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In-Depth Information
our three-dimensional example in Figure 5.9 induces an extra fourth dimension, it is not
possible to visually represent the marker and the plane
N
normal to the marker at this
point. However, the intersection spaces for the markers
µ
=
2, 3 and 4 of the embedded
axis of original variable
Y
can be shown, as is illustrated in Figure 5.12. Since the
embedded original axis
Y
is nonlinear, it follows that the intersection spaces
L
∩
N
as shown in Figure 5.12 are not parallel as was the case for PCA (Section 3.2.3). This,
in turn, leads to nonlinear prediction biplot trajectories.
Gower and Ngouenet (2005) give details of three different methods to obtain the
prediction biplot trajectories. The different trajectories are constructed by identifying
different 'optimal' points on the same intersection spaces
L
∩
N
. However, the
prediction methods identify the same intersection and hence all will yield the same
predictions.
In order to proceed we need the equation of the intersection space for a given marker
µ
whichwedenoteby
L
∩
N
(
µ)
. The space
N
(µ)
is normal to the embedded
k
th
ξ
k
biplot trajectory
at the point
µ
. Therefore, from (5.8) and (5.9), it passes through
the point
−
1
Y
d
n
+
1
(µ)
−
n
D1
,
1
ξ
k
(µ)
=
ξ
k
,
m
+
1
(µ)
where
2
k
,
m
n
2
1
D1
−
1
2
n
1
d
n
+
1
(µ)
−
ξ
k
(µ)
ξ
k
(µ)
ξ
(µ)
=
,
+
1
µ
ξ
k
(µ)
, which implies that
d
µ
ξ
k
(µ)
d
d
and is orthogonal to
y
−
−
1
Y
d
n
+
1
(µ)
−
n
D1
1
=
0,
y
m
+
1
−
ξ
k
,
m
+
1
(µ)
that is,
d
µ
ξ
k
(µ)
y
−
−
1
Y
(
d
n
+
1
(µ)
−
n
D1
)
+
1
d
d
µ
ξ
k
,
m
+
1
(µ)
{
y
m
+
1
−
ξ
k
,
m
+
1
(µ)
}=
0
.
(5.14)
The equation for
N
(µ)
is found by solving (5.14): now,
µ
ξ
k
(µ)
=
−
1
Y
d
µ
d
n
+
1
(µ)
d
d
and
d
µ
ξ
k
,
m
+
1
(µ)
n
1
d
µ
d
n
+
1
(µ)
d
µ
ξ
k
(µ)
2
−
2
ξ
k
(µ)
2
ξ
k
,
m
+
1
(µ)
=−
.
Substituting into (5.14) and simplifying leads to
d
µ
d
n
+
1
(µ)
y
y
m
+
1
Y
−
1
,
n
1
d
µ
d
n
+
1
(µ)
d
d
1
µ
ξ
k
,
m
+
1
(µ)
=−
.
(5.15)
