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not centred about the target. The smoothed trend line (see Figure 3.34) shows how the
process started in early 2000 with high levels for
C4
,
E5
,
C5
,
A1
,
A5
and
B5
and
low levels of
A3
,
A4
,
C6
,
C8
,
C7
,
D4
,
D7
,
D6
and
A2
. There was some movement
towards the target, especially on
C5
,
E5
,
C4
,
C7
and
D4
. However, these variables
changed too much and this, together with lower values for
A1
,
A5
,
B5
,
A3
,
A4
,
C6
,
C8
and higher values for
D6
,
D7
and
A2
, resulted in poor quality at the end of 2000. At
this stage the senior management requested a quality improvement plan which resulted
in a sharp turnaround, but mainly on
A3
,
A4
,
C6
and
C8
.
To assess what quality product is associated with each position in the biplot,
quality
regions
can be constructed according to company management's ruling: index values
above 80 are considered to be good quality, while index values below 50 are considered
poor quality. These quality regions are constructed as follows:
Start
: Overlay the biplot space,
L
, with a two-dimensional
m
×
m
grid, coded as
E
:
m
2
×
2
Initialize
:Set
i
=
1
Repeat:
•
Let
z
∗
be the
i
th row of
E
Find the predicted values of
z
∗
for all the variables using the prediction
formula (3.8) as
x
∗
=
z
∗
V
r
:1
×
p
•
x
∗
•
Black box
Index value
Red
if
0
≤
Index value
<
50
•
Colour grid point
Blue
if
50
≤
Index value
<
80
Green
if
80
≤
Index value
<
100
•
Set
i
=
i
+
1
Figure 3.34 shows the PCA biplot of the scaled process quality data with quality
regions added. Also shown is a smoothed trend line. This line is constructed by smoothing
each dimension separately. The R function
loess
is applied to the sample pairs (
z
1
j
,
t
1
)
,
(
z
2
j
,
t
2
)
,
...
,(
z
nj
,
t
n
)for
j
=
1, 2,
...
,
r
(in Figure 3.34,
r
=
2) and
t
i
=
i
. The resulting
loess
fitted values provide the coordinates
which are
then connected to trace a smooth path over time leading to the smooth yellow line shown
in Figure 3.34.
We see that
C6
for
Mar01
is quite high, 16.9, compared to the target of 13.5. This
sample is also located in the satisfactory (rather than good) quality region, indicating that
there is a problem of some kind for this month. We can see that some leeway around
the target is allowed, but the very narrow satisfactory quality area shows how steep the
penalty is when moving further away from the target. The current variability in
A2
,
A5
,
B5
,
D6
,
D7
,
A1
is such that these variables stay in the good quality regions while the
current variability in especially
C5
,
E5
and
C4
can easily lead to an overall poor quality
index value.
The overall quality for the PCA biplot in Figures 3.33 and 3.34 is 59.82%. This might
seem very low, 40% of the variability is not represented, but keeping in mind that the
number of dimensions has been reduced from 14 to 2, this is not too bad. The overall
(
z
11
,
z
12
)
,
(
z
21
,
z
22
)
,
...
,
(
z
n
1
,
z
n
2
)
