Particle filter

Particle filters or Sequential Monte Carlo (SMC) methods are a set of genetic-type particle Monte Carlo methodologies to solve the filtering problem. The term "particle filters" was first coined in 1996 by Del Moral^[1] in reference to mean field interacting particle methods used in fluid mechanics since the beginning of the 1960s. The terminology "sequential Monte Carlo" was proposed by Liu and Chen in 1998.

From the statistical and probabilistic point of view, particle filters can be interpreted as mean field particle interpretations of Feynman-Kac probability measures.^{[2][3][4][5][6]} These particle integration techniques were developed in molecular chemistry and computational physics by Theodore E. Harris and Herman Kahn in 1951, Marshall. N. Rosenbluth and Arianna. W. Rosenbluth in 1955^[7] and more recently by Jack H. Hetherington in 1984.^[8] In computational physics, these Feynman-Kac type path particle integration methods are also used in Quantum Monte Carlo, and more specifically Diffusion Monte Carlo methods.^[9][10][11] Feynman-Kac interacting particle methods are also strongly related to mutation-selection genetic algorithms currently used in evolutionary computing to solve complex optimization problems.

The particle filter methodology is used to solve Hidden Markov Chain (HMM) and nonlinear filtering problems arising in signal processing and Bayesian statistical inference. The filtering problem consists in estimating the internal states in dynamical systems when partial observations are made, and random perturbations are present in the sensors as well as in the dynamical system. The objective is to compute the conditional probability (a.k.a. posterior distributions) of the states of some Markov process, given some noisy and partial observations. With the notable exception of linear-Gaussian signal-observation models (Kalman filter) or wider classes of models (Benes filter^[12]) Mireille Chaleyat-Maurel and Dominique Michel proved in 1984 that the sequence of posterior distributions of the random states of the signal given the observations (a.k.a. optimal filter) have no finitely recursive recursion.^[13] Various numerical methods based on fixed grid approximations, Markov Chain Monte Carlo techniques (MCMC), conventional linearization, extended Kalman filters, or determining the best linear system (in expect cost-error sense) have never really coped with large scale systems, unstable processes or when the nonlinearities are not sufficiently smooth.

Particle filtering methodology uses a genetic type mutation-selection sampling approach, with a set of particles (also called individuals, or samples) to represent the posterior distribution of some stochastic process given some noisy and/or partial observations. The state-space model can be nonlinear and the initial state and noise distributions can take any form required. Particle filter techniques provide a well-established methodology^[1][14][15] 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 very high-dimensional systems.

Particle filters implement the prediction-updating transitions of the filtering equation directly by using a genetic type mutation-selection particle algorithm. The samples from the distribution are represented by a set of particles; each particle has a likelihood 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.^[16] In the resampling step, the particles with negligible weights are replaced by new particles in the proximity of the particles with higher weights.

Particle filters and Feynman-Kac particle methodologies find application in signal and image processing, Bayesian inference, machine learning, risk analysis and rare event sampling, engineering and robotics, artificial intelligence, bioinformatics, phylogenetics, computational science, Economics and mathematical finance, molecular chemistry, computational physics, pharmacokinetic and other fields.

History

Heuristic like algorithms

From the statistical and probabilistic viewpoint, particle filters belong to the class of branching/genetic type algorithms, and mean field type interacting particle methodologies. The interpretations of these particle methods depends on the scientific discipline. In Evolutionary Computing, mean field genetic type particle methodologies are often used as a heuristic and natural search algorithms (a.k.a. Metaheuristic). In computational physics and molecular chemistry they are used to solve Feynman-Kac path integration problems, or the compute Boltzmann-Gibbs measures, top eigenvalues and ground states of Schrödinger operators. In Biology and Genetics they also represent the evolution of a population of individuals or genes in some environment.

The origins of mean field type evolutionary computational techniques can be traced to 1950 and 1954 with the seminal work of Alan Turing on genetic type mutation-selection learning machines^[17] and the articles by Nils Aall Barricelli at the Institute for Advanced Study in Princeton, New Jersey.^[18][19] The first trace of particle filters in statistical methodology dates back to the mid-50's; the 'Poor Man's Monte Carlo',^[20] that was proposed by Hammersley et al., in 1954, contained hints of the genetic type particle filtering methods used today. In 1963, Nils Aall Barricelli simulated a genetic type algorithm to mimic the ability of individuals to play a simple game.^[21] In evolutionary computing literature, genetic type mutation-selection algorithms became popular through the seminal work of John Holland in the early 1970s, and particularly his book^[22] published in 1975.

