85 where
k i
w
is the normalized weight. Whenever
eff
N
is below a predefined threshold
T
N , it indicates severe degeneracy and to reduce it resampling is performed. Resampling is an
idea that eliminates particles with small weights and concentrates on particles with large weights. In principle, it is implemented as follows:
Generate a new set of particles
l i
x
by resampling from the previous set
k i
x
with probabilities
k i
k i
l i
w x
x P
Reset the weight
N w
l i
1
to the particles Next, we proceed to showing how these steps are applied to resolve the problem of
identifying the initial point of a random defect from vibrations of bearing data.
5.3.2 Implementation of Sequential Importance Sampling SIS for a Discrete Case
In this case study, both of the methods explained above are applied to our state prediction model. Note that
|
1 k
i i
x x
P
and
|
k i
i
x y
p
used in both algorithms follow the equations 3-8
– 3-16 and 3-17 respectively. First, we present an algorithm for the implementation of SIS sub-optimal to our problem.
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
2 1
, ~
x x
x
,
2 1
, ~
w w
w
represented by a set of
2
discrete states.
Step 2: Generate
| ~
1 k
i i
k i
x x
P x
, hence
2 1
, ~
i i
i
x x
x
. Step 3:
Assign the particle weight for each state,
k i
w
, where
|
1 k
i i
k i
k i
x y
p w
w
. Step 4: Normalize
the importance weights,
N k
k i
k i
i
w w
w
1
.
86
Step 5 : Calculate
|
i i
x P
N k
k i
i k
i
x x
w
1
.
Step 6: Repeat step 2 for
n
t i
, 2
, 1
, where
n
t
is the last monitoring point. End
Algorithm 2: SIS optimal importance
For time steps
, 2
, 1
,
i do the following:
Step 1 : Start with
the non-parametric prior distribution of the state
2 1
, ~
x x
x
,
2 1
, ~
w w
w
represented by a set of
2
discrete states.
Step 2: Generate
| ~
1 k
i i
k i
x x
p x
hence
2 1
, ~
i i
i
x x
x
.
Step 3: Assign the particle weight for each state,
|
1 1
k i
i k
i k
i
x y
p w
w
, where
|
1 k
i i
x y
p
=
N k
k i
i i
i
x k
X P
k X
y p
1 1
| |
. In the case of the initiation of a random defect, the transition matrix
|
1 k
i i
x x
P
used in Chapter 3, is normalized into 2 states.
Step 4: Normalize the importance weights,
N k
k i
k i
i
w w
w
1
.
Step 5: Calculate
|
i i
x p
N k
k i
i k
i
x x
w
1
.
Step 6: Repeat step 2 for
n
t i
, 2
, 1
where
n
t
is the last monitoring point. End
In the case of the initiation of a random defect, we have only two states, hence, degeneracy is not an issue; rather, it helps us to identify when the second stage starts. Thus, the
resampling approach is not applicable here. The grid-based approach and the simple version of the SIS approach described above are conceptually straightforward to implement.
However, because of the computational requirements of the method increased exponentially with the number of states, it can be very inefficient, except for problems with
fewer states Arulampalam et al., 2001; Doucet et al., 2000.
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5.4 Parameter Estimation
