Databases Reference
In-Depth Information
4 Degree of Granularity and Dependence
From Theorem 4, it seems that the increase of the degree of granularity gives
that of the dependence between two attributes.
However, our empirical observations are different from the above intuitive
analysis. Thus, there should be a strong constraint which suppress the above
effects on the degree of granularity.
Let us assume that the determinant of a give contingency matrix gives
the degree of the dependence of the matrix. Then, from the results of linear
algebra, we obtain the following theorem.
Theorem 5.
Let A denote a n
n contingency matrix, which includes N
samples. If the rank of A is equal to n, then there exists a matrix B (n
×
×
n)
which satisfies
⎛
⎞
ρ
1
⎝
⎠
ρ
2
O
BA
=
=
P,
.
.
.
O
n
where ρ
1
+
ρ
2
+
+
ρ
n
=
N.
It is notable that the value of determinants of P is larger than A:
···
detA
≤
detP
Example 2.
Let us consider
B
as an example (Example 1). Let
C
denote the
orthogonal matrix for transformation of
B
. Since the cardinality of
B
is equal
to 48, the diagonal matrix which gives the maximum determinant is equal to:
⎛
⎞
16 0 0
0160
0016
⎝
⎠
.
On the other hand, the determinant of
B
is equal to 18. Thus,
detB
=
18
<
16
3
= 4096
.
Then, C is obtained from the following equation.
⎛
⎞
⎛
⎞
123
456
7119
16 0 0
0160
0016
⎝
⎠
=
⎝
⎠
.
C
×
Thus,
⎛
⎞
−
56
/
340
/
3
−
8
/
3
⎝
⎠
C
=
16
/
3
−
32
/
316
/
3
8
8
/
3
−
8
/
3
It is notable that the determinant of
C
is equal to 2048
/
9. Also, since
detB
= 18, we do not have any diagonal matrix whose determinant is equal
to 18 and the sum of all the elements is equal to 48.
