Databases Reference
In-Depth Information
Example 3.
Let us consider the following contingency matrices
D
and
E
:
⎛
⎞
1230
4560
7
11
90
000
1
⎝
⎠
D
=
⎛
⎝
⎞
⎠
1230
4560
7
10
90
000
1
E
=
The numbers of examples of
D
and
E
are 49 and 48, respectively, which
can be comparable to that of
B
. Then, from Theorem 6,
detD
=18
<
(49
/
4)
4
=
5764801
256
∼
22518
detE
=12
<
(48
/
4)
4
= 20736
Thus, the maximum value of the determinant of
A
is at most
n
n
.Since
N
is constant for the given matrix
A
, the degree of dependence will decrease
very rapidly when
n
becomes very large. That is,
detA ∼ n
−n
.
Thus,
Corollary 2.
The determinant of A will converge into 0 when n increases
into infinity.
lim
n→∞
detA
=0
.
This results suggest that when the degree of granularity becomes higher, the
degree of dependence will become lower, due to the constraints on the sample
size.
However, it is notable that
N/n
is very important. If
N
is very large, the
rapid decrease will be observed
N
is close to
n
.Even
N
is 48 as shown in
Example 3,
n
=3
,
4 may give a strong dependency between two attributes.
For the behavior of (
N/n
)
n
, we can apply the technique of real analysis, which
will our future work.
5 Distribution of Determinant
As shown in the former section, the determinant of
D
and
E
is signifi-
cantly smaller than the maximum value of the determinant of a set of matrix
{
. Then, the next interest is how is the statistical
nature of the determinant for
M
(
m,n,N
).
M
(4
,
4
,
49)
}
or
{
M
(4
,
4
,
48)
}
