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1. 郑州大学 数学与统计学院, 河南 郑州 450001;
2. 河南财经政法大学, 河南 郑州 450046;
3. 河南师范大学 计算机与信息工程学院, 河南 新乡 453007;
4. 新乡学院 数学与信息科学学院, 河南 新乡 453007

Triangular fuzzy number decision-theoretic rough sets under incomplete information systems
LI Yage1,4, YANG Hongzhi2, XU Jiucheng3
1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China ;
2. Henan University of Economics and Law, Zhengzhou, Zhengzhou 450046, China ;
3. College of Computer and Information Engineering, Henan Normal University, Xinxiang 453007, China ;
4. Department of Mathematics and Information Science, Xinxiang University, Xinxiang 453007, China
Abstract: Aiming at the problems that when using an interval value to represent an unknown parameter in an incomplete information system, the opportunity to obtain the value over the whole interval is considered to be equal, but the result may cause an over-large error. In order to solve this problem, a triangular fuzzy number was introduced into decision-theoretic rough sets, and a triangular fuzzy decision-theoretic rough set under incomplete information systems is proposed. Firstly, a new similarity relation was defined to describe incomplete information systems. Then, in view of the missing values, a model of triangular fuzzy number decision-theoretic rough sets was constructed to obtain the loss function. Finally, examples show that the proposed method not only makes up for deficiency in representation of the interval value, but also highlights the main value most likely to reduce the classification error.
Key words: incomplete information system     interval value     triangular fuzzy number     decision-theoretic rough sets

1 基础知识 1.1 决策粗糙集

 (1)

P) 若R(aP|[x])≤R(aB|[x])和R(aP|[x])≤R(aN|[x]同时成立，那么x∈POS(X)；

B) 若R(aB|[x])≤R(aP|[x])和R(aB|[x])≤R(aN|[x])同时成立，那么x∈BND(X)；

N) 若R(aN|[x])≤R(aP|[x])和R(aN|[x])≤R(aB|[x]同时成立，那么x∈NEG(X)。

 (2)

P′)若P(X|[x])≥αP(X|[x])≥γ，则x∈POS(X)；

B′)若P(X|[x])≤αP(X|[x])≥β，则x∈BND(X)；

N′)若P(X|[x])≤βP(X|[x])≤γ，则x∈NEG(X)。

1.2 三角模糊数

 (3)

l=mm=u时，三角模糊数就转变为区间数，由此可见区间数是三角模糊数的一个特例。在区间数取值中，上下限的各个取值可以认为是机会均等的，而在三角模糊数区间取值中，主值a的取值机会最大，而由a靠近上限、下限的取值可能性递减。

1) a1+a1=(l1+ l2, m1+m2, u1+u2)；

2) a1-a2=(l1-l2, m1-m2, u1-u2)；

3) a1 a2=(l1 l2, m1 m2, u1 u2)；

4) a1/a2=(l1/u2, m1/m2, u1/l2)；

5) λ a2=(λ l2, λ m2, λ u2), λRλ >0。

2 基于IIS的三角模糊数决策粗糙集 2.1 不完备信息系统

2.2 相似度及相关知识

 (4)

 (5)

 (6)

 (7)
 (8)

 (9)
2.3 整数值排序法

 (10)

2.4 基于IIS的三角模糊数决策粗糙集的模型实现

 X(P) ﹁X(P) ap λPP=(lPP，mPP，uPP) λPN=(lPN，mPN，uPN) aB λBP=(lBP，mBP，uBP) λBN=(lBN，mBN，uBN) aN λNP=(lNP，mNP，uNP) λNN=(lNN，mNN，uNN)

 (11)

P1)若R(aP|[x])≤R(aB|[x])和R(aP|[x])≤R(aN|[x])同时成立，那么x∈POS(X)；

B1)若R(aB|[x])≤R(aP|[x])和R(aB|[x])≤R(aN|[x])同时成立，那么x∈BND(X)；

N1)若R(aN|[x])≤R(aP|[x])和R(aN|[x])≤R(aB|[x])同时成立，那么x∈NEG(X)。

P2)若r(R(aP|[x]SRL))≤r(R(aB|[x]SRL))和r(R(aP|[x]SRL))≤r(R(aN|[x]SRL))同时成立，那么x∈POS(X)；

B2)若r(R(aB|[x]SRL))≤r(R(aP|[x]SRL))和r(R(aB|[x]SRL))≤r(R(aN|[x]SRL))同时成立，那么x∈BND(X)；

N2)若r(R(aN|[x]SRL))≤r(R(aP|[x]SRL))和r(R(aN|[x]SRL))≤r(R(aB|[x]SRL))同时成立，那么x∈NEG(X)。

 (12)

