Particle filter

Particle filters or Sequential Monte Carlo (SMC) methods are a set of on-line posterior density estimation algorithms that estimate the posterior density of the state-space by directly implementing the Bayesian recursion equations. The term "particle filters" was first coined in 1996 by Del Moral, and the terminology "sequential Monte Carlo" was proposed by Liu and Chen in 1998. Particle filters and SMC methods use a sampling approach, with a set of particles to represent the posterior density. The state-space model can be non-linear and the initial state and noise distributions can take any form required. SMC methods provide a well-established methodology for generating samples from the required distribution without requiring assumptions about the state-space model or the state distributions. However, these methods do not perform well when applied to high-dimensional systems. SMC methods implement the Bayesian recursion equations directly by using an ensemble based approach. The samples from the distribution are represented by a set of particles; each particle has a weight assigned to it that represents the probability of that particle being sampled from the probability density function.

Weight disparity leading to weight collapse is a common issue encountered in these filtering algorithms; however it can be mitigated by including a resampling step before the weights become too uneven. Several adaptive resampling criteria can be used, including the variance of the weights and the relative entropy w.r.t. the uniform distribution. In the resampling step, the particles with negligible weights are replaced by new particles in the proximity of the particles with higher weights.

History

One trace of particle filters date back to the 50's; the 'Poor Man's Monte Carlo', that was proposed by Hammersley et al., in 1954, contained hints of the SMC methods used today. Later in the 70's, similar attempts were made in the control and genetic algorithms communities. However in the signal processing literature it was in 1993, that Gordon et al., published their seminal work 'A novel Approach to nonlinear/non-Gaussian Bayesian State estimation', that provided the first true heuristic-like implementation of the SMC methods used today in Bayesian statistical inference. The authors named their algorithm 'the bootstrap filter', and demonstrated that compared to other filtering methods, their bootstrap algorithm does not require any assumption about that state-space or the noise of the system. We also quote another pioneering article in this field of Genshiro Kitagawa , and the one by Himilcon Carvalho, Pierre Del Moral, André Monin and Gérard Salut published in the 1990s. All of these articles present particle filters as natural heuristic-like algorithms without a single proof of their consistency.

Particle filters belong to the class of genetic type algorithms and mean field type particle methods. We quote a pioneering article by Theodore E. Harris and Herman Kahn, published in 1951, using mean field but heuristic-like genetic methods for estimating particle transmission energies. Mean field genetic type particle methodologies are also used as heuristic natural search algorithms (a.k.a. Metaheuristic) in evolutionary computing. The origins of these mean field computational techniques can be traced to 1950 and 1954 with the work of Alan Turing on genetic type mutation-selection learning machines and the articles by Nils Aall Barricelli at the Institute for Advanced Study in Princeton, New Jersey. The Australian geneticist Alex Fraser also published in 1957 a series of papers on the genetic type simulation of artificial selection of organisms.

The conditional distribution of the random states of a signal given some partial and noisy observations is described by a Feynman-Kac probability on the random trajectories of the signal weighted by likelihood potential functions. The first rigorous mathematical analysis of these models are due to Pierre Del Moral in 1996. This article also contain a proof of the unbiased properties of particle approximations of likelihood functions and unnormalized conditional probability measures. Branching type particle methodologies with varying population sizes were also developed in the end of the 1990s by Dan Crisan, Jessica Gaines and Terry Lyons, and by Dan Crisan, Pierre Del Moral and Terry Lyons.

The theory on Feynman-Kac particle methodologies and related particle filters algorithms has been developed in 2000 and 2004 in the books. These probabilistic models encapsulate genetic type algorithms, particle and bootstrap filters, interacting Kalman filters (a.k.a. Rao–Blackwellized particle filter ), importance sampling and resampling style particle filter techniques, genealogical tree based and particle backward methodologies for solving filtering and smoothing problems. Other classes of particle filtering methodologies includes genealogical tree based models, backward particle models, adaptive mean field particle models, island type particle models, and particle Markov chain Monte Carlo methodologies. The first uniform convergence results with respect to the time parameter for particle filters were developed in the end of the 1990s by Pierre Del Moral and Alice Guionnet. The first rigorous analysis of genealogical tree based particle filter smoothers is due to P. Del Moral and L. Miclo in 2001

Objective

The objective of a particle filter is to estimate the posterior density of the state variables given the observation variables. The particle filter is designed for a hidden Markov Model, where the system consists of hidden and observable variables. The observable variables (observation process) are related to the hidden variables (state-process) by some functional form that is known. Similarly the dynamical system describing the evolution of the state variables is also known probabilistically.

A generic particle filter estimates the posterior distribution of the hidden states using the observation measurement process. Consider a state-space shown in the diagram (Figure 2).

Figure 2. The State-Space (The Hidden states given by the vector x, and the observation states given by the vector y)

The objective of the particle filter is to estimate the values of the hidden states x, given the values of the observation process y.

The particle filter aims to estimate the sequence of hidden parameters, x_k for k = 0,1,2,3,…, based only on the observed data y_k for k = 0,1,2,3,…. All Bayesian estimates of x_k follow from the posterior distribution p(x_k | y₀,y₁,…,y_k). In contrast, the MCMC or importance sampling approach would model the full posterior p(x₀,x₁,…,x_k | y₀,y₁,…,y_k).

Model

Particle methods assume $x_{k}$ and the observations $y_{k}$ can be modeled in this form:

• $x_{0},x_{1},\ldots$ is a first order Markov process that evolves according to the distribution $p_{x_{k}|x_{k-1}}$:

  $x_{k}|x_{k-1}\sim p_{x_{k}|x_{k-1}}(x|x_{k-1})$

and with an initial distribution $p(x_{0})$.

• The observations $y_{0},y_{1},\ldots$ are conditionally independent provided that $x_{0},x_{1},\ldots$ are known

In other words, each $y_{k}$ only depends on $x_{k}$. This conditional distribution for $y_{k}$ is written as

$y_{k}|x_{k}\sim p_{y|x}(y|x_{k})$

An example system with these properties is:

$x_{k}=g(x_{k-1})+w_{k}\,$

$y_{k}=h(x_{k})+v_{k}\,$

where both $w_{k}$ and $v_{k}$ are mutually independent and identically distributed sequences with known probability density functions and $g(\cdot )$ and $h(\cdot )$ are known functions. These two equations can be viewed as state space equations and look similar to the state space equations for the Kalman filter. The conditional distribution p(x₀,x₁,…,x_k | y₀,y₁,…,y_k) are given by the Bayes' rule

$$p(x_{0},...,x_{k}|y_{0},\dots ,y_{k})={\frac {p(y_{0},\dots ,y_{k}|x_{0},...,x_{k})\times p(x_{0},...,x_{k})}{p(y_{0},\dots ,y_{k})}}$$

with

$$p(y_{0},\dots ,y_{k}|x_{0},...,x_{k})=\prod _{0\leq l\leq k}p_{y|x}(y_{l}|x_{l})\quad and\quad p(x_{0},...,x_{k})=p_{0}(x_{0})\prod _{1\leq l\leq k}p_{l|l-1}(x_{l}|x_{l-1})$$

If the functions $g(\cdot )$ and $h(\cdot )$ are linear, and if both $w_{k}$ and $v_{k}$ are Gaussian, the Kalman filter finds the exact Bayesian filtering distribution. If not, Kalman filter based methods are a first-order approximation (EKF) or a second-order approximation (UKF in general, but if probability distribution is Gaussian a third-order approximation is possible). Particle filters are also an approximation, but with enough particles they can be much more accurate^..

