6 Fundamentals of Filtering
215
solved analytically. In addition, it can be shown that Eqs. (6.79)–(6.80), used in
the UKF derivation, are the result of a Gauss-Hermite cubature rule application to
Eq. (6.84) [28].
Recently, numerous novel filtering techniques have arisen from this general
moment matching formulation, as it allows to use any approximation rule for the
integral computation. Among the deterministic methods, Gauss-Hermite quadrature
and spherical cubature are efficient schemes [2, 28, 51, 65]. Also nondeterministic
methods can be used, such as the often employed Monte Carlo family sampling
techniques.
6.3.5 Particle Filter
The nonlinear model in Eq. (6.73), or its non-additive noise counterpart, can be
used to compute the conditional probabilities p(x k |x k−1 ) and p(y k |x k ), whose
distribution depends on the assumed probability density function of the process
and observation noises. Therefore, these quantities are assumed to be given in the
problem formulation tackled in this section [51].
The particle filter is a state estimation technique based on sequential Monte Carlo
methods (see Sect. 6.2.2.3). As the name implies, it relies on a set of weighted
random particles to approximate the posterior distribution [3, 24]:
p(x k |y 1:k ) ≈
i
w
(i)
k δ
x k − x
(i)
k
,
(6.86)
where δ(·) is the Dirac delta function. From this distribution, the expectation of a
generic function, and therefore its moments, can be computed by the weighted sum
[51]:
E{g(x k )|y 1:k } ≈
i
w
(i)
k g
x
(i)
k
.
(6.87)
The sample and weights are computed using the sequential importance sampling
technique adapted to account for the observations. With the help of the importance
distribution π , the weights at step k are defined by:
215
solved analytically. In addition, it can be shown that Eqs. (6.79)–(6.80), used in
the UKF derivation, are the result of a Gauss-Hermite cubature rule application to
Eq. (6.84) [28].
Recently, numerous novel filtering techniques have arisen from this general
moment matching formulation, as it allows to use any approximation rule for the
integral computation. Among the deterministic methods, Gauss-Hermite quadrature
and spherical cubature are efficient schemes [2, 28, 51, 65]. Also nondeterministic
methods can be used, such as the often employed Monte Carlo family sampling
techniques.
6.3.5 Particle Filter
The nonlinear model in Eq. (6.73), or its non-additive noise counterpart, can be
used to compute the conditional probabilities p(x k |x k−1 ) and p(y k |x k ), whose
distribution depends on the assumed probability density function of the process
and observation noises. Therefore, these quantities are assumed to be given in the
problem formulation tackled in this section [51].
The particle filter is a state estimation technique based on sequential Monte Carlo
methods (see Sect. 6.2.2.3). As the name implies, it relies on a set of weighted
random particles to approximate the posterior distribution [3, 24]:
p(x k |y 1:k ) ≈
i
w
(i)
k δ
x k − x
(i)
k
,
(6.86)
where δ(·) is the Dirac delta function. From this distribution, the expectation of a
generic function, and therefore its moments, can be computed by the weighted sum
[51]:
E{g(x k )|y 1:k } ≈
i
w
(i)
k g
x
(i)
k
.
(6.87)
The sample and weights are computed using the sequential importance sampling
technique adapted to account for the observations. With the help of the importance
distribution π , the weights at step k are defined by: