Some Applications and Progress of Set Pair Theory in Artificial Intelligence
-
摘要: 集对论(SPT)把事物所在的时空视为一个既确定又不确定(D-U)时空，把事物的确定性与不确定性作为一个确定不确定系统处理，对不确定性“客观承认、系统描述、定量刻画、具体分析、实践检验”，在应用中不断发展。该文综述集对(SP)及其联系数(CN)的来源与性质，集对论的成对原理与不确定原理、不确定性系统理论与同异反系统理论和基本算法之后，概述集对论在智能定义、航天数据快速评估和多雷达信号分选、复杂系统智能预测、不确定性智能决策、以及自然数的联系数化与群体智能测算等涉及人工智能基础方面的若干应用，简介集对论在智能算法创新方面的若干进展，包括偏联系数计算与联系数系统能守恒计算在内的绿色智能计算等；期待“集对论+非集对论”集成的绿色智能算法在新一代人工智能中得到更多应用。Abstract: Set Pair Theory(SPT) regards the spacetime of things as a Deterministic Uncertainty(D-U) spacetime which is both definite and uncertain, treats certainty and uncertainty of things as a system of certainty and uncertainty, and “Objective recognition, systematic description, quantitative description, concrete analysis and practical test” of uncertainty, in the application of continuous development. After reviewing the source and property of Set Pair (SP) and its Connection Number (CN), the pairwise principles and uncertainty principle of set pair theory, the uncertainty system theory and the theory of similarities and differences, and the basic algorithms; some applications of set pair theory in intelligent definition, space data rapid evaluation and multi-radar signal sorting, intelligent prediction of complex systems, intelligent decision-making under uncertainty, connection digitalization of natural numbers and intelligent calculation of groups are summarized. This paper briefly introduces some progresses of set pair theory in the field of intelligent algorithm innovation, including the green intelligent computation involving the calculation of partial connection coefficient and the conservation of system energy of connection number, etc.. It is expected that the green intelligent algorithm based on “set-to-theory non-set-to-theory” integration will be more applied in the new generation of artificial intelligence.
-
表 1 以2个2元联系数与3元联系数为例的普通四则运算
联系数 加法 减法 乘法 除法 2元联系数
$ u = A + Bi $$ \begin{gathered} \left( {A1 + B1i} \right) + \left( {A2 + B2i} \right) \\ = \left( {A1 + A2} \right) + \left( {B1 + B2} \right)i \\ \end{gathered} $ $ \begin{gathered} \left( {A1 + B1i} \right) - \left( {A2 + B2i} \right) \\ = \left( {A1 - A2} \right) + \left( {B1 - B2} \right)i \\ \end{gathered} $ $ \begin{aligned} & \left( {A1 + B1i} \right) \times \left( {A2 + B2i} \right) \\ & = \left( {A1A2} \right) \\ & \quad + \left( {A1B2 + A2B21} \right)i \\ & \quad + B1B2{i^2} \end{aligned} $ $ \begin{aligned}& \left( {A1 + B1i} \right)/\left( {A2 + B2i} \right) \\& = \left( {A1/A2} \right) + \\& \quad [\left( {A2B1 - A1B2} \right) \\& \quad /A2\left( {A2 + B2} \right)]i \end{aligned} $ 3元联系数
$ u = A + Bi + Cj $$ \begin{aligned} & \left( {A1 + B1i + C1j} \right) \\ & \quad+ \left( {A2 + B2i + C2j} \right) \\ & = \left( {A1 + A2} \right) + \left( {B1 + B2} \right)i \\& \quad + \left( {C1 + C2} \right)j \end{aligned} $ $ \begin{aligned}& \left( {A1 + B1i + C1j} \right) \\& \quad- \left( {A2 + B2i + C2j} \right) \\& = \left( {A1 - A2} \right) + \left( {B1 - B2} \right)i \\& \quad+ \left( {C1 - C2} \right)j \end{aligned} $ $ \begin{aligned}& \left( {A1 + B1i + C1j} \right) \\& \quad\times \left( {A2 + B2i + C2j} \right) \\& = A1A2 + \left( {A1B2 - A2B1} \right)i \\& \quad+ \left( {A1C2 + A2C1} \right)j \\& \quad+ B1B{\text{2}}{i^2} + \left( {B1C2 + B2C1} \right)ij \\& \quad + C1C2{j^2} \end{aligned} $ $ \begin{aligned} & \left( {A1 + B1i + C1j} \right) \\& \quad /\left( {A2 + B2i + C2j} \right) \\& = \left( {A3 + B3i + C3j} \right) \end{aligned} $ $ \begin{gathered} \left( {A1 + B1i + C1j} \right)/\left( {A2 + B2i + C2j} \right) = \left( {A3 + B3i + C3j} \right) \\ \left[ \begin{gathered} A3 \\ B3 \\ C3 \\ \end{gathered} \right] = \left[ \begin{gathered} A2,o,[B2A2/(A2 + C2)] + [C3A2/(A2 + B2)] \\ B2,1,[B2A2/(B2 + C2)] + [C2B2/(A2 + B2)] \\ C2,o,[A2C2/(B2 + C2)] + [B2C2/(A2 + C2)] \\ \end{gathered} \right].\left[ \begin{gathered} A1 \\ B1 \\ C1 \\ \end{gathered} \right] \\ \end{gathered} $ ··· ··· ··· ··· ··· 
下载: 导出CSV
表 2 以2元联系数为例的向量运算
联系数 三角函数表达式 模 辐角 乘法运算 2元联系数
$ u = A + Bi $$\mu = r(\cos \theta + i\sin \theta )$ $ r = \sqrt {{A^2} + {B^2}} $ $ \theta = \arctan \dfrac{B}{A} $ ${\mu _1} = {r_1}(\cos {\theta _1} + i\sin {\theta _1})$
${\mu _2} = {r_2}(\cos {\theta _2} + i\sin {\theta _2})$，则
$ \begin{aligned} & \mu 1\mu 2 = r1r2 \\& = [\cos \left( {\theta 1 + \theta 2} \right) + i\sin \left( {\theta 1 + \theta 2} \right)] \end{aligned} $
下载: 导出CSV
表 3 赵森烽克勤概率的常见运算
赵森烽-克勤概率的
一般表达式普通四则运算 期望值计算 基于赵森烽克勤概率的贝叶斯公式 $ \begin{gathered} P(A,\mathop A\limits^ - ) = P(A) + P(\mathop A\limits^ - )i \\ 0 \le p(A) \le 1 \\ 0 \le P(\mathop A\limits^ - ) \le 1 \\ P(A) + P(\mathop A\limits^ - ) = 1 \\ \end{gathered} $ 参照表1中2元联系数的
普通四则运算。$E(X,\mathop X\limits^ - ) = E(X) + E(\mathop X\limits^ - )i$当$ X $作为主事件时，则有
${\mathrm{Ec}}(X) = E(X) + E(\mathop X\limits^ - )i$；
当$ \mathop X\limits^ - $作为主事件时，则有
$ {\mathrm{Ec}}(\mathop X\limits^ - ) = E(\mathop X\limits^ - ) + E(X)i $$ \begin{gathered} {\mathrm{Pc}}(\left. {Ak} \right|B) = \frac{{P(Ak)P(B\left| {Ak} \right.)}}{{\sum\limits_j {P(Aj)P(B\left| {Aj} \right.)} }} \\ + \left\{ {1 - \frac{{P(Ak)P(B\left| {Ak} \right.)}}{{\sum\limits_j {P(Aj)P(B\left| {Aj} \right.)} }}} \right\}i \\ \end{gathered} $
下载: 导出CSV
表 4 集对分析对REEP的修改效果
个例序号 Ｙ 概率回归
预报概率回归
预报评定集对分析
概率预报集对分析
预报评定6 1 1 √ 0 × 9 0 1 × 0 √ 21 0 1 × 0 √ 22 0 1 × 0 √ 24 0 1 × 0 √ 30 0 1 × 0 √ 33 0 1 × 0 √ 42 1 0 × 1 √ 50 0 1 × 0 √ 54 0 1 × 0 √ 67 0 1 × 0 √ 89 0 1 × 0 √
下载: 导出CSV
表 5 3元联系数$ \mu = a + bi + cj,(a,b,c \in [0,1],a + b + c = 1,i \in [ - 1,1],j = - 1) $的态势排序
序号 $ a $, $ b $,$ c $ 大小关系 3元联系数的系统态势 同异反态势区 1 $ a > c $ $ a > b $ $ b > c $ 同势1级 强同势1级 整体由同势主导 同
势
区2 $ a > c $ $ a > b $ $ b = c $ 同势2级 强同势2级 整体由同势主导 3 $ a > c $ $ a > b $ $ b < c $ 同势3级 强同势3级 整体由同势主导 4 $ a > c $ $ a = b $ $ b > c $ 同势4级 弱同势 整体同势弱 $ a > c $ $ a = b $ $ b = c $ 非逻辑式 $ a > c $ $ a = b $ $ b < c $ 非逻辑式 5 $ a > c $ $ a < b $ $ b > c $ 同势5级 微同势 整体同势微 $ a > c $ $ a < b $ $ b = c $ 非逻辑式 $ a > c $ $ a < b $ $ b < c $ 非逻辑式 $ a = c $ $ a > b $ $ b > c $ 非逻辑式 均
势
