## A Tutorial on Particle Filters for On-line Nonlinear/Non-Gaussian Bayesian Tracking (2001)

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Venue: | IEEE TRANSACTIONS ON SIGNAL PROCESSING |

Citations: | 1160 - 2 self |

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@ARTICLE{Maskell01atutorial,

author = {Simon Maskell and Neil Gordon},

title = {A Tutorial on Particle Filters for On-line Nonlinear/Non-Gaussian Bayesian Tracking},

journal = {IEEE TRANSACTIONS ON SIGNAL PROCESSING},

year = {2001},

volume = {50},

pages = {174--188}

}

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### Citations

4288 | A tutorial on hidden markov models and selected applications in speech recognition
- Rabiner
- 1989
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Citation Context ...st be predefined and, therefore, cannot be partitioned unevenly to give greater resolution in high probability density regions, unless prior knowledge is used. Hidden Markov model (HMM) filters [30], =-=[35]-=-, [36], [39] are an application of such approximate grid-based methods in a fixed-interval smoothing context and have been used extensively in speech processing. In HMM-based tracking, a common approa... |

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Novel Approach to Nonlinear, and Non-Gaussian Gayesian State Estimation
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- 1993
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Citation Context ... method that forms the basis for most sequential Monte Carloslters developed over the past decades { see [13], [14]. This sequential Monte Carlo (SMC) approach is known variously as bootstrapsltering =-=[17]-=-, the condensation algorithm [29], particlesltering [6], interacting particle approximations [10], [11] and survival of thesttest [24]. It is a technique for implementing a recursive Bayesianslter by ... |

835 | An Introduction to Hidden Markov Models
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Citation Context ...predefined and, therefore, cannot be partitioned unevenly to give greater resolution in high probability density regions, unless prior knowledge is used. Hidden Markov model (HMM) filters [30], [35], =-=[36]-=-, [39] are an application of such approximate grid-based methods in a fixed-interval smoothing context and have been used extensively in speech processing. In HMM-based tracking, a common approach is ... |

747 | The Viterbi algorithm
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Citation Context ...f such approximate grid-based methods in asxed-interval smoothing context and have been used extensively in speech processing. In HMM based tracking, a common approach is to use the Viterbi algorithm =-=[18]-=- to calculate the maximum a-posteriori estimate of the path through the trellis, that is the sequence of discrete states that maximises the probability of the state sequence given the data. Another ap... |

709 |
Stochastic Processes and Filtering Theory
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Citation Context ...is that no algorithm can ever do better than a Kalman filter in this linear Gaussian environment. It should be noted that it is possible to derive the same results using a least squares (LS) argument =-=[22]-=-. All the distributions are then described by their means and covariances, and the algorithm remains unaltered, but are not constrained to be Gaussian. Assuming the means and covariances to be unbiase... |

667 | On sequential Monte Carlo sampling methods for Bayesian filtering
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- 2000
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Citation Context ...pproximated to be discrete. The algorithms are optimal if and only if the underlying state space is truly discrete in nature.s178 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 2, FEBRUARY 2002 =-=[14]-=-. This sequential MC (SMC) approach is known variously as bootstrap filtering [17], the condensation algorithm [29], particle filtering [6], interacting particle approximations [10], [11], and surviva... |

520 | Filtering via simulation: Auxiliary particle filters
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- 1999
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Citation Context ...; w i k ; g Ns i=1 ] = RESAMPLE [fx i k ; w i k g Ns i=1 ] B.2 Auxiliary Sampling Importance Resamplingslter The Auxiliary Sampling Importance Resampling (ASIR)slter was introduced by Pitt & Shephard =-=[34]-=- as a variant of the standard SIRslter. Thisslter can be derived from the SIS framework by introducing an importance density q(x k ; ijz 1:k ) which samples the pair fx j k ; i j g Ns j=1 , where i j ... |

