Particle filter
Particle filters, or sequential Monte Carlo methods, are a set of
Particle filtering uses a set of particles (also called samples) to represent the
Particle filters update their prediction in an approximate (statistical) manner. 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 uneven. Several adaptive resampling criteria can be used including the variance of the weights and the relative entropy concerning the uniform distribution.[6] In the resampling step, the particles with negligible weights are replaced by new particles in the proximity of the particles with higher weights.
From the statistical and probabilistic point of view, particle filters may be interpreted as
The particle filter methodology is used to solve Hidden Markov Model (HMM) and nonlinear filtering problems. With the notable exception of linear-Gaussian signal-observation models (Kalman filter) or wider classes of models (Benes filter[17]), Mireille Chaleyat-Maurel and Dominique Michel proved in 1984 that the sequence of posterior distributions of the random states of a signal, given the observations (a.k.a. optimal filter), has no finite recursion.[18] Various other numerical methods based on fixed grid approximations, Markov Chain Monte Carlo techniques, conventional linearization, extended Kalman filters, or determining the best linear system (in the expected cost-error sense) are unable to cope with large-scale systems, unstable processes, or insufficiently smooth nonlinearities.
Particle filters and Feynman-Kac particle methodologies find application in
History
Heuristic-like algorithms
From a statistical and probabilistic viewpoint, particle filters belong to the class of
The origins of mean-field type evolutionary computational techniques can be traced back to 1950 and 1954 with Alan Turing's work on genetic type mutation-selection learning machines[22] and the articles by Nils Aall Barricelli at the Institute for Advanced Study in Princeton, New Jersey.[23][24] The first trace of particle filters in statistical methodology dates back to the mid-1950s; the 'Poor Man's Monte Carlo',[25] 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.[26] In evolutionary computing literature, genetic-type mutation-selection algorithms became popular through the seminal work of John Holland in the early 1970s, particularly his book[27] published in 1975.
In Biology and
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.[7][8] 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.[7][8][9][13][14][31][32] 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,[33] 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.[13] One can also quote the 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.[34] 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.[12]
The use of genetic particle algorithms in advanced signal processing and Bayesian inference is more recent. In January 1993, Genshiro Kitagawa developed a "Monte Carlo filter",[35] a slightly modified version of this article appeared in 1996.[36] In April 1993, Gordon et al., published in their seminal work[37] an 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. Independently, the ones by Pierre Del Moral[2] and Himilcon Carvalho, Pierre Del Moral, André Monin, and Gérard Salut[38] on particle filters published in the mid-1990s. Particle filters were also developed in signal processing in 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.[39][40][41][42][43][44]
Mathematical foundations
From 1950 to 1996, all the publications on particle filters, and 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 genealogical and ancestral tree-based algorithms.
The mathematical foundations and the first rigorous analysis of these particle algorithms are due to Pierre Del Moral[2][4] in 1996. The article[2] also contains proof of the unbiased properties of a particle approximation 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.
Dan Crisan, Jessica Gaines, and Terry Lyons,[45][46][47] as well as Dan Crisan, Pierre Del Moral, and Terry Lyons,[48] created branching-type particle techniques with various population sizes around the end of the 1990s. P. Del Moral, A. Guionnet, and L. Miclo[8][49][50] made more advances in this subject in 2000. Pierre Del Moral and Alice Guionnet[51] proved the first central limit theorems in 1999, and Pierre Del Moral and Laurent Miclo[8] proved them in 2000. The first uniform convergence results concerning the time parameter for particle filters were developed at the end of the 1990s by Pierre Del Moral and Alice Guionnet.[49][50] The first rigorous analysis of genealogical tree-ased particle filter smoothers is due to P. Del Moral and L. Miclo in 2001[52]
The theory on Feynman-Kac particle methodologies and related particle filter algorithms was developed in 2000 and 2004 in the books.[8][5] These abstract probabilistic models encapsulate genetic type algorithms, particle, and bootstrap filters, interacting Kalman filters (a.k.a. Rao–Blackwellized particle filter[53]), 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 include genealogical tree-based models,[10][5][54] backward Markov particle models,[10][55] adaptive mean-field particle models,[6] island-type particle models,[56][57] particle Markov chain Monte Carlo methodologies,[58][59] Sequential Monte Carlo samplers [60][61][62] and Sequential Monte Carlo Approximate Bayesian Computation methods[63] and Sequential Monte Carlo ABC based Bayesian Bootstrap.[64]
The filtering problem
Objective
A particle filter's goal is to estimate the posterior density of state variables given observation variables. The particle filter is intended for use with a hidden Markov Model, in which the system includes both hidden and observable variables. The observable variables (observation process) are linked to the hidden variables (state-process) via a known functional form. Similarly, the probabilistic description of the dynamical system defining the evolution of the state variables is known.
A generic particle filter estimates the posterior distribution of the hidden states using the observation measurement process. With respect to a state-space such as the one 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 . 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 Markov Chain Monte Carlo or importance sampling approach would model the full posterior .