区$ a = c $ $ a > b $ $ b = c $ 非逻辑式 6 $ a = c $ $ a > b $ $ b < c $ 均势1级 强均势 对立同一相对持平 $ a = c $ $ a = b $ $ b > c $ 非逻辑式 7 $ a = c $ $ a = b $ $ b = c $ 均势2级 准均势 对立同一均势临界 $ a = c $ $ a = b $ $ b < c $ 非逻辑式 8 $ a = c $ $ a < b $ $ b > c $ 均势3级 微均势 不确定主导下的均势 $ a = c $ $ a < b $ $ b = c $ 非逻辑式 $ a = c $ $ a < b $ $ b < c $ 非逻辑式 $ a < c $ $ a > b $ $ b > c $ 非逻辑式
反
势
区$ a < c $ $ a > b $ $ b = c $ 非逻辑式 9 $ a < c $ $ a > b $ $ b < c $ 反势1级 微反势 整体微对立势 $ a < c $ $ a = b $ $ b > c $ 非逻辑式 $ a < c $ $ a = b $ $ b = c $ 非逻辑式 10 $ a < c $ $ a = b $ $ b < c $ 反势2级 弱反势 11 $ a < c $ $ a < b $ $ b > c $ 反势3级 强反势1级 ，强不确定主导下的对立势 12 $ a < c $ $ a < b $ $ b = c $ 反势4级 强反势2级，弱不确定主导下的对立势 13 $ a < c $ $ a < b $ $ b < c $ 反势5级 强反势3级，对立为主导趋势
下载: 导出CSV
-
[1] 赵克勤. 集对分析与熵的研究[J]. 浙江大学学报, 1992, 6(2): 65–72.ZHAO Keqin. Studies on set pair analysis and entropy[J]. Journal of Zhejiang University, 1992, 6(2): 65–72. [2] 赵克勤. 集对分析及其初步应用[J]. 大自然探索, 1994, 13(1): 67–72.ZHAO Keqin. Set pair analysis and its preliminary application[J]. Exploration of Nature, 1994, 13(1): 67–72. [3] 赵克勤. 集对分析对不确定性的描述和处理[J]. 信息与控制, 1995, 24(3): 162–166. doi: 10.13976/j.cnki.xk.1995.03.006.ZHAO Keqin. Disposal and description of uncertainties based on the set pair analysis[J]. Information and Control, 1995, 24(3): 162–166. doi: 10.13976/j.cnki.xk.1995.03.006. [4] 赵克勤, 宣爱理. 集对论——一种新的不确定性理论方法与应用[J]. 系统工程, 1996, 14(1): 18–23,72. doi: CNKI:SUN:GCXT.0.1996-01-003.ZHAO Keqin and XUAN Aili. Set pair theory—a new theory method of non-define and its applications[J]. Systems Engineering, 1996, 14(1): 18–23,72. doi: CNKI:SUN:GCXT.0.1996-01-003. [5] 赵克勤. 集对分析及其初步应用[M]. 杭州: 浙江科学技术出版社, 2000.ZHAO Keqin. Set Pair Analysis and Its Preliminary Application[M]. Hangzhou: Zhejiang Science and Technology Press, 2000. [6] https://kns.cnki.net/kns8/defaultresult/index. [7] 蒋云良, 赵克勤. 集对分析在人工智能中的应用与进展[J]. 智能系统学报, 2019, 14(1): 28–43. doi: 10.11992/tis.201803030.JIANG Yunliang and ZHAO Keqin. Application and development of set pair analysis in artificial intelligence: A survey[J]. CAAI Transactions on Intelligent Systems, 2019, 14(1): 28–43. doi: 10.11992/tis.201803030. [8] 蒋云良, 赵克勤. 人工智能集对分析[M]. 北京: 科学出版社, 2017: 45–60.JIANG Yunliang and ZHAO Keqin. Artificial Intelligence Set Pair Analysis[M]. Beijing: Science Press, 2017: 45–60. [9] 赵克勤. 二元联系数 A+ Bi的理论基础与基本算法及在人工智能中的应用[J]. 智能系统学报, 2008, 3(6): 476–486. doi: 10.3969/j.issn.1673-4785.2008.06.002.ZHAO Keqin. The theoretical basis and basic algorithm of binary connection A + Bi and its application in AI[J]. CAAI Transactions on Intelligent Systems, 2008, 3(6): 476–486. doi: 10.3969/j.issn.1673-4785.2008.06.002. [10] 赵克勤. 自然辩证法有数学模型吗?[J]. 自然辩证法报, 1988(10).ZHAO Keqin. Is there a mathematical model for Dialectic of nature?[J]. Dialectic of Nature, 1988(10). [11] 赵克勤. 咬住自然辩证法不放松[J]. 自然辩证法研究, 1997, 13(3): 71–72. doi: CNKI:SUN:ZRBZ.0.1997-03-020.ZHAO Keqin. Seize dialectic of nature and don't relax[J]. Studies in Dialectics of Nature, 1997, 13(3): 71–72. doi: CNKI:SUN:ZRBZ.0.1997-03-020. [12] 赵克勤. 自然辩证法可以称“联系科学”吗?——从《自然辩证法通讯》的副标题说起[J]. 自然辩证法通讯, 2008, 30(6): 99–101.ZHAO Keqin. Can Dialectic of nature be called linked science? -- Starting from the subtitle of Communication of Natural Dialectic[J]. Journal of Dialectics of Nature, 2008, 30(6): 99–101. [13] 赵克勤. 成对原理及其在集对分析(SPA)中的作用与意义[J]. 大自然探索, 1998, 17(4): 90.ZHAO Keqin. Pairing principle and its role and significance in set pair analysis (SPA)[J]. Exploration of Nature, 1998, 17(4): 90. [14] 赵克勤. 集对分析的不确定性系统理论在AI中的应用[J]. 智能系统学报, 2006, 1(2): 16–25. doi: 10.3969/j.issn.1673-4785.2006.02.004.ZHAO Keqin. The application of uncertainty systems theory of set pair analysis (SPU) in the artificial intelligence[J]. CAAI Transactions on Intelligent Systems, 2006, 1(2): 16–25. doi: 10.3969/j.issn.1673-4785.2006.02.004. [15] 赵克勤. SPA的同异反系统理论在人工智能研究中的应用[J]. 智能系统学报, 2007, 2(5): 20–35. doi: 10.3969/j.issn.1673-4785.2007.05.004.ZHAO Keqin. The application of SPA-based identical-discrepancy-contrary system theory in artificial intelligence research[J]. CAAI Transactions on Intelligent Systems, 2007, 2(5): 20–35. doi: 10.3969/j.issn.1673-4785.2007.05.004. [16] 赵克勤. 基于集对分析的对立分类、度量及应用[J]. 科学技术与辩证法, 1994, 11(2): 26–30.ZHAO Keqin. Classification, measurement and application of opposites based on set pair analysis[J]. Science, Technology and Dialectics, 1994, 11(2): 26–30. [17] 赵克勤. 联系数及其应用[J]. 吉林师范学院学报, 1996, 17(8): 50–53.ZHAO Keqin. Contact numbers and their applications[J]. Journal of Jilin Teachers College, 1996, 17(8): 50–53. [18] 赵克勤. 联系数学的基本原理与应用[J]. 安阳工学院学报, 2009, (2): 107–110. doi: 10.3969/j.issn.1000-5781.1999.02.002.ZHAO Keqin. Basic principles and applications of contact mathematics[J]. Journal of Anyang Institute of Technology, 2009, (2): 107–110. doi: 10.3969/j.issn.1000-5781.1999.02.002. [19] 黄德才, 赵克勤, 陆耀忠, 等. a+b i+c j型联系数的四则运算及其应用[J]. 机电工程, 2000, 17(3): 81–84. doi: 10.3969/j.issn.1001-4551.2000.03.030.HUANG Decai, ZHAO Keqin, LU Yaozhong, et al. The fundamental operation of arithmetic on connection number a+b i+c j and its application[J]. Mechanical & Electrical Engineering Magazine, 2000, 17(3): 81–84. doi: 10.3969/j.issn.1001-4551.2000.03.030. [20] 黄德才, 赵克勤, 陆耀忠. 联系数 a+ bi的运算及在网络计划中的应用[J]. 浙江工业大学学报, 2000, 28(3): 190–194. doi: 10.3969/j.issn.1006-4303.2000.03.002.HUANG Decai, ZHAO Keqin, and LU Yaozhong. Fundamental operation of arithmetic on connection number a+ bi and its application in network planning[J]. Journal of Zhejiang University of Technology, 2000, 28(3): 190–194. doi: 10.3969/j.issn.1006-4303.2000.03.002. [21] 赵克勤, 黄德才, 陆耀忠. 基于 a+ bi+ cj型联系数的网络计划方法初探[J]. 系统工程与电子技术, 2000, 22(2): 29–31. doi: 10.3321/j.issn:1001-506X.2000.02.009.ZHAO Keqin, HUANG Decai, and LU Yaozhong. A new network planning method based on the connection number a+ bi+ cj[J]. Systems Engineering and Electronics, 2000, 22(2): 29–31. doi: 10.3321/j.issn:1001-506X.2000.02.009. [22] 黄德才, 张丽君, 赵克勤. 基于 a+ bi型联系数的不确定网格静态调度算法[J]. 计算机科学, 2007, 34(8): 126–129,179. doi: 10.3969/j.issn.1002-137X.2007.08.034.HUANG Decai, ZHANG Lijun, and ZHAO Keqin. Static scheduling algorithms based on connective-number of type a+ bi for uncertain computing grid[J]. Computer Science, 2007, 34(8): 126–129,179. doi: 10.3969/j.issn.1002-137X.2007.08.034. [23] 刘秀梅, 赵克勤. 基于联系数复运算的区间数多属性决策方法及应用[J]. 数学的实践与认识, 2008, 38(23): 57–64.LIU Xiumei and ZHAO Keqin. Multiple attribute decision making and its applications based on complex number arithmetic operation of connection number with interval numbers[J]. Mathematics in Practice and Theory, 2008, 38(23): 57–64. [24] 刘秀梅, 赵克勤. 基于SPA的D-U空间的区间数多属性决策模型及应用[J]. 模糊系统与数学, 2009, 23(2): 167–174.LIU Xiumei and ZHAO Keqin. Multiple attribute decision making and its applications with interval numbers based on D-U space of SPA[J]. Fuzzy Systems and Mathematics, 2009, 23(2): 167–174. [25] 刘秀梅, 赵克勤. 区间数决策集对分析[M]. 北京: 科学出版社, 2014: 80–101.LIU Xiumei and ZHAO Keqin. Interval Number Decision Set Pair Analysis[M]. Beijing: Science Press, 2014: 80–101. [26] 赵森烽, 赵克勤. 概率联系数化的原理及其在概率推理中的应用[J]. 智能系统学报, 2012, 7(3): 200–205. doi: 10.3969/j.issn.1673-4785.201112014.ZHAO Senfeng and ZHAO Keqin. The principle of a connection number in probability and its application in probabilistic reasoning[J]. CAAI Transactions on Intelligent Systems, 2012, 7(3): 200–205. doi: 10.3969/j.issn.1673-4785.201112014. [27] 赵森烽, 赵克勤. 几何概型的联系概率(复概率)与概率的补数定理[J]. 智能系统学报, 2013, 8(1): 11–15. doi: 10.3969/j.issn.1673-4785.201208025.ZHAO Senfeng and ZHAO Keqin. Contact probability (complex probability) of the geometry probability and the complement number theorem of probability[J]. CAAI Transactions on Intelligent Systems, 2013, 8(1): 11–15. doi: 10.3969/j.issn.1673-4785.201208025. [28] 赵森烽, 赵克勤. 联系概率的由来及其在风险决策中的应用[J]. 数学的实践与认识, 2013, 43(4): 165–171. doi: 10.3969/j.issn.1000-0984.2013.04.024.ZHAO Senfeng and ZHAO Keqin. The contact probability in risk decision-making medium application[J]. Mathematics in Practice and Theory, 2013, 43(4): 165–171. doi: 10.3969/j.issn.1000-0984.2013.04.024. [29] 赵森烽, 赵克勤. 频率型联系概率与随机事件转化定理[J]. 智能系统学报, 2014, 9(1): 53–59. doi: 10.3969/j.issn.1673-4785.201305003.ZHAO Senfeng and ZHAO Keqin. Frequency-type contact probability and random events transformation theorem[J]. CAAI Transactions on Intelligent Systems, 2014, 9(1): 53–59. doi: 10.3969/j.issn.1673-4785.201305003. [30] 赵克勤, 赵森烽. 贝叶斯概率向赵森烽-克勤概率的转换与应用[J]. 智能系统学报, 2015, 10(1): 51–61. doi: 10.3969/j.issn.1673-4785.201405022.ZHAO Keqin and ZHAO Senfeng. Bayes probability transition to Zhao Senfeng-Keqin probability and its application[J]. CAAI Transactions on Intelligent Systems, 2015, 10(1): 51–61. doi: 10.3969/j.issn.1673-4785.201405022. [31] 赵克勤, 赵森烽. 赵森烽-克勤概率的赌本分配研究与期望值定理[J]. 智能系统学报, 2017, 12(5): 608–615. doi: 10.11992/tis.201604020.ZHAO Keqin and ZHAO Senfeng. Interval multiple attribute decision making based on geometric properties of connection number[J]. CAAI Transactions on Intelligent Systems, 2017, 12(5): 608–615. doi: 10.11992/tis.201604020. [32] 阎理, 阎滨. 相似系统集对分析[J]. 指挥技术学院学报, 2000, 11(3): 9–13.YAN Li and YAN Bin. The set pair analysis of similarity system[J]. Journal of Institute of Command and Technology, 2000, 11(3): 9–13. [33] 陆广地. 基于联系数几何特性的区间数多属性决策[J]. 数学的实践与认识, 2017, 47(18): 194–200.LU Guangdi. Interval multiple attribute decision making based on geometric properties of connection number[J]. Mathematics in Practice and Theory, 2017, 47(18): 194–200. [34] 李斌, 华亮, 徐蓉, 等. 基于银屑病疗效联系数几何特性的临床用药优选探讨[J]. 辽宁中医杂志, 2018, 45(2): 237–241. doi: 10.13192/j.issn.1000-1719.2018.02.004.LI Bin, HUA Liang, XU Rong, et al. Study on clinical drug selection scheme from geometrical characteristics based on connection number of clinical efficacy of psoriasis[J]. Liaoning Journal of Traditional Chinese Medicine, 2018, 45(2): 237–241. doi: 10.13192/j.issn.1000-1719.2018.02.004. [35] 赵克勤. 集对分析与熵的研究[J]. 浙江大学学报(社科版), 1992, 6(3): 65–72. doi: 10.13976/j.cnki.xk.2023.2392.ZHAO Keqin. Studies on set pair analysis and entropy[J]. Journal of Zhejiang University Sciences (social sciences edition), 1992, 6(3): 65–72. doi: 10.13976/j.cnki.xk.2023.2392. [36] 魏邦友. 载人航天器综合测试数据评估方法的研究[J]. 电子质量, 2017(7): 28–30. doi: 10.3969/j.issn.1003-0107.2017.07.008.WEI Bangyou. Research on evaluation method of comprehensive test data for manned spacecraft[J]. Electronics Quality, 2017(7): 28–30. doi: 10.3969/j.issn.1003-0107.2017.07.008. [37] 刘以安, 牛媛媛, 刘同明. 集对分析在多雷达数据融合中的应用研究[J]. 华东船舶工业学院学报:自然科学版, 2005, 19(2): 64–67. doi: 10.3969/j.issn.1673-4807.2005.02.015.LIU Yian, NIU Yuanyuan, and LIU Tongming. Application of set pair Analysis in multi-radar data fusion[J]. Journal of East China Shipbuilding Institute:Natural Science Edition, 2005, 19(2): 64–67. doi: 10.3969/j.issn.1673-4807.2005.02.015. [38] 张秀辉, 刘以安, 曹宁生, 等. 基于集对分析的雷达信号分选算法[J]. 现代雷达, 2010, 32(2): 35–37. doi: 10.3969/j.issn.1004-7859.2010.02.008.ZHANG Xiuhui, LIU Yian, CAO Ningsheng, et al. Radar signal sorting method based on set pair analysis[J]. Modern Radar, 2010, 32(2): 35–37. doi: 10.3969/j.issn.1004-7859.2010.02.008. [39] 黎蓉, 刘以安, 王刚. 基于改进集对分析聚类的雷达信号分选方法[J]. 现代电子技术, 2014, 37(9): 8–11. doi: 10.3969/j.issn.1004-373X.2014.09.003.LI Rong, LIU Yian, and WANG Gang. Radar signal sorting method based on modified set pair analysis clustering[J]. Modern Electronics Technique, 2014, 37(9): 8–11. doi: 10.3969/j.issn.1004-373X.2014.09.003. [40] 张萌萌, 刘以安, 宋萍. 偏联系数聚类和随机森林算法在雷达信号分选中的应用[J]. 激光与光电子学进展, 2019, 56(6): 062804. doi: 10.3788/LOP56.062804.ZHANG Mengmeng, LIU Yian, and SONG Ping. Applications of Partial connection clustering algorithm and random forest algorithm in radar signal sorting[J]. Laser & Optoelectronics Progress, 2019, 56(6): 062804. doi: 10.3788/LOP56.062804. [41] 杨承志, 肖卫华, 吴宏超, 等. 一种对多种重频调制类型雷达信号分选算法的研究[J]. 科学技术与工程, 2014, 14(34): 33–37. doi: 10.3969/j.issn.1671-1815.2014.34.007.YANG Chengzhi, XIAO Weihua, WU Hongchao, et al. Research on an improved sorting method for multiple PRI type radar signals[J]. Science Technology and Engineering, 2014, 14(34): 33–37. doi: 10.3969/j.issn.1671-1815.2014.34.007. [42] 殷志远, 彭涛, 沈铁元. 雷达估算和雨量站插值降水精度的对比分析[J]. 水利科技与经济, 2010, 16(9): 996–999. doi: 10.3969/j.issn.1006-7175.2010.09.015.YIN Zhiyuan, PENG Tao, and SHEN Tieyuan. The comparative study of radar estimation and grid interpolation of rainfall station[J]. Water Conservancy Science and Technology and Economy, 2010, 16(9): 996–999. doi: 10.3969/j.issn.1006-7175.2010.09.015. [43] 孟现海, 刘以安, 刘静. 基于联系度态势的图像边缘检测算法[J]. 计算机工程与设计, 2007, 28(10): 2364–2366,2370. doi: 10.3969/j.issn.1000-7024.2007.10.037.MENG Xianhai, LIU Yian, and LIU Jing. Algorithms of image edge detection based on degree connection situation[J]. Computer Engineering and Design, 2007, 28(10): 2364–2366,2370. doi: 10.3969/j.issn.1000-7024.2007.10.037. [44] 赵克勤. 