23
Identification from Wearable Device Brain Signals
in the outer approximation, but not in the lower approximation, are located in the boundary region. Let U be a set of objects. Rough sets were originally proposed using equivalence relations on U. However, it is possible to define a pair of lower and upper bounds
A C A C
(
)
( ), ( ) or a rough set for every set C U
⊆
as long as the properties specified by
Pawlak [20,21] are satisfied. Yao [22] described various generalizations of rough sets by
relaxing the assumptions of an underlying equivalence relation. Such a trend toward generalization is also evident in rough mereology proposed by Polkowski [23], and the use
of information granules in a distributed environment by Skowron [24]. The present study
uses a generalized view of rough sets. If one adopts a more restrictive view of rough set
theory, the rough sets developed in this chapter may have to be looked upon as interval
sets [25]. Let us consider a hypothetical clustering scheme
U P C C
C k
{
}
=
/
, ,...,
1
2
(2.1)
that partitions the set U based on an equivalence relation P. Let us assume that due to
insufficient knowledge, it is not possible to precisely describe the sets, C i ,1 ≤ i ≤k, in the
partition. However, it is possible to define each set C U P
i ∈ / using its lower A C i
( ) and
upper A C j
( ) bounds based on the available information. We will use vector representations
u v
, for objects and
ci for cluster C i
We are considering the upper and lower bounds of only a few subsets of U. Therefore, it
is not possible to verify all the properties of the rough sets [20,21]. However, the family of
upper and lower bounds of
c U P
i ∈ / are required to follow some of the basic rough set
properties such as:
(P1) An object
v can be part of at most one lower bound
(P2)
v A c
v A c
i
i
( )
( )
∈
⇒ ∈
(P3) An object
v is not part of any lower bound m
v belongs to two or more upper bounds.
Property (P1) emphasizes the fact that a lower bound is included in a set. If two sets are
mutually exclusive, their lower bounds should not overlap. Property (P2) confirms the
fact that the lower bound is contained in the upper bound. Property (P3) is applicable to
the objects in the boundary regions, which are defined as the differences between upper
and lower bounds. The exact membership of objects in the boundary region is ambiguous.
Therefore, property (P3) states that cannot belong to only a single boundary region. Note
that (P1) – (P3) are not necessarily independent or complete. However, enumerating them
will be helpful in understanding the rough set adaptation of evolutionary, neural, and
statistical clustering methods.
2.4.3 Genetic Algorithms
A genetic algorithm is a search process that follows the principles of evolution through
natural selection. The domain knowledge is represented using a candidate solution
Identification from Wearable Device Brain Signals
in the outer approximation, but not in the lower approximation, are located in the boundary region. Let U be a set of objects. Rough sets were originally proposed using equivalence relations on U. However, it is possible to define a pair of lower and upper bounds
A C A C
(
)
( ), ( ) or a rough set for every set C U
⊆
as long as the properties specified by
Pawlak [20,21] are satisfied. Yao [22] described various generalizations of rough sets by
relaxing the assumptions of an underlying equivalence relation. Such a trend toward generalization is also evident in rough mereology proposed by Polkowski [23], and the use
of information granules in a distributed environment by Skowron [24]. The present study
uses a generalized view of rough sets. If one adopts a more restrictive view of rough set
theory, the rough sets developed in this chapter may have to be looked upon as interval
sets [25]. Let us consider a hypothetical clustering scheme
U P C C
C k
{
}
=
/
, ,...,
1
2
(2.1)
that partitions the set U based on an equivalence relation P. Let us assume that due to
insufficient knowledge, it is not possible to precisely describe the sets, C i ,1 ≤ i ≤k, in the
partition. However, it is possible to define each set C U P
i ∈ / using its lower A C i
( ) and
upper A C j
( ) bounds based on the available information. We will use vector representations
u v
, for objects and
ci for cluster C i
We are considering the upper and lower bounds of only a few subsets of U. Therefore, it
is not possible to verify all the properties of the rough sets [20,21]. However, the family of
upper and lower bounds of
c U P
i ∈ / are required to follow some of the basic rough set
properties such as:
(P1) An object
v can be part of at most one lower bound
(P2)
v A c
v A c
i
i
( )
( )
∈
⇒ ∈
(P3) An object
v is not part of any lower bound m
v belongs to two or more upper bounds.
Property (P1) emphasizes the fact that a lower bound is included in a set. If two sets are
mutually exclusive, their lower bounds should not overlap. Property (P2) confirms the
fact that the lower bound is contained in the upper bound. Property (P3) is applicable to
the objects in the boundary regions, which are defined as the differences between upper
and lower bounds. The exact membership of objects in the boundary region is ambiguous.
Therefore, property (P3) states that cannot belong to only a single boundary region. Note
that (P1) – (P3) are not necessarily independent or complete. However, enumerating them
will be helpful in understanding the rough set adaptation of evolutionary, neural, and
statistical clustering methods.
2.4.3 Genetic Algorithms
A genetic algorithm is a search process that follows the principles of evolution through
natural selection. The domain knowledge is represented using a candidate solution