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where the second term on the right-hand-side vanishes because of the orthogonality
property
1
CV
=
0
. The meaning of (7.11) is that the row coordinates
R
−
1
/
2
U
are at
the centroid of the column coordinates, weighted by the relative row frequencies
R
−
1
X
.
Of course, given one set of coordinates, for the rows (say), it is simple to place column
points at their centroids with or without weights given by the relative row frequencies.
However, the centroid property is satisfied automatically for the CA scaling discussed
in this section; it does not apply for other forms of CA scaling. The centroid property
helps with interpreting the biplot because the column points for a row with high (low)
weights will be tightly (loosely) clustered around the corresponding row point.
7.2.4 Approximation to chi-squared distance
A third and frequently used variant of CA is expressed in terms of chi-squared distance.
This comes in two forms, (i) the chi-squared distance between the rows and (ii) the
chi-squared distance between the columns of
X
. The chi-squared distance
d
ii
between
the
i
th and
i
th rows of
X
is defined by
x
ij
x
i
.
−
2
q
x
i
j
x
i
.
1
x
.
j
d
ii
=
(7.12)
j
=
1
or, in matrix terms,
x
i
x
i
.
−
C
−
1
x
i
x
i
x
i
.
x
i
x
i
.
d
ii
=
x
i
.
−
.
(7.13)
Equation (7.13) is in the form of the square of a 'Mahalanobis distance', in the metric
of the column totals, between points whose coordinates are the row proportions. Thus,
when the
i
th and
i
th rows have the same proportions, the chi-squared distance is zero,
implying that we may amalgamate the rows without affecting the row chi-squared dis-
tances between any pairs of rows, including those that we have amalgamated. Rather less
obvious is that amalgamating columns with equal proportions also has no affect on the
row chi-squared distances. This statement may be verified by noting that proportionality
of columns
j
and
j
implies that
x
ij
=
τ
x
ij
,
i
=
1,
...
,
p
. Without amalgamation, these
two columns contribute to
d
ii
the quantity
x
ij
x
i
.
2
x
ij
x
i
.
2
x
i
j
x
i
.
2
x
i
j
x
i
.
1
x
.
j
τ
−
+
−
.
(7.14)
x
.
j
After amalgamation into a single column the contribution is
(
1
+
τ)
x
ij
x
i
.
2
−
(
1
+
τ)
x
i
j
x
i
.
1
(
1
+
τ)
x
.
j
.
(7.15)
But (7.14) can be written as
1
x
.
j
+
x
ij
x
i
.
−
1
x
.
j
+
x
ij
x
i
.
2
2
2
x
i
j
x
i
.
x
i
j
x
i
.
τ
x
.
j
=
−
=
(
.
).
7
15
x
.
j