461 | Sequential Monte Carlo methods for dynamic systems
- Liu, Chen
- 1998
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Citation Context ... updating particles whose contribution to the approximation to p(x k jz 1:k ) is almost zero. A suitable measure of degeneracy of the algorithm is the eective sample size N eff introduced in [3] and [=-=-=-28], and dened as N eff = N s 1 +Var(w k i ) ; (50) 106 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, NO. Y, MONTH 2001 where w k i = p(x i k jz 1:k )=q(x i k jx i k 1 ; z k ) is referred to as t... |

424 |
Stochastic Simulation
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Citation Context ... the weights are now reset to w i k = 1=N s . It is possible to implement this resampling procedure in O(N s ) operations by sampling N s ordered uniforms using an algorithm based on order statistics =-=[37]-=-, [6]. Note that other ecient (in terms of reduced MC variation) resampling schemes such as strati ed sampling and residual sampling [28] may be applied as alternatives to this algorithm. Systematic r... |

401 |
Monte Carlo filter and smoother for non-Gaussian nonlinear state space models
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Citation Context ...e that other ecient (in terms of reduced MC variation) resampling schemes such as strati ed sampling and residual sampling [28] may be applied as alternatives to this algorithm. Systematic resampling =-=[25]-=- is the scheme preferred by the authors (since it is simple to implement, takes O(N s ) time and minimises the MC variation) and its operation is described in algorithm 2, where U [a; b] is the Unifor... |

284 |
Bayesian forecasting and dynamic models
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- 1997
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Citation Context ... handling multivariate data and non-linear/non-Gaussian processes and it provides a significant advantage over traditional time-series techniques for these problems. A full description is provided in =-=[41]-=-. Also, many varied examples illustrating the application of nonlinear /non-Gaussian state space models are given in [26]. In order to analyse and make inference about a dynamic system at least two mo... |

230 |
A Tutorial on Hidden Markov Models and Selected
- Rabiner
- 1989
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Citation Context ...ce must be predened and therefore cannot be partitioned unevenly to give greater resolution in high probability density regions, unless prior knowledge is used. Hidden Markov model (HMM)slters [30], [=-=35]-=-, [36], [39] are an application of such approximate grid-based methods in asxed-interval smoothing context and have been used extensively in speech processing. In HMM based tracking, a common approach... |

224 |
An approach to time series smoothing and forecasting using the EM algorithm
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- 1982
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Citation Context ...ward passes to obtain overall smoothed estimates [20]. A different formulation implicitly calculates the backwards-time state estimates and covariances, recursively estimating the smoothed quantities =-=[38-=-]. Both techniques are prone to having to calculate matrix inverses that do not necessarily exist. Instead, it is preferable to propagate dierent quantities using an informationslter when carrying out... |

210 | A Probabilistic Exclusion Principle for Tracking Multiple Objects
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- 1999
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Citation Context ...most sequential Monte Carloslters developed over the past decades { see [13], [14]. This sequential Monte Carlo (SMC) approach is known variously as bootstrapsltering [17], the condensation algorithm =-=[29]-=-, particlesltering [6], interacting particle approximations [10], [11] and survival of thesttest [24]. It is a technique for implementing a recursive Bayesianslter by Monte Carlo simulations. The key ... |

210 | Multitarget-Multisensor Tracking: Principles and Techniques - Bar-Shalom, Li - 1995 |

161 | An improved particle filter for nonlinear problems
- Carpenter, Clifford, et al.
- 1999
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Citation Context ...ACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 2, FEBRUARY 2002 [14]. This sequential MC (SMC) approach is known variously as bootstrap filtering [17], the condensation algorithm [29], particle filtering =-=[6]-=-, interacting particle approximations [10], [11], and survival of the fittest [24]. It is a technique for implementing a recursive Bayesian filter by MC simulations. The key idea is to represent the r... |

150 | Stochastic simulation algorithms for dynamic probabilistic networks
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- 1995
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Citation Context ...onte Carlo (SMC) approach is known variously as bootstrapsltering [17], the condensation algorithm [29], particlesltering [6], interacting particle approximations [10], [11] and survival of thesttest =-=[24]-=-. It is a technique for implementing a recursive Bayesianslter by Monte Carlo simulations. The key idea is to represent the required posterior density function by a set of random samples with associat... |

