142 TBN of a used lubricant is a measurement of its ability to neutralize acids. A low TBN
will cause more corrosion and thus indicates oil contamination.
One way to reduce the size or the dimension of the input variable is by applying the PCA. However, it is notice that the measurement unit of each element in responsive
data varies, which prevents us from carrying out the PCA. Hence, we decided to use all the information and assumed that all elements are independent of each other. We
applied a concept called independent component analysis ICA, for two main reasons; the first is to normalize the unit of each individual oil performance measurement, and
the second is to identify the possibility of reducing the dimensions of the monitored variables for oil performance and contaminants as PCA is also used as a pre-processing
step for ICA. Normalization of the measurement unit of the monitored variables is
important, as each variable has its own unit and it was noted from the literature and discussions with persons familiar with similar types of data that these variables are very
much correlated among themselves, Hussin and Wang, 2006. Therefore, ICA can be used to calculate the contribution of each item of responsive data.
It is noted that we are applying PCA rather than ICA to the reflective data, even though ICA can also be used to reduce the dimensionality of the dataset, as all the elements in
reflective data are measured on the same scale and are assumed to be correlated Jolliffe, 1986. The literature also shows that the aim of PCA is variance maximization and
dimension reduction Fodor, 2002 , while ICA is more suited to ensuring the statistical independence of the input variables Lennon et al., 2001 and Fodor, 2002 .
7.4.1 Independent Component Analysis ICA
ICA of a random vector
Y
consists of finding a linear transformation WY
S
ˆ so that
the components
i
Sˆ
are as independent as possible, in the sense of maximizing some function
ˆ ,.....,
ˆ
1 n
s s
F
that measures independence. This is not very different from principal component analysis PCA; see Chapter 6, used in finding the transformation
UY Z
so that elements of a new set of
i
z are linearly uncorrelated and the variances of
i
z are ranked from the first to the last. To begin the discussion of ICA, we can express the entire system of n measured signals and m observations as
143
AS Y
:
n mn
m n
m
s s
a a
a a
y y
: :
: :
1 1
1 11
1
7-19
We refer to each independent signal as
S
. Each observed signal
Y
can be expressed as a linear combination of these independent signals.
A
is an n
m mixing matrix that
generates
Y
from
S
. Hence, the goal of ICA, given the observation
Y
, is to calculate matrix
A
and the set of independent
S
values. In order to perform this task, our intention is to find a matrix
W
the inverse of
A
such that, if it is applied to
Y
, it yields a dataset Sˆ that is as independent as possible and thus approaches the unknown signals
S
.
WY S
ˆ
:
m nm
n m
n
y y
w w
w w
s s
: :
: ˆ
: ˆ
1 1
1 11
1
7-20
It should be noted that each of these variables influences the others, so that applying ICA as a technique to separate linear mixed sources seems promising. Several
approaches for making ICA estimations are available. In this research, the FastICA package developed for Matlab applications and designed by Aapo Hyvärinen at
Helsinki University of Technology has been applied Hyvärinen et al., 1997. Thus, for every set of data, the transformation is made from the original value of lubricant
variables to a set of independent variables. These transformed data will be used for further analysis.
As noted above, ICA can reduce the dimensions of the variables, although this is not its main purpose. This analysis was conducted on our dataset, and while conducting the
whitening process, the eigenvalues of the first principal component of uncorrelated variables showed a range of values from 55 to 95. This strongly suggests that the
dimensionality could not be reduced, hence we decided to use the entire set of variables in the model.
144 Having all the data required in hand, we first estimated the parameter values for our
model. The 22 randomly selected dataset is again used for this purpose and the results are shown below in
Table 7-1
. The dataset chosen is similar to the data used in Chapter 6 for model comparison.
a ˆ
bˆ
ˆ
INS
Bˆ
KV
Bˆ
TBN
Bˆ
WC
Bˆ
0.00106 0.00021
1.19712 0.03321
0.01672 0.03516
0.06309
Table 7-1: Parameter estimation for responsive and reflective variables
The variance and covariance matrix was subsequently calculated to provide an indication of the accuracy of the estimates. The variance and covariance matrix for the
estimated parameters in shown in Table 7-2 below.
a ˆ
bˆ
ˆ
INS
Bˆ
KV
Bˆ
TBN
Bˆ
WC
Bˆ
a ˆ
3.809E-10 -3.42E-10
-2.38E-9 -2.34E-8
1.67E-8 -5.33E-10 -7.11E-8
bˆ
-3.42E-10 3.16E-10
-8.03E-8 -1.61E-8
-7.47E-8 -6.53E-8
2.25E-9
ˆ
-2.38E-9 -8.03E-8
2.46E-4 1.24E-5
-2.44E-5 7.79E-6
4.35E-5
INS
Bˆ
-2.34E-8 -1.61E-8
1.24E-5 0.00234
-1.86E-4 1.35E-4
-2.25E-5
KV
Bˆ
1.67E-8 -7.47E-8
-2.44E-5 -1.86E-4
0.00361 5.02E-4
-7.35E-4
TBN
Bˆ
-5.33E-10 -6.53E-8
7.79E-6 1.35E-4
5.02E-4 0.00498
2.13E-3
WC
Bˆ
-7.11E-8 2.25E-9
4.35E-5 -2.25E-5
-7.35E-4 2.13E-3
0.0013
Table 7-2: Variance and covariance of estimated parameters
The variances of the estimated parameters shown in Table 7-2 are relatively small for
all parameters, which tells us something about variability around the mean. Thus, we can say that the parameter estimates were good in terms of variances. The covariance of
each parameter is literally small, indicating that there are no relationships between estimated parameters.
Using the estimated parameters, we calculated our residual model
, |
i i
i
x p
using
the remaining dataset. Figure 7-1 and
Figure 7-2
below show two of the results for residual time distribution. The dotted straight line in the plot indicates the actual
145 residual life. Again, it is shown in both plots that the variance of the pdf. becomes
smaller as we have more information.
Figure 7-1: Case 1 – pdf and actual residual time with mixed monitoring information for
engine 83000128
Figure 7-2: Case 2 – pdf and actual residual time with mixed monitoring information for
engine 83000130
146
7.5 Model Comparison