In Biology and Genetics, the Australian geneticist Alex Fraser also published in 1957 a series of papers on the genetic type simulation of artificial selection of organisms.^[23] The computer simulation of evolution by biologists became more common in the early 1960s, and the methods were described in books by Fraser and Burnell (1970)^[24] and Crosby (1973).^[25] Fraser's simulations included all of the essential elements of modern mutation-selection genetic particle algorithms.

From the mathematical viewpoint, 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 a sequence of likelihood potential functions.^[2][3] Quantum Monte Carlo, and more specifically Diffusion Monte Carlo methods can also be interpreted as a mean field genetic type particle approximation of Feynman-Kac path integrals.^{[2][3][4][8][9][26][27]} The origins of Quantum Monte Carlo methods are often attributed to Enrico Fermi and Robert Richtmyer who developed in 1948 a mean field particle interpretation of neutron-chain reactions,^[28] but the first heuristic-like and genetic type particle algorithm (a.k.a. Resampled or Reconfiguration Monte Carlo methods) for estimating ground state energies of quantum systems (in reduced matrix models) is due to Jack H. Hetherington in 1984.^[8] We also quote an earlier seminal works of Theodore E. Harris and Herman Kahn in particle physics, published in 1951, using mean field but heuristic-like genetic methods for estimating particle transmission energies.^[29] In molecular chemistry, the use of genetic heuristic-like particle methodologies (a.k.a. pruning and enrichment strategies) can be traced back to 1955 with the seminal work of Marshall. N. Rosenbluth and Arianna. W. Rosenbluth.^[7]

The use of genetic particle algorithms in advanced signal processing and Bayesian inference is more recent. It was in 1993, that Gordon et al., published in their seminal work^[30] the first application of genetic type algorithm 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 on a related "Monte Carlo filter",^[31] and the ones by Pierre Del Moral^[1] and Himilcon Carvalho, Pierre Del Moral, André Monin and Gérard Salut^[32] on particle filters published in the mid-1990s. Particle filters were also developed in signal processing in the early 1989-1992 by P. Del Moral, J.C. Noyer, G. Rigal, and G. Salut in the LAAS-CNRS in a series of restricted and classified research reports with STCAN (Service Technique des Constructions et Armes Navales), the IT company DIGILOG, and the LAAS-CNRS (the Laboratory for Analysis and Architecture of Systems) on RADAR/SONAR and GPS signal processing problems.^{[33][34][35][36][37][38]}

Mathematical foundations

From 1950 to 1996, all the publications on particle filters, genetic algorithms, including the pruning and resample Monte Carlo methods introduced in computational physics and molecular chemistry, present natural and heuristic-like algorithms applied to different situations without a single proof of their consistency, nor a discussion on the bias of the estimates and on genealogical and ancestral tree based algorithms.

The mathematical foundations and the first rigorous analysis of these particle algorithms are due to Pierre Del Moral^[1][14] in 1996. The article^[1] also contains a proof of the unbiased properties of a particle approximations of likelihood functions and unnormalized conditional probability measures. The unbiased particle estimator of the likelihood functions presented in this article is used today in Bayesian statistical inference.

Branching type particle methodologies with varying population sizes were also developed toward the end of the 1990s by Dan Crisan, Jessica Gaines and Terry Lyons,^[39][40][41] and by Dan Crisan, Pierre Del Moral and Terry Lyons.^[42] Further developments in this field were developed in 2000 by P. Del Moral, A. Guionnet and L. Miclo.^[3][43][44] The first central limit theorems are due to Pierre Del Moral and Alice Guionnet^[45] in 1999 and Pierre Del Moral and Laurent Miclo^[3] in 2000. 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.^[43][44] The first rigorous analysis of genealogical tree based particle filter smoothers is due to P. Del Moral and L. Miclo in 2001^[46]