146 | The unscented particle filter
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- 2000
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Citation Context ...rization techniques [12]. Such linearizations use an importance density that is a Gaussian approximation to . Another approach is to estimate a Gaussian approximation to using the unscented transform =-=[40]-=-. The authors’ opinion is that the additional computational cost of using such an importance density is often more than offset by a reduction in the number of samples required to achieve a certain lev... |

143 | An Introduction to Sequential Monte Carlo Methods”, Sequential Monte Carlo Methods in Practice - Doucet, Freitas, et al. - 2001 |

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Following a moving target: Monte Carlo inference for dynamic Bayesian models
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- 2001
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Citation Context ...sample using algorithm 2: RESAMPLE END IF There have been some systematic techniques proposed recently to solve the problem of sample impoverishment. One such technique is the resample-move algorithm =-=[19]-=-, which is not be described in detail in this paper. Although this technique draws conceptually on the same technologies of importance sampling-resampling and MCMC sampling, it avoids sample impoveris... |

134 |
Combined parameter and state estimation in simulationbased filtering
- Liu, West
- 2001
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Citation Context ...articlesltering is a method well suited to the estimation of dynamic states. If static states, which can be regarded as parameters, need to be estimated then alternative approaches are necessary [7], =-=[27-=-]. of the paths of the particles is reduced, any smoothed estimates based on the particles' paths degenerate 6 . Schemes exist to counteract this eect. One approach considers the states for the partic... |

127 |
der Merwe. The unscented kalman filter for nonlinear estimation
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- 2000
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Citation Context ...F that retains further terms in the Taylor expansion exists, but the additional complexity has prohibited its widespread use. Recently, the unscented transform has been used in an EKF framework [23], =-=[42], [43-=-]. The resultingslter, known as the \Unscented Kalman Filter", considers a set of points that are deterministically selected from the Gaussian approximation to p(x k jz 1:k ). These points are al... |

126 | A Monte Carlo approach to nonnormal and nonlinear state-space modeling
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- 1992
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Citation Context ...determined by the forward lter and then obtains the smoothed estimates by re-calculating the particles' weights via a recursion from thesnal to thesrst time step [16]. Another approach is to use MCMC [5]. ALGORITHM 2: RESAMPLING ALGORITHM [fx j k ; w j k ; i j g Ns j=1 ] = RESAMPLE [fx i k ; w i k g Ns i=1 ] Initialise the CDF: c 1 = 0 FOR i = 2 : N s { Construct CDF: c i = c i 1 + w i k END ... |

121 | Particle filters for state estimation of jump markov linear systems
- Doucet, Gordon, et al.
- 2001
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Citation Context ...53) becomes a sum and sampling from p(x k jx i k 1 ; z k ) is possible. An example of an application when x k is a member of asnite set is a Jump-Markov Linear System for tracking maneuvering targets =-=[15-=-], whereby the discrete modal state (dening the maneuver index) is tracked using a particleslter and (conditioned on the maneuver index) the continuous base state is tracked using a Kalmanslter. Analy... |

91 | Recursive Bayesian Estimation: Navigation and Tracking Applications
- Bergman
- 1999
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Citation Context ... Ns X i=1 w i k (x 0:k x i 0:k ); (40) We therefore have a discrete weighted approximation to the true posterior, p(x 0:k jz 1:k ). The weights are chosen using the principle of Importance Sampling [=-=-=-3], [12]. This principle relies on the following: Suppose p(x) / (x) is a probability density from which it is dicult to draw samples, but for which (x) can be evaluated (and so p(x) up to 4 The Viter... |

85 |
W.: Smoothness Priors Analysis of Time Series
- Kitagawa, Gersch
- 1996
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Citation Context ...l time-series techniques for these problems. A full description is provided in [41]. Also, many varied examples illustrating the application of nonlinear /non-Gaussian state space models are given in =-=[26]-=-. In order to analyse and make inference about a dynamic system at least two models are required. Firstly, a model describing the evolution of the state with time (the system model), and secondly a mo... |

51 |
A Bayesian approach to problems in stochastic estimation and control
- Ho, Lee
- 1964
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Citation Context ...e step is Gaussian and hence parameterised by a mean and covariance. If p (x k 1 jz 1:k 1 ) is Gaussian, it can be proved that p (x k jz 1:k ) is also Gaussian, provided that certain assumptions hold =-=[-=-21]: v k 1 and n k are drawn from Gaussian distributions of known parameters f k (x k 1 ; v k 1 ) is known and is a linear function of x k 1 and v k 1 h k (x k ; n k ) is a known linear function of... |

30 | Improvement strategies for monte carlo particle
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- 2001
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Citation Context ... distribution than the likelihood, p(z k jx k ), then only a few particles will have a high weight. Methods exist for encouraging the particles to be in the right place; the use of bridging densities =-=[8]-=- and progressive correction [33] both introduce intermediate distributions between the prior and likelihood. The particles are then re-weighted according to these intermediate distributions and resamp... |

29 |
On sequential Monte Carlo methods for Bayesian filtering
- Doucet, Godsill, et al.
- 2000
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Citation Context ... X i=1 w i k (x 0:k x i 0:k ); (40) We therefore have a discrete weighted approximation to the true posterior, p(x 0:k jz 1:k ). The weights are chosen using the principle of Importance Sampling [3], =-=[1-=-2]. This principle relies on the following: Suppose p(x) / (x) is a probability density from which it is dicult to draw samples, but for which (x) can be evaluated (and so p(x) up to 4 The Viterbi and... |

29 |
Frequency line tracking using hidden markov models
- Streit, Barrett
- 1990
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Citation Context ...redened and therefore cannot be partitioned unevenly to give greater resolution in high probability density regions, unless prior knowledge is used. Hidden Markov model (HMM)slters [30], [35], [36], [=-=39]-=- are an application of such approximate grid-based methods in asxed-interval smoothing context and have been used extensively in speech processing. In HMM based tracking, a common approach is to use t... |

17 | Measure valued processes and interacting particle systems. Application to non linear filtering problems
- P
- 1997
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Citation Context ...tioned on the maneuver index) the continuous base state is tracked using a Kalmanslter. Analytic evaluation is possible for a second class of models for which p(x k jx i k 1 ; z k ) is Gaussian [12], =-=[-=-9]. This can occur if the dynamics are non-linear and the measurements linear. Such a system is given by x k =f k (x k 1 ) + v k 1 ; (54) z k =H k x k + n k ; (55) where v k 1 N (v k 1 ; 0nv1 ; Q k 1 ... |

17 |
Fixed-interval smoothing for markovian switching systems
- Helmick, Blair, et al.
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Citation Context ...ckwards-time Kalmanslter that recurses through the data sequence from thesnal data to thesrst and then combines the estimates from the forward and backward passes to obtain overall smoothed estimates =-=[20]-=-. A different formulation implicitly calculates the backwards-time state estimates and covariances, recursively estimating the smoothed quantities [38]. Both techniques are prone to having to calculat... |

12 |
Comparison of the particle filter with range parameterized and modified polar EKF’s for angle-only tracking
- Arulampalam, Ristic
- 2000
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Citation Context ... or heavily skewed) then a Gaussian can never describe it well. In such cases, approximate grid-basedslters and particleslters will yield an improvement in performance in comparison to that of an EKF =-=[1-=-]. 104 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, NO. Y, MONTH 2001 B. Approximate Grid-Based Methods If the state space is continuous, but can be decomposed into N s `cells', x i k : i = 1; : ... |

12 |
Moral, ”Nonlinear Filtering: Interacting Particle Solution
- Del
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Citation Context ...e [13], [14]. This sequential Monte Carlo (SMC) approach is known variously as bootstrapsltering [17], the condensation algorithm [29], particlesltering [6], interacting particle approximations [10], =-=[11]-=- and survival of thesttest [24]. It is a technique for implementing a recursive Bayesianslter by Monte Carlo simulations. The key idea is to represent the required posterior density function by a set ... |

12 |
Andrieu C., “On sequential Monte Carlo sampling methods for Bayesian filtering
- Doucet, Godsill
- 2000
(Show Context)
Citation Context ... (SIS) Algorithm The Sequential Importance Sampling (SIS) algorithm is a Monte Carlo (MC) method that forms the basis for most sequential Monte Carloslters developed over the past decades { see [13], =-=[14]-=-. This sequential Monte Carlo (SMC) approach is known variously as bootstrapsltering [17], the condensation algorithm [29], particlesltering [6], interacting particle approximations [10], [11] and sur... |

11 |
A Skewed Approach to Filtering
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Citation Context ...der EKF that retains further terms in the Taylor expansion exists, but the additional complexity has prohibited its widespread use. Recently, the unscented transform has been used in an EKF framework =-=[23], [42-=-], [43]. The resultingslter, known as the \Unscented Kalman Filter", considers a set of points that are deterministically selected from the Gaussian approximation to p(x k jz 1:k ). These points ... |

10 | Optimal estimation and Cramer-Rao bounds for partial non-Gaussian state-space model
- Bergman, Gordon
(Show Context)
Citation Context ...aving to calculate matrix inverses that do not necessarily exist. Instead, it is preferable to propagate different quantities using an information filter when carrying out the backward-time recursion =-=[4]-=-. 3 If – aH, then the problem reduces to the estimation of �@� �� A considered up to this point. B. Grid-Based Methods Grid-based methods provide the optimal recursion of the filtered density if the s... |

9 |
Methodology for Monte Carlo Smoothing with Application to Time-Varying Autoregressions
- Godsill, Doucet, et al.
- 2000
(Show Context)
Citation Context ...tates for the particles to be predetermined by the forward filter and then obtains the smoothed estimates by recalculating the particles’ weights via a recursion from the final to the first time step =-=[16]-=-. Another approach is to use MCMC [5]. Algorithm 2: Resampling Algorithm , RESAMPLE Initialize the CDF: FOR — Construct CDF: END FOR Start at the bottom of the CDF: Draw a starting point: FOR — Move a... |

8 |
Progressive correction for regularized particle filters
- Oudjane, Musso
- 2000
(Show Context)
Citation Context ...od, p(z k jx k ), then only a few particles will have a high weight. Methods exist for encouraging the particles to be in the right place; the use of bridging densities [8] and progressive correction =-=[33] both-=- introduce intermediate distributions between the prior and likelihood. The particles are then re-weighted according to these intermediate distributions and resampled. This \herds" the particles ... |

8 |
Nonlinear Filtering Using Branching and Interacting
- Crisan, Moral, et al.
- 1999
(Show Context)
Citation Context .... 2, FEBRUARY 2002 [14]. This sequential MC (SMC) approach is known variously as bootstrap filtering [17], the condensation algorithm [29], particle filtering [6], interacting particle approximations =-=[10]-=-, [11], and survival of the fittest [24]. It is a technique for implementing a recursive Bayesian filter by MC simulations. The key idea is to represent the required posterior density function by a se... |

7 |
A Marrs, Particle filters for tracking with out-of-sequence measurements
- Orton
- 2005
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Citation Context ...ents, the order of times to which the mesaurements relate can dier from the order in which the measurements are processed. For a particleslter solution to the problem of relaxing this assumption see [=-=32]-=-. and n z ; nn are dimensions of the measurement and measurement noise vectors, respectively. In particular, we seeksltered estimates of x k based on the set of all available measurements z 1:k = fz i... |

7 |
Moral, “ Measure valued processes and interacting particle systems. Application to nonlinear filtering problems
- Del
- 1998
(Show Context)
Citation Context ...le filter, and (conditioned on the maneuver index) the continuous base state is tracked using a Kalman filter. Analytic evaluation is possible for a second class of models for which is Gaussian [12], =-=[9]-=-. This can occur if the dynamics are nonlinear and the measurements linear. Such a system is given by (54) (55) where (56) (57) and : is a nonlinear function, is an observation matrix, and and are mut... |

5 |
Kalman Filtering and Neural Networks, chapter The Unscented Kalman Filter
- Wan, Merwe
- 2001
(Show Context)
Citation Context ... retains further terms in the Taylor expansion exists, but the additional complexity has prohibited its widespread use. Recently, the unscented transform has been used in an EKF framework [23], [42], =-=[43]. The-=- resultingslter, known as the \Unscented Kalman Filter", considers a set of points that are deterministically selected from the Gaussian approximation to p(x k jz 1:k ). These points are all prop... |

5 |
Data association and tracking using hidden markov models and dynamic programming
- Martinerie, Forster
- 1992
(Show Context)
Citation Context ...ace must be predefined and, therefore, cannot be partitioned unevenly to give greater resolution in high probability density regions, unless prior knowledge is used. Hidden Markov model (HMM) filters =-=[30]-=-, [35], [36], [39] are an application of such approximate grid-based methods in a fixed-interval smoothing context and have been used extensively in speech processing. In HMM-based tracking, a common ... |

5 |
Improving regularised particle filters,” in Sequential Monte Carlo Methods in Practice
- Musso, Oudjane, et al.
- 2001
(Show Context)
Citation Context ...r that ensures the particles asymtotically approximate samples from the posterior and, therefore, is the method of choice of the authors. An alternative solution to the same problem is regularization =-=[31]-=-, which is discussed in Section V-B3. This approach is frequently found to improve performance, despite a less rigorous derivation and is included here in preference to the resample-move algorithm sin... |

2 |
P Cliord & P Fearnhead, Improved Particle Filter for Non-linear Problems
- Carpenter
- 1999
(Show Context)
Citation Context ...arloslters developed over the past decades { see [13], [14]. This sequential Monte Carlo (SMC) approach is known variously as bootstrapsltering [17], the condensation algorithm [29], particlesltering =-=[6]-=-, interacting particle approximations [10], [11] and survival of thesttest [24]. It is a technique for implementing a recursive Bayesianslter by Monte Carlo simulations. The key idea is to represent t... |

2 |
Rabiner and B H Juang. An Introduction to Hidden Markov Models
- R
- 1986
(Show Context)
Citation Context ...t be predened and therefore cannot be partitioned unevenly to give greater resolution in high probability density regions, unless prior knowledge is used. Hidden Markov model (HMM)slters [30], [35], [=-=36]-=-, [39] are an application of such approximate grid-based methods in asxed-interval smoothing context and have been used extensively in speech processing. In HMM based tracking, a common approach is to... |

2 |
der Merwe, A Doucet, J F G de Freitas & E Wan, The Unscented Particle Filter
- van
- 2000
(Show Context)
Citation Context ...ions use an importance density that is a Gaussian approximation to p(x k jx k 1 ; z k ). Another approach is to estimate a Gaussian approximation to p(x k jx k 1 ; z k ) using the unscented transform =-=[40-=-]. The authors' opinion is that the additional computational cost of using such an importance density is often more than oset by a reduction in the number of samples required to achieve a certain leve... |

1 |
A Doucet & N Gordon, Optimal Estimation and Cramer-Rao Bounds for Partial Non-Gaussian State Space Models
- Bergman
(Show Context)
Citation Context ... having to calculate matrix inverses that do not necessarily exist. Instead, it is preferable to propagate dierent quantities using an informationslter when carrying out the backwards-time recursion [=-=4]-=-. 3 If ` = 0 then the problem reduces to the estimation of p(x k jz 1:k ) considered up to this point. ARULAMPALAM, MASKELL, GORDON AND CLAPP: A TUTORIAL ON PARTICLE FILTERS 103 B. Grid-based methods ... |

1 |
Statistical Methods for the Processing of Communications Data
- Clapp
- 2000
(Show Context)
Citation Context ...te. Particlesltering is a method well suited to the estimation of dynamic states. If static states, which can be regarded as parameters, need to be estimated then alternative approaches are necessary =-=[7-=-], [27]. of the paths of the particles is reduced, any smoothed estimates based on the particles' paths degenerate 6 . Schemes exist to counteract this eect. One approach considers the states for the ... |