Monte Carlo approximation

Particle methods, like all sampling-based approaches (e.g., MCMC), generate a set of samples that approximate the filtering distribution $p(x_{k}|y_{0},\dots ,y_{k})$. For example, we may have $N$ samples from the approximate posterior distribution of $x_{k}$, where the samples are labeled with superscripts as $x_{k}^{(1)},x_{k}^{(2)},\ldots ,x_{k}^{(N)}$. Then, expectations with respect to the filtering distribution are approximated by

$$\int f(x_{k})p(x_{k}|y_{0},\dots ,y_{k})\,dx_{k}\approx {\frac {1}{N}}\sum _{i=1}^{N}f\left(x_{k}^{(i)}\right)=\int f(x_{k})~{\widehat {p}}(dx_{k}|y_{0},\dots ,y_{k})\quad with\quad {\widehat {p}}(dx_{k}|y_{0},\dots ,y_{k})={\frac {1}{N}}\sum _{1\leq i\leq N}\delta _{x_{k}^{(i)}}(dx_{k})$$

In the above display ${\textstyle \delta _{a}}$ stands for the Dirac measure at a given state a. The function $f(\cdot )$, in the usual way for Monte Carlo, can give all the moments etc. of the distribution up to some degree of approximation. Particle filters can be interpreted as a genetic type particle algorithm evolving with mutation and selection transitions. If we keep track of the ancestral lines ${\textstyle (x_{0,k}^{i},x_{1,k}^{i},...,x_{k,k}^{i})}$ of the particles ${\textstyle i=1,...,N}$ (the lower indices ${\textstyle x_{l,k}^{i}}$ , with l=0,...,k, stands for the levels of the ancestors), then we have the approximation formula

$$\int F(x_{0},...,x_{k})p(x_{0},...,x_{k}|y_{0},\dots ,y_{k})\,dx_{0}...dx_{k}\approx {\frac {1}{N}}\sum _{i=1}^{N}F\left(x_{0,k}^{(i)},x_{1,k}^{(i)},...,x_{k,k}^{(i)}\right)$$

Here F stands for some function on the path space of the signal.

Mean field particle simulation

The mean field particle interpretation of the nonlinear filtering equation is better explained using the evolution of the one step optimal predictors

$${\boldsymbol {p(x_{k}|y_{0},\ldots ,y_{k-1})}}\quad \longrightarrow \quad p(x_{k+1}|y_{0},\ldots ,y_{k})=\int ~p(x_{k+1}|x'_{k})~\displaystyle {\frac {p(y_{k}|x_{k}')~{\boldsymbol {p(x'_{k}|y_{0},\dots ,y_{k-1})}}~dx'_{k}}{\displaystyle \int ~p(y_{k}|x''_{k})~{\boldsymbol {p(x''_{k}|y_{0},\ldots ,y_{k-1})}}~dx''_{k-1}}}$$

For k=0 we use the convention $p(x_{0}|y_{0},\dots ,y_{-1}):=p(x_{0})$.

One of the simplest way to approximate these probability measures is to start with ${\textstyle N}$ independent random variable ${\textstyle \left(\xi _{0}^{i}\right)_{1\leq i\leq N}}$ with common probability density ${\textstyle p(x_{0})}$ . By the law of large numbers, we have

$${\widehat {p}}(dx_{0})={\frac {1}{N}}\sum _{1\leq i\leq N}\delta _{\xi _{0}^{i}}(dx_{0})\approx _{N\uparrow \infty }~p(x_{0})dx_{0}\quad {\mbox{in the sense that}}\quad \int f(x_{0}){\widehat {p}}(dx_{0})={\frac {1}{N}}\sum _{1\leq i\leq N}f(\xi _{0}^{i})\approx _{N\uparrow \infty }\int f(x_{0})p(dx_{0})dx_{0}$$

for any bounded function $f$. We further assume that we have constructed a sequence of particles ${\textstyle \left(\xi _{k}^{i}\right)_{1\leq i\leq N}}$ at some rank k such that

$${\widehat {p}}(dx_{k}|y_{0},\ldots ,y_{k-1}):={\frac {1}{N}}\sum _{1\leq i\leq N}\delta _{\xi _{k}^{i}}(dx_{k})\approx _{N\uparrow \infty }~p(x_{k}~|~y_{0},\ldots ,y_{k-1})dx_{k}$$

in the sense that for any bounded function $f$ we have

$$\int f(x_{k}){\widehat {p}}(dx_{k}~|~y_{0},\ldots ,y_{k-1})={\frac {1}{N}}\sum _{1\leq i\leq N}f(\xi _{k}^{i})\approx _{N\uparrow \infty }\int f(x_{k})p(dx_{k}|y_{0},\ldots ,y_{k-1})dx_{k}$$

