﻿ 偏联系数的计算与应用研究
«上一篇
 文章快速检索 高级检索

 智能系统学报  2019, Vol. 14 Issue (5): 865-876  DOI: 10.11992/tis.201810022 0

### 引用本文

YANG Hongmei, 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.

### 文章历史

1. 山西广播电视大学 成人教育学院，山西 太原 030027;
2. 诸暨市联系数学研究所，浙江 诸暨 311800

The calculation and application of partial connection numbers
YANG Hongmei 1, ZHAO Keqin 2
1. Adult Education College, Shanxi Radio and TV University, Taiyuan 030027, China;
2. Institut of Zhuji Connection Mathematics, Zhuji 311800, China
Abstract: Partial connection numbers (PCNs) are a kind of adjoint function of connection numbers. Their computational process reflects a paradoxical movement on the micro level, and the result indicates that the phase result of such paradoxical movement is the main mathematical tool of the multi-layer approximation method of macro-state and micro-trend. This paper also systematically expounds the commonly used PCN algorithms from 2- to 5-element connection numbers and some ideas and establishes that the PCN algorithm is an intelligent algorithm from the aspects of intelligent technology innovation and information energy development and utilization.
Key words: set pair analysis    connection number    multi-connection number    partial connection number    full partial connection number    micro motion of system    multi-layer analysis method    information energy

1 联系数及其联系分量的示性系数

 $\mu = a + bi$ (1)
 $\mu = a + bi + cj$ (2)
 $\mu = a + bi + cj + d$ (3)
 $\mu = a + bi + cj + dk + el$ (4)

 $\left[ \begin{matrix} 1 & i & j & k & l \\ 1 & \left[ -1,1 \right] & {\rm{ }} & {\rm{ }} & {\rm{ }} \\ 1 & \left[ -1,1 \right] & \left[ -1 \right] & {\rm{ }} & {\rm{ }} \\ 1 & \left[ 0,1 \right] & \left[ -1,0 \right] & \left[ -1 \right] & {\rm{ }} \\ 1 & \left[ 0.33,1 \right] & \left[ -0.33,0.33 \right] & \left[ -1,-0.33 \right] & -1 \\ \end{matrix} \right]$ (5)

2 偏联系数 2.1 基本原理

2.2 二元联系数的偏联系数

 ${{\partial }^{+}}a=a/\left( a+b \right)$ (6)

 ${{\partial }^{+}}\mu ={{\partial }^{+}}a$ (7)

 ${{\partial }^{+}}\mu ={{\partial }^{+}}(a+bi)=a/(a+b)={{\partial }^{+}}a$ (8)

 ${{\partial }^{+}}\mu ={{\partial }^{+}}(a+bi)=a/(a+b)$ (9)

 ${{\partial}^{-}}b=b/\left( a+b \right)$ (10)

 ${{\partial}^{-}}\mu =i{{\partial}^{-}}b$ (11)

 ${{\partial}^{-}}\mu ={{\partial}^{-}}\left( a+bi \right)=i\left[ b/\left( a+b \right) \right]=i{{\partial}^{-}}b$ (12)

 ${{\partial}^{-}}\mu ={{\partial}^{-}}\left( a+bi \right)=i\left[ b/\left( a+b \right) \right]=i{{\partial}^{-}}b$ (13)

 ${{\partial }^{\pm }}\mu ={{\partial }^{+}}\mu +{{\partial }^{-}}\mu$ (14)

 \begin{align} & {{\partial }^{\pm }}\mu ={{\partial }^{+}}\mu +{{\partial }^{-}}\mu ={{\partial }^{+}}a+i{{\partial }^{-}}b= \\ & \ \ \ \ \ \ \ \ \ \ \ \ \frac{a}{a+b}+\frac{bi}{a+b}=a+bi \\ \end{align} (15)

 ${{\partial }^{+}}\mu ={{\partial }^{+}}\left( a+bi \right)={{\partial }^{+}}\left( 0.6+0.4i \right)= \frac{0.6}{0.4+0.6}=0.6$

 \begin{align} {{\partial }^{-}}\mu ={{\partial }^{-}}\left( a+bi \right)={{\partial }^{-}}\left( 0.6+0.4i \right)= \frac{0.4i}{0.6+0.4}=0.4i \end{align}

 ${{\partial }^{\pm }}\mu ={{\partial }^{+}}\mu +{{\partial }^{-}}\mu ={{\partial }^{+}}a+{{\partial }^{-}}b=0.6+0.4i$

$i$ 遍历[−1，1]时， ${{\partial }^{\pm }}\mu$ 遍历[0.2，1]，即当 $i=-1$ 时， ${{\partial }^{\pm }}\mu =0.2$ ；当 $i=1$ 时， ${{\partial }^{\pm }}\mu =1$ ；由于 $i$ 遍历 $\left[ -1,1 \right]$ 时，均有 ${{\partial }^{\pm }}\mu \geqslant 0$ ，所以二元联系数 $\mu =0.6+$ $0.4i$ 时系统在微观层上的演化趋势为正向趋势。

2.3 三元联系数的偏联系数

 ${{\partial }^{+}}\mu ={{\partial }^{+}}a+i{{\partial }^{+}}b=\frac{a}{a+b}+\frac{b}{b+c}i$ (16)

