6 Fundamentals of Filtering
195
p(y k |x k ) ∼ N y k (H (t k )x k , R k ) ,
(6.36)
where R k is the covariance of the measurement noise.
If the prior density function is Gaussian,
x 0 ∼ N (ˆ x 0 , P 0 ) ,
(6.37)
all the densities in the update step via Bayes’s rule (see Eq. (6.14)) are Gaussian as
well. Therefore, the first two moments’ evolutions between observations, and at an
observation update, completely characterise the distributions.
6.2.2 Nonlinear Transformation
In the majority of applications, the filtering model involves nonlinear dynamical
equations and observation relationships. The major drawback for filtering is that
when a Gaussian density is plugged in a nonlinear relationship, it loses its
Gaussianity. In general, for x a random variable with p x (x), the random variable
z = g(x) has density function [29]:
p z (z) = p x
g
−1 (z)
det
∂g −1 (z)
∂z
,
(6.38)
for invertible g. Generally, it is not possible to solve directly for this non-Gaussian
distribution. Often, numerical filter techniques rely on the Gaussian approximation
of this density to simplify the filtering computation. As a Gaussian distribution
is completely defined by its mean and covariance, a variety of methods exist to
compute directly these first two moments of the derived distribution. In this section,
the relation g(x) indicates an arbitrary function, which can represent both the
observation model and the discrete dynamical transition step. In the latter case, this
can be a state transition operator or the result of a numerical integration scheme.
This section will first present methods based on Taylor’s expansion of the nonlinear transformation, in Sect. 6.2.2.1. Then, methods based on sample propagation
will be shown, specifically with deterministic sampling in Sect. 6.2.2.2 and with
random sampling in Sect. 6.2.2.3.
6.2.2.1 Taylor Expansion
The nonlinear transformation g can be expanded in Taylor’s series about the
expected value ˆ
x = E{x}:
195
p(y k |x k ) ∼ N y k (H (t k )x k , R k ) ,
(6.36)
where R k is the covariance of the measurement noise.
If the prior density function is Gaussian,
x 0 ∼ N (ˆ x 0 , P 0 ) ,
(6.37)
all the densities in the update step via Bayes’s rule (see Eq. (6.14)) are Gaussian as
well. Therefore, the first two moments’ evolutions between observations, and at an
observation update, completely characterise the distributions.
6.2.2 Nonlinear Transformation
In the majority of applications, the filtering model involves nonlinear dynamical
equations and observation relationships. The major drawback for filtering is that
when a Gaussian density is plugged in a nonlinear relationship, it loses its
Gaussianity. In general, for x a random variable with p x (x), the random variable
z = g(x) has density function [29]:
p z (z) = p x
g
−1 (z)
det
∂g −1 (z)
∂z
,
(6.38)
for invertible g. Generally, it is not possible to solve directly for this non-Gaussian
distribution. Often, numerical filter techniques rely on the Gaussian approximation
of this density to simplify the filtering computation. As a Gaussian distribution
is completely defined by its mean and covariance, a variety of methods exist to
compute directly these first two moments of the derived distribution. In this section,
the relation g(x) indicates an arbitrary function, which can represent both the
observation model and the discrete dynamical transition step. In the latter case, this
can be a state transition operator or the result of a numerical integration scheme.
This section will first present methods based on Taylor’s expansion of the nonlinear transformation, in Sect. 6.2.2.1. Then, methods based on sample propagation
will be shown, specifically with deterministic sampling in Sect. 6.2.2.2 and with
random sampling in Sect. 6.2.2.3.
6.2.2.1 Taylor Expansion
The nonlinear transformation g can be expanded in Taylor’s series about the
expected value ˆ
x = E{x}: