Information Technology Reference
In-Depth Information
Myfanwy
Alisdair
Fair
Clerical
Dark
Scotland
Wales
Jane
F
University
School
Brown
M
Manual
Harriet
Professional
England
Grey
Ivor
George
Postgrad
Jeremy
Figure 8.1
Row chi-squared MCA biplot of the data in Table 8.2. First the SVD of
p
−
1
/
2
GL
−
1
/
2
is performed. The row points (the green filled circles) are then plotted using
the first two columns (after discarding the column associated with the singular value of
unity) of
Z
0
=
U
and the column points (the filled squares) are plotted as the projected
positions of the CLPs i.e. the first two columns of
Z
=
p
−
1
/
2
L
−
1
/
2
V
. The column points
are colour-coded such that the categories of any particular categorical variable appear in
the same colour. The quality of the display is 56.79%, which on adding a third dimension
increases to 77.50%, suggesting that a three-dimensional plot may be worthwhile.
Here,
L
k
is of size
L
×
L
with only the
L
k
×
L
k
diagonal block
L
k
nonzero; for com-
putational purposes the zero parts of
L
k
would be ignored. The first of the expressions
(8.5) follows immediately from the definition of
Z
, giving
GZ
=
(
p
−
1
/
2
GL
−
1
/
2
)
V
=
(
U
V
)
V
=
U
=
Z
0
.
The interpretation is that every row point is given by the vector-sum of the relevant
CLPs, as is consistent with the vector-sum method of interpolation.
The second expression in (8.5) is an extension of the orthogonality results. We have
V
)
L
1
/
2
1
k
p
−
1
/
2
GL
−
1
/
2
L
1
/
2
1
k
p
−
1
/
2
G1
k
p
−
1
/
2
1
,
(
U
=
(
)
=
=
where
1
k
denotes a vector of full size (
L
) with units in those parts pertaining to the
k
th
categorical variable, else zero. Premultiplication by
−
1
U
and using the orthogonality