ρ=0，对于悲观决策者，其阈值可以表达为

P′2)若Pr(X|[x]SRL)≥α，则x∈POS(X)；

B′2)若β < Pr(X|[x]SRL) < α，则x∈BND(X)；

N′2)若Pr(X|[x]SRL)≤β，则x∈NEG(X)。

OP′2)若Pr(X|[x]SRL)≥α，则x∈POS(X)；

OB′2)若β < Pr(X|[x]SRL) < α，则x∈BND(X)；

ON′2)若Pr(X|[x]SRL)≤β，则x∈NEG(X)。

PP′2)若Pr(X|[x]SRL)≥α，则x∈POS(X)；

PB′2)若β < Pr(X|[x]SRL) < α，则x∈BND(X)；

PN′2)若Pr(X|[x]SRL)≤β，则x∈NEG(X)。

P3)若Pr(X|[x]SRL)≥γ，则x∈POS(X)；

N3)若Pr(X|[x]SRL)≤γ，则x∈NEG(X)。

3 案例分析

 U a1 a2 a3 a4 a5 a6 a7 x1 1 1 * 1 1 1 3 x2 3 2 2 3 2 2 * x3 2 * 2 2 1 2 2 x4 2 1 2 * 1 1 3 x5 2 1 1 2 1 * 2 x6 2 2 2 * 2 2 * x7 1 1 2 1 * 2 3 x8 1 1 * 1 1 * 3 x9 2 1 1 1 1 1 * x10 3 2 * 2 2 2 3

 U λPP λBP λNP λPN λBN λNN x1 [0, u, 1.5u] [2u, 3u, 3.5u] [4u, 5u, 7u] [u1.5u, 2u] [2.5u, 3u, 4.5u] [5u, 7u, 8u] x2 [3u, 3.2u, 3.5u] [4u, 5u, 5.5u] [6u, 6.5u, 7.5u] [0, u, 2u] [2u, 3u, 3.5u] [4u, 6u, 7u] x3 [u, 2u, 3.5u] [3.5u, 4u, 4.5u] [4.5u, 5u, 6u] [0.5u, u, 2u] [2.5u, 3u, 4u] [4.5u, 5u, 7.5u] x4 [u, 2u, 2.5u] [3u, 3.5u, 4u] [5u, 6u, 7.5u] [u, 2u, 3u] [3.5u, 4u, 5u] [5.5u, 6u, 7u] x5 [0.5u, u, 1.5u] [2u, 3u, 4u] [4.5u, 5u, 6.5u] [1.5u, 2u, 3u] [3.5u, 3.8u, 4u] [4.5u, 5u, 7u] x6 [u, 2u, 3.5u] [3.5u, 4u, 5u] [5u, 6u, 7.5u] [0.5u, u, 2u] [2.5u, 3u, 4u] [2.5u, 3u, 4u] x7 [u, 1.5u, 2u] [3u, 3.2u, 3.5u] [4.5u, 5u, 7u] [u, 2u, 3.5u] [4u, 5u, 6u] [6.5u, 6.8u, 7u] x8 [0.5u, u, 1.5u] [2u, 3u, 3.5u] [4u, 5u, 7.5u] [1.5u, 2u, 3u] [4u, 5u, 5.5u] [6u, 7u, 7.5u] x9 [0, u, 1.5u] [2u, 3u, 4u] [5u, 6u, 7u] [u, 2u, 3.5u] [4u, 4.2u, 4.5u] [5u, 6u, 7.5u] x10 [3u, 3.5u, 4u] [4.5u, 5u, 5.5u] [6u, 6.5u, 7u] [0.5u, u, 2u] [2.5u, 3u, 3.5u] [4u, 5u, 6u]

 S(xi, xj) x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x1 1.00 0.14 0.28 0.69 0.43 0.19 0.71 0.85 0.64 0.28 x2 0.14 1.00 0.42 0.26 0.14 0.69 0.42 0.21 0.14 0.64 x3 0.28 0.42 1.00 0.55 0.71 0.62 0.43 0.36 0.43 0.50 x4 0.69 0.26 0.55 1.00 0.55 0.40 0.55 0.62 0.62 0.33 x5 0.43 0.14 0.71 0.55 1.00 0.33 0.28 0.43 0.64 0.36 x6 0.19 0.69 0.62 0.40 0.33 1.00 0.79 0.26 0.19 0.55 x7 0.71 0.42 0.43 0.55 0.28 0.48 1.00 0.79 0.36 0.50 x8 0.85 0.21 0.36 0.32 0.43 0.26 0.79 1.00 0.57 0.36 x9 0.64 0.14 0.43 0.62 0.64 0.19 0.36 0.57 1.00 0.21 x10 0.28 0.64 0.50 0.36 0.36 0.55 0.50 0.36 0.21 1.00

 U 乐观决策者 悲观决策者 α1 β1 γ1 α1 β1 γ1 x1 0.619 0 0.571 4 0.542 9 0.652 2 0.421 1 0.547 6 x2 0.428 6 0.428 6 0.588 2 0.631 1 0.500 0 0.578 0 x3 0.533 3 0.433 3 0.551 7 0.647 1 0.615 4 0.633 3 x4 0.666 7 0.666 7 0.515 2 0.571 4 0.400 0 0.470 6 x5 0.500 0 0.500 0 0.428 6 0.482 8 0.383 6 0.437 5 x6 0.457 8 0.457 8 0.500 0 0.631 6 0.470 6 0.555 6 x7 0.533 3 0.533 3 0.595 4 0.466 7 0.509 3 0.494 0 x8 0.645 2 0.645 2 0.558 8 0.500 0 0.478 3 0.487 2 x9 0.578 9 0.578 9 0.444 4 0.516 1 0.347 8 0.432 4 x10 0.464 3 0.464 3 0.555 6 0.600 0 0.538 5 0.571 4

