In this appendix, we describe succinctly two algorithms, the Kalman lter and the particle lter (henceforth KF and PF), as specic implementations of Bayesian lters.
A.1 The Kalman lter
If one assumes the tracking problem to be linear and Gaussian, one may prove that the optimal algorithm to estimate the state vector is the Kalman lter. The state space model is then given
by: ½
xk=ck+Fkxk−1+νk zk=dk+Hkxk+ηk
with νk ∼ N(0, Qk) and ηk ∼ N(0, Sk). Moreover, the initial distribution of the state vector is p(x0) =φ³x0,xˆ0,Pˆ0´, where φ(x, m, P) is the Gaussian pdf with argumentx, mean m and
covariance matrix P. The Bayes lter is then described by the following recursive equations: p(xk|z1:k−1) = φ ³ xk−1,ˆxk|k−1,Pˆk|k−1 ´ p(xk |z1:k) = φ³xk,ˆxk|k,Pˆk|k´ with: ˆ xk|k−1 =ck+Fkˆxk−1|k−1 ˆ Pk|k−1 =FkPˆk−1|k−1F⊤ k +Qk ˆ zk|k−1=dk+Hkˆxk|k−1 ˆ ek=zk−ˆzk|k−1 ˆ Vk=HkPˆk|k−1Hk⊤+Sk ˆ xk|k=ˆxk|k−1+Pˆk|k−1H⊤ kVˆk−1ˆek ˆ Pk|k =Pˆk|k−1−Pˆk|k−1H⊤ kVˆ −1 k HkPˆk|k−1 (A-1)
The set of equations (A-1) describes the Kalman lter algorithm. The previous quantities can be interpreted as follows:
• ˆxk|k−1 =E[xk|z1:k−1]is the estimate ofxk based on all available information until time
index k−1;
• Pˆk|k−1 is the covariance matrix of the estimatorxˆk|k−1: ˆ
Pk|k−1 =Eh¡xk−ˆxk|k−1¢ ¡xk−ˆxk|k−1¢⊤|z1:k−1i
• ˆzk|k−1=E[zk|z1:k−1]is the estimate of zk|z1:k−1;
• ˆek=zk−E[zk|z1:k−1]is the estimated tracking error ek =zk−zk|z1:k−1;
• Vˆk is the covariance matrix of the tracking error: Vˆk=E
£ eke⊤ k ¤ . We have: p(ek|z1:k−1) =φ³ek,ˆek,Vˆk´
• ˆxk|k =E[xk|z1:k]is an estimate ofxk based on all available information until time index
k;
• Finally,Pˆk|k is the covariance matrix of the estimatorxˆk|k: ˆ
Pk|k=Eh¡xk−ˆxk|k¢ ¡xk−ˆxk|k¢⊤|z1:k i
A.2 Particle lters
Particle ltering methods are techniques to implement recursive Bayesian lters using Monte- Carlo simulations. The key idea is to represent the posterior density function by a set of random samples with associated weights and to compute estimates based on these samples and weights [7, 20, 25, 26, 27, 28]. As the samples become very largeNs≫1, this Monte-Carlo approximation becomes an equivalent representation on the functional description of the posterior pdf. To clarify ideas14, let ©xik, wkiªNs
i=1 denotes a set of support points
©
xik, i= 1, . . . , Nsª and their
associated weights ©wi
k, i= 1, . . . , Ns
ª
characterizing the posterior density p(xk|z0:k). The
posterior density at timek can then be approximated as: p(xk|zk)≈
Ns
X
i=1
wkiδ¡xk−xik¢ (A-2)
We have thus a discrete weighted approximation to the true posterior distribution. One common way of choosing the weights is by way of importance sampling see for example [7, 20, 25, 28]. This principle relies on the following idea. In the general case, the probability densityp(xk|zk)
is such that it is dicult to draw samples from it. Assume for a moment thatp(x)∝π(x) is a probability density from which it is dicult to draw sample from, but for whichπ(x) is easy to evaluate. Hence, up to proportionality, so isp(x). Also, letxs∼q(x)be samples that are easily drawn from a proposal q(·), called an importance density. Then, similarly to A-2, a weighted
approximation of the densityp(·) can be obtained by using:
p(x)≈
Ns
X
i=1
wiδ¡x−xi¢
14Note that the succinct presentation given here of particle lters is adapted to our rst-order Markovian
where:
wi ∝ π
¡
xi¢ q(xi)
is the normalized weight of the i-th particle. Thus, if the samples {xi
k} were drawn from a proposal density q(xk|zk), then the weights in (A-2) are dened to be:
wi k∝ p¡xi k|zk ¢ q¡xi k |zk ¢ (A-3)
The PF sequential algorithm can thus be subsumed in the following steps. At each iteration, one has samples constituting an approximation of p¡xi
k−1|zk−1¢ and wants to approximate p¡xi
k |zk
¢
with a new set of samples. If the importance density can be chosen so as to factorize in the following way:
q(xk|zk) =q(xk|xk−1,zk)×q(xk−1|zk−1) (A-4)
then one can obtain samples {xi
k} by drawing samples from q
¡ xi
k|zk
¢
. To derive the weight update equation: p(xk|zk) = p(zk|xk,zk−1)×p(xk |zk−1) p(zk|zk−1) = p(zk|xk,zk−1)×p(xk |xk−1,zk−1) p(zk|zk−1) ×p(xk−1|zk−1) = p(zk|xk)×p(xk|xk−1) p(zk|zk−1) ×p(xk−1|zk−1) ∝ p(zk|xk)×p(xk |xk−1)×p(xk−1 |zk−1) (A-5)
By substituting (A-4) and (A-5) into (A-3), the weight equation can be derived to be:
wik∝wik−1 p¡zk|xi k ¢ ×p¡xi k|xik−1 ¢ q¡xi k|xik−1,zk ¢ (A-6)
and the posterior densityp(xk|zk) can be approximated using (A-2). We refer the reader to [7]
for a more detailed but concise exposé of the dierences between the dierent PF algorithms: sequential importance sampling (SIS), generic particle lter, sampling importance resampling (SIR), auxiliary particle lter (APF), and regularized particle lter (RPF). We provide a succinct exposé of the SIS, SIR algorithms as well as the generic particle lter's and the regularized particle lter's in Appendix C. One important feature of PF is that not one implementation is better than all the others. In dierent contexts, dierent PFs may have wildly dierent performances.