This shows that the mean of a weighted particle random variable can be calculated as the weighted sum of particles.
The above development can be applied identically to the general expectation function case with the following result:
(36.79)
This can easily be extended to represent a set of sufficient statistics for an arbitrary density function. As a result, it can be shown that any density function can be represented to arbitrary accuracy, given enough particles. Because we seek estimation methods that are computationally feasible, we are searching for methods that give us “good enough” performance (e.g. accuracy and stability) with limited computational resources.
In the next section, we investigate one approach, known as the grid particle filter, to representing the location of our particle collection.
Figure 36.10 Importance sampling used to represent arbitrary density functions. The density function is represented by a combination of particle locations and weights (represented by arrows), which can be varied independently.
Figure 36.11 Visualization of nonlinear transformation on a random variable. Given uniform random variables x, y, the effects of three nonlinear transformations show that the density can change significantly during transformation.
36.3.5 Grid Particle Filtering
One approach to addressing the generalized nonlinear estimation requirement to maintain the full probability density is the so‐called grid particle filter. The grid particle filter maintains a discrete collection of possible system states and associates a probability with each of those states (i.e. particles). This approach is optimal given systems with the following conditions:
1 The state vector is truly discrete or can be accurately approximated using a discretization of the state space.
2 The number of discrete states is computationally tractable.
Given these conditions, the state density function can be expressed as a weighted collection of particles (repeated from Eq. 36.72)
(36.80)
where the particle weights, w[j], must sum to one. Because the particle locations are assumed to be static, the filtering operation is performed over the collection of weights. This allows the filter to maintain the density function as the collection of propagation and update steps are applied. At this point, it is relatively straightforward to derive the propagation and update relations for the collection of particles.
We begin with the propagation from time k – 1 to k. Assume that the posterior density function at time k – 1 is given by
Substituting Eq. 36.81 into the Chapman–Kolmogorov equation (Eq. 36.12) and simplifying:
(36.82)
(36.83)
(36.84)
The rightmost density in Eq. 36.85 is the transition probability function, which can be rewritten as
Substituting Eq. 36.86 into Eq. 36.85 and simplifying
(36.87)
(36.88)
where the new particle weight is given by
(36.90)
Conceptually, this can be calculated as the sum of all posterior weights at time k – 1 multiplied by the specific transition probability into state l from all possible prior states.
The development of the measurement update function proceeds in a similar fashion. Recalling our definition of the a priori density function (repeated from Eq. 36.89 for clarity):
(36.91)
and substituting into the update equation (Eq. 36.16) yields
(36.92)
which can be simplified by changing the order of integration and using the properties of the delta function
(36.93)
(36.94)
Finally, recalling the grid particle filter form of the posterior density function
(36.96)