The theory on Feynman-Kac particle methodologies and related particle filters algorithms has been developed in 2000 and 2004 in the books.^[3][15] These abstract probabilistic models encapsulate genetic type algorithms, particle and bootstrap filters, interacting Kalman filters (a.k.a. Rao–Blackwellized particle filter^[47]), importance sampling and resampling style particle filter techniques, including genealogical tree based and particle backward methodologies for solving filtering and smoothing problems. Other classes of particle filtering methodologies includes genealogical tree based models,^[5][15][48] backward Markov particle models,^[5][49] adaptive mean field particle models,^[16] island type particle models,^[50][51] and particle Markov chain Monte Carlo methodologies.^[52][53]

The filtering problem

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 below.

The filtering problem is to estimate sequentially the values of the hidden states , given the values of the observation process at any time step k.

All Bayesian estimates of follow from the posterior density p(x_k | y₀,y₁,…,y_k). The particle filter methodology provides an approximation of these conditional probabilities using the empirical measure associated with a genetic type particle algorithm. In contrast, the MCMC or importance sampling approach would model the full posterior p(x₀,x₁,…,x_k | y₀,y₁,…,y_k).

An illustration in Tracking and RADAR processing

In a 3d-simplified Singer's tracking model^[54] the target is represented by a Markov chain with 3 coordinates

The first coordinate represents the acceleration of the target at time n, the second and the third stands for the speed and the position of the target at time n.

The maneuver equation are often derived from a time discretization of a continuous time model given by the equations

for some viscosity coefficients , some time step . In the above display, the unknown acceleration jump times are modeled by a sequence of independent and identically distributed {0, 1}-valued Bernoulli variables . The unknown speed-up changes are modeled by a sequence of independent and identically distributed variables . The RADAR sensor delivers at each time step some noisy information on the position of the target. This observation model is given the equation

The random perturbations of the sensors coming from the thermic noise of the uncertainty of the model are represented by a sequence of independent random variable with a prescribed probability density. The objective of the filtering problem consists of computing sequentially the conditional distribution of the signal given an observation sequence , at any time step n. A more complex problem is to compute the conditional distribution of the random paths of the signal given an observation sequence at any time step n. In filtering literature, these state estimation problems can be given different names:

• The optimal filter is the conditional probability distribution of the signal given the observation sequence .
• The one step optimal predictor is the conditional probability distribution of the signal given the observation sequence .
• The smoothing problem consists of estimating the conditional probability distribution of the signal given the observation sequence , for some .

The Signal-Observation Model

Particle methods often assume and the observations can be modeled in this form:

• is a Markov process on (for some ) that evolves according to the transition probability density . This model is also often written in a synthetic way as

  

with an initial probability density .

• The observations take values in some state space on (for some ) are conditionally independent provided that are known. In other words, each only depends on . In addition, we assume conditional distribution for given are absolutely continuous, and in a synthetic way we have

  

An example of system with these properties is:

where both and are mutually independent sequences with known probability density functions and g and h 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. If the functions g and h in the above example are linear, and if both and 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).

The assumption that the initial distribution and the transitions of the Markov chain are absolutely continuous with respect to the Lebesgue measure can be relaxed. To design a particle filter we simply need to assume that we can sample the transitions of the Markov chain and to compute the likelihood function (see for instance the genetic selection mutation description of the particle filter given below). The absolutely continuous assumption on the Markov transitions of are only used to derive in an informal (and rather abusive) way different formulae between posterior distributions using the Bayes' rule for conditional densities.

Approximate Bayesian Computation models

In some important problems, the conditional distribution of the observations given the random states of the signal may fail to have a density or may be impossible or too complex to compute. In this situation, we need to resort to an additional level of approximation. One strategy is to replace the signal by the Markov chain and to introduce a virtual observation of the form

for some sequence of independent sequences with known probability density functions. The central idea is to observe that

The particle filter associated with the Markov process given the partial observations is defined in terms of particles evolving in with a likelihood function given with some obvious abusive notation by . These probabilistic techniques are closely related to Approximate Bayesian Computation (ABC). In the context of particle filters, these ABC particle filtering techniques were introduced in 1998 by P. Del Moral, J. Jacod and P. Protter in the article.^[55] They were further developed by P. Del Moral, A. Doucet and A. Jasra.^[56][57]

The nonlinear filtering equation

The Bayes' rule for conditional probability gives:

where

Particle filters are also an approximation, but with enough particles they can be much more accurate.^{[1][14][15][43][44]} The nonlinear filtering equation is given by the recursion

(Eq. 1)

with the convention for k = 0. The nonlinear filtering problem consists in computing sequentially these sequence of conditional distributions.