 ${{\partial }^{-}}\mu =\left( {{\partial }^{-}}b \right)i+\left( {{\partial }^{-}}c \right)j=\frac{b}{a+b}i+\frac{c}{b+c}j$ (17)

 ${{\partial }^{+}}\mu =\frac{a}{a+b}+\frac{b}{b+c}i$

 ${{\partial }^{-}}\mu =\frac{b}{a+b}i+\frac{c}{b+c}j$

 $\begin{split} &\qquad\quad{{\partial }^{\pm }}\mu ={{\partial }^{+}}\mu +{{\partial }^{-}}\mu = \\ & \frac{a}{a+b}+\frac{b}{b+c}{{i}^{+}}+\frac{b}{a+b}{{i}^{-}}+\frac{c}{b+c}j \end{split}$ (18)

 ${{\partial }^{\pm }}\mu =\frac{a+b{{i}^{-}}}{a+b}+\frac{b{{i}^{+}}+cj}{b+c}$ (19)

 $\partial \mu =\frac{a+b{{i}^{-}}}{a+b}+\frac{b{{i}^{+}}+cj}{b+c}$ (20)

 ${{\partial }^{+}}\mu ={{\partial }^{+}}a+i{{\partial }^{+}}b=\frac{a}{a+b}+\frac{b}{b+c}i$ (21)

 ${{\partial }^{2+}}\mu =\partial ({{\partial }^{+}}\mu )=\dfrac{\dfrac{a}{a+b}}{\dfrac{a}{a+b}+\dfrac{b}{b+c}}$ (22)

 ${{\partial }^{2-}}\mu ={{\partial }^{-}}\left( {{\partial }^{-}}\mu \right)=\dfrac{\dfrac{c}{b+c}}{\dfrac{b}{a+b}+\dfrac{c}{b+c}}j$ (23)

 $\begin{split} {{\partial }^{2{\pm }}}\mu = & {{\partial }^{2+}}\mu +{{\partial }^{2-}}\mu = \dfrac{\dfrac{a}{a+b}}{\dfrac{a}{a+b}+\dfrac{b}{b+c}}+\dfrac{\dfrac{b}{a+b}}{\dfrac{b}{a+b}+\dfrac{c}{b+c}}j= \\ & \dfrac{\dfrac{a}{a+b}}{\dfrac{a}{a+b}+\dfrac{b}{b+c}}-\dfrac{\dfrac{b}{a+b}}{\dfrac{b}{a+b}+\dfrac{c}{b+c}} \\[-28pt] \end{split}$ (24)

 \begin{aligned} &\qquad\quad{{\partial }^{\pm }}\left( {{\partial }^{\pm }}\mu \right)= {{\partial }^{\pm }}\left( {{\partial }^{+}}\mu +{{\partial }^{-}}\mu \right) = \\ & \frac{{{\partial }^{+}}\mu }{{{\partial }^{+}}\mu +{{\partial }^{-}}\mu }+\frac{{{\partial }^{-}}\mu }{{{\partial }^{+}}\mu +{{\partial }^{-}}\mu } =\frac{{{\partial }^{+}}\mu +{{\partial }^{-}}\mu }{{{\partial }^{+}}\mu +{{\partial }^{-}}\mu }=1 \end{aligned}

 ${{\partial }^{2+}}\mu =\dfrac{\dfrac{0.5}{0.5+0.3}}{\dfrac{0.5}{0.5+0.3}+\dfrac{0.3}{0.3+0.2}}=\dfrac{25}{49}=0.510\;2$

 \begin{align} {{\partial }^{2-}}\mu =\dfrac{\dfrac{0.2}{0.3+0.2}}{\dfrac{0.3}{0.5+0.3}+\dfrac{0.2}{0.3+0.2}}j =-\dfrac{16}{31}=-0.516\;1 \end{align}

 \begin{align} {{\partial }^{2}}^{\pm }\mu ={{\partial }^{2+}}\mu +{{\partial }^{2-}}\mu =0.510\;2-0.516\;1=-0.006 \end{align}

2.4 四元联系数的偏联系数

 ${{\partial }^{+}}\mu ={{\partial }^{+}}a+i{{\partial }^{+}}b+j{{\partial }^{+}}c$ (25)

 $\begin{split} {{\partial }^{2+}}\mu = & {{\partial }^{+}}\left( {{\partial }^{+}}\mu \right)={{\partial }^{+}}\left( {{\partial }^{+}}a+i{{\partial }^{+}}b+j{{\partial }^{+}}c \right) = \\ &\frac{{{\partial }^{+}}a}{{{\partial }^{+}}a+{{\partial }^{+}}b}+\frac{{{\partial }^{+}}b}{{{\partial }^{+}}b+{{\partial }^{+}}c}i \end{split}$ (26)

$\,\mu$ 的三阶偏正联系数为 ${{\partial }^{3+}}\mu$ ，则

 \begin{aligned} &{{\partial }^{3+}}\mu = {{\partial }^{+}}\left( {{\partial }^{2+}}\mu \right)={{\partial }^{+}}\left[ {{\partial }^{+}}\left( {{\partial }^{+}}\mu \right) \right]= {{\partial }^{+}}\left[ {{\partial }^{+}}\left( {{\partial }^{+}}a+i{{\partial }^{+}}b+j{{\partial }^{+}}c \right) \right] =\\ & {{\partial }^{+}}\left( \frac{{{\partial }^{+}}a}{{{\partial }^{+}}a+{{\partial }^{+}}b}+\frac{{{\partial }^{+}}b}{{{\partial }^{+}}b+{{\partial }^{+}}c}i \right)= \dfrac{\dfrac{{{\partial }^{+}}a}{{{\partial }^{+}}a+{{\partial }^{+}}b}}{\dfrac{{{\partial }^{+}}a}{{{\partial }^{+}}a+{{\partial }^{+}}b}+\dfrac{{{\partial }^{+}}b}{{{\partial }^{+}}b+{{\partial }^{+}}c}} \end{aligned} (27)

