![](https://csdnimg.cn/release/download_crawler_static/15902993/bg4.jpg)
Proof. Pð:Xj½x
A
Þ¼1 PðXj½x
A
Þ. Thus,
I
POS
a
;b
A
ðXÞ¼fxjPðXj½x
A
Þ P
a
g;
I
POS
a
;b
A
ð:XÞ¼fxjPð:Xj½x
A
Þ P
a
g¼fxjPðXj½x
A
Þ 6 1
a
g;
I
BND
a
;b
A
ðXÞ¼
fxjb < PðXj½x
A
Þ <
a
g;
I
BND
a
;b
A
ð:XÞ¼fxj1
a
< PðXj½x
A
Þ < 1 bg.
Furthermore, (1) if
a
þ b P 1, then 0 1
a
6 b < 0:5 <
1 b 6
a
1. Thus,
I
BND
a
;b
A
ðDÞ¼fxj1
a
< PðXj½x
A
Þ <
a
g;
I
POS
a
;b
A
ðDÞ\
I
BND
a
;b
A
ðDÞ¼;. (2) If
a
þ b < 1, then 0 6 b < 1
a
<
0:5 <
a
< 1 b 1. Thus,
I
BND
a
;b
A
ðDÞ¼fxjb < PðXj½x
A
Þ < 1 bg;
I
POS
a
;b
A
ðDÞ\
I
BND
a
;b
A
ðDÞ – ;. Hence, the C-Region results are
obtained. h
Corollary 3.2 (Type I’s Structure).
I
POS
a
;b
A
ðDÞ[
I
BND
a
;b
A
ðDÞ¼U;
I
POS
a
;b
A
ðDÞ\
I
BND
a
;b
A
ðDÞ¼
;; if
a
þ b P 1;
fxjb < PðXj½x
A
Þ1
a
g[fxj
a
6 PðXj½x
A
Þ < 1bg – ;; if
a
þ b < 1:
8
>
<
>
:
Corollary 3.3 (Type I’s Expansion).
(1)
I
POS
a
;b
A
ðDÞPOS
A
ðDÞ,
I
BND
a
;b
A
ðDÞ # BND
A
ðDÞ.
(2) When
a
¼ 1;
I
NEG
a
;b
A
ðDÞ¼/,
I
POS
a
;b
A
ðDÞ¼POS
A
ðDÞ;
I
BND
a
;b
A
ðDÞ¼BND
A
ðDÞ.
For Type I, Theorem 3.1 provides a basic formula, Corollary 3.2
reflects systemic structure, while Corollary 3.3 exhibits expansion
for Qualitative Type, where POS
A
ðDÞ¼fxjPðXj½x
A
Þ¼1g[
fxjPð Xj½x
A
Þ¼0g and BND
A
ðDÞ¼fxj0 < PðXj½x
A
Þ < 1g. Next, its
improvement spaces are analyzed.
(1) Type I is directly constructed by its basic element – S-Regions,
thus exhibiting a structural characteristic. However, it relates
NEG to C-NEG rather than C-POS, but NEG more yields classi-
fication certainty. Thus, C-NEG has vague classification
semantics, while C-POS completeness can be improved.
(2) Both C-POS and C-BND are related to only an independent
threshold –
a
or b. In fact, DTRS is a double-threshold system
and the double-threshold representation can exhibit sys-
tematicness and stability. Thus, we need a more systematic
and rational description based on
a
and b.
(3) Type I has an improvement space for systemic structure
because
I
POS
a
;b
A
ðDÞ\
I
BND
a
;b
A
ðDÞ – ; when
a
þ b < 1.
(4) Based on S-Regions, DTRS-Model exhibits point expansion
regarding ð
a
; bÞ¼ð1; 0Þ for Pawlak-Model. However, Type I
corresponds to wider plane expansion regarding
a
¼ 1.
Thus, the essential point expansion is worth inheriting for
C-Regions.
Next, we provide Type II’s degenerate results and improvement
spaces.
Theorem 3.4 (Type II’s Formula).
II
POS
a
;b
A
ðDÞ¼fxjPðXj½x
A
Þ P
a
g[fxjPðXj½x
A
Þ1
a
g;
II
BND
a
;b
A
ðDÞ¼fxj1
a
< PðXj½x
A
Þ <
a
g;
II
NEG
a
;b
A
ðDÞ¼;:
8
>
<
>
:
Proof. Type II is mainly described by MAX ¼ maxðPðXj½x
A
Þ;
Pð:Xj½x
A
ÞÞ ¼ maxðPðXj½x
A
Þ; 1 PðXj½x
A
ÞÞ.
