Information Technology Reference
In-Depth Information
Figure 9.2
Three-dimensional map of the one continuous and the single categorical
variable given in Table 9.1.
with the interpolation formula. The value
φ
e
k
is a pseudo-sample having zero values,
representing the means, for all variables except the
k
th. As
τ
varies, the interpolated
values trace out the trajectory for the
k
th variable. For a categorical variable the concept
of a centred data matrix with zero representing the mean value of the variable is invalid.
To make progress, for each admissible value
τ
of the
k
th variable we need a set of
n
pseudo-samples that do not require the zero mean. Thus, we consider
(
x
11
,
x
12
,
...
,
x
1
;
k
−
1
,
τ
,
x
1
;
k
+
1
,
...
,
x
1
p
)
.
(
x
i
1
,
x
i
2
,
...
,
x
i
;
k
−
1
,
τ
,
x
i
;
k
+
1
,
...
,
x
ip
)
.
(
x
n
1
,
x
n
2
,
...
,
x
n
;
k
−
1
,
τ
,
x
n
;
k
+
1
,
...
,
x
np
).
It follows that the pseudo-samples differ from the original
X
only in the common value
τ
assigned to the
k
th variable. If the
k
th variable is continuous then
−∞
<τ <
∞
.and
if it is categorical then
τ
will take the
L
k
distinct values representing the category levels
for that variable.
We may interpolate the
n
pseudo-samples and find their centroid. It is the locus of
this centroid as
τ
varies that defines a nonlinear trajectory for a continuous variable