In this situation, replacing $p(x_{k}~|~y_{0},\ldots ,y_{k-1})~dx_{k}$ by ${\widehat {p}}(dx_{k}~|~y_{0},\ldots ,y_{k-1})$ in the evolution equation of the one-step optimal filter stated above, we find that

$$p(x_{k+1}|y_{0},\ldots ,y_{k})\approx _{N\uparrow \infty }\int ~p(x_{k+1}|x'_{k})~\displaystyle {\frac {p(y_{k}|x_{k}')~{\widehat {p}}(x'_{k}|y_{0},\dots ,y_{k-1})~dx'_{k}}{\displaystyle \int ~p(y_{k}|x''_{k})~{\widehat {p}}(x''_{k}|y_{0},\ldots ,y_{k-1})dx''_{k-1}}}=\sum _{1\leq i\leq N}~{\frac {p(y_{k}~|~\xi _{k}^{i})}{\sum _{1\leq j\leq N}p(y_{k}~|~\xi _{k}^{j})}}~~p(x_{k+1}~|~\xi _{k}^{i}):={\widehat {q}}(x_{k+1}|y_{0},\ldots ,y_{k})$$

where $p(y_{k}~|~\xi _{k}^{i})$ stands for the density $p(y_{k}~|~x_{k})$ evaluated at $x_{k}=\xi _{k}^{i}$, and $p(x_{k+1}~|~\xi _{k}^{i})$ stands for the density $p(x_{k+1}~|~x_{k})$ evaluated at $x_{k}=\xi _{k}^{i}$., for $i=1,\ldots ,N.$ Then, we sample ${\textstyle N}$ independent random variable ${\textstyle \left(\xi _{k+1}^{i}\right)_{1\leq i\leq N}}$ with common probability density ${\textstyle {\widehat {q}}(x_{k+1}|y_{0},\ldots ,y_{k})}$ so that

$${\widehat {p}}(dx_{k+1}|y_{0},\ldots ,y_{k}):={\frac {1}{N}}\sum _{1\leq i\leq N}\delta _{\xi _{k+1}^{i}}(dx_{k+1})\approx _{N\uparrow \infty }~{\widehat {q}}(x_{k+1}~|~y_{0},\ldots ,y_{k})dx_{k+1}\approx _{N\uparrow \infty }~p(x_{k+1}~|~y_{0},\ldots ,y_{k})dx_{k+1}$$

Iterating this procedure, we design a Markov chain ${\textstyle \left(\xi _{k}^{i}\right)_{1\leq i\leq N}}$ such that

$${\widehat {p}}(dx_{k}|y_{0},\ldots ,y_{k-1}):={\frac {1}{N}}\sum _{1\leq i\leq N}\delta _{\xi _{k}^{i}}(dx_{k})\approx _{N\uparrow \infty }~p(x_{k}~|~y_{0},\ldots ,y_{k-1})dx_{k}$$

Notice that the optimal filter is approximated at each time step k using the Bayes' formulae

$$p(x_{k}|y_{0},\ldots ,y_{k})dx_{k}\approx _{N\uparrow \infty }\displaystyle {\frac {p(y_{k}|x_{k})~{\widehat {p}}(dx_{k}|y_{0},\ldots ,y_{k-1})}{\displaystyle \int p(y_{k}|x'_{k}){\widehat {p}}(dx'_{k}|y_{0},\ldots ,y_{k-1})}}=\sum _{1\leq i\leq N}~{\frac {p(y_{k}~|~\xi _{k}^{i})}{\sum _{1\leq j\leq N}p(y_{k}~|~\xi _{k}^{j})}}~\delta _{\xi _{k}^{i}}(dx_{k})$$

The terminology "mean field approximation" comes from the fact that we replace at each time step the probability measure $p(x_{k}~|~y_{0},\ldots ,y_{k-1})~dx_{k}$ by the empirical approximation ${\widehat {p}}(dx_{k}~|~y_{0},\ldots ,y_{k-1})$.

