PARTICLE FILTERING - Adaptive Signal Processing: Next Generation Solutions - page 277

where / signifies proportionality. The first factor on the right of the proportionality
sign is the likelihood function of the unknown state, and the second factor is the
predictive density of the state. For the predictive density we have

ð f (x( n ) j x( n 1)) f (x( n 1) j y(1 : n 1)) d x( n 1) :

f (x( n ) j y(1 : n 1)) ¼

(5 : 8)

In writing (5.8) we used the property of the state that given x( n 1), x( n ) does
not depend on y(1 : n 1). Now, the required recursive equation for the update of
the filtering density is obtained readily by combining (5.7) and (5.8), that is, we
formally have

f (x( n ) j y(1 : n )) / f ( y( n ) j x( n ))

ð f (x( n ) j x( n 1)) f (x( n 1) j y(1 : n 1)) d x( n 1) :

Thus, on the left of the proportionality sign we have the filtering PDF at time instant n ,
and on the right under the integral, we see the filtering PDF at time instant n 1.
There are at least two problems in carrying out the above recursion, and they may
make the recursive estimation of the filtering density very challenging. The first
one is the solving of the integral in (5.8) and obtaining the predictive density
f (x( n ) j y(1 : n 1)). In some cases it is possible to obtain the solution analytically,
which considerably simplifies the recursive algorithm and makes it more accurate.
The second problem is the combining of the likelihood and the predictive density in
order to get the updated filtering density. These problems may mean that it is imposs-
ible to express the filtering PDF in a recursive form. For instance, in Example 5.1, we
cannot obtain an analytical solution due to the nonlinearities in the observation
equation. It will be seen in the sequel that the smoothing and the predictive densities
suffer from analogous problems.
We can safely state that in many problems the recursive evaluation of the densities
of the state-space model cannot be done analytically, and consequently we have to
resort to numerical methods. As already discussed, an important class of systems
which allows for exact analytical recursions is the one represented by linear state-
space models with Gaussian noises. These recursions are known as Kalman filtering
[3]. When analytical solutions cannot be obtained, particle filtering can be employed
with elegance and with performance characterized by high accuracy.
Particle filtering has been used in many different disciplines. They include surveil-
lance guidance, and obstacle avoidance systems, robotics, communications, speech
processing, seismic signal processing, system engineering, computer vision, and
econometrics. During the past decade, in practically all of these fields, the number
of contributions has simply exploded. To give a perspective of how the subject has
been vibrant with activities, we provide a few examples of developments in target
tracking, positioning, and navigation.