Some Applications and Progress of Set Pair Theory in Artificial Intelligence
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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.
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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} $ ··· ··· ··· ··· ··· 
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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 $$\mu = r(\cos \theta + i\sin \theta )$ $ r = \sqrt {{A^2} + {B^2}} $ $ \theta = \arctan \dfrac{B}{A} $ ${\mu _1} = {r_1}(\cos {\theta _1} + i\sin {\theta _1})$
${\mu _2} = {r_2}(\cos {\theta _2} + i\sin {\theta _2})$，则
$ \begin{aligned} & \mu 1\mu 2 = r1r2 \\& = [\cos \left( {\theta 1 + \theta 2} \right) + i\sin \left( {\theta 1 + \theta 2} \right)] \end{aligned} $
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表 3 赵森烽克勤概率的常见运算
赵森烽-克勤概率的
一般表达式普通四则运算 期望值计算 基于赵森烽克勤概率的贝叶斯公式 $ \begin{gathered} P(A,\mathop A\limits^ - ) = P(A) + P(\mathop A\limits^ - )i \\ 0 \le p(A) \le 1 \\ 0 \le P(\mathop A\limits^ - ) \le 1 \\ P(A) + P(\mathop A\limits^ - ) = 1 \\ \end{gathered} $ 参照表1中2元联系数的
普通四则运算。$E(X,\mathop X\limits^ - ) = E(X) + E(\mathop X\limits^ - )i$当$ X $作为主事件时，则有
${\mathrm{Ec}}(X) = E(X) + E(\mathop X\limits^ - )i$；
当$ \mathop X\limits^ - $作为主事件时，则有
$ {\mathrm{Ec}}(\mathop X\limits^ - ) = E(\mathop X\limits^ - ) + E(X)i $$ \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} $
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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 √
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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
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