152 Approximate grid, as the name reveals, approximates the posterior density by a set of
points on a predefined grid. On the other hand, particle filter is also approximation by a set of points; however the grid is chosen in a stochastic way. The same algorithms were
also used in Chapter 4 but they are targeted to solve the discrete case of the underlying state of a system to identify the initiation of a random defect.
8.4.1 Approximated Grid Method
In Arulampalam et al. 2001, the continuous state space can be used within a grid- based method to approximate the posterior density by composing it into
N
‗cells‘, where
N k
w
k i
, ,
2 ,
1 :
. With this approach, equation 8-3, can be rewritten as
N k
k i
i k
i i
i i
w w
v w
p
1 |
|
8-4
where
is the Dirac delta function and
k i
i
v
|
is given by
N j
w w
w i
i i
j k
k w
w w
i i
i k
i i
k i
i
j i
j i
i k
i k
i i
dw w
y p
v dw
w y
p v
v
1 ,
1 |
, 1
| |
1 1
| |
8-5
and
k i
i
v
1 |
is given by
N j
w w
w i
j i
i j
i i
k i
i
k i
k i
i
dw w
w p
v v
1 ,
1 1
| 1
1 |
1
| 8-6
where
j i
w
1
denotes the centre of the jth cell at time index
1
i
. The integration in equation 8-6 is reduced by another approximation to simplify the computation. This
can be done by assuming that the weight is computed at the centre of the cell corresponding to
k i
v
where it can be written as
2
1 k
i k
i k
i
w w
w
. Thus, equation 8-5 and 8-6 can be rewritten as
153
N j
j i
k i
j i
i k
i i
w w
p w
v
1 1
1 |
1 1
|
| 8-7
N j
k i
i j
i i
k i
i k
i i
k i
i
w y
p v
w y
p v
v
1 1
| 1
| |
| |
8-8
In order to have a good approximation to the continuous state,
N
must be sufficiently large. However, in our case, depending on the parameters value
p
and
q
, singularity may occur particularly when
i
w approaches
1
i
w
or 1. One might consider numerical methods where the current value of the pdf. is divided by their cumulative probability,
to overcome these limitations; hence the approximation can be calculated.
8.4.2 Particle Filtering
The overview of particle filtering has already been explained in Chapter 5 but it is applied to two states of a discrete case. Generally, the particle filter approximates the
posterior density by using sets of state particles specified by
N k
k i
k i
i
w x ,
, where
N
is the number of particles denoted by
N k
k i
x
1
and
N k
k i
w
1
are the associated weights. In the case of continuous states,
N
is chosen to have a large value, 100
, .
N
g e
. Here, we present an algorithm for the implementation of SIS sub-optimal to the continuous
case.
Algorithm 1: SIS prior-importance
For time steps
, 2
, 1
,
i do the following:
Step 1: Start with the non-parametric prior distribution of the state
N
w w
w w
2 1
, ,
~
,
N
v v
v v
2 1
, ,
~
represented by a set of N discrete states.
Step 2: Generate
| ~
1 k
i i
k i
w w
p w
, hence
N i
i i
i
w w
w w
,
, ~
2 1
. Step 3:
Assign the particle weight for each state,
k i
v
, where
|
1 k
i i
k i
k i
w y
p v
v
. Step 4: Normalise
the importance weights,
N k
k i
k i
i
v v
v
1
.
154
Step 5 : Calculate
|
i i
w p
N k
k i
i k
i
w w
v
1
.
Step 6: Repeat step 2 for
n
t i
, 2
, 1
, where
n
t
is the last monitoring point. End
It has already been explained in Chapter 5 that degeneracy is a common problem found in particle filtering. The solution to this problem is to use the re-sampling step, mainly
to discard particles with small weights and multiply particles with large weights. Unfortunately, the re-sampling step also introduces other problems. Theoretically, a
particle that has large weights that will be selected many times could give rise to the problem known as sample impoverishment Arulampalam et al., 2001, which implies
that the effective number of samples is reduced. To explain this problem suppose we have a particle in a particular state before the probability of that state can be evaluated.
If the state has no particles in it, the assumption is that its probability of being the true state of the system is zero. Without a particle in a certain state, that state will continue to
be given zero probability of having occurred and this implies that the particles will not reflect the true density.
The simplest solution to the sample impoverishment problem is to increase the number of particles being used but this is probably unrealistic. In the literature, several ideas
have been proposed to reduce the problem Schon, 2003. One such idea is referred as roughening Arulampalam et al., 2001 or jittering Fearnhead, 1998, which introduces
an additional noise to make the particles different from each other. With all the negative effects of re-sampling steps, one needs to provide a measure for degeneracy and avoid
re-sampling if it is not necessary. Such measurement is already introduced in Chapter 5.
8.5 Parameter Estimation
