Databases Reference
In-Depth Information
In the example of the above subsection,
d
1
(
A
)=1,
d
2
(
A
)=3and
d
3
(
A
)=0.
Example 1.
Let us consider the following corresponding matrix:
⎛
⎞
123
456
7
11
9
⎝
⎠
.
B
=
Calculation gives:
d
1
(
B
)=1,
d
2
(
B
)=3and
d
3
(
B
) = 18.
It is notable that a simple change of a corresponding matrix gives a sig-
nificant change to the determinant, which suggests a change of structure in
dependence/independence.
The relation between
d
k
(
A
) gives a interesting constraint.
Proposition 2.
Since d
k
(
A
)
|
d
k
+1
(
A
)
, the sequence of the devisors is mono-
tonically increasing one:
d
1
(
A
)
≤
d
2
(
A
)
···≤
d
r
(
A
)
,
where r denotes the rank of A.
The sequence of
B
illustrates this: 1
<
3
<
18.
Let us define a ratio of
d
k
(
A
)to
d
k−
1
(
A
), called
elementary divisors
,where
C
denotes a corresponding matrix and
k
≤
rankA
:
e
k
(
C
)=
d
k
(
C
)
d
k−
1
(
C
)
(
d
0
(
C
)=0)
.
The elementary divisors may give the increase of dependency between two
attributes. For example,
e
1
(
B
)=1,
e
2
(
B
) = 3, and
e
3
(
B
)=6.Thus,a
transition from 2
×
2to3
×
3 have a higher impact on the dependency of two
attributes.
It is trivial to see that
det
(
B
)=
e
1
e
2
e
3
,
which can be viewed as a decom-
position of the determinant of a corresponding matrix.
3.2 Divisors and Degree of Dependence
Since the determinant can be viewed as the degree of dependence, this result
is very important. If values of all the subdeterminants (size
r
) are very small
(nearly equal to 0) and
d
r
(
A
)
1, then the values of the subdeterminants
(size
r
+ 1) are very small. This property may hold until the
r
reaches the
rank of the corresponding matrix. Thus, the sequence of the divisors of a
corresponding matrix gives a hidden structure of a contingency table.
Also, this results show that
d
1
(
A
)and
d
2
(
A
) are very important to estimate
the rank of a corresponding matrix. Since
d
1
(
A
) is only given by the greatest
common divisor of all the elements of
A
,
d
2
(
A
) are much more important
