Information Technology Reference
In-Depth Information
Ta b l e 8 . 4
MCA equivalents for the CA notation used in Chapter 7.
CA
MCA
X
: Two-way contingency table with
p
rows and
q
columns representing two
categorical variables with
p
and
q
categories, respectively.
G
: Indicator matrix with
n
rows and
L
columns.
G
,
G
p
]
G
j
:
n
×
L
j
. The indicator matrix for the
j
th (
j
=
[
G
1
,
G
2
,
...
,
p
) categorical variable
having
L
j
categories.
=
1,
...
n
: Total number of samples, i.e.
n
=
1
X1
n
: Total number of samples (number of
rows of
G
), i.e.
n
=
1
G1
/
p
R
:
p
×
p
=
diag
(
X1
)
diag
(
G1
)
=
diag
(
p
1
)
=
p
I
:
n
×
n
1
X
1
G
1
L
C
:
q
×
q
=
diag
(
)
diag
(
)
=
diag
(
)
=
L
:
L
×
L
Also,
L
=
diag
(
diag
(
L
1
)
,diag
(
L
2
)
,
...
,
diag
,where
L
k
is a diagonal
matrix giving the frequencies
(
l
k
1
,
l
k
2
,
...
,
l
kL
k
)
of the
k
th variable.
(
L
p
))
R11
C
p
11
L
11
L
E
:
p
×
q
=
/
n
n
(A subtle point to note is that the sum of
all elements of the input matrix is
needed here, not the number of
samples.)
/
np
=
/
11
/
11
/
Column-centred
X
=
(
I
−
p
)
X
Column-centred
G
=
(
I
−
n
)
G
W
−
1
1
(
X
−
E
)
C
−
1
/
2
W
−
1
2
W
−
1
1
(
p
−
1
/
2
I
)(
G
−
11
L
/
n
)
L
−
1
/
2
W
−
1
2
R
−
1
/
2
=
W
−
1
(
p
−
1
/
2
I
)(
G
−
11
G
/
n
)
L
−
1
/
2
W
−
1
1
2
=
W
−
1
1
)
}
W
−
1
2
where the expression within the braces
denotes the column-centred matrix
p
−
1
/
2
GL
−
1
/
2
{
(
I
−
11
/
n
)(
p
−
1
/
2
GL
−
1
/
2
8.9.3 Function
CATPCAbipl
CATPCAbipl
is our main function for constructing the two-dimensional categorical PCA
biplot discussed in Section 8.8.
Usage
CATPCAbipl
uses the same calling conventions as
PCAbipl
and shares the following
arguments with it:
alpha
line.width
ort.lty
ax
max.num
parplotmar
ax.name.size
offset
pos
c.hull.n
orthog.transx
predict.sample
exp.factor
orthog.transy
select.origin
