6 Fundamentals of Filtering
217
distribution. Obviously, good properties are the simplicity to draw samples from it
and the ease to evaluate the probability density associated to a particle.
The Bootstrap filter is a particle filter which employs the transitional density as
importance distribution [24]:
π = p(x k |x k−1 ) .
(6.90)
This particular choice leads to the simplification of the weight update equation to:
w
(i)
k = w
(i)
k−1 p
y k |x
(i)
k
.
(6.91)
The resulting algorithm, the first particle filter ever, is simple, intuitive and modular.
Indeed, the samples are simple to draw from the transitional density, and the
weight update requires the evaluation of the observation’s conditional density,
given by problem formulation. On the other hand, it draws samples according to
the dynamical information only. Hence, when there is little overlap between the
predicted and the observation distributions, most of the particles will be associated
to small importance weights, and the posterior distribution’s approximation will
be dominated by a very limited number of particles with large weights [48].
Since each particle requires the same amount of computational load, this filter
implementation is often inefficient as it requires a high number of particles for
accurate approximations.
It is clear how a trade-off between the resemblance of the importance distribution
to the true posterior and the computational efficiency is the key of particle filters.
Plenty of research has been, and still is, focused on the selection of optimal importance distributions. As a general rule, it is advantageous to retain the conditionality
on the last measurement [48].
One alternative which minimises the variance of the importance weights is [19]:
π = p(x k |x k−1 , y k ) .
(6.92)
This importance density leads to the weight update equation:
w
(i)
k−1 = w
(i)
k−1 p
y k |x
(i)
k−1
.
(6.93)
However, both the equations cannot be directly used. When this is the case, local
linearisation techniques, e.g. EKF or UKF, can be employed to create suitable
importance distribution [19, 51, 61].
As introduced in Sect. 6.2.2.3, and discussed for the Bootstrap filter, one issue
often encountered is the weight degeneracy as a result of sampling from an
inappropriate importance distribution. Therefore, another way to improve computational efficiency is to introduce resampling techniques, which remove low-weighted
particles and replace them with duplicates of the high-weighted samples. As it
is often a matter of heuristics when and how this resampling step should be
217
distribution. Obviously, good properties are the simplicity to draw samples from it
and the ease to evaluate the probability density associated to a particle.
The Bootstrap filter is a particle filter which employs the transitional density as
importance distribution [24]:
π = p(x k |x k−1 ) .
(6.90)
This particular choice leads to the simplification of the weight update equation to:
w
(i)
k = w
(i)
k−1 p
y k |x
(i)
k
.
(6.91)
The resulting algorithm, the first particle filter ever, is simple, intuitive and modular.
Indeed, the samples are simple to draw from the transitional density, and the
weight update requires the evaluation of the observation’s conditional density,
given by problem formulation. On the other hand, it draws samples according to
the dynamical information only. Hence, when there is little overlap between the
predicted and the observation distributions, most of the particles will be associated
to small importance weights, and the posterior distribution’s approximation will
be dominated by a very limited number of particles with large weights [48].
Since each particle requires the same amount of computational load, this filter
implementation is often inefficient as it requires a high number of particles for
accurate approximations.
It is clear how a trade-off between the resemblance of the importance distribution
to the true posterior and the computational efficiency is the key of particle filters.
Plenty of research has been, and still is, focused on the selection of optimal importance distributions. As a general rule, it is advantageous to retain the conditionality
on the last measurement [48].
One alternative which minimises the variance of the importance weights is [19]:
π = p(x k |x k−1 , y k ) .
(6.92)
This importance density leads to the weight update equation:
w
(i)
k−1 = w
(i)
k−1 p
y k |x
(i)
k−1
.
(6.93)
However, both the equations cannot be directly used. When this is the case, local
linearisation techniques, e.g. EKF or UKF, can be employed to create suitable
importance distribution [19, 51, 61].
As introduced in Sect. 6.2.2.3, and discussed for the Bootstrap filter, one issue
often encountered is the weight degeneracy as a result of sampling from an
inappropriate importance distribution. Therefore, another way to improve computational efficiency is to introduce resampling techniques, which remove low-weighted
particles and replace them with duplicates of the high-weighted samples. As it
is often a matter of heuristics when and how this resampling step should be