Information Technology Reference
In-Depth Information
points. This gives approximations to the row and column chi-squared distance, but the
relationships between the two sets of points seem to have no simple interpretation. Fur-
thermore, their inner product seems to be of little interest. Nevertheless, this seems to be
the most commonly occurring form of correspondence analysis.
Before leaving chi-squared distance we establish a result that goes some way towards
justifying the name. Let
w
i
,
i
=
1, 2,
...
,
n
, be a set of nonnegative quantities, called
weights. Define the weighted mean
i
=
1
w
i
x
i
x
=
i
=
1
w
i
.
(7.19)
Then
n
n
n
2
2
2
w
i
(
x
i
−
a
)
=
w
i
(
x
i
−¯
x
)
+
(
¯
x
−
a
)
w
i
,forany
a
.
(7.20)
i
=
1
i
=
1
i
=
1
Setting
a
=
x
i
in (7.20) leads to
2
n
w
i
n
n
n
n
n
2
2
w
i
w
i
(
x
i
−
x
i
)
=
w
i
w
i
(
x
i
−
x
)
+
w
i
(
x
−
x
i
)
i
=
i
=
i
=
1
i
=
1
1
i
=
1
1
i
=
1
2
n
,
n
=
2
w
i
(
x
i
−
x
)
w
i
i
=
1
i
=
1
that is,
2
n
n
n
2
w
i
(
x
i
−
x
)
w
i
=
w
i
w
i
(
x
i
−
x
i
)
.
(7.21)
i
i
=
1
i
=
1
i
<
Using weights
w
i
=
1
/
n
,
i
=
1, 2,
...
,
n
, in (7.21) gives the well-known identity
n
n
i
=
1
(
x
i
−
x
)
2
2
n
=
i
(
x
i
−
x
i
)
.
(7.22)
i
<
The total sum of squares of the Pearson residuals is given by
p
q
2
(
x
ij
−
x
i
.
x
.
j
/
n
)
2
χ
=
,
x
i
.
x
.
j
/
n
i
=
1
j
=
1
which may be written as
p
2
p
q
q
2
x
i
.
(
x
ij
/
x
i
.
−
x
.
j
/
n
)
1
x
.
j
2
χ
=
=
n
x
i
.
(
x
ij
/
x
i
.
−
x
.
j
/
n
)
.
(7.23)
x
.
j
/
n
i
=
1
j
=
1
j
=
1
i
=
1
Replacing in (7.19)
w
i
=
x
i
.
,
x
i
=
x
ij
/
x
i
.
and
n
=
p
,then
i
=
1
x
i
.
x
ij
/
x
i
.
i
=
1
x
i
.
x
.
j
n
x
¯
=
=
,
