t Cov(x         T ˆ t+1 , yt+1 ) ˆ t+1 ) Cov(x Var(y ˆ t+1 , yt+1 ) and Var(y ˆ t+1 ) are given in (2.20), (2.23) and (2.21) where yˆt+1|t , Cov(x respectively The posterior distribution xt+1 |yt+1 is normal, its expectation and variance is given by: xˆt+1|t+1 = xˆt+1|t + Pˆt+1|t JgT (xˆt+1|t )[Var(yˆt+1|t )]−1 (yt − yˆt+1|t ), (A.0.13) Pˆt+1|t+1 = Pˆt+1|t − Pˆt+1|t JgT (xˆt+1|t )[Var(yˆk|k−1 )]−1 JgT (xˆt+1|t )Pˆt+1|t (A.0.14) Comparing with the extended Kalman filter algorithm in Section 3.2.3, we see that xˆt+1|t+1 and Pˆt+1|t+1 are estimates of mean and variance of the posterior distribution p(xt+1 |y1:t+1 ) 64 Appendix B Proof of A.0.6 and A.0.7 It suffices to show the following: Suppose y is a k-dimensional vector having a multivariate normal distribution N(0, Σ), where Σ is a k × k matrix, symmetric and positive definite If A is a k × k symmetric positive definite matrix, B is a k × vector, then: Cov(yT Ay, By) = Proof: denote the components of y as yi : y = (y1 , y2 , , yk )T If yi are independent, i.e., Σ is a diagonal matrix, then the result is trivial Since the covariance will be a linear combination of E[ym yn yl ], where ≤ m ≤ n ≤ l ≤ k, which is no matter what m,n,l are chosen If yi are not independent, we can make a transformation: Consider the eigen decomposition of Σ: Σ = QT ΛQ Where Q is orthogonal, i.e., 65 QT Q = I and Λ is a diagonal matrix Qy will have a normal distribution with mean and variance=QΣQT = Λ Therefore the components of Qy are independent Since: Cov(yT Ay, By) = Cov[(Qy)T (QAQT )(Qy), (BQT )Qy] we have the desired result 66 Appendix C R Code for Example C.1 Initialize set.seed(143) x.t[...]... Kalman filter and its generalization, extended Kalman filter In addition, we derive extended Kalman filter based on second order approximation (the standard approach uses linear approximation only) In Chapter 4, we give review on particle filter We build up the algorithm step by step, starting from the simplest Monte Carlo method The next two chapters are devoted to the development of two new methods, locally- linearized. .. given by tr[Hg(i) (xˆt|t−1 )Pˆt|t−1 Hg( j) (xˆt|t−1 )Pˆt|t−1 ] For derivation of the multidimensional extended Kalman filter, see Appendix A 22 Chapter 4 Particle Filter Another approach to estimate the unobserved variable in a non-linear state-space model is particle filter (or bootstrap filter, sequential Monte Carlo method), proposed by Gordon et al (1993) It is a simulation based method which uses... step, starting from the simplest Monte Carlo method The next two chapters are devoted to the development of two new methods, locally- linearized particle filters In Chapter 5, we introduce our first algorithm, LLPF I and in Chapter 6, we introduce a modified particle filter which we call LLPF II Two simulation studies have been done and the results are summarized in Chapter 7 and Chapter 8 respectively... Pˆt|t−1 Comparing with the Kalman filter algorithm, we see that in the algorithm xˆt|t and Pˆt|t are exactly the posterior mean and variance of xt |y1:t Since p(xt |y1:t ) has a normal distribution, we have shown that Kalman filter gives exact estimates of the posterior distribution, under the assumption of linear system and Gaussian noises 3.2 Extended Kalman Filter Kalman filter requires the dynamic system,... more complex systems that are highly non-linear, in which case Kalman filters are not directly applicable Extended Kalman filter is an generalization of Kalman filter that can be applied to these problems It is an algorithm based on linear (or higher-order) approximations of the system In this section, we 12 derive the extended Kalman filter based on second-order approximation For sake of simplicity,... 0.0052 ) 7 xt [3] ) + vt , xt [1] Chapter 3 Extended Kalman Filter This chapter aims at introducing extended Kalman filter and deriving its algorithm based on second-order approximation For illustration purpose, we first introduce Kalman filter and show why it is optimal for linear systems with Gaussian noise The derivation of extended Kalman filter is thus straight forward, except that at certain points... Kt × g (xˆt|t−1 ))Pˆt|t−1 where f and f denote the first and second derivative of f 3.2.2 Derivation of Extended Kalman Filter In this subsection, we derive the one-dimensional extended Kalman filter based on second-order approximations.The derivation is similar to that of Kalman filter We make two crucial assumptions First, the posterior distribution p(xt |y1:t−1 ) can be approximated by a normal... p(x0:t |y1:t ) at every time t Information from previous runs is not fully utilized In the next chapter, we will introduce the extended Kalman filter which approximate the above distribution with one that is easier to handle In Chapter 4, we introduce particle filter that approximate this distribution by drawing representative sample from it The following is an real example of such system 6 Example 1... w¯ t+1 = wt × p(Yt+1 |X t+1 ) 4.3 Particle Filter Although SIS method works decently for some models, a major issue associated with SIS is the weight degeneracy problem In most scenarios, the variance of the weights increases exponentially with the time parameter (Chopin 2004) A common situation is that after a certain period of time, only a few (or even one) of the particles has non-zero weights; the... demonstrate and derive the one-dimensional extended Kalman filter This is a special case where xt and yt are both scalars We then generalize our algorithms to higher dimensional spaces later 3.2.1 One-dimensional EKF Consider the model described in Chapter 2, assume that xt and yt are of dimension 1 (k = 1) The one-dimensional extended Kalman filter can be implemented through the following steps: 1 Initializing: ... ii 4.3 Particle Filter 28 4.4 Mixture Kalman Filter (Rao-Blackwellized PF) 30 Locally-linearized Particle Filter I 32 Locally-linearized Particle Filter. .. Extended Kalman filter (EKF) and particle filter (PF) are two traditional methods in estimating these systems In the thesis, two algorithms called locally-linearized particle filters are proposed... extended Kalman filter, see Appendix A 22 Chapter Particle Filter Another approach to estimate the unobserved variable in a non-linear state-space model is particle filter (or bootstrap filter, sequential