6 Fundamentals of Filtering
185
However, this posterior is expensive to compute as it is a joint distribution over
all x k , conditionally dependent on all the observations y 1:l . Furthermore, in realtime scenarios, every time a new observation is available, the full joint posterior
distribution needs to be computed again.
6.1.1.1 State Marginalization
The major computational complexity in Eq. (6.4) stems from the joint nature of
the posterior distribution. Indeed, whenever a new observation is available for
a different time step, the posterior distribution’s dimensionality will increase,
degrading the computational efficiency severely. If the main interest concerns the
computation of the state estimate x k at a specific time step, like in real-time
applications, this excessive numerical burden can be reduced by computing the
marginal distribution instead:
p(x k |y 1:l ) .
(6.5)
This approach reduces dramatically the posterior dimensionality and, therefore,
improves the computational performance.
6.1.1.2 Markov and Independence Assumptions
Given the knowledge of x k , a Markov process links the future process probability
law for t > t k only to x k , independently of how this state was reached. This is
analogous to an ordinary differential equation which links the state rate ˙
x k only
to simultaneous time t k , state x k (and possibly parameters) [29]. If we restrict
to this nevertheless wide class of models, the system dynamics in Eq. (6.1) is a
Markov process and the time-ordered collections of states x 1:T a Markov sequence.
More formally, this class respects the Markov property which simplifies the state
dependency:
p(x k | x 0:k−1 ) = p(x k | x k−1 ) .
(6.6)
Also measurements can be included in the Markov model. Then, the previous
property can be generalised as [51]:
p(x k | x 0:k−1 , y 1:k−1 ) = p(x k | x k−1 ) .
(6.7)
As a consequence of these two assumptions, x k does not depend on anything which
happened before the time t k−1 given x k−1 .
Another traditional assumption is to consider the measurement y k to be conditionally independent of previous state history or observations:
185
However, this posterior is expensive to compute as it is a joint distribution over
all x k , conditionally dependent on all the observations y 1:l . Furthermore, in realtime scenarios, every time a new observation is available, the full joint posterior
distribution needs to be computed again.
6.1.1.1 State Marginalization
The major computational complexity in Eq. (6.4) stems from the joint nature of
the posterior distribution. Indeed, whenever a new observation is available for
a different time step, the posterior distribution’s dimensionality will increase,
degrading the computational efficiency severely. If the main interest concerns the
computation of the state estimate x k at a specific time step, like in real-time
applications, this excessive numerical burden can be reduced by computing the
marginal distribution instead:
p(x k |y 1:l ) .
(6.5)
This approach reduces dramatically the posterior dimensionality and, therefore,
improves the computational performance.
6.1.1.2 Markov and Independence Assumptions
Given the knowledge of x k , a Markov process links the future process probability
law for t > t k only to x k , independently of how this state was reached. This is
analogous to an ordinary differential equation which links the state rate ˙
x k only
to simultaneous time t k , state x k (and possibly parameters) [29]. If we restrict
to this nevertheless wide class of models, the system dynamics in Eq. (6.1) is a
Markov process and the time-ordered collections of states x 1:T a Markov sequence.
More formally, this class respects the Markov property which simplifies the state
dependency:
p(x k | x 0:k−1 ) = p(x k | x k−1 ) .
(6.6)
Also measurements can be included in the Markov model. Then, the previous
property can be generalised as [51]:
p(x k | x 0:k−1 , y 1:k−1 ) = p(x k | x k−1 ) .
(6.7)
As a consequence of these two assumptions, x k does not depend on anything which
happened before the time t k−1 given x k−1 .
Another traditional assumption is to consider the measurement y k to be conditionally independent of previous state history or observations: