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Ta b l e 3 . 1 8
Scaled three-dimensional data from Table 3.3.
Sample no
X
Y
Z
1
0
.
5551
0
.
2612
0
.
2566
2
0
.
0538
1
.
9743
0
.
0441
3
0
.
6399
0
.
2094
0
.
2906
4
0 . 8074
0 . 9122
1 . 7062
5
0 . 7096
0 . 7555
1 . 0198
6
0 . 4078
0 . 0277
0 . 2680
7
0 . 5903
0 . 3849
0 . 1994
8
0 . 5219
0 . 1746
1 . 4854
9
1 . 0736
0 . 0074
1 . 7367
10
1 . 3290
1 . 1505
0 . 3994
11
0 . 0589
0 . 9242
0 . 4274
12
1 . 0993
1 . 9248
0 . 1797
13
0 . 2867
1 . 0882
0 . 0014
14
1 . 1371
0 . 0050
0 . 3374
15
0 . 1913
0 . 5562
0 . 9729
16
0 . 5794
0 . 5444
0 . 0564
17
0 . 4894
0 . 5545
1 . 0888
18
0 . 2207
0 . 9735
0 . 5100
0 . 6303
1 . 1597
0 . 4484
19
.
.
.
20
1
3652
0
2527
0
5663
21
1
.
9663
0
.
1531
1
.
7111
22
1
.
2357
0
.
1971
0
.
5802
23
0
.
9944
0
.
5163
0
.
2290
24
2
.
1728
2
.
2770
2
.
7218
25
1
.
3439
1
.
5993
0
.
6573
This apparent ineffectiveness of scaling is not true in general. Consider a data set
from a normal distribution with mean vector
µ
and covariance matrix
,where
7
7
6
10
.
20
.
6
,
.
µ =
=
0
.
210
.
36
(3.30)
0
.
60
.
36
4
The resulting clouds of three-dimensonal data points before and after scaling each column
to unit variance are expected to be like Figure 3.28. Again the correlation structure does
not change with scaling but the ellipsoids have quite different shapes. This is in contrast
with what happens in Figure 3.26 and can be explained by considering the correlation
matrix associated with (3.30),
. 2
. 3
1
,
.
.
0
210
18
0
.
30
.
18
1
which points to very little correlation between the three variables, therefore the cloud
for the scaled data is close to a sphere in three dimensions. The cloud for the unscaled
data is stretched in the z-direction since this variable has a much larger variance than for
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