6 Fundamentals of Filtering
219
This chapter introduced both the fundamental concepts of state estimation in
general, and filtering theory in particular, through its probabilistic development, and
practical techniques for computing its solution.
In Sect. 6.1, the general mathematical statement was introduced with the building
blocks which are necessary for the filtering problem discussed in this chapter:
time-continuous dynamical equations, an observation model and a known initial
distribution of the state. Within the class of state estimation, the main focus was
filtering theory, which aims at computing the state distribution at the time of the
last received observation. Hence, filtering is appropriate for real-time applications.
The inference step needed to combine dynamical and measurement information
was solved by Bayes’ rule. Its application to the current setting was presented,
and two mathematically equivalent update rules were derived. One of them suits
a sequential scheme convenient for real-time applications (sequential filtering),
while the other processes a whole set of observations at once (batch processor).
As the chapter focused on the former, key importance was given on analytical and
numerical techniques to compute the corresponding update step. The section ended
with a discussion on which estimate should be used as representative of the state
probability conditional distribution. A general method to compute optimal statistical
estimators via loss functions was presented. Among the alternatives, specific loss
functions allow to select the conditional distribution mean, mode, median, etc.
For any filtering algorithm, one necessary step is to be able to describe, or
approximate, how the state distribution evolves through the dynamical equations
and measurement model. In Sect. 6.2, the methods for propagating probability
distributions through a transformation were presented. Initially, the exact solution
for linear dynamics, linear observation model and zero-mean Gaussian noises was
presented: if the input random variable is normally distributed, the transformed
variable is still Gaussian with mean and variance analytically computed. Moreover,
the case of a generic nonlinear function was considered and methods to approximate
the transformed distribution introduced. Specifically, the Taylor expansion and the
unscented transform approximate, respectively, the transformation and the posterior
to have a final normal distribution, whereas Monte Carlo methods are able to
describe generic posteriors using sampling-based discrete distributions.
Lastly, in Sect. 6.3, practical algorithms to compute or approximate the filtering
solution were derived, described and schematised. Specifically, when linear dynamics, linear observation model and Gaussian prior and noises are considered, the
Kalman filter is the closed-form solution of the filtering problem. However, most
state estimation problems involve nonlinear transformations, and a general analytical solution is not available. When the Gaussian assumption (or approximation)
is retained, a family of methods exists to obtain a normal posterior distribution.
In detail, the Taylor expansion approximation results in the extended Kalman
filter, while the unscented transform is the basis for the unscented Kalman filter.
To conclude the family of Gaussian filtering methods, the general framework of
219
This chapter introduced both the fundamental concepts of state estimation in
general, and filtering theory in particular, through its probabilistic development, and
practical techniques for computing its solution.
In Sect. 6.1, the general mathematical statement was introduced with the building
blocks which are necessary for the filtering problem discussed in this chapter:
time-continuous dynamical equations, an observation model and a known initial
distribution of the state. Within the class of state estimation, the main focus was
filtering theory, which aims at computing the state distribution at the time of the
last received observation. Hence, filtering is appropriate for real-time applications.
The inference step needed to combine dynamical and measurement information
was solved by Bayes’ rule. Its application to the current setting was presented,
and two mathematically equivalent update rules were derived. One of them suits
a sequential scheme convenient for real-time applications (sequential filtering),
while the other processes a whole set of observations at once (batch processor).
As the chapter focused on the former, key importance was given on analytical and
numerical techniques to compute the corresponding update step. The section ended
with a discussion on which estimate should be used as representative of the state
probability conditional distribution. A general method to compute optimal statistical
estimators via loss functions was presented. Among the alternatives, specific loss
functions allow to select the conditional distribution mean, mode, median, etc.
For any filtering algorithm, one necessary step is to be able to describe, or
approximate, how the state distribution evolves through the dynamical equations
and measurement model. In Sect. 6.2, the methods for propagating probability
distributions through a transformation were presented. Initially, the exact solution
for linear dynamics, linear observation model and zero-mean Gaussian noises was
presented: if the input random variable is normally distributed, the transformed
variable is still Gaussian with mean and variance analytically computed. Moreover,
the case of a generic nonlinear function was considered and methods to approximate
the transformed distribution introduced. Specifically, the Taylor expansion and the
unscented transform approximate, respectively, the transformation and the posterior
to have a final normal distribution, whereas Monte Carlo methods are able to
describe generic posteriors using sampling-based discrete distributions.
Lastly, in Sect. 6.3, practical algorithms to compute or approximate the filtering
solution were derived, described and schematised. Specifically, when linear dynamics, linear observation model and Gaussian prior and noises are considered, the
Kalman filter is the closed-form solution of the filtering problem. However, most
state estimation problems involve nonlinear transformations, and a general analytical solution is not available. When the Gaussian assumption (or approximation)
is retained, a family of methods exists to obtain a normal posterior distribution.
In detail, the Taylor expansion approximation results in the extended Kalman
filter, while the unscented transform is the basis for the unscented Kalman filter.
To conclude the family of Gaussian filtering methods, the general framework of