Particulate filter (statistical)

The particulate filters , so known like sequential Methods of Monte Carlo , are sophisticated techniques of estimate of Modèle S based on the Simulation.

The particulate filters are generally used to estimate models Bayésiens and constitute the “in line” methods analogues with the Méthodes of Monte Carlo by Chaînes of Markov which they is methods “out-line” (thus a posteriori) and often similar to the methods of sampling of importance.

If they are conceived correctly, the particulate filters can be faster than the Methods of Monte Carlo by Chaînes of Markov. They often constitute an alternative to the filters of wide $kalmans with the advantage that with sufficient samples, they approach estimated optimal Bayésien. They can thus be made more precise than the filters of $kalman. The approaches can also be combined by using a Filtre of $kalman like a proposal for a distribution for the particulate filter .

Objective

The purpose of a particulate filter is to estimate the sequence of hidden parameters, $x\_k$ for $k\; =\; 0,1,2,3,\; \backslash \; cdots$, by basing only on the data observed $y\_k$ for $k\; =\; 0,1,2,3,\; \backslash \; cdots$. All the estimated parameters bayésiens of $x\_k$ come from the distribution a posteriori, but rather than to use the probability united a posteriori $p\; (x\_0,\; x\_1,\; \backslash \; dowries,\; x\_k\; |\; y\_0,\; y\_1,\; \backslash \; dowries,\; y\_k)$, which would result from one MCMC usual or a sampling of importance, the methods particulate estimate the distribution of filtering $p\; (x\_k|y\_0,\; y\_1,\; \backslash \; dowries,\; y\_k)$.

Modeling

The particulate filters make the assumption that the states $x\_k$ and the observations $y\_k$ can be modelled in the following form:

• the continuation of the parameters $x\_0,\; x\_1,\; \backslash \; cdots$ form a Chaîne of Markov of first order, such as $x\_k\; |\; x\_\{k-1\}\; \backslash sim\; p\_\{x\_k\; |\; x\_\; \{k-1\}\}\; (X\; |\; x\_\; \{k-1\})$ and with an initial distribution $p\; (x\_0)$.
• the observations $y\_0,\; y\_1,\; \backslash \; cdots$ are independent conditionally provided the $x\_0,\; x\_1,\; \backslash \; cdots$ are known. In other words, each observation $y\_k$ depends only on the parameter $x\_k$

$\{y\_k\; |\; x\_k\}\; \backslash \; sim\; \{p\_\; \{there\; |\; X\}\; (there\; |\; x\_k)\}$

An example of this scenario is $\backslash \; \{\backslash \; begin\; \{matrix\}\; x\_k=f\; (x\_\; \{k-1\})\; +\; v\_k\; \backslash \; \backslash \; y\_k\; =\; H\; (x\_k)\; +\; w\_k\; \backslash \; end\; \{matrix\}$

where at the same time $v\_k$ and $w\_k$ are sequences mutually independent and distributed to identical with functions of density of known probability and where $f\; ()$ and $h\; ()$ are known functions. These two equations can be seen like equations of the Espace of state and resemble those of the Filtre of $kalman.

If the functions $f\; (\backslash \; cdot)$ and $h\; (\backslash \; cdot)$ were linear, and so at the same time $v\_k$ and $w\_k$ were Gaussian , then the Filtre of $kalman finds the distribution of filtering Bayésien exact. In the contrary case, the methods containing filter of $kalman give an estimate of first order. The particulate filters also give approximations, but with sufficient particles, the results can be even more precise.

Approximation of Monte Carlo

The methods with particles, like all the methods containing samplings (such as the MCMC ), generate a whole of samples which approximate the distribution of filtering $p\; (x\_k|y\_0,\; \backslash \; dowries,\; y\_k)$. Thus, with $P$ samples, the values hoped with respect to the distribution of filtering are approximated by: $\backslash \; int\; F\; (x\_k)\; p\; (x\_k|y\_0,\; \backslash \; dowries,\; y\_k)\; dx\_k\; \backslash \; approx\; \backslash \; frac1P\; \backslash \; sum\_\; \{L=1\}\; ^Pf\; (x\_k^\; \{(L)\})$ and $f\; (\backslash \; cdot)$, in the usual way of the methods Monte Carlo, can give all the data of the distribution (moments, etc) until a certain degree of approximation.

In general, the algorithm is repeated repeatedly for a given number of values $k$ (which we will note $N$).

To initialize $x\_k=0|\_\; \{k=0\}$ for all the particles provides a starting position to generate $x\_1$, which can be used to generate $x\_2$, who can be used to generate $x\_3$, and so on until $k=N$.

Once this carried out, the Average of the $x\_k$ on all the particles (or $\backslash \; frac\; \{1\}\; \{P\}\; \backslash \; sum\_\; \{L=1\}\; ^P\; x\_k^\; \{(L)\}$) is roughly the true value of $x\_k$.

Sampling with rééchantillonnage by importance (SIR)

Sampling with Rééchantillonnage by importance (Sampling Resampling Importance or SIR) is an algorithm of filtering used very usually. It approximates the distribution of filtering $p\; (x\_k|y\_0,\; \backslash \; ldots,\; y\_k)$ by a whole of balanced particles: $\backslash \; \{(w^\; \{(L)\}\_k,\; x^\; \{(L)\}\_k)\; ~:\; ~L=1,\; \backslash \; ldots,\; P\; \backslash \}$.

The weight of importance $w^\; \{(L)\}\_k$ is approximations of the probabilities (or densities) a posteriori relative of the particles such as $\backslash \; sum\_\; \{L=1\}\; ^P\; w^\; \{(L)\}\_k\; =\; 1$.

Algorithm SIR is a recursive version of the sampling by importance. As in sampling by importance, hoped for function $f\; (\backslash \; cdot)$ can be approximated like a weighted average: $\backslash \; int\; F\; (x\_k)\; p\; (x\_k|y\_0,\; \backslash \; dowries,\; y\_k)\; dx\_k\; \backslash \; approx\; \backslash \; sum\_\; \{L=1\}\; ^P\; w^\; \{(L)\}\; F\; (x\_k^\; \{(L)\}).$

The performance of the algorithm is dependant on the choice of the distributions of importances : $\backslash \; pi\; (x\_k|x\_\; \{0:\; k-1\},\; y\_\; \{0:\; K\})$.

The distribution of optimal importance is given like: $\backslash \; pi\; (x\_k|x\_\; \{0:\; k-1\},\; y\_\; \{0:\; K\})\; =\; p\; (x\_k|x\_\; \{k-1\},\; y\_\; \{K\}).$

However, the probability of transition is often used like function of importance, as it is easier to calculate, and that also simplifies calculations of the subsequent weights of importance: $\backslash \; pi\; (x\_k|x\_\; \{0:\; k-1\},\; y\_\; \{0:\; K\})\; =\; p\; (x\_k|x\_\; \{k-1\}).$

The rééchantillonnage filters by importance (SIR) with probabilities of transitions as function from importance are known commonly like filters starting (Bootstrap filters) or Algorithme of condensation.

The rééchantillonnage makes it possible to avoid the problem of the degeneration of the algorithm. One avoids the situations thus where all the weights of importance except one are close to zero. The performance of the algorithm can also be affected by the choice of the suitable method of rééchantillonnage. The rééchantillonnage laminated proposed by Kitagawa (1996) is optimal in terms of variance.

Only one sequential step of rééchantillonnage of importance proceeds in the following way:

1. For $L=1,\; \backslash \; ldots,\; P$, one draws the samples from the distributions of importances : $x^\; \{(L)\}\_k\; \backslash \; sim\; \backslash \; pi\; (x\_k|x^\; \{(L)\}\_\; \{0:\; k-1\},\; y\_\; \{0:\; K\})$
2. For $L=1,\; \backslash \; ldots,\; P$, one evaluates the weights of importance with a constant of standardization: $$

\ hat {W} ^ {(L)}_k = w^ {(L)}_ {k-1} \ frac {p (y_k|x^ {(L)}_k) p (x^ {(L)}_k|x^ {(L)}_ {k-1})} {\ pi (x_k^ {(L)}|x^ {(L)}_ {0: k-1}, y_ {0: K})}.