Feynman-Kac formulation

We fix a time horizon n and a sequence of observations , and for each k = 0, ..., n we set:

In this notation, for any bounded function F on the set of trajectories of from the origin k = 0 up to time k = n, we have the Feynman-Kac formula

These Feynman-Kac path integration models arise in a variety of scientific disciplines, including in computational physics, biology, information theory and computer sciences.^[3][5][15] Their interpretations depend on the application domain. For instance, if we choose the indicator function of some subset of the state space, they represent the conditional distribution of a Markov chain given it stays in a given tube; that is, we have:

and

as soon as the normalizing constant is strictly positive.

Particle filters

A Genetic type particle algorithm

Initially we start with N independent random variable with common probability density . The genetic algorithm selection-mutation transitions

mimic/approximate the updating-prediction transitions of the optimal filter evolution (Eq. 1):

• During the selection-updating transition we sample N (conditionally) independent random variables with common (conditional) distribution

• During the mutation-prediction transition, from each selected particle we sample independently a transition

In the above displayed formulae stands for the likelihood function evaluated at , and stands for the conditional density evaluated at .

At each time k, we have the particle approximations

and

A detailed proof of these convergence results can be found in,^[1][14] see also the more recent developments provided in the books.^[5][15] In Genetic algorithms and Evolutionary computing community, the mutation-selection

Markov chain described above is often called the genetic algorithm with proportional selection. Several branching variants, including with random population sizes have also been proposed in the articles.^[15][39][42]

Monte Carlo principles

Particle methods, like all sampling-based approaches (e.g., MCMC), generate a set of samples that approximate the filtering density

For example, we may have N samples from the approximate posterior distribution of , where the samples are labeled with superscripts as

Then, expectations with respect to the filtering distribution are approximated by

(Eq. 2)

with

where stands for the Dirac measure at a given state a. The function f, in the usual way for Monte Carlo, can give all the moments etc. of the distribution up to some degree of approximation. When the approximation equation (Eq. 2) is satisfied for any bounded function f we write

Particle filters can be interpreted as a genetic type particle algorithm evolving with mutation and selection transitions. We can keep track of the ancestral lines

of the particles . The random states , with the lower indices l=0,...,k, stands for the ancestor of the individual at level l=0,...,k. In his situation, we have the approximation formula

(Eq. 3)

with the empirical measure

Here F stands for any founded function on the path space of the signal. In a more synthetic form (Eq. 3) is equivalent to

Particle filters can be interpreted in many different ways. From the probabilistic point of view they coincide with a mean field particle interpretation of the nonlinear filtering equation. The updating-prediction transitions of the optimal filter evolution can also be interpreted as the classical genetic type selection-mutation transitions of individuals. The sequential importance resampling technique provides another interpretation of the filtering transitions coupling importance sampling with the bootstrap resampling step. Last, but not least, particle filters can be seen as an acceptance-rejection methodology equipped with a recycling mechanism.^[5][15]

Mean field particle simulation

The general probabilistic principle

The nonlinear filtering evolution can be interpreted as a dynamical system in the set of probability measures of the following form where stands for some mapping from the set of probability distribution into itself. For instance, the evolution of the one-step optimal predictor

satisfies a nonlinear evolution starting with the probability distribution . One of the simplest way to approximate these probability measures is to start with N independent random variables with common probability distribution . Suppose we have defined a sequence of N random variables such that

At the next step we sample N (conditionally) independent random variables with common law .

A particle interpretation of the filtering equation

We illustrate this mean field particle principle in the context of the evolution of the one step optimal predictors

(Eq. 4)

For k = 0 we use the convention .