集对分析在系统智能预测中的应用综述[J]. 智能系统学报, 2022, 17(2): 233–247. doi: 10.11992/tis.202103023.ZHAO Keqin. Application overview of set pair analysis in intelligent prediction system[J]. CAAI Transactions on Intelligent Systems, 2022, 17(2): 233–247. doi: 10.11992/tis.202103023. [45] 王国强. 不确定性理论——集对分析在MOS概率天气预报中的应用[J]. 浙江气象科技, 1999, 20(1): 1–6. doi: 10.16000/j.cnki.zjqx.1999.01.001.WANG Guoqiang. Uncertainty theory — application of set pair analysis in MOS probabilistic weather forecast[J]. Zhejiang Meteorological Science and Technology, 1999, 20(1): 1–6. doi: 10.16000/j.cnki.zjqx.1999.01.001. [46] 薛根元, 王国强. 不确定性理论集对分析在预报模型建立中的应用研究[J]. 气象学报, 2003, 61(5): 592–599. doi: 10.3321/j.issn:0577-6619.2003.05.008.XUE Genyuan and WANG Guoqiang. Application of set pair analysis to fuzzy predictors of multiple regression weather forcast models[J]. Acta Meteorologica Sinica, 2003, 61(5): 592–599. doi: 10.3321/j.issn:0577-6619.2003.05.008. [47] 王国强, 赵克勤, 郑选军. 天气预报多元回归模型中模糊因子的集对分析[J]. 科技通报, 2004, 20(2): 151–155. doi: 10.3969/j.issn.1001-7119.2004.02.014.WANG Guoqiang, ZHAO Keqin, and ZHENG Xuanjun. Application of set pair analysis to fuzzy predictors of multiple regression weather forcast models[J]. Bulletin of Science and Technology, 2004, 20(2): 151–155. doi: 10.3969/j.issn.1001-7119.2004.02.014. [48] 刘晓, 唐辉明, 刘瑜. 基于集对分析的滑坡变形动态建模研究[J]. 岩土力学, 2009, 30(8): 2371–2378. doi: 10.3969/j.issn.1000-7598.2009.08.031.LIU Xiao, TANG Huiming, and LIU Yu. Landslide deformation dynamic modeling research based on set pair analysis[J]. Rock and Soil Mechanics, 2009, 30(8): 2371–2378. doi: 10.3969/j.issn.1000-7598.2009.08.031. [49] 许增光, 线美婷, 熊伟, 等. 基于集对分析模型的岩溶区浅埋穿河隧道突涌水危险性评价[J]. 应用力学学报, 2023, 40(1): 135–145. doi: 10.11776/j.issn.1000-4939.2023.01.017.XU Zengguang, XIAN Meiting, XIONG Wei, et al. Risk assessment of water inrush in karst shallow tunnel under river based on SPA model[J]. Chinese Journal of Applied Mechanics, 2023, 40(1): 135–145. doi: 10.11776/j.issn.1000-4939.2023.01.017. [50] 赵克勤. 基于集对分析的不确定性多属性决策模型与算法[J]. 智能系统学报, 2010, 5(1): 41–50.ZHAO Keqin. Uncertain attribute decision making model and algorithm based on the Set pair analysis[J]. Journal of Intelligent Systems, 2010, 5(1) : 41–50. [51] 刘秀梅, 赵克勤. 集对分析在不确定性智能决策中的应用[J]. 智能系统学报, 2020, 15(1): 121–135. doi: 10.11992/tis.201910025.LIU Xiumei and ZHAO Keqin. Application of set pair analysis in the uncertainty intelligent decision making[J]. CAAI Transactions on Intelligent Systems, 2020, 15(1): 121–135. doi: 10.11992/tis.201910025. [52] 刘秀梅, 赵克勤. 区间数决策集对分析[M]. 北京: 科学出版社, 2014.LIU Xiumei and ZHAO Keqin. Interval Number Decision Set Pair Analysis[M]. Beijing: Science Press, 2014. [53] 汪新凡, 隆丽兰, 周欢. 多型异构数据下准则具有优先级别的双边匹配决策方法[J]. 信息与控制, 2023, 52(3): 405–416. doi: 10.13976/j.cnki.xk.2023.2392WANG Xinfan, LONG Lilan, and ZHOU Huan. Two-sided matching decision making approach under multi-type heterogeneous data considering different priority levels of criteria[J]. Information and Control, 2023, 52(3): 405–416. doi: 10.13976/j.cnki.xk.2023.2392. [54] 钟义信. 高等人工智能原理—观念、方法、模型、理论[M]. 北京: 科学出版社, 2014.ZHONG Yixin. Principles of Advanced Artificial Intelligence — Concepts, Methods, Models, theories[M]. Beijing: Science Press, 2014. [55] 黄德才, 赵克勤. 用联系数描述和处理网络计划中的不确定性[J]. 系统工程学报, 1999, 14(2): 112–117. doi: 10.3969/j.issn.1000-5781.1999.02.002.HUANG Decai and ZHAO Keqin. Using the connection number of the spa to express and process the uncertainties in network planning[J]. Journal of Systems Engineering, 1999, 14(2): 112–117. doi: 10.3969/j.issn.1000-5781.1999.02.002. [56] 赵克勤, 黄德才, 陆耀忠. 基于a+bi+cj型联系数的网络计划方法初探[J]. 系统工程与电子技术, 2000, 22(2): 29–31. doi: 10.3321/j.issn:1001-506X.2000.02.009ZHAO Keqin, HUANG Decai, and LU Yaozhong. A new network planning method based on the connection number a+bi+cj[J]. Systems Engineering and Electronics, 2000, 22(2): 29–31. doi: 10.3321/j.issn:1001-506X.2000.02.009. [57] 赵克勤, 黄德才, 朱艺华, 等. 含有突发性的网络关键路线问题[J]. 管理工程学报, 2000, 14(2): 33–34. doi: 10.3969/j.issn.1004-6062.2000.02.010.ZHAO Keqin, HUANG Decai, ZHU Yihua, et al. Analysis on critical path of a network planning in which accidents be in volved[J]. Journal of Industrial Engineering/ Engineering Management, 2000, 14(2): 33–34. doi: 10.3969/j.issn.1004-6062.2000.02.010. [58] 黄德才, 龚卫华, 张丽君, 等. 基于联系数的网格任务动态调度算法[J]. 计算机工程, 2009, 35(8): 112–115. doi: 10.7666/d.d093185.HUANG Decai, GONG Weihua, ZHANG Lijun, et al. Dynamic grid task Scheduling Algorithm based on connection number[J]. Computer Engineering, 2009, 35(8): 112–115. (in Chinese). doi: 10.7666/d.d093185. [59] 赵克勤, 黄德才, 陆耀忠. 同异反网络计划的不确定性分类与分析[J]. 系统工程与电子技术, 2000, 22(11): 72–74. doi: 10.3321/j.issn:1001-506X.2000.11.023.ZHAO Keqin, HUANG Decai, and LU Yaozhong. Forecasting and controlling method of the time limit for a project of identical-discrepancy-contrary network planning[J]. Systems Engineering and Electronics, 2000, 22(11): 72–74. doi: 10.3321/j.issn:1001-506X.2000.11.023. [60] 黄德才, 赵克勤, 陆耀忠. 同异反网络计划的工期预测方法[J]. 系统工程与电子技术, 2001, 23(5): 24–27. doi: 10.7666/d.D617408.HUANG Decai, ZHAO Keqin, LU Yaozhong. A method for predicting the construction period of the same-different and counter-network planning[J]. Systems Engineering and electronics, 2001, 23(5): 24–27. doi: 10.7666/d.D617408. [61] 郭瑞林. 作物育种同异理论与方法[M]. 北京, 中国农业科学技术出版社, 2011.GUO Ruilin. Theory and method of similarity and difference in crop breeding[M] . Beijing, China Agricultural Science and Technology Press, 2011. [62] 郭瑞林, 王占中. 作物同异育种智能决策系统及其应用[M]. 北京, 科学出版社, 2014: 1–343.GUO ruilin, WANG Zhanzhong. Intelligent decision system and its application in crop breeding[M]. Beijing, Science Press, January, 2014: 1–343. [63] 郑建青. 观察数据用联系数表示的最小二乘法及应用[J]. 宁波大学学报(理工版), 2013, 26(1): 57–59.ZHENG Jianqing. The least squares and application of observed data in terms of connection numbers[J]. Journal of Ningbo University Science and Technology, 2013, 26(1): 57–59. [64] 张玲, 张亚飞, 张立舒. 联系数四则运算的证明与联系数群[J]. 数学学习与研究, 2018(7): 18–20.ZHANG Ling, ZHANG Yafei, ZHANG Lishu. Proof of four operations of connection number and connection number group[J]. Mathematics Learning and research, 2018(7): 18–20. [65] 赵森烽, 赵克勤. 