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As mentioned in Section 36.3, the key requirement of a nonlinear filter is the ability to accurately represent arbitrary probability density functions. Particle filters accomplish this by representing density functions by using collections of discrete, weighted state vectors instances. These state vectors and associated weights are referred to as particles.

Figure 36.7 MMAE position error and one‐sigma uncertainty. Note that the error uncertainty collapses once sufficient information is available to resolve the integer ambiguity.

The development of the theory related to a particle‐based representation of density functions begins by reviewing the essential properties of both the probability density function and the cumulative distribution function. An example cdf and pdf are shown in Figure 36.9. The cumulative distribution function is a monotonically increasing function which represents the probability of a random variable realization that is less than the operand and can be defined as the integral of the density function [11]:

(36.64)

(36.65)

Additionally, the probability of a random variable realization between a range x_a and x_b is expressed by

(36.66)

(36.67)

As a result, the density and cumulative distribution functions must have the following properties:

(36.68)

(36.69)

(36.70)

Figure 36.8 MMAE integer ambiguity particle weights (subset). The correct ambiguity particle (N=7) likelihood increases over time while the outliers are determined to be less likely.

Figure 36.9 Probability density function (PDF) and cumulative density function example (CDF).

(36.71)

The particle filter uses a collection of weighted delta functions to represent the pdf:

(36.72)

where w^[j] is a scalar weighting value for the j‐th particle with location x^[j]. As mentioned previously, the sum of weights must be unity:

(36.73)

An example pdf represented by a collection of weighted particles is shown in Figure 36.10. This importance sampling strategy allows us to represent any pdf with a desired level of fidelity, given enough particles. Additional details regarding importance sampling are provided in Section 36.3.7.

In addition to representing arbitrary pdfs of random vectors, successful nonlinear estimation requires the ability to determine the resulting pdfs after applying nonlinear transformations to random vectors. In general, this can be intractable; however, representing the pdf using the collection of weighted particles makes the transformation relatively straightforward. An example of the effect of some sample nonlinear transformations is shown in Figure 36.11.

One of the most common functions necessary for filtering applications is calculation using the expectation operator. The expectation operator is defined as

(36.74)

where E[·] is the expectation operator, g(x) is an arbitrary function of the random vector x, and p(x) is the pdf of the random vector.

Based on this definition, we can easily calculate some common expectations of the weighted particle pdf. The first is the mean, which is defined as E[x]:

(36.75)

Substituting the pdf of x from Eq. 36.72, rearranging the summation and integral, and then applying the sifting property:

(36.76)

(36.77)

(36.78)

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.