By the law of large numbers, we have

in the sense that

for any bounded function . We further assume that we have constructed a sequence of particles at some rank k such that

in the sense that for any bounded function we have

In this situation, replacing by the empirical measure in the evolution equation of the one-step optimal filter stated in (Eq. 4) we find that

Notice that the right hand side in the above formula is a weighted probability mixture

where stands for the density evaluated at , and stands for the density evaluated at for

Then, we sample N independent random variable with common probability density so that

Iterating this procedure, we design a Markov chain such that

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

The terminology "mean field approximation" comes from the fact that we replace at each time step the probability measure by the empirical approximation . The mean field particle approximation of the filtering problem is far from being unique. Several strategies are developed in the books.^[5][15]

Some convergence results

The analysis of the convergence of particle filters has been started in 1996^[1][14] and in 2000 in the book^[3] and the series of articles.^{[42][43][44][45][46][58][59]} More recent developments can be found in the books,^[5][15] When the filtering equation is stable (in the sense that it corrects any erroneous initial condition), the bias and the variance of the particle particle estimates

are controlled by the non asymptotic uniform estimates

for any function f bounded by 1, and for some finite constants In addition, for any :

for some finite constants related to the asymptotic bias and variance of the particle estimate, and some finite constant c. The same results are satisfied if we replace the one step optimal predictor by the optimal filter approximation.

Genealogical trees and Unbiasedness properties

Genealogical tree based particle smoothing

Tracing back in time the ancestral lines

of the individuals and at every time step k, we also have the particle approximations

These empirical approximations are equivalent to the particle integral approximations

for any bounded function F on the random trajectories of the signal. As shown in^[48] the evolution of the genealogical tree coincides with a mean field particle interpretation of the evolution equations associated with the posterior densities of the signal trajectories. For more details on these path space models, we refer to the books.^[5][15]

Unbiased particle estimates of likelihood functions

We use the product formula

with

and the conventions and for k = 0. Replacing by the empirical approximation

in the above displayed formula, we design the following unbiased particle approximation of the likelihood function

with

where stands for the density evaluated at . The design of this particle estimate and the unbiasedness property has been proved in 1996 in the article.^[1] Refined variance estimates can be found in^[15] and.^[5]

Backward particle smoothers

Using Bayes' rule, we have the formula

Notice that

This implies that

Replacing the one-step optimal predictors by the particle empirical measures

we find that

We conclude that

with the backward particle approximation

The probability measure

is the probability of the random paths of a Markov chain running backward in time from time k=n to time k=0, and evolving at each time step k in the state space associated with the population of particles

• Initially (at time k=n) the chain chooses randomly a state with the distribution

• From time k to the time (k-1), the chain starting at some state for some at time k moves at time (k-1) to a random state chosen with the discrete weighted probability

In the above displayed formula, stands for the conditional distribution evaluated at . In the same vein, and stand for the conditional densities and evaluated at and These models allows to reduce integration with respect to the densities in terms of matrix operations with respect to the Markov transitions of the chain described above.^[49] For instance, for any function we have the particle estimates

where

This also shows that if

then

Some convergence results

We shall assume that filtering equation is stable, in the sense that it corrects any erroneous initial condition.

In this situation, the particle approximations of the likelihood functions are unbiased and the relative variance is controlled by

for some finite constant c. In addition, for any :

for some finite constants related to the asymptotic bias and variance of the particle estimate, and for some finite constant c.

The bias and the variance of the particle particle estimates based on the ancestral lines of the genealogical trees

are controlled by the non asymptotic uniform estimates

for any function F bounded by 1, and for some finite constants In addition, for any :

for some finite constants related to the asymptotic bias and variance of the particle estimate, and for some finite constant c. The same type of bias and variance estimates hold for the backward particle smoothers. For additive functionals of the form

with

with functions bounded by 1, we have

and

for some finite constants More refined estimates including exponentially small probability of errors are developed in.^[5]

Sequential importance sampling (SIS)

The bootstrap filter

Sequential importance sampling (SIS), the original bootstrap filtering algorithm (Gordon et al. 1993), is also a very commonly used filtering algorithm, which approximates the filtering probability density by a weighted set of N samples

The importance weights are approximations to the relative posterior probabilities (or densities) of the samples such that

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

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

.

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

This particular choice of proposal transition has been proposed by P. Del Moral in^[14] in 1996 and 1998. When it is difficult to sample transitions according to the distribution one natural strategy is to use the following particle approximation

with the empirical approximation

associated with N (or any other large number of samples) independent random samples with the conditional distribution of the random state given . The consistency of the resulting particle filter of this approximation and other extensions are developed in.^[14] In the above display 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:

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 draw samples from the proposal distribution

2) For update the importance weights up to a normalizing constant:

Note that when we use the transition prior probability distribution as the importance function,

this simplifies to the following :

3) For compute the normalized importance weights:

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

This criteria reflects the variance of the weights, other criteria can be found in the article,^[16] including their rigorous analysis and central limit theorems.

5) If the effective number of particles is less than a given threshold , 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 set

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 :

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

2) Uniformly choose an index i from the range

3) Generate a test from the distribution with

4) Generate the probability of using from where is the measured value

5) Generate another uniform u from where

6) Compare u and

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

6b) If u is smaller then save as and increment n

7) If n == N then quit

The goal is to generate P "particles" at k using only the particles from . This requires that a Markov equation can be written (and computed) to generate a based only upon . This algorithm uses composition of the P particles from 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, would be the i^th particle at and can also be written (as done above in the algorithm). Step 3 generates a potential based on a randomly chosen particle () at time and rejects or accepts it in step 6. In other words, the values are generated using the previously generated .

Other particle filters

• Exponential Natural Particle Filter^[60]
• Auxiliary particle filter^[61]
• Regularized auxiliary particle filter^[62]
• Gaussian particle filter
• Unscented particle filter
• Gauss–Hermite particle filter
• Cost Reference particle filter
• Hierarchical/Scalable particle filter^[63]
• Rao–Blackwellized particle filter^[47]
• Rejection-sampling based optimal particle filter^[64][65]
• Feynman-Kac and mean field particle methodologies^[1][5][15]

See also

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

References

Bibliography

• Del Moral, Pierre (1996). "Non Linear Filtering: Interacting Particle Solution" (PDF). Markov Processes and Related Fields 2 (4): 555–580. 
• Del Moral, Pierre (2004). Feynman-Kac formulae. Genealogical and interacting particle approximations. Springer. p. 575. Series: Probability and Applications
• Del Moral, Pierre (2013). Mean field simulation for Monte Carlo integration. Chapman & Hall/CRC Press. p. 626. Monographs on Statistics & Applied Probability
• Cappe, O.; Moulines, E.; Ryden, T. (2005). Inference in Hidden Markov Models. Springer. 
• Liu, J.S.,; Chen, R. (1998). "Sequential Monte Carlo methods for dynamic systems" (PDF). Journal of the American Statistical Association (Taylor & Francis Group) 93 (443): 1032–1044. doi:10.1080/01621459.1998.10473765. 

• 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 (Taylor & Francis Group) 89 (425): 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 (Taylor & Francis Group) 90 (430): 567–576. doi:10.2307/2291068. 

• Ristic, B.; Arulampalam, S.; Gordon, N. (2004). Beyond the Kalman Filter: Particle Filters for Tracking Applications. Artech House. 

• Doucet, A.; Johansen, A.M. (December 2008). "A tutorial on particle filtering and smoothing: fifteen years later" (PDF). Technical report (Department of Statistics, University of British Columbia). 

• Doucet, A.; Godsill, S.; Andrieu, C. (2000). "On sequential Monte Carlo sampling methods for Bayesian filtering". Statistics and Computing 10 (3): 197–208. doi:10.1023/A:1008935410038. 

• Arulampalam, M.S.; Maskell, S.; Gordon, N.; Clapp, T. (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. 

• Cappe, O.; Godsill, S.; Moulines, E. (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. 

• 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. 

• Kotecha, J.H.; Djuric, P. (2003). "Gaussian Particle filtering". IEEE Transactions Signal Processing 51 (10). 

• Haug, A.J. (2005). "A Tutorial on Bayesian Estimation and Tracking Techniques Applicable to Nonlinear and Non-Gaussian Processes" (PDF). The MITRE Corporation, USA, Tech. Rep., Feb. Retrieved 2008-05-06. 

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

• Gordon, N. J.; Salmond, D. J.; Smith, A. F. M. (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. 

