JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (9): 94-104.doi: 10.6040/j.issn.1671-9352.0.2022.430

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Incomplete neighborhood weighted multi-granularity decision-theoretic rough sets and three-way decision

Qian WANG1,2(),Xianyong ZHANG1,2,*()   

  1. 1. School of Mathematical Sciences, Sichuan Normal University, Chengdu 610066, Sichuan, China
    2. Institute of Intelligent Information and Quantum Information, Sichuan Normal University, Chengdu 610066, Sichuan, China
  • Received:2022-08-19 Online:2023-09-20 Published:2023-09-08
  • Contact: Xianyong ZHANG E-mail:2055778250@qq.com;xianyongzh@sina.com

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Abstract:

Concerning the differences and imbalances of granularities, the classification capacity of granularity is considered to mine the granularity weight, and two rough set models based on the granularity weighting strategy are constructed to implement three-way decision. At first, the granularity significance is defined by the influence of boundary region for knowledge classification, and it induces the granularity weight; by fusing the granularity weights and condition probabilities, incomplete neighborhood weighted multi-granularity decision-theoretic rough sets are modeled to establish three-way decision. Then considering the specific restriction of important attribute, incomplete neighborhood weighted-restrictive multi-granularity decision-theoretic rough sets are further proposed, and relevant properties and mutual relationships are acquired. At last, example demonstrations and data experiments are performed by variable three-way decision, and the rationality and superiority of new models are verified. Regarding incomplete neighborhood multi-granularity decision-theoretic rough sets, the two weighted models optimally improve and systematically extend corresponding basic models, and they facilitate relevant data analysis and decision making.

Key words: incomplete neighborhood information system, multi-granularity rough set, decision-theoretic rough set, three-way decision, granularity significance

CLC Number: 

  • TP18

Table 1

Information sheet of three good student assessment"

U A B C class
a1 a2 b1 b2 c1 c2
u1 0.47 0.66 0.76 0.61 0.43 0.31 0
u2 0.73 0.62 0.54 0.76 0.56 0.44 1
u3 0.21 * 0.35 0.26 0.46 0.59 0
u4 0.76 0.61 0.87 0.67 0.26 0.19 1
u5 0.45 0.54 0.67 * 0.75 0.66 0
u6 0.88 0.78 0.34 0.47 0.56 0.62 1
u7 0.17 0.23 0.45 0.36 0.45 0.52 0
u8 0.95 0.86 0.76 0.64 0.67 * 1
u9 0.65 0.57 * 0.57 0.34 0.27 0
u10 0.87 0.81 0.88 0.76 0.47 0.58 1
u11 * 0.64 0.62 0.59 0.54 0.62 0
u12 0.31 0.46 0.46 0.71 0.74 0.65 1

Table 2

Cost function of three good student assessment"

Γ X ~X
aP $\tilde{\lambda}_{\mathrm{PP}}=[0，0.2]$ $\tilde{\lambda}_{\mathrm{PN}}=[0.8，1.2]$
aB $\tilde{\lambda}_{\mathrm{BP}}=[0.2，0.6]$ $\tilde{\lambda}_{\mathrm{BN}}=[0.3，0.6]$
aN $\tilde{\lambda}_{\mathrm{NP}}=[0.8，1.2]$ $\tilde{\lambda}_{\mathrm{NN}}=[0.2，0.4]$

Table 3

Three-way regions of optimistic, pessimistic and average models of multi-granularity decision-theoretic rough sets"

