Civil Engineering Reference
In-Depth Information
The change between the two input vectors is equal to the difference Z
2
Z
1
, which is equal to the
ij
deviations from the mean vector:
sum of the
11
21
11
21
2
11
Z
2
Z
1
.
(1.43)
2
21
Since
X
i
Z
i
X
a
, we can see that
X
2
X
1
Z
2
Z
1
, which indicates the change between the two
input vectors.
Predicting Fault Patterns in Signals
To demonstrate the ability of the Karhunen-Loève method to obtain a reconstruction of the original
signal for error prediction purposes, we again revisit the simple case of linear vectors. The reconstruc-
tion ability of the method will be demonstrated further in a later section using numerically-generated
signals [69, 70]. The original data vectors
X
1
, and
X
2
can be reconstructed as a linear combination of the
eigenvector
T
T
[
1
2
]
[
y
1
y
2
]
and the coefficient vector
Y
in the transform domain. The zero-
y
1
y
2
mean data matrices
Z
1
, and
Z
2
are reconstructed using
Z
1
,
and
Z
2
, which become:
11
--------------------------
1
2
2
2
11
1
2
2
2
Z
1
(
)
(1.44)
21
21
--------------------------
1
2
2
2
11
--------------------------
11
21
1
2
2
2
1
2
2
2
Z
2
(
)
.
(1.45)
21
--------------------------
1
2
2
2
When the mean vector
X
a
is added to the zero-mean vectors
Z
1
and
Z
2
, we obtain the original data vectors
X
1
and
X
2
.
Monitoring and Prediction of Numerically Generated Surface Signals
Up to this point, we have discussed the potential uses of the Karhunen-Loève transform and presented
the mathematical extension to demonstrate its mechanics. It is now necessary to show that the method
can be used effectively with actual signals. In this light, the purpose of this section is to demonstrate the
ability of the Karhunen-Loève method to decompose complex signals into physically meaningful patterns
which can be monitored individually. For this purpose, we present two examples using numerically-
generated signals to simulate manufacturing signals [69].
The first example is the case of a multi-component signal, composed of three deterministic compo-
nents: a linear function and two sinusoidal functions with different amplitudes and frequencies. From
this study, we expect two things: (1) the multi-component signal can be decomposed into a few funda-
mental patterns, which can be used to reconstruct an estimate of the original signals; and (2) the multi-
component signal can be decomposed into individual eigenfunctions which can be monitored separately
by means of the corresponding coefficient vectors.
The second example investigates the case where nonstationary changes occur in the signal of the first
example, with added noise. These changes include: (1) a change in the mean-square value simulated by
a sudden change in the amplitude of one of the sinusoidal components; and (2) a change in the mean