${{\partial }^{+}}a=\dfrac{a}{a+b},{{\partial }^{+}}b=\dfrac{b}{b+c},{{\partial }^{+}}c=\dfrac{c}{c+d}$ 代入式(27)得

 ${{\partial }^{3+}}\mu =\dfrac{\dfrac{\dfrac{a}{a+b}}{\dfrac{a}{a+b}+\dfrac{b}{b+c}}}{\dfrac{\dfrac{a}{a+b}}{\dfrac{a}{a+b}+\dfrac{b}{b+c}}+\dfrac{\dfrac{b}{b+c}}{\dfrac{b}{b+c}+\dfrac{c}{c+d}}}$ (28)

 ${{\partial }^{-}}\mu =i{{\partial }^{-}}b+j{{\partial }^{-}}c+k{{\partial }^{-}}d$ (29)

 $\begin{split} {{\partial }^{2-}}\mu = & {{\partial }^{-}}\left( {{\partial }^{-}}\mu \right)={{\partial }^{-}}\left( i{{\partial }^{-}}b+j{{\partial }^{-}}c+k{{\partial }^{-}}d \right) =\\ & \frac{{{\partial }^{-}}c}{{{\partial }^{-}}b+{{\partial }^{-}}c}j+\frac{{{\partial }^{-}}d}{{{\partial }^{-}}c+{{\partial }^{-}}d}k \end{split}$ (30)

$\, \mu$ 的三阶偏负联系数为 ${{\partial }^{{3-}}}\mu$ ，则

 $\begin{split} &{{\partial }^{{3-}}}\mu= {{\partial }^{-}}\left( {{\partial }^{2-}}\mu \right)={{\partial }^{-}}\left[ {{\partial }^{-}}\left( {{\partial }^{-}}\mu \right) \right]=\\ & \quad{{\partial }^{-}}\left[ {{\partial }^{-}}\left( i{{\partial }^{-}}b+j{{\partial }^{-}}c+k{{\partial }^{-}}d \right) \right]= \\ & {{\partial }^{-}}\left( \frac{{{\partial }^{-}}c}{{{\partial }^{-}}b+{{\partial }^{-}}c}j+\frac{{{\partial }^{-}}d}{{{\partial }^{-}}c+{{\partial }^{-}}d}k \right)= \\ & \qquad\dfrac{\dfrac{{{\partial }^{-}}d}{{{\partial }^{-}}c+{{\partial }^{-}}d}k}{\dfrac{{{\partial }^{-}}c}{{{\partial }^{-}}b+{{\partial }^{-}}c}+\dfrac{{{\partial }^{-}}d}{{{\partial }^{-}}c+{{\partial }^{-}}d}} \\ \end{split}$ (31)

${{\partial }^{-}}b=\dfrac{b}{a+b},{{\partial }^{-}}c=\dfrac{c}{b+c},{{\partial }^{-}}d=\dfrac{d}{c+d}$ 代入式(31)得

 ${{\partial }^{3-}}\mu =\frac{\dfrac{\dfrac{d}{c+d}}{\dfrac{c}{b+c}+\dfrac{d}{c+d}}k}{\dfrac{\dfrac{c}{b+c}}{\dfrac{b}{a+b}+\dfrac{c}{b+c}}+\dfrac{\dfrac{d}{c+d}}{\dfrac{c}{b+c}+\dfrac{d}{c+d}}}$ (32)

 $\begin{split} {{\partial }^{3\pm }}\mu \!=\! \dfrac{\dfrac{\dfrac{a}{a\!+\!b}}{\dfrac{a}{a\!+\!b}\!+\!\dfrac{b}{b\!+\!c}}}{\dfrac{\dfrac{a}{a\!+\!b}}{\dfrac{a}{a\!+\!b}\!+\!\dfrac{b}{b\!+\!c}}\!+\!\dfrac{\dfrac{b}{b\!+\!c}}{\dfrac{b}{b\!+\!c}\!+\!\dfrac{c}{c\!+\!d}}} - \dfrac{\dfrac{\dfrac{d}{c\!+\!d}}{\dfrac{c}{b\!+\!c}\!+\!\dfrac{d}{c\!+\!d}}}{\dfrac{\dfrac{c}{b\!+\!c}}{\dfrac{b}{a\!+\!b}\!+\!\dfrac{c}{b\!+\!c}}\!+\!\dfrac{\dfrac{d}{c\!+\!d}}{\dfrac{c}{b\!+\!c}\!+\!\dfrac{d}{c\!+\!d}}} \\ \end{split}$ (33)