Sequential importance resampling (SIR)

Sequential importance resampling (SIR), the original particle filtering algorithm (Gordon et al. 1993), is a very commonly used particle filtering algorithm, which approximates the filtering distribution $p(x_{k}|y_{0},\ldots ,y_{k})$ by a weighted set of P particles

$\{(w_{k}^{(i)},x_{k}^{(i)})~:~i\in \{1,\ldots ,N\}\}.$

The importance weights $w_{k}^{(i)}$ are approximations to the relative posterior probabilities (or densities) of the particles such that $\sum _{i=1}^{N}w_{k}^{(i)}=1$.

SIR is a sequential (i.e., recursive) version of importance sampling. As in importance sampling, the expectation of a function $f(\cdot )$ can be approximated as a weighted average

$\int f(x_{k})p(x_{k}|y_{0},\dots ,y_{k})dx_{k}\approx \sum _{i=1}^{N}w_{k}^{(i)}f(x_{k}^{(i)}).$

For a finite set of particles, the algorithm performance is dependent on the choice of the proposal distribution

$\pi (x_{k}|x_{0:k-1},y_{0:k})\,$.

The "optimal" proposal distribution is given as the target distribution

$\pi (x_{k}|x_{0:k-1},y_{0:k})=p(x_{k}|x_{k-1},y_{k})={\frac {p(y_{k}|x_{k})}{\int ~p(y_{k}|x_{k})p(x_{k}|x_{k-1})dx_{k}}}~p(x_{k}|x_{k-1}).\,$

This particular choice of proposal transition has been proposed by P. Del Moral in in 1996 and 1998. When it is difficult to sample transitions according to the distribution $p(x_{k}|x_{k-1},y_{k})$ one natural strategy is to use the following particle approximation

${\frac {p(y_{k}|x_{k})}{\int ~p(y_{k}|x_{k})p(x_{k}|x_{k-1})dx_{k}}}~p(x_{k}|x_{k-1})dx_{k}\simeq _{N\uparrow \infty }{\frac {p(y_{k}|x_{k})}{\int ~p(y_{k}|x_{k}){\widehat {p}}(dx_{k}|x_{k-1})}}~{\widehat {p}}(dx_{k}|x_{k-1})=\sum _{1\leq i\leq N}{\frac {p(y_{k}|X_{k}^{i}(x_{k-1}))}{\sum _{1\leq j\leq N}p(y_{k}|X_{k}^{j}(x_{k-1}))}}~\delta _{X_{k}^{i}(x_{k-1})}(dx_{k})$

with the empirical approximation

$${\widehat {p}}(dx_{k}|x_{k-1})={\frac {1}{N}}\sum _{i=1}^{N}\delta _{X_{k}^{i}(x_{k-1})}(dx_{k})~\simeq _{N\uparrow \infty }~p(x_{k}|x_{k-1})dx_{k}$$

associated with $N$ (or any other large number of samples) independent random samples $X_{k}^{i}(x_{k-1}),~i=1,...,N$with the conditional distribution of the random state $X_{k}$ given $X_{k-1}=x_{k-1}$. The consistency of the resulting particle filter of this approximation and other extensions are developed in. In the above display ${\textstyle \delta _{a}}$ stands for the Dirac measure at a given state a.

However, the transition prior probability distribution is often used as importance function, since it is easier to draw particles (or samples) and perform subsequent importance weight calculations:

$\pi (x_{k}|x_{0:k-1},y_{0:k})=p(x_{k}|x_{k-1}).\,$

Sequential Importance Resampling (SIR) filters with transition prior probability distribution as importance function are commonly known as bootstrap filter and condensation algorithm.

Resampling is used to avoid the problem of degeneracy of the algorithm, that is, avoiding the situation that all but one of the importance weights are close to zero. The performance of the algorithm can be also affected by proper choice of resampling method. The stratified sampling proposed by Kitagawa (1996) is optimal in terms of variance.

