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1. 北京工业大学 信息学部, 北京 100124;
2. 计算智能与智能系统北京市重点实验室, 北京 100124;
3. 曼彻斯特大学 计算机科学学院, 曼彻斯特 M13 9PL

Computing and performance analysis of similarity between fuzzy rules
LI Wei1,2, QIAO Junfei1,2, HAN Honggui1,2, ZENG Xiaojun3
1. Faculty of Information Technology, Beijing University of Technology, Beijing 100124, China;
2. Beijing Key Laboratory of Computational Intelligence and Intelligent System, Beijing 100124, China;
3. School of Computer Science, the University of Manchester, Manchester M13 9PL, UK
Abstract: Facing the weaknesses of the existing analysis and computing methods for the similarity between fuzzy rules (FRs), this paper investigated the computing methods for the similarity between FRs. First, the similarity between FSs was transferred equivalently into the similarity between multivariable fuzzy sets, and then three application based performance criterions-distinguishability, dimension dependency, and computing complexity were proposed to evaluate the computing methods of the similarity between FRs. Second, four new methods were proposed based on the two existing methods for computing the similarity between FRs, and then the performance analysis and comparison between these new and existing methods were performed. Next, a simulation example for the similarity computing between FRs was provided, and the simulation shows effectiveness of the proposed performance criteria, feasibility of the computing methods, and correctness of the analysis conclusions. The results obtained in this paper provide powerful tools and guides for the similarity analysis and computing of FRs. Inparticular, they establish the methodological foundation and provide a new design approach for the merging of similar FRs in the structure simplification of fuzzy systems and fuzzy neural networks.
Key words: fuzzy rules     similarity computing     distinguishability     dimension dependency     computing complexity

1 模糊规则相似性计算问题描述 1.1 问题描述

 (1)

 (2)
 (3)

 (4)
 (5)

 (6)

1.2 性能评价指标

1) 正则性：0≤S(A, B)≤1。

2) 对称性：S(A, B)=S(B, A)。

3) 不相交模糊规则的相似度应为0，即

S(A, B)=0 uA (x)uB(x)=0, xU

4) 相交模糊规则的相似度应大于0，即

S(A, B)>0 xU, uA(x)uB(x)>0。

S(A, B)=1 uA(x)=uB(x), xU

5) 缩放或移位下的不变性，即

S(A′, B′)=S(A, B) uA (l+kx)=uA(x),

uB(l+kx)=uB(x), lRn, kR, k>0。

1.2.1 可区分性

1.2.2 维数依赖性

1.2.3 计算复杂性

2 计算方法与性能分析 2.1 最小值方法

 (7)

2.2 乘积方法及其改进方法

 (8)

 (9)

2.3 交并面积和比值法

 (10)

 (11)
 (12)

M(A1B1)=M(A2B2)=0.4

M(A1B1)=M(A2B2)=0.8

M(A1B1)=0.4, M(A1B1)=0.8

M(A2B2)=0.35, M(A2B2)=0.8

2.4 交并总面积比值法及其改进方法

 (13)

 (14)
 (15)
 (16)

 (17)
 (18)

 (19)
 (20)

M(|A~B|) 如式 (16) 所示。由于σA>σB >0，推出

 (21)

 (22)

 (23)

 (24)

 (25)

 (26)

 (27)

2.5 6种方法的性能比较

 方法 可区分性 维度依赖性 计算复杂性 最小值方法 最差 否 简单 乘积方法 较好 是 简单 改进的乘积方法 较好 否 简单 交并面积和比值法 较好 否 简单 交并总面积比值法 较好 是 复杂 改进的交并总面积比值法 较好 否 复杂

3 仿真实验

3.1 实验1

 c/σ x1 x2 x3 R1 2.547 6/1.334 2 1.204 4/4.813 8 2.786 2/1.564 0 R2 3.821 2/3.694 9 1.034 4/ 4.613 8 2.691 1/2.740 7 R3 1.395 2/5.591 3 1.020 4/5.272 2 1.654 0/3.359 7 R4 2.952 3/1.833 6 1.877 4/2.249 8 2.171 8/1.397 7 R5 3.541 6/1.623 0 1.653 0/4.074 5 2.014 4/4.410 1 R6 2.037 7/1.068 5 1.905 7/1.243 1 3.237 6/3.626 4

 M1 M2 M3 M4 M5 M6 S12 0.343 0.186 0.571 0.658 0.210 0.594 S13 0.237 0.094 0.455 0.534 0.100 0.465 S14 0.465 0.200 0.586 0.549 0.357 0.710 S15 0.352 0.137 0.515 0.580 0.234 0.616 S16 0.257 0.068 0.409 0.374 0.195 0.580 S23 0.535 0.318 0.682 0.690 0.402 0.738 S24 0.482 0.118 0.490 0.488 0.123 0.497 S25 0.439 0.227 0.610 0.647 0.387 0.729 S26 0.258 0.051 0.371 0.408 0.084 0.438 S34 0.320 0.056 0.383 0.380 0.058 0.387 S35 0.274 0.158 0.541 0.582 0.237 0.619 S36 0.191 0.027 0.299 0.323 0.042 0.349 S45 0.317 0.119 0.495 0.479 0.174 0.558 S46 0.373 0.095 0.456 0.446 0.243 0.624 S56 0.274 0.059 0.389 0.472 0.109 0.478 时间/s 0.085 0.084 0.092 0.084 376.69 379.32

 x1 x2 x3 S12 0.343 0.951 0.571 S15 0.465 0.836 0.352 S25 0.439 0.840 0.615

3.1.1 可区分性

3.1.2 维数依赖性

3.1.3 计算复杂性

3.2 实验2

 c/σ R1 R2 x1 0.587 0/0.364 3 0.413 9/0.532 3 x2 0.309 1/0.711 7 0.263 8/0.871 5 x3 0.758 8/0.328 7 0.995 2/0.650 1 x4 0.186 6/0.974 8 0.781 1/0.076 0

 S12 时间/s M1 0.066 0.036 M2 0.015 0.037 M3 0.352 0.038 M4 0.463 0.038 M5 0.067 1 359.807 M6 0.508 1 366.569

3.3 实验3

 p1/q/p2 R1 R2 x1 3.5/4/4.5 3/4/5 x2 2.5/4/5.5 2/4/6 x3 1.5/4/6.5 1/4/7

 S12 时间/s M1 0.500 0.037 M2 0.313 0.043 M3 0.679 0.043 M4 0.750 0.050 M5 0.312 1 536.931 M6 0.679 1 564.911

4 结束语

 (A.1)

 (A.2)

 (A.3)

 (A.4)

 (A.5)

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DOI: . 10.11992/tis.201512040

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#### 文章信息

LI Wei, QIAO Junfei, HAN Honggui, ZENG Xiaojun

Computing and performance analysis of similarity between fuzzy rules

CAAI Transactions on Intelligent Systems, 2017, 12(1): 124-131
. http://dx.doi.org/10.11992/tis.201512040