2.5 五元联系数的偏联系数

 ${{\partial }^{+}}\mu ={{\partial }^{+}}a+i{{\partial }^{+}}b+j{{\partial }^{+}}c+k{{\partial }^{+}}d$ (34)

 \begin{align} & {{\partial }^{2+}}\mu ={{\partial }^{^{+}}}({{\partial }^{+}}a+i{{\partial }^{+}}b+j{{\partial }^{+}}c+k{{\partial }^{+}}d)= \\ & \frac{{{\partial }^{+}}a}{{{\partial }^{+}}a+{{\partial }^{+}}b}+\frac{{{\partial }^{+}}b}{{{\partial }^{+}}b+{{\partial }^{+}}c}+\frac{{{\partial }^{+}}c}{{{\partial }^{+}}c+{{\partial }^{+}}d} \\ \end{align} (35)

$\, \mu$ 的三阶偏正联系数为 ${{\partial }^{3+}}\mu$ ，则

 \begin{align} & \qquad\qquad\qquad{{\partial }^{3+}}\mu ={{\partial }^{+}}({{\partial }^{2+}}\mu ) =\\ &\qquad\quad{{\partial }^{^{+}}}[{{\partial }^{^{+}}}({{\partial }^{+}}a+i{{\partial }^{+}}b+j{{\partial }^{+}}c+k{{\partial }^{+}}d)] =\\ &\quad{{\partial }^{^{+}}}[\frac{{{\partial }^{+}}a}{{{\partial }^{+}}a+{{\partial }^{+}}b}+\frac{i{{\partial }^{+}}b}{{{\partial }^{+}}b+{{\partial }^{+}}c}+\frac{j{{\partial }^{+}}c}{{{\partial }^{+}}c+{{\partial }^{+}}d}] =\\ &\dfrac{\dfrac{{{\partial }^{+}}a}{{{\partial }^{+}}a+{{\partial }^{+}}b}}{\dfrac{{{\partial }^{+}}a}{{{\partial }^{+}}a+{{\partial }^{+}}b}+\dfrac{{{\partial }^{+}}b}{{{\partial }^{+}}b+{{\partial }^{+}}c}}+\dfrac{\dfrac{{{\partial }^{+}}b}{{{\partial }^{+}}b+{{\partial }^{+}}c}i}{\dfrac{{{\partial }^{+}}b}{{{\partial }^{+}}b+{{\partial }^{+}}c}+\dfrac{{{\partial }^{+}}c}{{{\partial }^{+}}c+{{\partial }^{+}}d}} \end{align} (36)

$\, \mu$ 的四阶偏正联系数为 ${{\partial }^{4+}}\mu$ ，则

 \begin{align} & \qquad\qquad\qquad\qquad\quad{{\partial }^{4+}}\mu ={{\partial }^{+}}\left( {{\partial }^{3+}}\mu \right)= \\ & {{\partial }^{+}}\left( \dfrac{\dfrac{{{\partial }^{+}}a}{{{\partial }^{+}}a+{{\partial }^{+}}b}}{\dfrac{{{\partial }^{+}}a}{{{\partial }^{+}}a+{{\partial }^{+}}b}+\dfrac{{{\partial }^{+}}b}{{{\partial }^{+}}b+{{\partial }^{+}}c}} + \dfrac{\dfrac{{{\partial }^{+}}b}{{{\partial }^{+}}b+{{\partial }^{+}}c}i}{\dfrac{{{\partial }^{+}}b}{{{\partial }^{+}}b+{{\partial }^{+}}c}+\dfrac{{{\partial }^{+}}c}{{{\partial }^{+}}c+{{\partial }^{+}}d}} \right) = \\ & \quad\dfrac{\dfrac{\dfrac{{{\partial }^{+}}a}{{{\partial }^{+}}a+{{\partial }^{+}}b}}{\dfrac{{{\partial }^{+}}a}{{{\partial }^{+}}a+{{\partial }^{+}}b}+\dfrac{{{\partial }^{+}}b}{{{\partial }^{+}}b+{{\partial }^{+}}c}}}{\dfrac{\dfrac{{{\partial }^{+}}a}{{{\partial }^{+}}a+{{\partial }^{+}}b}}{\dfrac{{{\partial }^{+}}a}{{{\partial }^{+}}a+{{\partial }^{+}}b}+\dfrac{{{\partial }^{+}}b}{{{\partial }^{+}}b+{{\partial }^{+}}c}}+\dfrac{\dfrac{{{\partial }^{+}}b}{{{\partial }^{+}}b+{{\partial }^{+}}c}}{\dfrac{{{\partial }^{+}}b}{{{\partial }^{+}}b+{{\partial }^{+}}c}+\dfrac{{{\partial }^{+}}c}{{{\partial }^{+}}c+{{\partial }^{+}}d}}} \\ \end{align} (37)

 ${{\partial }^{-}}\mu =i{{\partial }^{-}}b+j{{\partial }^{-}}c+k{{\partial }^{-}}d+l{{\partial }^{-}}e$ (38)

$\, \mu$ 的二阶偏负联系数为 ${{\partial }^{2-}}\mu$ ，则

 \begin{aligned} & \qquad\qquad{{\partial }^{2-}}\mu ={{\partial }^{-}}\left( {{\partial }^{-}}\mu \right)= \\ & \;\;{{\partial }^{-}}\left( i{{\partial }^{-}}b \right.+j{{\partial }^{-}}c+k{{\partial }^{-}}d +\left. l{{\partial }^{-}}e \right)= \\ & \frac{j{{\partial }^{-}}c}{{{\partial }^{-}}b+{{\partial }^{-}}c}+\frac{k{{\partial }^{-}}d}{{{\partial }^{-}}c+{{\partial }^{-}}d}+\frac{l{{\partial }^{-}}e}{{{\partial }^{-}}d+{{\partial }^{-}}e} \end{aligned} (39)

