42 can set
| x
P x
P
. In our case it is assumed that x
P = 1 as we start our
system from new. Since we assume that the system starts from new, at
i
, the true state
x
can only be a normal state that is 1, hence equation 3-6 for
1
i
can be written as
|
1 1
x
P
2 1
1 1
1 1
1 1
1
| 1
| |
| 1
| |
x
x P
X x
P x
y p
x P
X x
P x
y p
3-7
Hence, for every ,....
3 ,
2
i |
i i
x P
can be calculated from equation 3-6 recursively.
3.4 Formulation of the Transition Probabilities
Now, we go into the details of the formulation of
|
1
k X
m X
P
i i
and
|
i i
x y
p
. First, we adopt the concept of delay-time modelling Christer and Waller, 1984 to
model the transitional probability. Using a Markov model, it is assumed that
i
x must always be in one of a finite number of discrete states,
3 ,
2 ,
1
i
X
. We define
1 1
l F
and
2 2
l F
as the cumulative probability of random variables
1
L and
2
L . The transition probabilities that state m will occur at time
i
t
, given that the current state is
k
at time
1
i
t , are as follows, see Christer et al. 2001.
1 |
1
1
i
i
X X
P
=
|
1 1
1
i i
t L
t L
P
=
1
1 1
1 1
1 1
i i
t t
dl l
f dl
l f
=
1 1
1 1
1
i i
t F
t F
3-8
1 |
2
1
i
i
X X
P =
| ,
1 1
1 1
2
i i
i
t L
t L
l t
L P
1 1
1
1 1
1 1
2 2
2 1
1
i i
i i
t t
t l
t
dl l
f dl
dl l
f l
f
=
1 1
1 1
1 1
2 1
1
1
i t
t i
t F
dl l
t F
l f
i i
3-9
43
1 |
3
1
i
i
X X
P
=
| ,
1 1
1 1
2
i i
i
t L
t L
l t
L P
1
1 1
1 2
2 2
1 1
1 1
i t
t l
t
t F
dl dl
l f
l f
i i
i
= 1
1 1
1 1
2 1
1
1
i t
t i
t F
dl l
t F
l f
i i
Equation 3-10 assumed that if the item failed before
i
t
, it will stay failed until
i
t
. 3-10
2 |
1
1
i
i
X X
P = 0
3-11
2 |
2
1
i
i
X X
P
=
, ,
| ,
1 1
1 1
2 1
1 2
i i
i i
t L
l t
L t
L l
t L
P
=
1 1
1 1
1
1 2
2 2
1 1
1 2
2 2
1 1
i i
i i
t l
t t
l t
dl dl
l f
l f
dl dl
l f
l f
=
1 1
1 1
1 2
1 1
1 1
2 1
1
1 1
i i
t i
t i
dl l
t F
l f
dl l
t F
l f
3-12
2 |
3
1
i
i
X X
P
=
, ,
| ,
1 1
1 1
2 1
1 2
i i
i i
t L
l t
L t
L l
t L
P
=
1 1
1 1
1 1
1
1 2
2 2
1 1
1 2
2 2
1 1
i i
i i
i
t l
t t
l t
l t
dl dl
l f
l f
dl dl
l f
l f
=
1 1
1 1
1 2
1 1
1 1
1 2
1 2
1 1
1
i i
t i
t i
i
dl l
t F
l f
dl l
t F
l t
F l
f
3-13
3 |
1
1
i
i
X X
P
= 0 3-14
3 |
2
1
i
i
X X
P
= 0 3-15
3 |
3
1
i
i
X X
P
= 1 3-16
Note that the stationary assumption is no longer valid, as at every time
i
t
, the transition probability is clearly time-dependent. This is the property we needed, since most
existing applications of Markov models are time-independent.
44
3.5 Formulation of the Relationship between the Observed Data and
