# 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 "sequential Monte Carlo" was first coined in Liu and Chen (1998). 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. In the resampling step, the particles with negligible weights are replaced by new particles in the proximity of the particles with higher weights.

## History

The first traces 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 community. However 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 implementation of the SMC methods used today. The authors named their algorithm 'the bootstrap filter', and demonstrated that compared to other filtering methods, their algorithm does not require any assumption about that state-space or the noise of the system.

## 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).

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} | *y*_{0},*y*_{1},…,*y*_{k}). In contrast, the MCMC or importance sampling approach would model the full posterior *p*(*x*_{0},*x*_{1},…,*x*_{k} | *y*_{0},*y*_{1},…,*y*_{k}).

## Model

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

- is a first order Markov process that evolves according to the distribution :

and with an initial distribution .

An example system with these properties is:

where both and are mutually independent and identically distributed sequences with known probability density functions and and 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 and 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). 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 . For example, we may have 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

and , in the usual way for Monte Carlo, can give all the moments etc. of the distribution up to some degree of approximation.

## 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 by a weighted set
of P particles

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

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

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

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

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:

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

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

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To generate a single sample at from :

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

- 2) Uniformly choose an index L from the range

- 5) Generate another uniform u from where

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

- 7) If n == P then quit

The goal is to generate P "particles" at 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 and repeats (steps 2–6) until P particles are generated at .

This can be more easily visualized if is viewed as a two-dimensional array.
One dimension is and the other dimensions is the particle number.
For example, would be the L^{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

- Auxiliary particle filter
^{[1]} - Regularized auxiliary particle filter
^{[2]} - Gaussian particle filter
- Unscented particle filter
- Gauss–Hermite particle filter
- Cost Reference particle filter
- Hierarchical/Scalable particle filter
^{[3]} - Rao–Blackwellized particle filter
^{[4]} - Rejection-sampling based optimal particle filter
^{[5]}^{[6]}

## See also

- Ensemble Kalman filter
- Generalized filtering
- Moving horizon estimation
- Recursive Bayesian estimation
- Monte Carlo localization

## References

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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