Mathematical Problems in Engineering
Sample
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Hyperedge
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Attribute
F : An example of neighborhood hypergraph.
spaces and proposed the neighborhood-based rough set
model. Hu et al. [] employed the neighborhood rough set
to classify data by using the Minkowski distance to calculate
the samples neighborhood threshold which involves all the
attribute values of the one calculated. However, for incom-
plete information system, it is dicult to compute the
distance for the sake of the missing value. us, we present
an extension neighborhood of sample in incomplete informa-
tion system by combining with relevant degree.
Given arbitrary
𝑖
∈and ⊆, the neighborhood
𝐵
(
𝑖
) of
𝑖
on attribute set is dened as
𝐵
𝑖
=|∈,
𝐵
,
𝑖
≥,
()
where
𝐵
(,
𝑖
)is the relevant degree between and
𝑖
.
𝐵
(
𝑖
)
denotes the sample set within the neighborhood of
𝑖
.
According to the denition, we can nd out easily that
() ∀
𝑥
𝑖
∈𝑈
𝐵
(
𝑖
) =;
() ∀
𝑥
𝑖
∈𝑈
∀
𝑥
𝑗
∈𝑈
(
𝑗
∈
𝐵
(
𝑖
)↔
𝑖
∈
𝐵
(
𝑗
));
()
𝑥
𝑖
∈𝑈
𝐵
(
𝑖
)=.
Combining with the neighborhood rough set theory, we
dene the neighborhood hypergraph as follows.
Denition 4 (the neighborhood hypergraph of IIS). Let
=,be a neighborhood hypergraph of incomplete
information system, referred to as neighborhood hypergraph.
en ={
1
,
2
,...,
𝑛
} is the vertex set of , indicating
that it has vertices, ={
1
,
2
,...,
𝑛
} is hyperedge set,
and each
𝑖
in is a hyperedge which connects vertices
(
𝑖1
,
𝑖2
,...,
𝑖𝑘
). ={
1
,
2
,...,
𝑚
} is the attribute set, and
is the decision set, where
𝑖
denotes sample.
Vertices of hypergraph represent the attribution of sam-
ples in some literatures like []andsoon.However,inthis
paper, vertices of hypergraph are denoted as samples and
dierentsamplesononehyperedgehavethesameattributes
set (see Figure ).
Denition 5 (the sample in neighborhood hypergraph).
Given =,, ={
1
,
2
,...,
𝑚
}denotes the attribute set
of hyperedge, ={
1
(),
2
(),
3
(),...,
𝑝
(),(),
𝐵
} is a
sample, where
𝑖
() denotes the values of on the attribute
𝑖
(
𝑖
∈), () denotes the decisions of ,and
𝐵
is the
threshold of a neighborhood.
Denition 6 (the neighborhood hyperedge set of a sample).
Given =,and the attribute set ( ⊆ ),the
hyperedge set which is included by sample is dened as
𝐵
(
)
=
{
|
(
∈
)
∧
(
,
)
≥
}
.
()
Denition 7 (the sample set related to a hyperedge). Given
=,, ∀ ∈ ,andattributesset(⊆), for arbitrary
∈,thesamplesetrelatedto is dened as
𝐵
() = { |
∈
𝐵
(), ∈ }. Given arbitrary ⊆and attributes set
(⊆),thesamplesetrelatedto is dened as
𝐵
(
)
=
𝐵
(
)
|∈.
()
Denition 8 (the condence degree of a hyperedge). Given
=,, for arbitrary ∈, assume that () = { |
∈},where denotes decision set.
𝐵
() is the sample set
related to hyperedge on attributes set (⊆). According
to the decisions ,
𝐵
()is divided into equivalence classes:
1
,
2
,...,
𝑝
;when
𝐵
() =, the condence degree of
is dened as follows:
Conf
𝐵
(
)
=
| ∈
𝐵
(
)
,
(
)
=
(
)
| ∈
𝐵
(
)
. ()
Denition 9 (the upper approximation, lower approximation,
boundary region, and negative domains of hyperedge set
for the sample decision set). Given =,, is the
attitudes set of samples and is the decision set of samples.
For arbitrary hyperedge set
(
⊆), according to
the decisions , the hyperedge set
is divided into
equivalence classes:
1
,
2
,...,
𝑝
. For arbitrary ⊆,
the upper approximation, lower approximation, boundary
region, and negative domains of decision related to set of
attributes are, respectively, dened as
𝐵
(
)
=
𝑝
𝑖=1
|
𝐵
(
)
∩
𝐵
𝑖
=∨∈
𝑖
,∈
,
𝐵
(
)
=
𝑝
𝑖=1
| Conf
𝐵
(
)
>, ∈
𝑖
,
(
0 ≤≤1
)
,
𝐵
(
)
=
𝐵
(
)
−
𝐵
(
)
Neg
𝐵
(
)
=−
𝐵
(
)
.
()
e lower approximation of decisions related to
attribute set is also called positive domain. e size of
positive domain reects the separable degree of classication
probleminagivenattributespaceandthebiggerthepositive
region is, the smaller the border is.