A single step of sequential importance resampling is as follows:

1) For $i=1,\ldots ,N$ draw samples from the proposal distribution

$x_{k}^{(i)}\sim \pi (x_{k}|x_{0:k-1}^{(i)},y_{0:k})$

2) For $i=1,\ldots ,N$ update the importance weights up to a normalizing constant:

${\hat {w}}_{k}^{(i)}=w_{k-1}^{(i)}{\frac {p(y_{k}|x_{k}^{(i)})p(x_{k}^{(i)}|x_{k-1}^{(i)})}{\pi (x_{k}^{(i)}|x_{0:k-1}^{(i)},y_{0:k})}}.$

Note that when we use the transition prior probability distribution as the importance function, $\pi (x_{k}^{(i)}|x_{0:k-1}^{(i)},y_{0:k})=p(x_{k}^{(i)}|x_{k-1}^{(i)})$, this simplifies to the following  :

${\hat {w}}_{k}^{(i)}=w_{k-1}^{(i)}p(y_{k}|x_{k}^{(i)}),$

3) For $i=1,\ldots ,N$ compute the normalized importance weights:

$w_{k}^{(i)}={\frac {{\hat {w}}_{k}^{(i)}}{\sum _{j=1}^{N}{\hat {w}}_{k}^{(j)}}}$

4) Compute an estimate of the effective number of particles as

${\hat {N}}_{\mathit {eff}}={\frac {1}{\sum _{i=1}^{N}\left(w_{k}^{(i)}\right)^{2}}}$

This criteria reflects the variance of the weights, other criteria can be found in the article, including their rigorous analysis and central limit theorems.

5) If the effective number of particles is less than a given threshold ${\hat {N}}_{\mathit {eff}}<N_{thr}$, then perform resampling:

a) Draw $N$ particles from the current particle set with probabilities proportional to their weights. Replace the current particle set with this new one.

b) For $i=1,\ldots ,N$ set $w_{k}^{(N)}=1/N.$

The term Sampling Importance Resampling is also sometimes used when referring to SIR filters.

Sequential importance sampling (SIS)

• Is the same as sequential importance resampling, but without the resampling stage.

"direct version" algorithm

The "direct version" algorithm is rather simple (compared to other particle filtering algorithms) and it uses composition and rejection. To generate a single sample $x$ at $k$ from $p_{x_{k}|y_{1:k}}(x|y_{1:k})$:

1) Set n=0 (This will count the number of particles generated so far)

2) Uniformly choose an index i from the range $\{1,...,N\}$

3) Generate a test ${\hat {x}}$ from the distribution $p_{x_{k}|x_{k-1}}(x|x_{k-1|k-1}^{(i)})$

4) Generate the probability of ${\hat {y}}$ using ${\hat {x}}$ from $p_{y|x}(y_{k}|{\hat {x}})$ where $y_{k}$ is the measured value

5) Generate another uniform u from $[0,m_{k}]$ where $m_{k}=\sup _{x}p_{y|x}(y_{k}|x)$

6) Compare u and $p\left({\hat {y}}\right)$

6a) If u is larger then repeat from step 2

6b) If u is smaller then save ${\hat {x}}$ as $x_{k|k}^{(i)}$ and increment n

7) If n == N then quit

The goal is to generate P "particles" at $k$ using only the particles from $k-1$. This requires that a Markov equation can be written (and computed) to generate a $x_{k}$ based only upon $x_{k-1}$. This algorithm uses composition of the P particles from $k-1$ to generate a particle at $k$ and repeats (steps 2–6) until P particles are generated at $k$.

This can be more easily visualized if $x$ is viewed as a two-dimensional array. One dimension is $k$ and the other dimensions is the particle number. For example, $x(k,i)$ would be the i^th particle at $k$ and can also be written $x_{k}^{(i)}$ (as done above in the algorithm). Step 3 generates a potential $x_{k}$ based on a randomly chosen particle ($x_{k-1}^{(i)}$) at time $k-1$ and rejects or accepts it in step 6. In other words, the $x_{k}$ values are generated using the previously generated $x_{k-1}$.