联系概率的由来及其在风险决策中的应用[J]. 数学的实践与认识, 2013, 43(4): 165–171. doi: 10.3969/j.issn.1000-0984.2013.04.024.ZHAO Senfeng and ZHAO Keqin. The contact probability in risk decision-making medium application[J]. Mathematics in Practice and Theory, 2013, 43(4): 165–171. doi: 10.3969/j.issn.1000-0984.2013.04.024. [66] 黎振宇, 陈晓国, 宋永超, 等. 二元联系数-投影灰靶决策理论在电网应急能力评估中的应用[J]. 浙江大学学报(工学版), 2021, 55(5): 927–934+975. doi: 10.3785/j.issn.1008-973X.2021.05.013.LI Zhenyu, CHEN Xiaoguo, SONG Yongchao et al. Application of binary connection-projected gray target decision theory in power grid emergency capability evaluation[J]. Journal of Zhejiang University (Engineering and Technology), 2021, 55(5): 927–934+975. doi: 10.3785/j.issn.1008-973X.2021.05.013. [67] 柯丽华, 唐华倩, 王其虎, 等. 基于二元联系数可能度函数的区间数排序方法及应用[J]. 系统科学与数学, 2023, 43(2): 417–430.KE Lihua, TANG Huaqian, WANG Qihu, et al. Ranking method of interval numbers based on possibility function of binary connection number and its application[J]. Systems science and mathematics, 2023, 43(2): 417–430. [68] 徐宗本. 人工智能的10个重大数理基础问题[J]. 中国科学:信息科学, 2021, 51(12): 1967–1978. doi: 10.1360/SSI-2021-0254.XU Zongben. Ten fundamental problems for artificial intelligence: Mathematical and physical aspects[J]. Scientia Sinica Informationis, 2021, 51(12): 1967–1978. doi: 10.1360/SSI-2021-0254. [69] 张春英, 郭景峰. 集对社会网络 α关系社区及动态挖掘算法[J]. 计算机学报, 2013, 36(8): 1682–1692. doi: 10.3724/SP.J.1016.2013.01682.ZHANG Chunying and GUO Jingfeng. The α relationship communities of set pair social networks and its dynamic mining algorithms[J]. Chinese Journal of Computers, 2013, 36(8): 1682–1692. doi: 10.3724/SP.J.1016.2013.01682. [70] 张春英, 高瑞艳, 王佳昊, 等. 面向不完备分类型矩阵数据的集对k-modes聚类算法[J]. 小型微型计算机系统, 2021, 42(9): 1837–1844. doi: 10.3969/j.issn.1000-1220.2021.09.007.ZHANG Chunying, GAO Ruiyan, WANG Jiahao, et al. Set Pair k-modes Clustering algorithm for incomplete categorical matrix data[J]. Journal of Chinese Computer Systems, 2021, 42(9): 1837–1844. doi: 10.3969/j.issn.1000-1220.2021.09.007. [71] WANG Jing, WANG Jing, GUO Jingfeng, et al. Research progress of complex network modeling methods based on uncertainty theory[J]. Mathematics, 2023, 11(5): 1212. doi: 10.3390/math11051212. [72] ZHANG Chunying, REN Jing, LIU Lu, et al. Set pair three-way overlapping community discovery algorithm for weighted social internet of things[J]. Digital Communications and Networks, 2023, 9(1): 3–13. doi: 10.1016/j.dcan.2022.04.004. [73] 胡波, 王汝传, 王海艳. 基于集对分析的P2P网络安全中的信誉度改进算法[J]. 电子学报, 2007, 35(2): 244–247. doi: 10.3321/j.issn:0372-2112.2007.02.012.HU Bo, WANG Ruchuan, and WANG Haiyan. A modified security solution based on SPA for Servents' reputations in P2P Systems[J]. Acta Electronica Sinica, 2007, 35(2): 244–247. doi: 10.3321/j.issn:0372-2112.2007.02.012. [74] HE Chaokai and WU Meng. A new reputation model for P2P network based on set pair analysis[j]. The Open Cybernetics & Systemics Journal, 2015, 9(1): 1393–1398. doi: 10.2174/1874110X01509011393. [75] PENG Xindong, GARG H, and LUO Zhigang. When content-centric networking meets multi-criteria group decision-making: Optimal cache placement policy achieved by MARCOS with q-rung orthopair fuzzy set pair analysis[J]. Engineering Applications of Artificial Intelligence, 2023, 123: 106231. doi: 10.1016/j.engappai.2023.106231. [76] 陈晓. 网络中顶点间相似性度量方法研究[D]. [硕士论文] 燕山大学, 2018.CHEN Xiao. Research on similarity measurement between vertices in networks[D]. [Master dissereation]. Yanshan University, 2018. [77] 孙勇, 李宝聚, 孙志博, 等. 融合RBF神经网络和集对分析的风电功率超短期预测[J]. 昆明理工大学学报:自然科学版, 2020, 45(5): 49–58. doi: 10.16112/j.cnki.53-1223/n.2020.05.008.SUN Yong, LI Baoju, SUN Zhibo, et al. Ultra-short-term wind power forecasting integrated RBF neural network and set pair analysis[J]. Journal of Kunming University of Science and Technology:Natural Science, 2020, 45(5): 49–58. doi: 10.16112/j.cnki.53-1223/n.2020.05.008. [78] 耿鹏, 郑中团. 基于集对分析-RBF神经网络的生态文明建设评价指标体系构建[J]. 智能计算机与应用, 2020, 10(12): 86–90. doi: 10.3969/j.issn.2095-2163.2020.12.020.GENG Peng and ZHENG Zhongtuan. Construction of ecological civilization construction evaluation index system based on set pair analysis-RBF neural network[J]. Intelligent Computer and Applications, 2020, 10(12): 86–90. doi: 10.3969/j.issn.2095-2163.2020.12.020. [79] 赵鹏丽, 顾伟红. 基于BP神经网络与SPA的地铁TBM施工安全风险评估[J]. 建筑安全, 2019, 34(11): 43–49. doi: 10.3969/j.issn.1004-552X.2019.11.013.ZHAO Pengli and GU Weihong. Safety risk assessment of subway TBM construction based on BP neural network and SPA[J]. Construction Safety, 2019, 34(11): 43–49. doi: 10.3969/j.issn.1004-552X.2019.11.013. [80] 陈笑, 胡宏祥, 戚王月, 等. 基于集对分析和GA-BP神经网络的地下水埋深预测研究[J]. 华北水利水电大学学报:自然科学版, 2019, 40(4): 57–64. doi: 10.19760/j.ncwu.zk.2019052.CHEN Xiao, HU Hongxiang, QI Wangyue, et al. Groundwater depth prediction based on set pair analysis and GA-BP neural network[J]. Journal of North China University of Water Resources and Electric Power:Natural Science Edition, 2019, 40(4): 57–64. doi: 10.19760/j.ncwu.zk.2019052. [81] 陈晶, 王文圣, 李跃清. 集对分析径向基函数神经网络预测模型[J]. 水文, 2011, 31(2): 11–14. doi: 10.3969/j.issn.1000-0852.2011.02.003.CHEN Jing, WANG Wensheng, and LI Yueqing. Prediction model of radial basis function neural network based on set pair analysis[J]. Journal of China Hydrology, 2011, 31(2): 11–14. doi: 10.3969/j.issn.1000-0852.2011.02.003. [82] ZHANG Rui, WANG Yan, WANG Kaibo, et al. An evaluating model for smart growth plan based on BP neural network and set pair analysis[J]. Journal of Cleaner Production, 2019, 226: 928–939. doi: 10.1016/j.jclepro.2019.03.053. [83] 赵志峰, 文虎, 高炜欣, 等. 同异反模式的管道土壤腐蚀综合评价[J]. 西安科技大学学报, 2017, 37(3): 352–357. doi: 10.13800/j.cnki.xakjdxxb.2017.0308.ZHAO Zhifeng, WEN Hu, GAO Weixin, et al. Integrated evaluation of the soil corrosion in pipeline in contrary identical discrepancy model[J]. Journal of Xi'an University of Science and Technology, 2017, 37(3): 352–357. doi: 10.13800/j.cnki.xakjdxxb.2017.0308. [84] 李德顺, 许开立, 崔岳峰. 同异反最优模式识别模型及其在危险性评价中应用[C]. 2010(沈阳)国际安全科学与技术学术研讨会论文集, 沈阳, 2010.LI Deshun, XU Kaili, and CUI Yuefeng. Heterogeneous inverse optimal pattern recognition model and its application in risk assessment[C]. Proceedings of 2010 (Shenyang) International Colloquium on Safety Science and Technology, Shenyang, China, 2010. [85] 白扬文. 