6 Fundamentals of Filtering
199
sigma points of this variant faithfully capture the mean and covariance, while the
transformed probability distribution reconstructed from the propagated samples has
exact mean for polynomial g(x) up to degree three and exact covariance for g(x)
linear [51].
The main advantage of the unscented transformation is that it does not require
differentiability, or derivative information, of the nonlinear mapping g but only
the propagation of a limited number of deterministic samples. It is worth stressing
that the Gaussianity approximation on the prior and updated distributions is not a
required assumption of the unscented transformation. Nonetheless, the majority of
practical filters employ this transformation with Gaussian distributions only, mainly
because of the simplification in the Bayes’ step.
6.2.2.3 Monte Carlo Methods
Another large family of techniques to approximate the posterior density is Monte
Carlo methods. Unlike unscented transformation methods, the set of samples is
generated randomly according to a given distribution. Therefore, this method does
not require any linearity or Gaussian assumption on the model. Furthermore, unlike
deterministic methods, the number of samples required for the mean to converge is
theoretically independent of the problem’s dimensionality [38]. With the propagated
samples, the moments are estimated as [27]:
ˆ
z ≈
1
N
N
i=1
z i =
1
N
N
i=1
g(x i )
(6.48)
P z ≈
1
N − 1
N
i=1
(z i − ˆ
z)(z i − ˆ
z)
T .
(6.49)
In this conventional Monte Carlo, it is evident to infer how crucial it is to properly
select the random samples x i in accordance to the original probability distribution
p(x). This is numerically straightforward when p(x) is Gaussian or belongs to any
simple distribution family. However, in Bayesian filtering, it is usually numerically
demanding to sample directly from the required density because of its complex
functional form (see Eq. (6.19)).
Markov chain Monte Carlo techniques are a class of efficient methods to generate
the random samples from a distribution p(x). The basic concept is to replace
the target density sampling by a Markov chain which has p(x) as equilibrium
distribution, and sample a realisation of this instead. In the literature, there is an
abundance of algorithms, mainly differing by the transition Kernel used for the
Markov chain. The first example is the notable Metropolis algorithm [44]. Extensive
references are provided by Gilks et al. [23] or Brooks et al. [8].
Another class of methods is importance sampling, which draws from an approximated density π(x), simpler to sample, instead of the original p(x). Then, the
199
sigma points of this variant faithfully capture the mean and covariance, while the
transformed probability distribution reconstructed from the propagated samples has
exact mean for polynomial g(x) up to degree three and exact covariance for g(x)
linear [51].
The main advantage of the unscented transformation is that it does not require
differentiability, or derivative information, of the nonlinear mapping g but only
the propagation of a limited number of deterministic samples. It is worth stressing
that the Gaussianity approximation on the prior and updated distributions is not a
required assumption of the unscented transformation. Nonetheless, the majority of
practical filters employ this transformation with Gaussian distributions only, mainly
because of the simplification in the Bayes’ step.
6.2.2.3 Monte Carlo Methods
Another large family of techniques to approximate the posterior density is Monte
Carlo methods. Unlike unscented transformation methods, the set of samples is
generated randomly according to a given distribution. Therefore, this method does
not require any linearity or Gaussian assumption on the model. Furthermore, unlike
deterministic methods, the number of samples required for the mean to converge is
theoretically independent of the problem’s dimensionality [38]. With the propagated
samples, the moments are estimated as [27]:
ˆ
z ≈
1
N
N
i=1
z i =
1
N
N
i=1
g(x i )
(6.48)
P z ≈
1
N − 1
N
i=1
(z i − ˆ
z)(z i − ˆ
z)
T .
(6.49)
In this conventional Monte Carlo, it is evident to infer how crucial it is to properly
select the random samples x i in accordance to the original probability distribution
p(x). This is numerically straightforward when p(x) is Gaussian or belongs to any
simple distribution family. However, in Bayesian filtering, it is usually numerically
demanding to sample directly from the required density because of its complex
functional form (see Eq. (6.19)).
Markov chain Monte Carlo techniques are a class of efficient methods to generate
the random samples from a distribution p(x). The basic concept is to replace
the target density sampling by a Markov chain which has p(x) as equilibrium
distribution, and sample a realisation of this instead. In the literature, there is an
abundance of algorithms, mainly differing by the transition Kernel used for the
Markov chain. The first example is the notable Metropolis algorithm [44]. Extensive
references are provided by Gilks et al. [23] or Brooks et al. [8].
Another class of methods is importance sampling, which draws from an approximated density π(x), simpler to sample, instead of the original p(x). Then, the