Civil Engineering Reference
In-Depth Information
TABLE 1.11
Dominant Eigenvalues for Examples 1 and 2
Index
Example 1
Example 2
1
1873.525
555.528
2
103.499
283.348
3
95.351
259.521
4
25.411
25.997
5
24.183
25.322
FIGURE 1.11
Decomposition of profiles from Example 1.
value simulated by a sudden change in the offset level of the signal. We expect to decompose the signal
into fundamental eigenfunctions, and monitor their corresponding coefficient vectors. The coefficient
vectors must indicate where the nonstationary change occurred and what the characteristics of the
nonstationary change are. Finally, an estimate of the original signals will be reconstructed to demonstrate
that the effect of added noise is reduced.
We assume that snapshots of the manufacturing process are collected on a regular basis, allowing the
continual monitoring of the process. We expect to use these snapshots to detect the occurance of faults.
M
represents the number of snapshots collected,
N
is the number of points sampled per snapshot.
Table 1.11
presents the resulting dominant eigenvalues for the two cases.
Example One: Decomposition of Multi-Component Signals
The first example involves a multi-component signal composed of two sinusoidal patterns and a linear
pattern [69]. When standard Fourier-based techniques are used to analyze data, linear trends typically
result in misleading information. As a result, linear patterns are typically removed prior to analyzing
signal characteristics. However, the temporal or spatial occurrence of linear trends can provide valuable
information about the status of the manufacturing process.
In this example, we generate numerical signals simulating a manufacturing signal composed of two
sinusoidal functions and linear function. The multi-component signal has the form of
A
1
sin(
F
l
j
)
A
2
sin(
F
2
j
)
B
1
j
B
2
: two sinusoidal components with one high frequency (
F
1
0.9 rad/sec), small
amplitude (
A
1
1 mm), and the other low frequency (
F
2
0.2 rad/sec) and large amplitude (
A
2
2 mm),
and a linear component with slope
B
1
0.0. We want to decompose the
multi-component signal into eigenfunctions representing a linear trend and two sinusoidal functions.
The eigenfunctions can then be monitored individually to detect potential changes in the manufacturing
system.
M
0.005 and
y
-intercept
B
2
30 snapshots of the manufacturing process are assumed to be collected; each shapshot
contains
N
100 sampled points.
The Karhunen-Loève decomposition of the data in this example results in five fundamental eigenfunc-
tions, as shown in
Fig. 1.11
.
The corresponding eigenvalues are shown in
Table 1.11
.
The first eigenfunction
corresponds to a function representing a linear trend. The second and third eigenfunctions correspond
to the low-frequency sinusoidal component. Notice than, instead of one eigenfunction, we obtain two
