Discovery of Positive and Negative Rules from Medical Databases Based on Rough Sets - Advanced Techniques in Knowledge Discovery and Data Mining - page 234

236 Shusaku Tsumoto

The atomic formulas over B ⊆ A ∪{ d } and V are expressions of the form

[a = v], called descriptors over B, where a V a .ThesetF (B, V )

of formulas over B is the least set containing all atomic formulas over B and
closed with respect to disjunction, conjunction, and negation. For example,
[location = occular] is a descriptor of B.
For each f

∈ B and v ∈

F (B, V ), f A denotes the meaning of f in A, i.e., the set of
all objects in U with property f, defined inductively as follows:
1. If f is of the form [a = v], then f A =

∈

{ s ∈ U | a(s)=v } .

2. (f ∧ g) A = f A ∩ g A ;(f ∨ g) A = f A ∨ g A ;( ¬ f) A = U − f a .

For example, f =[location = whole]andf A = { 2, 4, 5, 6 } . As an example of

a conjunctive formula, g =[location = whole] ∧ [nausea = no] is a descriptor

of U and f A is equal to g location,nausea = { 2, 5 } .

9.3.2 Classification Accuracy and Coverage

Definition of Accuracy and Coverage. By use of the preceding frame-
work, classification accuracy and coverage, or true positive rate are defined
as follows.

Definition 9.3.1. Let R and D denote a formula in F (B, V ) and a set of
objects that belong to a decision d. Classification accuracy and coverage(true
positive rate) for R

→ d is defined as:

α R (D)= |

R A ∩ D |

(= P (D | R)) and

| R A |

κ R (D)= |

R A ∩ D |

(= P (R | D)),

| D |

where , α R (D), κ R (D),andP (S) denote the cardinality of a set S,a

classification accuracy of R as to classification of D,andcoverage(atrue
positive rate of R to D), and probability of S, respectively.

| S |

Figure 9.2 depicts the Venn diagram of relations between accuracy and cov-
erage. Accuracy views the overlapped region from the meaning of

arelationR. On the other hand, coverage views the overlapped region from
the meaning of a concept D.
In the preceding example, when R and D are set to [nau = yes]and
[class = migraine], α R (D)=2/3=0.67 and κ R (D)=2/2=1.0.
It is notable that α R (D) measures the degree of the su ciency of a propo-
sition, R

| R A ∩ D |

D,andthatκ R (D) measures the degree of its necessity. For
example, if α R (D) is equal to 1.0, then R

→

→ D is true. On the other hand, if

κ R (D) is equal to 1.0, then D → R is true. Thus, if both measures are 1.0,

then R ↔ D.