$\, \mu$ 的三阶偏负联系数为 ${{\partial }^{3-}}\mu$ ，则

 \begin{align} &\qquad\quad{{\partial }^{3-}}\mu ={{\partial }^{-}}\left( {{\partial }^{2-}}\mu \right)={{\partial }^{-}}\left[ {{\partial }^{-}}\left( {{\partial }^{-}}\mu \right) \right] = \\ & \quad\;\;{{\partial }^{-}}\left( \frac{j{{\partial }^{-}}c}{{{\partial }^{-}}b+{{\partial }^{-}}c}+\frac{k{{\partial }^{-}}d}{{{\partial }^{-}}c+{{\partial }^{-}}d}+\frac{l{{\partial }^{-}}e}{{{\partial }^{-}}d+{{\partial }^{-}}e} \right)= \\ & \dfrac{\dfrac{k{{\partial }^{-}}d}{{{\partial }^{-}}d+{{\partial }^{-}}c}}{\dfrac{{{\partial }^{-}}d}{{{\partial }^{-}}d+{{\partial }^{-}}c}+\dfrac{{{\partial }^{-}}c}{{{\partial }^{-}}b+{{\partial }^{-}}c}}+\dfrac{\dfrac{l{{\partial }^{-}}e}{{{\partial }^{-}}d+{{\partial }^{-}}e}}{\dfrac{{{\partial }^{-}}e}{{{\partial }^{-}}d+{{\partial }^{-}}e}+\dfrac{{{\partial }^{-}}d}{{{\partial }^{-}}d+{{\partial }^{-}}c}} \end{align} (40)

$\,\mu$ 的四阶偏负联系数为 ${{\partial }^{4-}}\mu$ ，则

 \begin{align} & \qquad\qquad\qquad\qquad {{\partial }^{4-}}\mu ={{\partial }^{-}}\left( {{\partial }^{3-}}\mu \right)= \\ & {{\partial }^{-}}\left( \dfrac{\dfrac{k{{\partial }^{-}}d}{{{\partial }^{-}}c+{{\partial }^{-}}d}}{\dfrac{{{\partial }^{-}}c}{{{\partial }^{-}}b+{{\partial }^{-}}c}+\dfrac{{{\partial }^{-}}d}{{{\partial }^{-}}c+{{\partial }^{-}}d}} +\dfrac{\dfrac{l{{\partial }^{-}}e}{{{\partial }^{-}}d+{{\partial }^{-}}e}}{\dfrac{{{\partial }^{-}}d}{{{\partial }^{-}}c+{{\partial }^{-}}d}+\dfrac{{{\partial }^{-}}e}{{{\partial }^{-}}d+{{\partial }^{-}}e}} \right) =\\ &\quad \dfrac{\dfrac{\dfrac{l{{\partial }^{-}}e}{{{\partial }^{-}}d+{{\partial }^{-}}e}}{\dfrac{{{\partial }^{-}}d}{{{\partial }^{-}}c+{{\partial }^{-}}d}+\dfrac{{{\partial }^{-}}e}{{{\partial }^{-}}d+{{\partial }^{-}}e}}}{\dfrac{\dfrac{{{\partial }^{-}}d}{{{\partial }^{-}}c+{{\partial }^{-}}d}}{\dfrac{{{\partial }^{-}}c}{{{\partial }^{-}}b+{{\partial }^{-}}c}+\dfrac{{{\partial }^{-}}d}{{{\partial }^{-}}c+{{\partial }^{-}}d}}+\dfrac{\dfrac{{{\partial }^{-}}e}{{{\partial }^{-}}d+{{\partial }^{-}}e}}{\dfrac{{{\partial }^{-}}d}{{{\partial }^{-}}c+{{\partial }^{-}}d}+\dfrac{{{\partial }^{-}}e}{{{\partial }^{-}}d+{{\partial }^{-}}e}}} \\ \end{align} (41)

