133 lubricant and contaminants will influence the wear rate or residual life, then they are
types of what we call responsive variables, which implies that they are, to some extent, responsive or contributing to the wear process.
In a parallel study, Chapter 6 showed how the residual time of marine engines can be modelled using total metal concentrations based on SOAP analysis. However, SOAP
analysis can detect metal particles only up to 10 microns in diameter, which implies that larger metal particles are not detected. Hence, it might not capture information that
could tell us about the exact accumulated metal concentration in engines. However, it does provide some indication of machine condition. SOAP provides lubricant
performance and contaminants measures as indicators of lubricant performance, which could influence the engine wear. Hence, we are interested to develop a residual
prediction model that combines the information given by metal concentration, the lubricant data, and contaminants, i.e. both responsive and reflective variables.
7.2 Model Developments
This development of the model is similar to other work Wang et al., 2000, 2002, 2005; Wang, 2002, 2003, 2004; Zhang, 2004 but with some new extensions, as we embed
lubricant and contaminant data as a responsive factor that could increase or decrease the residual time, which has not been studied before in this context. The notation used in
our model building is as follows:
7.2.1 Notation
1.
i
X is a random variable representing the residual time at time
i
t
with
i
x as its realization.
2.
i
Y is a type of reflective monitoring information which reflects the condition of the engine state.
i
Y is random and its realization is
i
y
. The relationship between
i
X and
i
Y is described by
|
i i
x y
p
. 3.
i
z is not a random variable hence it is treated as a covariable, which indicates a type of responsive monitoring information which may influence
i
X , but is otherwise assumed not to have an influence on
i
Y . This is because, if the measurements are
134 correlated, it is sufficient to use only one measurement. Thus, at this early stage, a
simple analysis has been carried out that supports our assumption that there is no correlation between
i
Y and
i
z . 4.
i
X
is the residual time before information
i
z is taken at time
i
t
. In our case, both information
i
y
and
i
z
are available at the same time. 5.
i
X
is the residual life after the information
i
z
is taken at time
i
t
. 6.
i
and
i
are the monitoring history for
i
y
and
i
z
, that is
i i
y y
y ,
, ,
2 1
and
i i
z z
z
, ,
2 1
.
7.2.2 Model Formulations
Our objective is to find the residual time distribution, given both responsive and reflective data, i.e.
, |
i i
i
x p
at time
i
t
. To start with, we defined the residual time adopted from Wang and Christer 2000 as
defined not
1 1
i i
i i
t t
x x
otherwise ,
if
1 1
i i
i
t t
x
no maintenance intervention, 7-1
The relationship between
i
x and
i
x is assumed to be
i
x
=
i
z B
i
e x
7-2
where
B
is a parameter that defines the influence of covariate
i
z
on
i
x
and
i
z
=
1
i i
z z
. Note that if
i
z
is a vector, then this relationship can be presented as
i
x
=
M s
s i
s
z B
i
e x
,
7-3 where
M
is the size of the vector. For the sake of simplicity, we will use a scalar vector to explain the influence of
i
z
on the model. We will deal with this vector while fitting the model to data. Hence, equations 7-2 and 7-3 can be explained as follows:
1. if
i
z
, then
i i
x x
, which means the residual time remains the same.
135 2. if
i
z
, then
i i
x x
, which means the residual time becomes shorter and the system is deteriorating.
3. if
i
z
, then
i i
x x
, which means the residual life increases as the system improves.
We seek to establish the
, |
i i
i
x p
. From Wang and Christer 2000,
, |
i i
i
x p
can be written as
, |
i i
i
x p
=
1 1
1 1
, |
| ,
| |
i i
i i
i i
i i
i i
i
dx x
p x
y p
x p
x y
p
7-4
By directly using the relationship of
|
i
i
x y
p
and equations 7-1 and 7-2, equation 7-4 can have the following form
, |
i i
i
x p
=
1 1
1 1
1 1
1 1
, |
| ,
| |
i i
i i
i z
B i
i i
i i
i i
i z
B i
i i
i
dx t
t e
x x
p x
y p
t t
e x
x p
x y
p 7-5
Equation 7-5 is important since it explains how the condition-monitoring data is taken into account in the model, which needs further clarification.
|
i
i
x y
p
is a measurement model specifying the relationship between observed reflective and
responsive data and residual time immediately after time
i
t
. The rationale behind this approach is that we observe the information
i
y
and
i
z
at the same time. Two key issues must be discussed here. The first is, what is the distribution of
, |
x p
at time
t
. Since and
are not available at
t
in most cases, we could set-up
, |
x p
x p
, which is the pdf of the machine life.
The second issue is to establish the relationship between the observed information,
i
y
and
i
z
, with the residual time,
i
x
, which can be established by a probability distribution Wang et al., 2000. Generally, we expect that a short residual time,
i
x
, will generate a
136 high reading in
i
y
. However, the covariate,
i
z
, could increase or decrease the residual time,
i
x . This relationship is modelled in equation 7-2. Thus, the forms of
x p
and
|
i
i
x y
p
can be chosen from a distribution and it can be shown that 7-5 can be determined recursively if
|
x p
and
|
i
i
x y
p
are known. As an example, the Weibull distribution is chosen to represent both
x p
and
|
i
i
x y
p
. Given
1
|
i
y i
i i
e y
x y
p
7-6
where
i i
t x
b
ae
and
, |
x p
x p
=
1 x
e x
7-7 we start with
1
i
,
, |
1 1
1
x p
=
1 1
1 1
1 1
1 1
1
| |
1 1
dx t
t e
x x
p x
y p
t t
e x
x p
x y
p
z B
z B
=
1 1
1 1
1 1
1
1 1
1 1
1 1
1 1
1 1
1 1
1 1
1 1
1 1
dx e
t e
x e
e e
t e
x e
e
t e
x z
B y
ae t
x b
t e
x z
B y
ae t
x b
z B
t x
b z
B t
x b
7-8
then
2
i ,
|
2 2
2
x p
=
2 1
1 1
2 2
1 2
2 1
1 1
2 2
1 2
2
, |
| ,
| |
2 2
dx t
t e
x x
p x
y p
t t
e x
x p
x y
p
z B
z B
2 1
1 1
2 2
1 1
1 2
2
1 1
1 2
2 2
1 2
1 2
2 2
2 2
2 2
2 2
2 2
1 1
1 2
2 2
1 2
1 2
2 2
2 2
2 2
2 2
2 2
dx e
t e
t t
e x
e e
e e
e t
e t
t e
x e
e e
e
t e
t t
e x
z B
z B
y ae
t e
x b
y ae
t x
b t
e t
t e
x z
B z
B y
ae t
e x
b y
ae t
x b
z B
z B
t z
B e
x b
z B
t x
b z
B z
B t
z B
e x
b z
B t
x b
7-9
137 Generalizing to the
ith
term,
, |
i i
i
x p
can be written as
, |
i i
i
x p
=
1 1
1 1
1 2
1 2
1 2
1 2
i x
i y
e a
x b
y ae
t x
b x
i y
e a
x b
y ae
t x
b
dx e
x e
e e
e e
x e
e e
e
i i
j j
i x
j b
i j
i j
i i
t i
x b
i i
i i
j j
i x
j b
i j
i j
i i
t i
x b
i i
where
i j
t e
I t
t e
x x
j i
j k
z B
i k
k k
z B
i i
j
k j
m m
i j
k k
,
1
1 ,
1
1
and
otherwise i
k if
I I
i k
i k
1
, ,
. 7-10
This completes our formulation for predicting the conditional residual time, given that we have all the information.
7.3 Parameter Estimation