Other particle filters

• Auxiliary particle filter
• Regularized auxiliary particle filter
• Gaussian particle filter
• Unscented particle filter
• Gauss–Hermite particle filter
• Cost Reference particle filter
• Hierarchical/Scalable particle filter
• Rao–Blackwellized particle filter
• Rejection-sampling based optimal particle filter
• Feynman-Kac and mean field particle methodologies

See also

• Ensemble Kalman filter
• Generalized filtering
• Moving horizon estimation
• Recursive Bayesian estimation
• Monte Carlo localization
• Mean field particle methods

Bibliography

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• Liu, J.S.,; Chen, R. (1998). "Sequential Monte Carlo methods for dynamic systems" (PDF). Journal of the American Statistical Association. 93 (443). Taylor & Francis Group: 1032–1044. doi:10.1080/01621459.1998.10473765.{{cite journal}}: CS1 maint: extra punctuation (link) CS1 maint: multiple names: authors list (link)

• Liu, J.S. (2001). Monte Carlo strategies in Scientific Computing. Springer.

• Kong, A.; Liu, J.S.; Wong, W.H. (1994). "Sequential imputations and Bayesian missing data problems" (PDF). Journal of the American Statistical Association. 89 (425). Taylor & Francis Group: 278–288. doi:10.1080/01621459.1994.10476469.

• Liu, J.S.; Chen, R. (1995). "Blind deconvolution via sequential imputations" (PDF). Journal of the American Statistical Association. 90 (430). Taylor & Francis Group: 567–576. doi:10.2307/2291068.

• Ristic, B. (2004). Beyond the Kalman Filter: Particle Filters for Tracking Applications. Artech House. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Doucet, A. (December 2008). "A tutorial on particle filtering and smoothing: fifteen years later" (PDF). Technical report. Department of Statistics, University of British Columbia. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)CS1 maint: extra punctuation (link)

• Doucet, A. (2000). "On sequential Monte Carlo sampling methods for Bayesian filtering". Statistics and Computing. 10 (3): 197–208. doi:10.1023/A:1008935410038. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)CS1 maint: extra punctuation (link)

• Arulampalam, M.S. (2002). "A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking". IEEE Transactions on Signal Processing. 50 (2): 174–188. doi:10.1109/78.978374. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)CS1 maint: extra punctuation (link)

• Cappe, O. (2007). "An overview of existing methods and recent advances in sequential Monte Carlo". Proceedings of IEEE. 95 (5): 899–924. doi:10.1109/JPROC.2007.893250. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)CS1 maint: extra punctuation (link)

• Kitagawa, G. (1996). "Monte carlo filter and smoother for non-Gaussian nonlinear state space models". Journal of Computational and Graphical Statistics. 5 (1): 1–25. doi:10.2307/1390750. JSTOR 1390750.

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• Pitt, M.K.; Shephard, N. (1999). "Filtering Via Simulation: Auxiliary Particle Filters". Journal of the American Statistical Association. 94 (446). Journal of the American Statistical Association, Vol. 94, No. 446: 590–591. doi:10.2307/2670179. JSTOR 2670179. Retrieved 2008-05-06.

• Gordon, N. J. (1993). "Novel approach to nonlinear/non-Gaussian Bayesian state estimation". IEE Proceedings F on Radar and Signal Processing. 140 (2): 107–113. doi:10.1049/ip-f-2.1993.0015. Retrieved 2009-09-19. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Chen, Z. (2003). "Bayesian Filtering: From Kalman Filters to Particle Filters, and Beyond". CiteSeer^x: 10.1.1.107.7415. {{cite journal}}: |access-date= requires |url= (help); Cite journal requires |journal= (help)

• Vaswani, N. (2007). "Tracking deforming objects using particle filtering for geometric active contours". IEEE Trans. on Pattern Analysis and Machine Intelligence. 29 (8): 1470–1475. doi:10.1109/tpami.2007.1081. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

External links

• Feynman–Kac models and interacting particle algorithms (a.k.a. Particle Filtering) Theoretical aspects and a list of application domains of particle filters
• Sequential Monte Carlo Methods (Particle Filtering) homepage on University of Cambridge
• Dieter Fox's MCL Animations
• Rob Hess' free software
• SMCTC: A Template Class for Implementing SMC algorithms in C++
• Java applet on particle filtering
• vSMC : Vectorized Sequential Monte Carlo