The Signal-Observation model
Particle methods often assume and the observations can be modeled in this form:
- is a Markov processon (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 ) and 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
The assumption that the initial distribution and the transitions of the Markov chain are continuous for 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 continuous assumption on the Markov transitions of is 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 certain problems, the conditional distribution of observations, given the random states of the signal, may fail to have a density; the latter may be impossible or too complex to compute.[19] In this situation, an additional level of approximation is necessitated. 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 random variables 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
The nonlinear filtering equation
where
Particle filters are also an approximation, but with enough particles they can be much more accurate.[2][4][5][49][50] The nonlinear filtering equation is given by the recursion
-
(Eq. 1)
with the convention for k = 0. The nonlinear filtering problem consists in computing these conditional distributions sequentially.
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
Feynman-Kac path integration models arise in a variety of scientific disciplines, including in computational physics, biology, information theory and computer sciences.[8][10][5] Their interpretations are dependent 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, such an algorithm starts with N independent random variables with common probability density . The genetic algorithm selection-mutation transitions[2][4]
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
where stands for the Dirac measure at a given state a.
- 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
In Genetic algorithms and
Monte Carlo principles
Particle methods, like all sampling-based approaches (e.g., Markov Chain Monte Carlo), 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 approximation error. 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 this 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.[10][5]
Mean-field particle simulation
This section may be too technical for most readers to understand.(June 2017) |
The general probabilistic principle
The nonlinear filtering evolution can be interpreted as a dynamical system in the set of probability measures of the 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 ways 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.[10][5]
Some convergence results
The analysis of the convergence of particle filters was started in 1996[2][4] and in 2000 in the book[8] and the series of articles.[48][49][50][51][52][68][69] More recent developments can be found in the books,[10][5] 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
This section may be too technical for most readers to understand.(June 2017) |
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[54] 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.[10][5]
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.[2] Refined variance estimates can be found in[5] and.[10]
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.[55] 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.[10]
Sequential Importance Resampling (SIR)
Monte Carlo filter and bootstrap filter
Sequential importance Resampling (SIR), Monte Carlo filtering (Kitagawa 1993[35]), bootstrap filtering algorithm (Gordon et al. 1993[37]) and single distribution resampling (Bejuri W.M.Y.B et al. 2017[70]), are also commonly applied filtering algorithms, which approximate 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
Sequential importance sampling (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 1996 and 1998.[4] 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.[4] 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 the 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 (1993[35]) 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 criterion reflects the variance of the weights. Other criteria can be found in the article,[6] 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, but the term Importance Resampling is more accurate because the word "resampling" implies that the initial sampling has already been done.[71]
Sequential importance sampling (SIS)
- Is the same as sequential importance resampling, but without the resampling stage.
"Direct version" algorithm
This section may be confusing or unclear to readers. (October 2011) |
The "direct version" algorithm [citation needed] 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) Uniformlychoose 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 uniformu 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 the 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 dimension is the particle number. For example, would be the ith 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 .
Applications
Particle filters and Feynman-Kac particle methodologies find application in several contexts, as an effective mean for tackling noisy observations or strong nonlinearities, such as:
- risk analysis and rare event sampling
- Bioinformatics[19]
- Computational science
- financial mathematics and mathematical finance: particle filters can perform simulations which are needed to compute the high-dimensional and/or complex integrals related to problems such as dynamic stochastic general equilibrium models in macro-economics and option pricing[72]
- Engineering
- Infectious disease epidemiology where they have been applied to a number of epidemic forecasting problems, for example predicting seasonal influenza epidemics[73]
- Fault detection and isolation: in observer-based schemas a particle filter can forecast expected sensors output enabling fault isolation[74][75][76]
- Molecular chemistry and computational physics
- Pharmacokinetics[77]
- Phylogenetics
- Robotics, artificial intelligence: Monte Carlo localization is a de facto standard in mobile robot localization[78][79][80]
- Signal and image processing: visual localization, tracking, feature recognition[81]
Other particle filters
- Auxiliary particle filter[82]
- Cost Reference particle filter
- Exponential Natural Particle Filter[83]
- Feynman-Kac and mean-field particle methodologies[2][10][5]
- Gaussian particle filter
- Gauss–Hermite particle filter
- Hierarchical/Scalable particle filter[84]
- Nudged particle filter[85]
- Particle Markov-Chain Monte-Carlo, see e.g. pseudo-marginal Metropolis–Hastings algorithm.
- Rao–Blackwellized particle filter[53]
- Regularized auxiliary particle filter[86]
- Rejection-sampling based optimal particle filter[87][88]
- Unscented particle filter
See also
- Ensemble Kalman filter
- Generalized filtering
- Genetic algorithm
- Mean-field particle methods
- Monte Carlo localization
- Moving horizon estimation
- Recursive Bayesian estimation
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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
- Particle filter explained in the context of self driving car