 \begin{align} & \qquad\qquad\qquad{{\partial}^{4\pm }}\mu ={{\partial}^{4+}}\mu +{{\partial}^{4-}}\mu= \\ & \dfrac{\dfrac{\dfrac{{{\partial }^{+}}a}{{{\partial }^{+}}a+{{\partial }^{+}}b}}{\dfrac{{{\partial }^{+}}a}{{{\partial }^{+}}a+{{\partial }^{+}}b}+\dfrac{{{\partial }^{+}}b}{{{\partial }^{+}}b+{{\partial }^{+}}c}}}{\dfrac{\dfrac{{{\partial }^{+}}a}{{{\partial }^{+}}a+{{\partial }^{+}}b}}{\dfrac{{{\partial }^{+}}a}{{{\partial }^{+}}a+{{\partial }^{+}}b}+\dfrac{{{\partial }^{+}}b}{{{\partial }^{+}}b+{{\partial }^{+}}c}}+\dfrac{\dfrac{{{\partial }^{+}}b}{{{\partial }^{+}}b+{{\partial }^{+}}c}}{\dfrac{{{\partial }^{+}}b}{{{\partial }^{+}}b+{{\partial }^{+}}c}+\dfrac{{{\partial }^{+}}c}{{{\partial }^{+}}c+{{\partial }^{+}}d}}}+ \\ & \dfrac{\dfrac{\dfrac{-{{\partial }^{-}}e}{{{\partial }^{-}}d+{{\partial }^{-}}e}}{\dfrac{{{\partial }^{-}}d}{{{\partial }^{-}}c+{{\partial }^{-}}d}+\dfrac{{{\partial }^{-}}e}{{{\partial }^{-}}d+{{\partial }^{-}}e}}}{\dfrac{\dfrac{{{\partial }^{-}}d}{{{\partial }^{-}}c+{{\partial }^{-}}d}}{\dfrac{{{\partial }^{-}}c}{{{\partial }^{-}}b+{{\partial }^{-}}c}+\dfrac{{{\partial }^{-}}d}{{{\partial }^{-}}c+{{\partial }^{-}}d}}+\dfrac{\dfrac{{{\partial }^{-}}e}{{{\partial }^{-}}d+{{\partial }^{-}}e}}{\dfrac{{{\partial }^{-}}d}{{{\partial }^{-}}c+{{\partial }^{-}}d}+\dfrac{{{\partial }^{-}}e}{{{\partial }^{-}}d+{{\partial }^{-}}e}}} \\ \end{align} (42)

${{\partial }^{+}}a=\dfrac{a}{a+b}{\text{、}}{{\partial }^{+}}b=\dfrac{b}{b+c}{\text{、}}{{\partial }^{+}}c=\dfrac{c}{c+d}$ ${{\partial }^{+}}d=\dfrac{d}{d+e}$ ${{\partial }^{-}}b=\dfrac{b}{a+b}{\text{、}}{{\partial }^{-}}c=\dfrac{c}{b+c}$ ${{\partial }^{-}}d=\dfrac{d}{c+d}{\text{、}}$ ${{\partial }^{-}}e=\dfrac{e}{d+e}$ 代入式(42)就得到具体数值。也就是说，式(42)是一个没有不确定示性系数的实数，其物理意义是：当 ${{\partial }^{4}}^{\pm }\mu >0$ 时，表明五元联系数 $\mu$ 的系统在微观层次上的演化趋势是正向趋势；当 ${{\partial }^{4}}^{\pm }\mu <0$ 时，表明五元联系数 $\mu$ 的系统在微观层次上的演化趋势是负向趋势；当 ${{\partial }^{4}}^{\pm }\mu =0$ 时，表明五元联系数 $\mu$ 的系统在微观层次上的演化趋势处在正负临界状态。

3 反偏联系数

4 偏联系数算法研究的若干新思路 4.1 加权偏联系数

4.2 基于相互作用的全偏联系数

5 应用举例

 \begin{align} & u\left( x_1 \right)=19+8i+0j+3k+0l \\ & u\left( x_2 \right)=3+15i+7j+5k+0l \\ & u\left( x_3 \right)=2+22i+5j+1k+0l \\ & u\left( x_4 \right)=0+29i+1j+0k+0l \\ & u\left( x_5 \right)=28+1i+0j+0k+1l \\ & u\left( x_6 \right)=0+5i+19j+4k+2l \\ & u\left( x_7 \right)=21+6i+1j+0k+2l \\ \end{align}

2)把上述五元联系数归一化处理，得：

 \begin{aligned} & \mu \left( x_1 \right)=0.6333+0.2667i+0j+0.1k+0l \\ & \mu \left( x_2 \right)=0.1+0.5i+0.2333j+0.1667k+0l \\ & \mu \left( x_3 \right)=0.0667+0.7333i+0.1667j+0.0333k+0l \\ & \mu \left( x_4 \right)=0+0.9667i+0.0333j+0k+0l \\ & \mu \left( x_5 \right)=0.9333+0.0333i+0j+0k+0.0333l \\ & \mu \left( x_6 \right)=0+0.1667i+0.6333j+0.1333k+0.0667l \\ & \mu \left( x_7 \right)=0.7+0.2i+0.0333j+0k+0.0667l \\ \end{aligned}

3)按式(42)计算全偏联系数，并对全偏联系数的大小作出排序，得

 \begin{align} & {{\partial}^{4\pm }}\mu \left( x_1 \right)=0.226\;2 \\ & {{\partial}^{4\pm }}\mu \left( x_2 \right)=0.310\;2 \\ & {{\partial}^{4\pm }}\mu \left( x_3 \right)=0.323\;2 \\ & {{\partial}^{4\pm }}\mu \left( x_4 \right)=0 \\ & {{\partial}^{4\pm }}\mu \left( x_5 \right)=0.247\;8 \\ & {{\partial}^{4\pm }}\mu \left( x_6 \right)=-0.775\;4 \\ & {{\partial}^{4\pm }}\mu \left( x_7 \right)=-0.383\;5 \\ \end{align}