模型 k值 POS(X)；NEG(X)；BND(X)
乐观 0.0 {u2, u4, u6, u8, u10}；{u1, u3, u5, u7, u9, u11, u12}；Ø
0.2 {u2, u4, u6, u8, u10}；{u1, u3, u5, u7, u9, u11, u12}；Ø
0.4 {u2, u4, u6, u8, u10}；{u1, u3, u5, u7, u9, u11, u12}；Ø
0.6 {u2, u4, u6, u8, u10}；{u1, u3, u5, u7, u9, u11, u12}；Ø
0.8 {u2, u4, u6, u8, u10}；{u1, u3, u5, u7, u9, u11, u12}；Ø
1.0 {u2, u4, u6, u8, u10}；{u1, u3, u5, u7, u9, u11, u12}；Ø
悲观 0.0 Ø；{u2, u4, u6, u8, u10, u12}；{u1, u3, u5, u7, u9, u11}
0.2 Ø；{u2, u4, u6, u8, u10, u12}；{u1, u3, u5, u7, u9, u11}
0.4 {u4}；{u2, u6, u8, u10, u12}；{u1, u3, u5, u7, u9, u11}
0.6 {u4}；{u2, u6, u8, u10, u12}；{u1, u3, u5, u7, u9, u11}
0.8 {u4}；{u2, u6, u8, u10, u12}；{u1, u3, u5, u7, u9, u11}
1.0 {u4}；{u2, u6, u8, u10, u12}；{u1, u3, u5, u7, u9, u11}
平均 0.0 {u2, u4, u6, u10}；{u1, u5, u8, u9, u11, u12}；{u3, u7}
0.2 {u2, u4, u6, u10}；{u5, u8, u9, u11, u12}；{u1, u3, u7}
0.4 {u2, u4, u6, u10}；{u5, u8, u9, u11, u12}；{u1, u3, u7}
0.6 {u2, u4, u6, u8, u10}；{u5, u9, u11, u12}；{u1, u3, u7}
0.8 {u2, u4, u6, u8, u10}；{u5, u11, u12}；{u1, u3, u7, u9}
1.0 {u2, u4, u6, u8, u10}；{u5, u11, u12}；{u1, u3, u7, u9}

Table 4

Three-way regions of two weighted models of multi-granularity decision-theoretic rough sets"

模型 k值 $\left(\operatorname{Sig}_{A}(X)，\operatorname{Sig}_{B}(X)，\operatorname{Sig}_{C}(X)\right)$ $\operatorname{POS}(X) ; \operatorname{NEG}(X) ; \operatorname{BND}(X)$
加权 0.0 (0.5, 0.5, 0.333) {u2, u4, u6, u10}；{u3, u5, u8, u9, u11, u12}；{u1, u7}
0.2 (0.5, 0.5, 0.333) {u2, u4, u6, u10}；{u5, u8, u9, u11, u12}；{u1, u3, u7}
0.4 (0.583, 0.5, 0.667) {u2, u4, u6, u8, u10}；{u5, u9, u11, u12}；{u1, u3, u7}
0.6 (0.583, 0.5, 0.667) {u2, u4, u6, u8, u10}；{u5, u11, u12}；{u1, u3, u7, u9}
0.8 (0.333, 0.286, 0.381) {u2, u4, u6, u8, u10}；{u5, u11, u12}；{u1, u3, u7, u9}
1.0 (0.32, 0.32, 0.36) {u2, u4, u6, u8, u10}；{u5, u11, u12}；{u1, u3, u7, u9}
加权限制 0.0 (0.5, 0.5, 0.333) {u2, u4, u10}；{u5, u8, u12}；{u1, u3, u6, u7, u9, u11}
0.2 (0.5, 0.5, 0.333) {u2, u4, u10}；{u5, u8, u12}；{u1, u3, u6, u7, u9, u11}
0.4 (0.583, 0.5, 0.667) {u2, u4, u8, u10}；{u5, u12}；{u1, u3, u6, u7, u9, u11}
0.6 (0.583, 0.5, 0.667) {u2, u4, u8, u10}；{u5, u12}；{u1, u3, u6, u7, u9, u11}
0.8 (0.333, 0.286, 0.381) {u2, u4, u8, u10}；{u5, u12}；{u1, u3, u6, u7, u9, u11}
1.0 (0.32, 0.32, 0.36) {u2, u4, u8, u10}；{u5, u12}；{u1, u3, u6, u7, u9, u11}

Table 5

UCI datasets description"

编号 数据集 对象数 条件属性数 决策类数
(a) Wine 178 13 3
(b) Wpbc 198 33 2
(c) Wdbc 569 30 2
(d) Sonar 208 60 2

Table 6

Cost function"

Γ X ~X
aP $\tilde{\lambda}_{\mathrm{PP}}=[0，0]$ $\tilde{\lambda}_{\mathrm{PN}}=[0.9，1.1]$
aB $\tilde{\lambda}_{\mathrm{BP}}=[0.15，0.35]$ $\tilde{\lambda}_{\mathrm{BN}}=[0.02，0.22]$
aN $\tilde{\lambda}_{\mathrm{NP}}=[0.26，0.46]$ $\tilde{\lambda}_{\mathrm{NN}}=[0，0]$

Fig.1

Positive region size variation of three models of multi-granularity decision-theoretic rough sets"

Fig.2

Boundary region size variation of three models of multi-granularity decision-theoretic rough sets"

Fig.3

Negative region size variation of three models of multi-granularity decision-theoretic rough sets"

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