1. For $L=1,\; \backslash \; ldots,\; P$ one calculates the standardized weights of importance: $$

w^ {(L)}_k = \ frac {\ hat {W} ^ {(L)}_k} {\ sum_ {J=1} ^P \ hat {W} ^ {(J)}_k}

1. One calculates an estimate of the effective number of particles like $$

\ hat {NR} _ \ mathit {EFF} = \ frac {1} {\ sum_ {L=1} ^P \ left (w^ {(L)}_k \ right) ^2}

1. If the effective number of particles is smaller than a threshold given , then one carries out the rééchantillonnage:
   1. To draw $P$ particles from the whole of particles running with the probabilities proportional to their weight then to replace the whole of the current particles with this new unit.
   2. For $L=1,\; \backslash \; ldots,\; P$ the unit $w^\; \{(L)\}\_k\; =\; 1/P$.

The term sequential Rééchantillonnage of importance (Sequential Resampling Importance) is also used sometimes to refer to filters SIR.

Sequential sampling by importance (LOCATED)

Sampling sequential by importance (or LOCATED for Sequential Sampling Importance) is similar to Sampling with rééchantillonage by importance (SIR) but without the stage of rééchantillonnage.

Direct version of the algorithm

The direct version of the algorithm is relatively simple compared to other algorithm of particulate filtering and uses the composition and the rejection. To generate a simple sample $x$ with $k$ of $p\_\; \{x\_k|y\_\; \{1:\; K\}\}\; (X|y\_\; \{1:\; K\})$:

(1) To fix p=1

(2) To uniformly generate L since $\backslash \; \{1,\dots ,\; P\; \backslash \}$

(3) To generate a test $\backslash \; hat\; \{X\}$ since its distribution $p\_\; \{x\_k|x\_\; \{k-1\}\}\; (X|x\_\; \{k-1|k-1\}\; ^\; \{(L)\})$

(4) To generate the probabilities of $\backslash \; hat\; \{there\}$ by using $\backslash \; hat\; \{X\}$ since $p\_\; \{there|X\}\; (y\_k|\backslash \; hat\; \{X\})$ where $y\_k$ is the value measured

(5) Générer another uniformly U since $m\_k$

(6) Comparer U and $\backslash \; hat\; \{there\}$

(A) If U is larger then to repeat since the stage (2)

(b) If U is smaller then to save $\backslash \; hat\; \{X\}$ like $x\; \{K|K\}\; ^\; \{(p)\}$ and to increment p

(c) If p > P then to stop

The objective is to generate P particles with the step $k$ by using only the particles of the step $k-1$. That requires that a Markovian equation can be written (and calculated) to generate a $x\_k$ while being based only on $x\_\; \{k-1\}$. This algorithm uses the composition of P particles since $k-1$ to generate with $k$.

That can be more easily visualized if $x$ is seen like a two-dimensional board. A dimension is $k$ and other dimension corresponds to the number of particles. For example, $x\; (K,\; L)$ would be L^ème particle at the stage $k$ and can be thus written $x\_k^\; \{(L)\}$ (as carried out higher in the algorithm).

The stage (3) generates a potential $x\_k$ based on a particle chosen by chance ($x\_\; \{k-1\}\; ^\; \{(L)\}$) has time $k-1$ and rejects or accepts this particle at the stage (6). In other words, the $x\_k$ values are generated by using the $x\_\; \{k-1\}$ generated previously.

See too

• Filter of $kalman, an analytical estimator for the Gaussian distributions
• recursive Estimate bayésienne

Refer

• Sequential Monte Carlo Methods in Practice , by has Doucet, NR of Freitas and NR Gordon. Published by Springer.
• One Sequential Monte Carlo Sampling Methods for Bayesian Filtering , by has Doucet, C Andrieu and S. Godsill, Statistics and Computing, vol. 10, No 3, pp. 197-208, 2000 CiteSeer link
• Tutorial one Particle Filters for On-line Nonlinear/Non-Gaussian Bayesian Tracking (2001) ; S. Arulampalam, S. Maskell, NR. Gordon and T. Clapp; CiteSeer link
• F. Dellaert, D. Fox, W. Burgard, and S. Thrun, " Mobile Monte Carlo Localization for Robots, " International IEEE Conference one Robotics and Automation (ICRA99), May, 1999.

External bonds

• Sequential Methods of Monte Carlo (Particulate Filtering) at the University of Cambridge
• Animations MCL of Dieter Fox

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