6 Fundamentals of Filtering
191
6.1.4 Batch Processor vs. Sequential Filtering
As seen in the previous section, one alternative in the inverse problem solution
is to consider a set of observations y 1:k at once instead of sequentially (see
Eq. (6.13)). This approach, called batch processor, employs the dynamical model
to map observations at different times to t k , usually by means of the state transition
matrix and linearized observation models [58]. In smoothing applications, one
advantage of this procedure is that all the available information is exploited, also
the knowledge coming from possible measurements later in time. On the other
hand, the batch processor becomes intractable when the number of observations
becomes sufficiently high, leading to a high-dimensional and highly overdetermined
inversion problem if we consider all the measurements at once. By a probabilistic
perspective, it requires the computation of a new full posterior distribution for each
instant of time t k when the state estimate is desired [51]. When a new batch of data is
available, algorithms for using the previous computed estimate as prior are available
[58].
This concept can be generalised to the case when the observation batch has
dimension 1 in the so-called sequential filtering approach, which will be the focus
of the remainder of the chapter. The state probability distribution is updated after
each new observation y k , using the previous knowledge of p(x k |y 1:k−1 ). The update
formula after a new observation is reported in Eq. (6.14). Therefore, it is directly the
pre-computed conditional state distribution at t k−1 to be mapped at t k , accordingly
propagated with the dynamical equations, rather than the observations at different
times as in the batch processor. Hence, in filtering applications, this approach
results in numerical schemes, as it will be shown in detail in Sect. 6.3, able to
employ efficient rules to compute the posterior p(x k |y 1:k ) using the only current
observation y k and the estimate at a previous time p(x k−1 |y 1:k−1 ), i.e. without
directly taking into account the set of observations y 1:k−1 and without the need to
update p(x k−1 |y 1:k−1 ) with y k . Therefore, the inversion problem dimension depends
only on the number of new observations. This characteristic dramatically alleviates
the computational burden associated to the computation of the posterior distribution,
resulting in an efficient approach for dynamic estimation problems.
6.1.5 Optimal Estimate
The posterior conditional distribution is the solution of the filtering problem
combining the a priori dynamical knowledge with the obtained measurements.
According to the selected approach, this posterior could be a joint distribution, as in
Eq. (6.4), or the marginal probability, in Eq. (6.5). In the general nonlinear case, this
solution is infinite-dimensional.
191
6.1.4 Batch Processor vs. Sequential Filtering
As seen in the previous section, one alternative in the inverse problem solution
is to consider a set of observations y 1:k at once instead of sequentially (see
Eq. (6.13)). This approach, called batch processor, employs the dynamical model
to map observations at different times to t k , usually by means of the state transition
matrix and linearized observation models [58]. In smoothing applications, one
advantage of this procedure is that all the available information is exploited, also
the knowledge coming from possible measurements later in time. On the other
hand, the batch processor becomes intractable when the number of observations
becomes sufficiently high, leading to a high-dimensional and highly overdetermined
inversion problem if we consider all the measurements at once. By a probabilistic
perspective, it requires the computation of a new full posterior distribution for each
instant of time t k when the state estimate is desired [51]. When a new batch of data is
available, algorithms for using the previous computed estimate as prior are available
[58].
This concept can be generalised to the case when the observation batch has
dimension 1 in the so-called sequential filtering approach, which will be the focus
of the remainder of the chapter. The state probability distribution is updated after
each new observation y k , using the previous knowledge of p(x k |y 1:k−1 ). The update
formula after a new observation is reported in Eq. (6.14). Therefore, it is directly the
pre-computed conditional state distribution at t k−1 to be mapped at t k , accordingly
propagated with the dynamical equations, rather than the observations at different
times as in the batch processor. Hence, in filtering applications, this approach
results in numerical schemes, as it will be shown in detail in Sect. 6.3, able to
employ efficient rules to compute the posterior p(x k |y 1:k ) using the only current
observation y k and the estimate at a previous time p(x k−1 |y 1:k−1 ), i.e. without
directly taking into account the set of observations y 1:k−1 and without the need to
update p(x k−1 |y 1:k−1 ) with y k . Therefore, the inversion problem dimension depends
only on the number of new observations. This characteristic dramatically alleviates
the computational burden associated to the computation of the posterior distribution,
resulting in an efficient approach for dynamic estimation problems.
6.1.5 Optimal Estimate
The posterior conditional distribution is the solution of the filtering problem
combining the a priori dynamical knowledge with the obtained measurements.
According to the selected approach, this posterior could be a joint distribution, as in
Eq. (6.4), or the marginal probability, in Eq. (6.5). In the general nonlinear case, this
solution is infinite-dimensional.