Databases Reference
In-Depth Information
8 Missing Information
Missing information is a common problem in data mining. One of possibilities
how to deal with missing information is secured X-extension introduced in [2].
It deals with data matrices with missing information. We assume that there
is a special symbol
X
that we interpret as the fact “the value of the corre-
sponding attribute is not known for the corresponding object”. An example
of data matrix
X
with missing information is in Fig. 2.
The principle of secured X-extension is to extend the set
M
{
0
,
1
}
of values
of Boolean attributes and values of association rules to the set
{
0
,
1
,X
}
such
that the below given conditions are satisfied.
We denote the value of Boolean attribute
ϕ
in row
o
of data matrix
as
ϕ
(
o,M
). It can be
ϕ
(
o,M
) = 1 (i.e.
ϕ
is true in row
o
of
M
)or
ϕ
(
o,M
)=0
(i.e.
ϕ
is false in row
o
of
M
X
M
). If we have data matrix
M
with missing
X
)=1,
ϕ
(
o,
X
)=0or
ϕ
(
o,
X
)=
X
.
information then it can be
ϕ
(
o,
M
M
M
X
The secured X-extension deals with completions of data matrix
M
with
X
missing information. The
completion of the data matrix
M
with missing
information
is each data matrix
with the same rows and columns such
that each symbol
X
is replaced by one of possible values of the corresponding
attribute (i.e. the column of
M
).
The principle of
secured X-extension for Boolean attributes
means that val-
ues of each Boolean attribute
ϕ
in data matrix
M
X
with missing information
M
are defined such that
⎨
X
ξ
∈{
0
,
1
}
if
ϕ
(
o,
M
)=
ξ
in each completion
M
of
M
X
)=
ϕ
(
o,
M
⎩
X
otherwise.
X
) of basic Boolean attribute
A
(
α
)inrow
o
of data
The value
A
(
α
)(
o,
M
X
matrix
M
is according to the principle of secured X-extension defined such
that
•
X
)=1if
A
(
o,
X
)
A
(
α
)(
o,
M
M
∈
α
X
)=0if
A
(
o,
X
)
X
)
•
A
(
α
)(
o,
M
M
∈
α
∧
A
(
o,
M
=
X
X
)=
X
otherwise
•
A
(
α
)(
o,
M
X
.In
Fig. 2 there are some examples of values of basic Boolean attributes in data
matrix with missing information.
X
) is the value of attribute
A
in row
o
of data matrix
Here
A
(
o,
M
M
object
ABCD
...
Z
A
(
a
1
)
B
(
b
3
,b
4
)
o
1
a
1
b
8
c
16
X. . .
z
14
1
0
o
2
a
5
X
c
7
d
2
...
X
0
X
.
.
.
.
.
.
.
.
.
.
.
o
n
X
b
3
c
4
d
1
...
z
6
X
1
X
Fig. 2.
An example of data matrix
M