• Chen, Z. (2003). "Bayesian Filtering: From Kalman Filters to Particle Filters, and Beyond". CiteSeerX: 10.1.1.107.7415.  |access-date= requires |url= (help)

• Vaswani, N.; Rathi, Y. Yezzi, A., Tannenbaum, A. (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 uses deprecated parameter |coauthors= (help)

External links

Stochastic processes Discrete time Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Continuous time Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage Both Branching process Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process White noise Fields and other Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph Time series models Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model Financial models Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie Actuarial models Bühlmann Cramér–Lundberg Risk process Sparre–Anderson Queueing models Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c Properties Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible Limit theorems Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem Inequalities Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe Tools Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract Disciplines Actuarial mathematics Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Statistics Stochastic analysis Time series analysis Machine learning List of topics Category

Statistics Outline Index   Descriptive statistics Continuous data Location Mean arithmetic geometric harmonic Median Mode Dispersion Range Standard deviation Coefficient of variation Percentile Interquartile range Shape Variance Skewness Kurtosis Moments L-moments Count data Index of dispersion Summary tables Grouped data Frequency distribution Contingency table Dependence Pearson product-moment correlation Rank correlation Spearman's rho Kendall's tau Partial correlation Scatter plot Statistical graphics Bar chart Biplot Box plot Control chart Correlogram Fan chart Forest plot Histogram Pie chart Q–Q plot Run chart Scatter plot Stem-and-leaf display Radar chart   Data collection Study design Effect size Standard error Statistical power Sample size determination Survey methodology Sampling stratified cluster Opinion poll Questionnaire Controlled experiments Design control optimal Controlled trial Randomized Random assignment Replication Blocking Factorial experiment Uncontrolled studies Observational study Natural experiment Quasi-experiment   Statistical inference Statistical theory Sampling distribution Order statistic Scan statistic Record value Sufficiency Completeness Exponential family Permutation test (Randomization test) Empirical distribution Bootstrap U statistic Efficiency Asymptotics Robustness Frequentist inference Confidence interval Testing hypotheses Power Unbiased estimators Mean unbiased minimum-variance Median unbiased Biased estimators Maximum likelihood Method of moments Minimum distance Density estimation Parametric tests Likelihood-ratio Wald Score Specific tests Z (normal) Student's t-test F Shapiro–Wilk Kolmogorov–Smirnov Goodness of fit Chi-squared G Sample source (Anderson–Darling) Sample normality (Shapiro–Wilk) Skewness / kurtosis normality (Jarque-Bera) Model comparison (Likelihood-ratio) Model quality (Akaike criterion) Signed-rank 1-sample (Wilcoxon) 2-sample (Mann–Whitney U) 1-way anova (Kruskal–Wallis) Bayesian inference Bayesian probability prior posterior Credible interval Bayes factor Bayesian estimator Maximum posterior estimator   Correlation Regression analysis Correlation Pearson product–moment Partial correlation Confounding variable Coefficient of determination Regression analysis Errors and residuals Regression model validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS) Linear regression Simple linear regression Ordinary least squares General linear model Bayesian regression Non-standard predictors Nonlinear regression Nonparametric Semiparametric Isotonic Robust Heteroscedasticity Homoscedasticity Generalized linear model Exponential families Logistic (Bernoulli) / Binomial / Poisson regressions Partition of variance Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom   Categorical / Multivariate / Time-series / Survival analysis Categorical Cohen's kappa Contingency table Graphical model Log-linear model McNemar's test Multivariate Multivariate regression Principal components Factor analysis Cluster analysis Classification Copulas Time-series General Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality Specific tests Dickey–Fuller Johansen Q-statistic (Ljung–Box) Durbin–Watson Breusch–Godfrey Time domain Autocorrelation (ACF) partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model (Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR) Frequency domain Spectral density estimation Fourier analysis Wavelet Survival Survival function Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time Hazard function Nelson–Aalen estimator Test Log-rank test   Applications Biostatistics Bioinformatics Clinical trials / studies Epidemiology Medical statistics Engineering statistics Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification Social statistics Actuarial science Census Crime statistics Demography Econometrics National accounts Official statistics Population Psychometrics Spatial statistics Cartography Environmental statistics Geographic information system Geostatistics Kriging Category Portal Commons WikiProject