Database Reference
In-Depth Information
250
Shusaku Tsumoto
Interestingly, from the set-theoretic point of view, sets of examples sup-
porting these rules correspond to the lower and upper approximations in
rough sets. Furthermore, from the viewpoint of propositional logic, both in-
clusive and exclusive rules are defined as classical propositions, or determin-
istic rules with two probabilistic measures, classification accuracy and cover-
age. Total covering rules have several interesting relations with inclusive and
exclusive rules, which reflects the characteristics of medical reasoning.
Originally, a total covering rule is defined as a set of symptoms that can
be observed in at least one case of a target disease. That is, this rule is defined
as a collection of attribute-value pairs whose accuracy is larger than 0:
∨
j
[a
j
= v
k
], α
R
(D) > 0.
From the definition of accuracy and coverage, this formula can be transformed
into:
R
→
d
s.t.
R =
∨
j
[a
j
= v
k
],
R
(D) > 0.
For each attribute, the attribute-value pairs form a partition of U. Thus,
for each attribute, total covering rules include a covering of all the positive
examples. According to this property, the preceding formula is redefined as:
R
→
d
s.t.
R =
∨
k
[a
j
= v
k
] s.t.κ
R
(D)=1.0.
It is notable that this definition is an extension of exclusive rules and this
total covering rule can be written as:
R
→
d
s.t.
R =
∨
j
R(a
j
),
R(a
j
)=
→∨
j
∨
k
[a
j
= v
k
]s.t.κ
[a
j
=v
k
]
(D)=1.0.
Let S(R) denote a set of attribute-value pairs of rule R. For each class d,
let R
pos
(d), R
ex
(d), and R
tc
(d) denote the positive exclusive rule and total
covering rules, respectively. Then
d
S(R
tc
(d)),
because a total covering rule can be viewed as an upper approximation of
exclusive rules. It is also notable that this relation will hold in the relation
between the positive rule (inclusive rule) and total covering rule. That is,
S(R
ex
(d))
⊆
S(R
tc
(d)).
Thus, the total covering rule can be viewed as an upper approximation of
inclusive rules. This relation also holds when R
pos
(d) is replaced with a prob-
abilistic rule, which shows that total covering rules are the weakest form of
diagnostic rules.
In this way, rules that reflect the diagnostic reasoning of medical experts
have sophisticated background from the viewpoint of set theory. Especially,
the rough set framework provides a good tool for modeling such focusing
mechanisms. Our future work will investigate the relation between these three
types of rules from the viewpoint of rough set theory.
S(R
pos
(d))
⊆
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