平面图象的同异反模式识别技术[C]//1996年中国智能自动化学术会议论文集(下册), 呼和浩特, 1996.BAI Yangwen. Anti-pattern recognition of identical and different planar images[C]. Proceedings of the Academic Conference of Intelligent Automation Committee of Chinese Society of Automation, Hohhot, China, 1996. [86] 杨静, 李文平, 张健沛. 一种基于SPA的多数据流同异反分析法[J]. 武汉大学学报:信息科学版, 2011, 36(1): 92–97. doi: 10.13203/j.whugis2011.01.025.YANG Jing, LI Wenping, and ZHANG Jianpei. A data mining approach based on identical-different-contrary analysis[J]. Geomatics and Information Science of Wuhan University, 2011, 36(1): 92–97. doi: 10.13203/j.whugis2011.01.025. [87] 魏冬慧, 吕英, 蒯仂, 等. 基于同异反分析的寻常型银屑病用药优选[J]. 中华中医药学刊, 2018, 36(10): 2445–2447. doi: 10.13193/j.issn.1673-7717.2018.10.036.WEI Donghui, LYU Ying, KUAI Le, et al. Optimization of psoriasis drugs based on same and similar back analysis[J]. Chinese Archives of Traditional Chinese Medicine, 2018, 36(10): 2445–2447. doi: 10.13193/j.issn.1673-7717.2018.10.036. [88] 徐忆琳. 用SPA同异反系统理论研究知识创新规律[J]. 科学学研究, 2002, 20(3): 327–329. doi: 10.3969/j.issn.1003-2053.2002.03.022.XU Yilin. Research on the law of knowledge innovation using IDC-SPA theory[J]. Studies in Science of Science, 2002, 20(3): 327–329. doi: 10.3969/j.issn.1003-2053.2002.03.022. [89] 余国祥. 默会知识和显性知识的同异反集对分析[J]. 襄樊学院学报, 2008, 29(5): 84–88. doi: 10.3969/j.issn.1009-2854.2008.05.021.YU Guoxiang. A SPA of tacit knowledge and explicit knowledge in identity, discrepancy and contrary[J]. Journal of Xiangfan University, 2008, 29(5): 84–88. doi: 10.3969/j.issn.1009-2854.2008.05.021. [90] 黄大荣, 黄丽芬. 基于集对分析联系数故障树的BA系统可靠性分析[J]. 计算机应用研究, 2010, 27(1): 111–113. doi: 10.3969/j.issn.1001-3695.2010.01.033.HUANG Darong and HUANG Lifen. Reliability analysis of BA system based on connection number of set pair analysis and FTA[J]. Application Research of Computers, 2010, 27(1): 111–113. doi: 10.3969/j.issn.1001-3695.2010.01.033. [91] 刘英, 陈宇, 陈志恒. 基于集对分析理论的金刚滚轮转动系统故障树分析[J]. 机械科学与技术, 2014, 33(9): 1335–1339. doi: 10.13433/j.cnki.1003-8728.2014.0911.LIU Ying, CHEN Yu, and CHEN Zhiheng. Fault tree analysis of diamond roller rotation system based on set pair analysis theory[J]. Mechanical Science and Technology for Aerospace Engineering, 2014, 33(9): 1335–1339. doi: 10.13433/j.cnki.1003-8728.2014.0911. [92] XIE Hongtao, LI Bo, and ZHAO Yunsheng. Study on risk trend assessment of metro tunnel crossing underground pipeline based on partial connection number[J]. Applied Mechanics and Materials, 2014, 580–583: 1283–1287. doi: 10.4028/www.scientific.net/amm.580-583.1283. [93] XIE Xuecai and GUO Deyong. Human factors risk assessment and management: Process safety in engineering[J]. Process Safety and Environmental Protection, 2018, 113: 467–482. doi: 10.1016/j.psep.2017.11.018. [94] 崔铁军, 李莎莎. 多因素集对分析的系统故障模式识别方法[J]. 智能系统学报, 2022, 17(2): 387–392. doi: 10.11992/tis.202011006.CUI Tiejun and LI Sasha. System fault-pattern recognition based on set pair analysis with multiple factors[J]. CAAI Transactions on Intelligent Systems, 2022, 17(2): 387–392. doi: 10.11992/tis.202011006. [95] 崔铁军, 李莎莎. 系统多功能状态表达式构建及其置信度研究[J]. 智能系统学报, 2023, 18(1): 124–130. doi: 10.11992/tis.202111022.CUI Tiejun and LI Sasha. Construction of a system multi-function state expression and its confidence[J]. CAAI Transactions on Intelligent Systems, 2023, 18(1): 124–130. doi: 10.11992/tis.202111022. [96] GHOLAMIZADEH K, ZAREI E, YAZDI M, et al. A hybrid model for dynamic analysis of domino effects in chemical process industries[J]. Reliability Engineering & System Safety, 2024, 241: 109654. doi: 10.1016/j.ress.2023.109654. [97] 孟庆刚, 王连心, 赵世初, 等. 浅谈集对分析在证候规范化研究中的应用[J]. 北京中医药大学学报, 2005(4): 9–13. doi: 10.3321/j.issn:1006-2157.2005.04.002.MENG Qinggang, WANG Lianxin, ZHAO shichu et al. Primary the application of set pair analysis in the study of syndrome standardization [J]. Journal of Beijing University of Chinese Medicine, 2005(4): 9–13. doi: 10.3321/j.issn:1006-2157.2005.04.002. [98] 李斌, 徐蓉,李伦, 等. 基于联系数的痛风性关节炎血瘀证辨证因子研究[J]. 西医结合学报, 2009,7(8): 724–728. doi: 10.3736/jcim20090804.LI Bin, XU Rong, LI Fulun, et al. Study on the factors of blood stasis syndrome differentiation of gouty arthritis based on connection number[J]. Journal of Integrated Chinese and Western medicine, 2009,7(8): 724–728. doi: 10.3736/jcim20090804. [99] 李斌, 李福伦, 赵克勤. 慢性皮肤溃疡中医辨证论治规律数学建模探析[J]. 中国中西医结合皮肤性病学杂志, 2010, 9(1): 4–7.LI Bin, LI Fulun, ZHAO Keqin. Mathematical Modeling of TCM syndrome differentiation and treatment of chronic skin ulcer[J]. Chinese Journal of integrated traditional and Western medicine of dermatology and Venereology, 2010, 9(1): 4–7. [100] 李欣, 徐蓉, 周敏, 等. 基于集对分析的寻常型银屑病方证相关性研究[J]. 辽宁中医杂志, 2012, 39(6) : 974–978. doi: CNKI:SUN:LNZY.0.2012-06-008.LI Xin, XU Rong, ZHOU Min, et al. Study on the correlation between prescriptions and syndromes of psoriasis vulgaris based on set pair analysis[J]. Liaoning Journal of Traditional Chinese medicine, 2012, 39(6) : 974–978. doi: CNKI:SUN:LNZY.0.2012-06-008. [101] 蒯仂, 赵克勤, 李斌. 基于集对分析偏联系数的寻常型银屑病对症用药优选探讨[J]. 上海医药, 2018, 39(23): 9–14+67. doi: CNKI:SUN:SYIY.0.2018-23-004.KUAI Le, ZHAO Keqin, Li Bin. Discussion on optimal drug use for psoriasis vulgaris based on set pair analysis of partial correlation number[J]. Shanghai Medicine, 2018, 39(23): 9–14+67. (in Chinese) doi: CNKI:SUN:SYIY.0.2018-23-004. [102] 许逊哲, 茹意, 蒯仂, 等. 四元联系数在土槐菝葜汤治疗血热型银屑病疗效研究中的应用[J]. 中国中西医结合皮肤性病学杂志, 2018, 17(6): 489–492.XU Xun ZHE, RU Yi, KUAI Le, et al. Application of four-element correlation number in the treatment of psoriasis of hematrexia[J]. Chinese Journal of Dermatology and Venereology of Integrated Traditional and Western Medicine, 2018, 17(6): 489–492. [103] 茹意, 蒯仂, 许逊哲, 等. 基于集对分析的疗效曲线在银屑病血热证典型方剂选优中的应用[J]. 中华中医药学刊, 2019, 37(2): 322–325. doi: CNKI:SUN:ZYHS.0.2019-02-014.RU Yi, KUAI Le, XU Xunzhe et al. Application of curative effect curve based on set pair analysis in the selection of typical prescriptions for blood-heat syndrome of psoriasis[J]. Chinese Journal of Traditional Chinese Medicine, 2019, 37(2): 322–325. doi: CNKI:SUN:ZYHS.0.2019-02-014. [104] 迮侃, 陈曦, 赵淮波, 等. 基于集对分析成果的寻常型银屑病血热证诊疗方案的临床研究[J]. 