Implicit sampling for particle filters Alexandre Chorin and Xuemin Tu

26 Pages·2009·0.17 MB·English

Implicit sampling for particle filters
Alexandre Chorin and Xuemin Tu
University of California at Berkeley
2/26
Example:
Try to rescue people in a boat in the middle of the ocean from
approximate knowledge of their initial position and from uncer-
tain observations of a later position.
Equations of motion : dx = f(x, ω)dt + gdW, or, in a discrete
approximation, x = x + δF (x ) + GdW,
n+1 n n
n n
Observations: b = b(nδ) = g(x ) + V , V random.
3/26
Special case: equations linear, pdf Gaussian, Kalman filter.
⇒
Extensions: extended Kalman filter, ensemble Kalman filter, try
to fit a non-Gaussian situation into a Gaussian framework.
4/26
Simple particle filter:
Follow a bunch of ”particles” (samples, replicas) whose empir-
ical density at time t = nδ approximates a pdf P of a set of
n
systems moved by the equations of motion constrained by the
observations.
Given P :
n
First evolve the particles by the equations of motion alone: (gen-
erates a “prior” density).
Take the observations into account by weighting the particles.
(generates a “posterior” density).
5/26
To avoid following irrelevant particles, resample, so that you have
again a bunch of particles with equal weights. (for θ [0, 1],
k
∼
n+1 n+1
pick xˆ = x such that
1 i 1 1 i
A W <θ < A W , where A = W .
− 1− j k − 1 j j
! ! !
Final (important) step: go back, smooth the past, and resample.
6/26
Bayes theorem:
n+1 n n+1 n+1
n+1 n n+1 P(x x )P(b x )
P(x x , b ) =
| |
| P(bn+1 bn)
|
7/26
Fails, in particular when there are many variables: (example from
Bickel at al.):
Model: x N(0, 1) Observation: b = x + N(0, 1)
∼
dim(x) = 100, 1000 particles, 1000 runs for max weight distri-
butions
300
250
200
150
100
50
0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
8/26
Partial remedy: better choice of prior in linear problems (special
case of what we offer).
Our remedy: sampling by interpolation and iteration, special case
of chainless sampling.
9/26
Simple example of sampling by interpolation and iteration: Non-
linear Brownian bridge.
dx = f(x, t)dt + √βdw, x(0) = 0, x(1) = X.
N/2
X
0 1
n+1 n n n n+1 n n n n+1
Discretize: x = x +f(x , t )δ+(x x )f (x , t )δ+W ,
%
−
n n ∂f n n
where f (x , t ) = (x , t )
% ∂xn
1 N 1
The joint probability density of the variables x , . . . , x is
−
1 N 1
Z exp( V ), where Z is the normalization constant and
− 0 − n
−
! 2
n+1 n
(1 δf )(x x ) δf
%
− − −
V =
n
" #
2βδ
2
n+1 n
x x δf/(1 δf )
%
− − −
= ,
" 2β #
n
n n 2
where f, f are functions of the x , t , and β = βδ/(1 δf ) .
% n %
−