MAX ¼
P ðXj½x
A
Þ; if PðXj½x
A
Þ P 0:5;
1 PðXj½x
A
Þ; if PðXj½x
A
Þ0:5:
Thus, MAX P 0:5 > b,so
II
NEG
a
;b
A
ðDÞ¼fxjMAX 6 bg¼;. Moreover,
II
POS
a
;b
A
ðDÞ¼fxjMAX P ag¼fxjPðXj½x
A
Þ P ag[fxjPðXj½x
A
Þ1 ag,
II
BND
a
;b
A
ðDÞ¼fxjb < MAX < ag¼fxj0:5 6 PðXj½x
A
Þ < ag[fxj1 a
< PðXj½x
A
Þ0:5g¼fxj1 a < PðXj½x
A
Þ < ag. h
Corollary 3.5 (Type II’s Structure).
II
POS
a
;b
A
ðDÞ[
II
BND
a
;b
A
ðDÞ¼U;
II
POS
a
;b
A
ðDÞ\
II
BND
a
;b
A
ðDÞ¼;:
8
<
:
Corollary 3.6 (Type II’s Expansion).
(1)
II
POS
a
;b
A
ðDÞPOS
A
ðDÞ,
II
BND
a
;b
A
ðDÞ # BND
A
ðDÞ.
(2) When
a
¼ 1;
II
NEG
a
;b
A
ðDÞ¼;,
II
POS
a
;b
A
ðDÞ¼POS
A
ðDÞ,
II
BND
a
;b
A
ðDÞ¼BND
A
ðDÞ.
For Type II, Theorem 3.4 provides a basic formula, Corollary 3.5
reflects systemic structure (where C-POS and C-BND construct a
universe division), while Corollary 3.6 exhibits expansion for Qual-
itative Type. Next, the improvement spaces are simply provided by
referring to Type I’s relevant analyzes.
(1) Type II adopts an optimal strategy on decisions and thus has
application optimization. However, it lacks a necessary link
with essential S-Regions; thus, its classification semantics
(especially C-NEG’s) becomes weak and ambiguous.
(2) Both C-POS and C-BND are related to only one independent
threshold –
a
, and this result implies a flaw due to the
neglect of b. Thus, a systemic double-threshold description
is needed.
(3) The essential point expansion is also expected in view of the
same plane expansion.
Types I and II are fundamental for relevant DTRS-Reduction, and
the above studies provide their degenerate results at the two-
category level. First, C-NEG is always empty (though it may be
non-empty in the multi-category case), so there are only two
C-Regions. Second, Type I concerns only
a
if
a
þ b P 1, and
I
POS
a
;b
A
ðDÞ\
I
BND
a
;b
A
ðDÞ – ; if
a
þ b < 1; in contrast, Type II concerns
only
a
. Moreover, they exhibit plane expansion for Qualitative
Type by virtue of only
a
. In view of quantitative essence of DTRS,
a double-threshold description becomes more reasonable; more-
over, a structural division is also needed for C-Regions. According
to these improvement spaces, we next propose an improvement
type at the two-category level by completely considering the
S-Region system.
Definition 3.7 (Type III). Type III of C-Regions is defined by
following C-POS, C-BND, and C-NEG:
POS
a
;b
A
ðDÞ¼POS
a
;b
A
ðXÞ[NEG
a
;b
A
ðXÞ;
BND
a
;b
A
ðDÞ¼BND
a
;b
A
ðXÞ;
NEG
a
;b
A
ðDÞ¼U ðPOS
a
;b
A
ðDÞ[BND
a
;b
A
ðDÞÞ:
8
>
>
>
<
>
>
>
:
ð6Þ
Proposition 2.3 provides S-Region-based C-Regions for Pawlak-
Model, where C-NEG does not exist. In contrast, Definition 3.7 pro-
poses S-Region-based C-Regions for DTRS-Model, so Type III natu-
rally simulates Qualitative Type by promoting S-Regions. Herein,
three S-Regions are completely used and NEG information is
related to C-POS; moreover, C-NEG is always empty but is also
set up for multi-category generalization.
X. Zhang, D. Miao / Knowledge-Based Systems 71 (2014) 146–161
149