6 讨论

1)关于事物微观运动的数学刻画。众所周知，客观事物处于相互联系和运动变化之中，如何定量刻画事物的相互联系和运动变化，是包括人工智能学者在内的众多科技人员的研究课题。文献[160]和本文的工作表明，基于集对分析理论的联系数及其偏联系数是定量刻画事物相互联系和运动的一个新数学工具，其理由：首先，偏联系数把联系数中的各个联系分量不再看作相互独立的量，而是假设成一定条件下相互生成的量，理论上，这种假设成立；其次，借助偏联系数的算法，揭示联系数中联系分量的相互生成是在微观层次上的一对矛盾运动，这也可以接受；因为哲学、物理学和无数事实告诉我们，矛盾普遍存在，运动成对进行，“作用力与反作用力大小相等，方向相反，作用在2个不同的物体上”已是一种科学常识；再次，偏联系数着眼于事物的运动在微观层次上的定量刻画。科学史表明，牛顿的微积分在刻画事物宏观层次上的运动已取得巨大成功,但人们对事物在微观层次上的运动观测和测量则受制于海森堡的“测不准原理”；正是在这一点上，集对分析借助联系数对不确定性“客观承认、系统描述、定量刻画、具体分析”[59]，使得基于联系数的偏联系数算法能够刻画出事物在微观层次上的矛盾运动。当然，微观与宏观是一个相对的划分，文献[19]中指出，“在生物学中，全体是宏观，个体就是微观；个体是宏观，细胞就是微观；细胞是宏观，基因就是微观；在物理化学中，肉眼直接见到的是宏观，要在显微镜下看到的是微观；在低倍显微镜下看到的是宏观，在高倍显微镜下看到的是微观；在时间序列中，世纪是宏观，年度就是微观；年度是宏观，月度是微观，小时是宏观，分钟就是微观；分钟是宏观，秒是微观，如此等等”。

2)关于全偏联系数。由第2章可知，计算一个给定 $n\left( n\geqslant 2 \right)$ 元联系数的偏联系数时，需要同时计算其 $n-1$ 阶偏正联系数和 $n-1$ 阶偏负联系数及其代数和，才能如实反映该 $n$ 元联系数的 $n$ 个联系分量在微观层次上的矛盾运动及其结果，这里说的代数和就是给定 $n$ 元联系数的全偏联系数，概念清晰，意义明确。文献[52]把 $\partial c=c/\left( a+c \right)$ 看成 $\mu$ 的全偏联系数，错误地理解全偏联系数，诱导出错误的结论，这说明对基本概念的正确理解极为重要。

3)关于偏联系数的生成机制和时态。在偏联系数计算过程中，需要注意各阶偏联系数中各联系分量的生成机制和时态。一般地说，用分式表示的某阶偏正(负)联系数中的联系分量，其分子的状态指过程完成时所处的状态，分母的状态则是过去进行时的状态，例如三元联系数 $\mu =a+bi+cj$ ，其一阶偏正联系数为

 ${{\partial }^{+}}\mu ={{\partial }^{+}}a+i{{\partial }^{+}}b=\frac{a}{a+b}+\frac{b}{b+c}i$

4)不难推知，第4章中有关加权偏联系数和效应全偏联系数的算法，以及基于相互作用的偏联系数算法，要比第2章中介绍的偏联系数基本算法复杂，由此推知反加权偏联系数、反效应全偏联系数、反相互作用偏联系数的算法更复杂，限于篇幅，本文没有展开介绍，特此说明。

5)偏联系数算法是一种新的智能算法。首先，从信息利用的角度看，偏联系数算法有效地挖掘了联系数中联系分量的动态信息，这种动态信息反映出联系数所刻画的研究对象的本质。因为客观事物总是处于动态变化之中，某一时刻相对静止的宏观状态与这种状态在微观层次上的变化趋势共存在一个系统中是所有研究对象的共同属性，借助联系数的偏联系数计算，能够看到系统在宏观静态下的微观动态，显然是一种智能；其次，从系统的角度看，偏联系数算法揭示了对象系统线性与非线性的关系，因为从形式上看，联系数中的各个联系分量可以有序地放置在一根水平轴上，具有明显的线性特征，但式(6)～式(42)表明，偏联系数所展示的图象是一幅非线性图象；再次，从人工智能技术创新的角度看，基于偏联系数的聚类、模式识别、系统综合评价决策与风险防控以及社交网络中的隐私保护研究，也在一定意义上属于智能技术的范畴，偏联系数算法因而是一种新的智能算法，需要作深入系统研究。

6)运动需要能量，无论这种运动处在宏观层次还是微观层次。偏联系数及其算法既然刻画了联系数中联系分量之间的矛盾运动，人们自然会问，驱使这种运动的能量又是什么性质的能量？回答是“信息能”。“信息能”是赵克勤在2015年7月在杭州举办的第3期非传统安全集对分析研学班上提出的一个概念，认为信息是物质和能量相互作用的产物，信息具有能量，称为信息能[15, 60]。联系数是刻画研究对象某个特定状态的一个信息系统，本身蕴含着一定的信息能，且具体蕴含在联系数中联系分量所在不同层次的系统结构中；偏联系数及其算法在一定程度上开发了这种“信息能”，得到的结果让人们从系统的一组宏观状态参数中认识和掌握这种状态在微观层次上的演化趋势，从而把联系数中的“信息能”在一定程度上转化成“智能”；但更多关于“信息能”转化成“智能”的问题待深入研究。