 U 乐观决策者 悲观决策者 x1 POS(X) POS(X) x2 NEG(X) NEG(X) x3 BND(X) NEG(X) x4 POS(X) POS(X) x5 POS(X) POS(X) x6 NEG(X) NEG(X) x7 POS(X) POS(X) x8 POS(X) POS(X) x9 POS(X) POS(X) x10 NEG(X) NEG(X)

 L值 乐观决策者 悲观决策者 0.5 POS(X) x1, x4, x5, x7, x8, x9 x1, x4, x5, x7, x8, x9 NEG(X) x2, x6, x10 x2, x3, x6, x10 BND(X) x3 0.6 POS(X) x1, x4, x7, x8, x9 x1, x4, x7, x8, x9 NEG(X) x2, x3, x5, x6, x10 x2, x3, x5, x6, x10 BND(X) 0.7 POS(X) x1, x4, x5, x7, x8 x1, x4, x5, x7, x8 NEG(X) x2, x9, x10 x2, x9, x10 BND(X) x3, x6 x3, x6 0.8 POS(X) x1, x4, x5, x7, x8 x1, x4, x5, x7, x8 NEG(X) x2, x3, x6, x9, x10 x2, x3, x6, x9, x10 BND(X) 0.9 POS(X) x1, x4, x5, x7, x8 x1, x4, x5, x7, x8 NEG(X) x2, x3, x6, x9, x10 x2, x3, x6, x9, x10 BND(X) 1.0 POS(X) x1, x4, x5, x7, x8 x1, x4, x5, x7, x8 NEG(X) x2, x3, x6, x9, x10 x2, x3, x6, x9, x10 BND(X)

 U 乐观决策者 悲观决策者 α1 β1 γ1 α1 β1 γ1 x1 0.500 0 0.250 0 0.400 0 0.500 0 0.250 0 0.333 3 x2 0.833 3 0.333 3 0.583 3 0.833 3 0.333 3 0.625 0 x3 0.285 7 0.500 0 0.333 3 0.285 7 0.500 0 0.470 3 x4 0.666 7 0.555 6 0.600 0 0.666 7 0.555 6 0.500 0 x5 0.566 7 0.800 0 0.554 5 0.566 7 0.800 0 0.600 0 x6 0.574 2 0.500 0 0.533 3 0.574 2 0.500 0 0.666 7 x7 0.855 6 0.666 7 0.611 1 0.855 6 0.666 7 0.400 0 x8 0.250 0 0.200 0 0.222 2 0.250 0 0.200 0 0.350 0 x9 0.872 9 0.750 0 0.636 3 0.872 9 0.750 0 0.571 4 x10 0.333 3 0.142 8 0.200 0 0.333 3 0.142 8 0.333 3

 U 乐观决策者 悲观决策者 x1 POS(X) POS(X) x2 NEG(X) NEG(X) x3 POS(X) POS(X) x4 POS(X) POS(X) x5 BND(X) NEG(X) x6 NEG(X) NEG(X) x7 POS(X) POS(X) x8 POS(X) POS(X) x9 POS(X) POS(X) x10 NEG(X) NEG(X)

 L 乐观决策者 悲观决策者 0.5 POS(X) x1, x3, x4, x7, x8, x9 x1, x3, x4, x7, x8, x9 NEG(X) x2, x6, x10 x2, x5, x6, x10 BND(X) x5 0.6 POS(X) x1, x4, x5, x7, x8, x9 x1, x4, x5, x7, x8, x9 NEG(X) x2, x6, x10 x2, x3, x6, x10 BND(X) x3 0.7 POS(X) x1, x4, x7, x8, x9 x1, x3, x4, x5, x7, x8 NEG(X) x2, x6, x10 x2, x6, x10 BND(X) x3, x5 x9 0.8 POS(X) x1, x4, x5, x7, x8, x9 x1, x4, x5, x7, x8, x9 NEG(X) x2, x3, x6, x10 x2, x3, x6, x10 BND(X) 0.9 POS(X) x1, x4, x5, x7, x8, x9 x1, x4, x5, x7, x8, x9 NEG(X) x2, x3, x6, x10 x2, x3, x6, x10 BND(X) 1.0 POS(X) x1, x4, x5, x7, x8, x9 x1, x4, x5, x7, x8, x9 NEG(X) x2, x3, x6, x10 x2, x3, x6, x10 BND(X)

4 结束语

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DOI: 10.11992/tis.201606016

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#### 文章信息

LI Yage, YANG Hongzhi, XU Jiucheng

Triangular fuzzy number decision-theoretic rough sets under incomplete information systems

CAAI Transactions on Intelligent Systems, 2016, 11(4): 449-458
http://dx.doi.org/10.11992/tis.201606016