中医杂志, 2019, 60(10): 849–852.ZE Kan, CHEN Xi, ZHAO Huaibo, et al. Clinical study on diagnosis and treatment of blood-heat syndrome of psoriasis vulgaris based on set pair analysis results[J]. Chinese Journal of Traditional Chinese Medicine, 2019, 60(10): 849–852. [105] 罗月, 蒯仂, 茹意, 等. 皮肤病脏腑辨证的联系数学模型在临床中的应用初探[J]. 时珍国医国药, 2019, 30(5): 1247–1248. doi: CNKI:SUN:SZGY.0.2019-05-079.LUO Yue, KUAI Le, RU Yi, et al. Clinical application of the relational mathematical model for differentiation of viscera syndrome in skin diseases[J]. Chinese Medicine, 2019, 30(5): 1247–1248. doi: CNKI:SUN:SZGY.0.2019-05-079. [106] 华亮, 蒯仂, 陈洁, 等. 联系数在银屑病治疗中医单方研究中的应用[J]. 上海医药, 2019, 41(3): 17–19+28.HUA Liang, KUAI Le, CHEN Jie, et al. Application of correlation number in the treatment of psoriasis[J]. Shanghai Journal of Medicine, 2019, 41(3): 17–19+28. [107] 马天, 范斌, 王一飞, 等. 凉血潜阳法治疗寻常型银屑病血热证的临床观察[J]. 世界临床药物, 2020, 41(7): 524–529+561. doi: 10.13683/j.wph.2020.07.006.MA Tian, FAN Bin, WANG Yifei et al. Clinical observation of cooling blood and suppressing Yang in the treatment of blood-heat syndrome of psoriasis vulgaris[J]. World Clinical Drugs, 2020, 41(7): 524–529+561. doi: 10.13683/j.wph.2020.07.006. [108] 华亮, 蒯仂, 李苏, 等. 基于六元联系数方程的六经辨治皮肤病模型[J]. 辽宁中医杂志, 2020, 47(12): 12–15.HUA Liang, KUAI Le, LI Su, et al. Model of skin disease differentiation and treatment by six channels based on six-element correlation equation[J]. Liaoning Journal of Traditional Chinese Medicine, 2020, 47(12): 12–15. [109] 卢怡, 蒯仂, 茹意, 等. 集对分析阴阳平衡方程在皮肤病诊治中的应用[J]. 世界中医药, 2020, 15(14): 2170–2174. doi: 10.3969/j.issn.1673-7202.2020.14.032.LU Yi, KUAI Le, RU Yi, et al. Application of yin-yang balance equation in diagnosis and treatment of dermatosis[J]. World Journal of Chinese Medicine, 2020, 15(14): 2170–2174. doi: 10.3969/j.issn.1673-7202.2020.14.032. [110] 邢梦, 蒯仂, 丁晓杰, 等. 基于集对分析偏联系数探讨银屑病的复发因素及预后趋势[J]. 辽宁中医杂志, 2021, 48(5): 19–22. doi: 10.13192/j.issn.1000-1719.2021.05.004.XING Meng, KUAI Le, DING Xiaojie, et al. Study on recurrence factors and prognosis trend of psoriasis based on set pair analysis partial correlation number[J]. Liaoning Journal of Traditional Chinese Medicine, 2021, 48(5): 19–22. (in Chinese) doi: 10.13192/j.issn.1000-1719.2021.05.004. [111] 华亮, 魏冬慧, 蒯仂, 等. 基于集对分析势值与疗效曲线的银屑病血热证中药选优[J]. 中国中西医结合皮肤性病学杂志,2022, 21(5): 412–416.HUA Liang, WEI Donghui, KUAI Le, et al. Selection of traditional Chinese medicine for blood-heat syndrome of psoriasis based on set pair analysis potential value and curative effect curve[J]. Chinese Journal of Dermatology and Venereology of Integrated Traditional and Western Medicine, 2022, 21(5): 412–416. [112] KUAI Le, FEI Xiaoya, XING Jiaqi, et al. An efficacy predictive method for diabetic ulcers based on higher-order markov chain-set pair analysis[J]. Hindawi, Evidence-Based Complementary and Alternative Medicine Volume 2020, Article ID 5091671, 19 pages. [113] 李斌, 李欣, 蒯仂, 等. 中医辨证论治集对分析[M]. 科学出版社, 2021年.LI Bin, LI Xin, KUAI Le, et al. Set pair analysis of tcm syndrome differentiation and treatment [M]. Science Press, 2021. [114] PAN Fuquan, WU Qiudie, WANG Zhaoqiang, et al. Effectiveness evaluation of optical illusion deceleration markings for a V-shaped undersea tunnel based on the set pair analysis method and the technique for order preference by similarity to ideal solution theory[J]. Transportation Research Record:Journal of the Transportation Research Board, 2023, 2677(5): 308–324. doi: 10.1177/03611981221130326. [115] WANG Wensheng, JIN Juliang, DING Jing, et al. A new approach to water resources system assessment——set pair analysis method[J]. Science in China Series E:Technological Sciences, 2009, 52(10): 3017–3023. doi: 10.1007/s11431-009-0099-z. [116] 王文圣, 金菊良, 丁晶, 等. 水文水资源集对分析[M]. 北京: 科学出版社, 2010.WANG Wensheng, JIN Juliang, DING Jing, et al. Set Pair Analysis for Hydrology and Water Resources Systems[M]. Beijing: Science Press, 2010. [117] 潘争伟, 吴成国, 金菊良. 水资源系统评价与预测的集对分析方法[M]. 北京: 科学出版社, 2016.PAN Zhengwei, WU Chengguo, and JIN Juliang. Set Pair Analysis Method for Water Resources System Evaluation and Prediction[M]. Beijing: Science Press, 2016. [118] WANG Dong, BORTHWICK A G, HE Handan, et al. A hybrid wavelet de-noising and Rank-Set Pair Analysis approach for forecasting hydro-meteorological time series[J]. Environmental Research, 2018, 160: 269–281. doi: 10.1016/j.envres.2017.09.033. [119] XU Feng, ZHENG Xiaoping, ZHANG Jian, et al. A hybrid reasoning mechanism integrated evidence theory and set pair analysis in Swine-Vet[J]. Expert Systems with Applications, 2010, 37(10): 7086–7093. doi: 10.1016/j.eswa.2010.03.008. [120] GARG H and KUMAR K. An advanced study on the similarity measures of intuitionistic fuzzy sets based on the set pair analysis theory and their application in decision making[J]. Soft Computing, 2018, 22(15): 4959–4970. doi: 10.1007/s00500-018-3202-1. [121] 龚士良. 集对分析及其在城市地面沉降研究中的应用[J]. 上海地质, 1997(4): 43–47.GONG Shiliang. Set pair analysis and its application in study of urban land subsidence[J]. Shanghai Geology, 1997(4): 43–47. [122] 郑逸加. 基于高斯混合模型的模仿学习算法的优化与评价[D]. [硕士论文], 北京工业大学, 2017.ZHENG Yijia. Optimization and evaluation of imitation learning algorithm based on Gauss mixture model[D]. [Master dissertation], Beijing University of Technology, 2017. [123] XIANG Weiqi, YANG Xiaohua, and LI Yuqi. A set pair analysis model for suitability evaluation of human settlement environment[J]. Thermal Science, 2021, 25(3): 2109–2116. doi: 10.2298/TSCI191001095X. [124] LI Lianhui, LEI Bingbing, and MAO Chunlei. Digital twin in smart manufacturing[J]. Journal of Industrial Information Integration, 2022, 26: 100289. doi: 10.1016/j.jii.2021.100289. [125] ZHANG Peng, ZHANG Xuemei, YUAN Peng, et al. Performance optimization of geopolymer mortar blending in nano-SiO2 and PVA fiber based on set pair analysis[J]. e-Polymers, 2023, 23(1): 20230015. doi: 10.1515/epoly-2023-0015. [126] XIANG Weiqi, YANG Xiaohua, BABUNA P, et al. Development, application and challenges of set pair analysis in environmental science from 1989 to 2020: A bibliometric review[J]. Sustainability, 2021, 14(1): 153. doi: 10.3390/su14010153. [127] 汪明武, 金菊良, 周玉良. 集对分析耦合方法与应用[M]. 