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C. Greco and M. Vasile
performed, numerous alternatives have been studied in the literature. The Bootstrap
filter performs a resampling step after each observation update, which substitutes
the particles with a new set of particles according to the current discrete density
in Eq. (6.86) and re-initialises the weights to w
(i)
k
= 1/N . Alternatively, the
resampling step can be performed after n weight updates. Another alternative
requires the introduction of a check step in which the variance of the weights is
assessed, leading to the so-called adaptive resampling. If this variance becomes too
high, or equivalently its inverse too low, the particles are resampled [39]. In general,
this approach helps to better distribute the samples in the zones where the weights
are higher, and therefore the resampling step is always present in particle filters [51].
However, along with its advantages, the resampling step introduces an undesired
phenomenon called sample impoverishment [49]. Indeed, the resampling technique
replicates, possibly several times, particles associated with high weights after a
filter iteration. Then, these samples are propagated via the importance distribution,
and they are supposed to diversify as a result of the process noise. However, if
the process noise is small, the same initial particles will end up to be close after
propagation. Eventually, in degenerate cases, all the particles will collapse to the
same point [55]. Several techniques exist to mitigate this issue: roughening, which
adds random noise to the particle just after the resampling process [20, 24]; prior
editing, using roughening on the prior samples with small weights [24, 55]; regularized particle filtering, which performs resampling from a continuous approximated
auxiliary density function [20, 49]; Markov chain Monte Carlo resampling [22, 49]
and auxiliary particle filtering [46].
The particle filter suffers from the so-called curse of dimensionality, as several
studies have shown that the number of particles needed for a successful filtering
process scales exponentially with the state dimension [5, 57, 62]. When some
components of the state vector follow a linear evolution and they are Gaussian,
while the others are non-Gaussian, the computational burden can be reduced by
evaluating part of the filtering equations analytically, while the rest still require
sampling techniques [26, 51]. The resulting algorithm is called Rao-Blackwellized
particle filter [1, 11, 45].
Although relatively recent, there exists an extensive literature on particle filters,
its variants and associated heuristics. Mainly, this topic can be found in the literature
focused on sequential Monte Carlo methods in general [20, 38], or on filtering
techniques in particular [10, 49, 55].
6.4 Conclusions
State estimation theory is of crucial importance for a great variety of fields. In
this framework, the time-varying state of a hidden dynamical system is sought by
combining uncertain evolution knowledge with noisy observations.
C. Greco and M. Vasile
performed, numerous alternatives have been studied in the literature. The Bootstrap
filter performs a resampling step after each observation update, which substitutes
the particles with a new set of particles according to the current discrete density
in Eq. (6.86) and re-initialises the weights to w
(i)
k
= 1/N . Alternatively, the
resampling step can be performed after n weight updates. Another alternative
requires the introduction of a check step in which the variance of the weights is
assessed, leading to the so-called adaptive resampling. If this variance becomes too
high, or equivalently its inverse too low, the particles are resampled [39]. In general,
this approach helps to better distribute the samples in the zones where the weights
are higher, and therefore the resampling step is always present in particle filters [51].
However, along with its advantages, the resampling step introduces an undesired
phenomenon called sample impoverishment [49]. Indeed, the resampling technique
replicates, possibly several times, particles associated with high weights after a
filter iteration. Then, these samples are propagated via the importance distribution,
and they are supposed to diversify as a result of the process noise. However, if
the process noise is small, the same initial particles will end up to be close after
propagation. Eventually, in degenerate cases, all the particles will collapse to the
same point [55]. Several techniques exist to mitigate this issue: roughening, which
adds random noise to the particle just after the resampling process [20, 24]; prior
editing, using roughening on the prior samples with small weights [24, 55]; regularized particle filtering, which performs resampling from a continuous approximated
auxiliary density function [20, 49]; Markov chain Monte Carlo resampling [22, 49]
and auxiliary particle filtering [46].
The particle filter suffers from the so-called curse of dimensionality, as several
studies have shown that the number of particles needed for a successful filtering
process scales exponentially with the state dimension [5, 57, 62]. When some
components of the state vector follow a linear evolution and they are Gaussian,
while the others are non-Gaussian, the computational burden can be reduced by
evaluating part of the filtering equations analytically, while the rest still require
sampling techniques [26, 51]. The resulting algorithm is called Rao-Blackwellized
particle filter [1, 11, 45].
Although relatively recent, there exists an extensive literature on particle filters,
its variants and associated heuristics. Mainly, this topic can be found in the literature
focused on sequential Monte Carlo methods in general [20, 38], or on filtering
techniques in particular [10, 49, 55].
6.4 Conclusions
State estimation theory is of crucial importance for a great variety of fields. In
this framework, the time-varying state of a hidden dynamical system is sought by
combining uncertain evolution knowledge with noisy observations.