北京: 科学出版社, 2014: 1–188.WANG Mingwu, JIN Juliang, and ZHOU Yuliang. Set Pair Analysis Based Coupling Methods and Applications[M]. Beijing: Science Press, 2014: 1–188. [128] 赵克勤. 偏联系数[C]. 中国人工智能进展2005, 北京, 2005: 884–886.ZHAO Keqin. Partial linkage number[C]. Progress in Artificial Intelligence in China, Beijing, China, 2005: 884–886. [129] 杨红梅, 赵克勤. 偏联系数的计算与应用研究[J]. 智能系统学报, 2019, 14(5): 865–876. doi: 10.11992/tis.201810022.YANG Hongmei and ZHAO Keqin. The calculation and application of partial connection numbers[J]. CAAI Transactions on Intelligent Systems, 2019, 14(5): 865–876. doi: 10.11992/tis.201810022. [130] 杨红梅. 偏联系数的哲学原理与应用[M]. 北京: 国家开放大学出版社, 2020.YANG Hongmei. The Philosophical Principle and Application of partial connectives[M]. Beijing: National Open University Press, 2020. [131] 杨红梅. 基于偏联系数的系统在临界点附近的变化趋势研究[J]. 山西广播电视大学学报, 2019, 24(1): 77–81. doi: CNKI:SUN:SXGB.0.2019-01-018.YANG Hongmei. Study on the change trend of the system near the critical point based on the partial connection number[J]. Journal of Shanxi Radio and Television University, 2019, 24(1): 77–81. doi: CNKI:SUN:SXGB.0.2019-01-018. [132] LI Zheng, JIN Juliang, CUI Yi, et al. Dynamic evaluation of regional water resources carrying capacity based on set pair analysis and partial connection number[J]. Water Supply, 2022, 22(3): 2407–2423. doi: 10.2166/ws.2021.371. [133] SHEN Qing, ZHANG Xiongtao, LOU Jungang, et al. Interval-valued intuitionistic fuzzy multi-attribute second-order decision making based on partial connection numbers of set pair analysis[J]. Soft Computing, 2022, 26(19): 10389–10400. doi: 10.1007/s00500-022-07314-2. [134] 申情, 蒋云良, 张雄涛. 属性权重未知情况下犹豫模糊多属性决策方法[J]. 智能系统学报, 2022, 17(4): 728–736. doi: 10.11992/tis.202107038.SHEN Qing, JIANG Yunliang, and ZHANG Xiongtao. A hesitant fuzzy multi-attribute decision-making method with unknown attribute weights[J]. CAAI Transactions on Intelligent Systems, 2022, 17(4): 728–736. doi: 10.11992/tis.202107038. [135] SHEN Qing, HUANG Xu, LIU Yong, et al. Multiattribute decision making based on the binary connection number in set pair analysis under an interval-valued intuitionistic fuzzy set environment[J]. Soft Computing, 2020, 24(10): 7801–7809. doi: 10.1007/s00500-019-04398-1. [136] 赵克勤. 反偏联系数[C]. 中国人工智能学会第12届全国学术年会论文汇编, 哈尔滨, 2007: 66–67.ZHAO Keqin. Anti-partial linkage number[C]. Progress in Artificial Intelligence in China, Harbin, China, 2007: 66–67. [137] 王万军. 一种基于偏联系数的区间数排序方法及其应用[J]. 甘肃联合大学学报:自然科学版, 2008, 22(1): 48–50. doi: 10.13804/j.cnki.2095-6991.2008.01.007.WANG Wanjun. Ranking and application of interval numbers based on partial connection numbers[J]. Journal of Gansu Lianhe University:Natural Sciences, 2008, 22(1): 48–50. doi: 10.13804/j.cnki.2095-6991.2008.01.007. [138] 王万军. 一种基于集对决策的偏联系数方法[J]. 甘肃联合大学学报:自然科学版, 2009, 23(3): 43–45. doi: 10.3969/j.issn.1672-691X.2009.03.013.WANG Wanjun. A decision method in parital connection number based on SPA[J]. Journal of Gansu Lianhe University:Natural Sciences, 2009, 23(3): 43–45. doi: 10.3969/j.issn.1672-691X.2009.03.013. [139] 王万军, 李恒杰, 胡建军, 等. 一种Vague值转化Fuzzy值的偏联系数方法[J]. 计算机工程与应用, 2013, 49(1): 134–136. doi: 10.3778/j.issn.1002-8331.1112-0329.WANG Wanjun, LI Hengjie, HU Jianjun, et al. Partial connection number method for transforming Vague value into Fuzzy value[J]. Computer Engineering and Applications, 2013, 49(1): 134–136. doi: 10.3778/j.issn.1002-8331.1112-0329. [140] 晏燕, 王万军. 偏联系数隐私风险态势评估方法[J]. 计算机工程与应用, 2018, 54(10): 143–148. doi: 10.3778/j.issn.1002-8331.1612-0444.YAN Yan and WANG Wanjun. Privacy risk situation assessment method based on partial connection numbers[J]. Computer Engineering and Applications, 2018, 54(10): 143–148. doi: 10.3778/j.issn.1002-8331.1612-0444. [141] 沈定珠. 体育用联系数学[M]. 北京: 中国教育文化出版社, 2007: 1–192.SHEN Dingzhu. Contact Mathematics for Sports[M]. Beijing: China Education and Culture Press, 2007: 1–192. [142] 陆广地, 吴陈. 基于联系数伴随函数的区间数多属性决策[J]. 模糊系统与数学, 2018, 32(1): 182–190.LU Guangdi and WU Chen. Interval number multiple-attribute decision-making based on adjoint functions of connection number[J]. Fuzzy Systems and Mathematics, 2018, 32(1): 182–190. [143] 金菊良, 张浩宇, 崔毅, 等. 联系数伴随函数的若干问题探讨[J]. 黑龙江大学工程学报, 2020, 11(2): 1–10. doi: 10.13524/j.2095-008x.2020.02.015.JIN Juliang, ZHANG Haoyu, CUI Yi, et al. Discussions on some problems for adjoint function of connection number[J]. Journal of Engineering of Heilongjiang University, 2020, 11(2): 1–10. doi: 10.13524/j.2095-008x.2020.02.015. [144] 赵克勤, 赵森烽. 奇妙的联系数[M]. 北京: 知识产权出版社, 2014.ZHAO Keqin and ZHAO Senfeng. Fantastic Contact Number[M]. Beijing: Intellectual Property Publishing House, 2014. [145] 蒋云良, 赵克勤, 刘以安, 等. 信息处理集对分析[M]. 北京: 清华大学出版社, 2015.JIANG Yunliang, ZHAO Keqin, LIU Yian, et al. Set Pair Analysis of Information Processing[M]. Beijing: Tsinghua University Press, 2015. [146] DATTA B K and HAIDER M R. The double burden of overweight or obesity and anemia among women married as children in India: A case of the Simpson's paradox[J]. Obesity Research & Clinical Practice, 2022, 16(5): 364–372. doi: 10.1016/j.orcp.2022.09.002. [147] 李国重, 许伟, 韩松辉, 等. 辛普森悖论产生机理的数学解析[J]. 信息工程大学学报, 2019, 20(2): 242–245. doi: CNKI:SUN:XXGC.0.2019-02-020.LI Guozhong, XU Wei, HAN Songhui et al. Mathematical analysis of Simpson Paradox[J]. Journal of Information Engineering University, 2019, 20(2): 242–245. doi: CNKI:SUN:XXGC.0.2019-02-020. [148] 中国逻辑学会辩证逻辑研究会编, 辩证逻辑研究[C]. 上海人民出版社, 1981年.Dialectical Logic Research Institute, Chinese Logic Society, Eds. Research on Dialectical Logic[M]. Shanghai People's Publishing House, 1981. [149] 林达华. 集合论: 现代数学的共同基础[J]. 高等数学研究, 2019, 22(1): 83. doi: CNKI:SUN:XUSJ.0.2019-01-024.LIN Dahua. Set Theory: The Common Foundation of Modern Mathematics[J]. Research in Advanced Mathematics, 2019, 22(1): 83. doi: CNKI:SUN:XUSJ.0.2019-01-024. [150] WANG Jing, LAN Siwu, LI Xiangyu, et al. Research on the method of hypergraph construction of information systems based on set pair distance measurement[J]. Electronics, 2023, 12(20): 4375. doi: 10.3390/electronics12204375. [151] TENG Zhijun, LI Mingzhe, YU Libo, et al. Sinkhole attack defense strategy integrating SPA and jaya algorithms in wireless sensor networks[J]. Sensors, 2023, 23(24). -
下载:
图(2) / 表(5)
计量
- 文章访问数: 463
- HTML全文浏览量: 110
- PDF下载量: 180
- 被引次数: 0