7 结束语

 [1] 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. ZHAO Keqin. Disposal and description of uncertainties based on the set pair analysis[J]. Information and control, 1995, 24(3): 162-166. (0) [2] 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. (0) [3] 王文圣, 李跃清, 金菊良, 等. 水文水资源集对分析[M]. 北京: 科学出版社, 2010. (0) [4] 杨红梅. 基于联系数的梯形直觉与非直觉模糊决策算法与应用[J]. 中北大学学报(自然科学版), 2012, 33(6): 688-694. YANG Hongmei. Operation and application of trapezoidal intuitionistic fuzzy number and unintuitionistic fuzzy decision method based on correlate[J]. Journal of North University of China (Natural Science Edition), 2012, 33(6): 688-694. (0) [5] 杨红梅. 集对分析在我国经济增长模糊规则提取中的应用[J]. 运筹与管理, 2013, 22(3): 194-200. YANG Hongmei. The application of set pair analysis in fuzzy rule extraction of domestic economy growth[J]. Operations research and management science, 2013, 22(3): 194-200. DOI:10.3969/j.issn.1007-3221.2013.03.027 (0) [6] 杨红梅. 基于二元联系数的三角模糊数直觉模糊集多属性决策[J]. 山西师范大学学报(自然科学版), 2015, 29(2): 13-19. YANG Hongmei. Multiple attribute decision making of triangular fuzzy number intuitionistic fuzzy set based on two-element connection number[J]. Journal of Shanxi Normal University (Natural Science Edition), 2015, 29(2): 13-19. (0) [7] 赵克勤, 赵森烽. 奇妙的联系数[M]. 北京: 知识产权出版社, 2014. (0) [8] 刘秀梅, 赵克勤. 区间数决策集对分析[M]. 北京: 科学出版社, 2014. (0) [9] 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. (0) [10] 王万军, 晏燕. 不确定信息处理的集对分析研究与应用[M]. 兰州: 兰州大学出版社, 2015. (0) [11] 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. (0) [12] 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. (0) [13] 刘保相. 粗糙集对分析理论与决策模型[M]. 北京: 科学出版社, 2010. (0) [14] 汪明武, 金菊良. 联系数理论与应用[M]. 北京: 科学出版社, 2017. (0) [15] 蒋云良, 赵克勤. 人工智能集对分析[M]. 北京: 科学出版社, 2017. (0) [16] 赵克勤, 米红. 非传统安全与集对分析[M]. 北京: 知识产权出版社, 2010. (0) [17] 蒋云良, 徐从富. 集对分析理论及其应用研究进展[J]. 计算机科学, 2006, 33(1): 205-209. JIANG Yunliang, XU Congfu. Advances in set pair analysis theory and its applications[J]. Computer science, 2006, 33(1): 205-209. DOI:10.3969/j.issn.1002-137X.2006.01.057 (0) [18] 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. 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. (0) [19] 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. 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. (0) [20] 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. The function and meaning of Pair principle in the Set Pair Analysis[J]. Discovery of nature, 1998, 17(4): 90. (0) [21] 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. 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. (0) [22] PAN Zhengwei, WANG Yanhua, JIN Juliang, et al. Set pair analysis method for coordination evaluation in water resources utilizing conflict[J]. Physics and chemistry of the earth, parts A/B/C, 2017, 101: 149-156. DOI:10.1016/j.pce.2017.05.009 (0) [23] YU Furong, QU Jihong, LI Zhiping, et al. Application of set pair analysis based on the improved five-element connectivity in the evaluation of groundwater quality in Xuchang, Henan Province, China[J]. Water science and technology: water supply, 2017, 17(3): 632-642. DOI:10.2166/ws.2016.135 (0) [24] YAN Fang, XU Kaili. Application of a cloud model-set pair analysis in hazard assessment for biomass gasification stations[J]. PLoS one, 2017, 12(1): e0170012. DOI:10.1371/journal.pone.0170012 (0) [25] YAN Fang, XU Kaili, LI Deshun, et al. Hazard assessment for biomass gasification station using general set pair analysis[J]. Bioresources technology, 2016, 11(4): 8307-8324. (0) [26] TAN Chong, SONG Yi, CHE Heng. Application of set pair analysis method on occupational hazard of coal mining[J]. Safety science, 2017, 92: 10-16. DOI:10.1016/j.ssci.2016.09.005 (0) [27] LI Chunhui, SUN Lian, JIA Junxiang, et al. Risk assessment of water pollution sources based on an integrated k-means clustering and set pair analysis method in the region of Shiyan, China [J]. Science of the total environment, 2016, 557/558: 307-316. DOI:10.1016/j.scitotenv.2016.03.069 (0) [28] WANG Ya, ZHOU Lihua. Assessment of the coordination ability of sustainable social-ecological systems development based on a set pair analysis: a case study in Yanchi County, China[J]. Sustainability, 2016, 8(8): 733. DOI:10.3390/su8080733 (0) [29] WANG Mingwu, XU Xinyu, LI Jian, et al. A novel model of set pair analysis coupled with extenics for evaluation of surrounding rock stability[J]. Mathematical problems in engineering, 2015, 2015: 892549. (0) [30] ZHANG Jian, YANG Xiaohua, LI Yuqi. A refined rank set pair analysis model based on wavelet analysis for predicting temperature series[J]. International journal of numerical methods for heat and fluid flow, 2015, 25(5): 974-985. (0) [31] YANG Xiaohua, SUN Boyang, ZHANG Jian, et al. Hierarchy evaluation of water resources vulnerability under climate change in Beijing, China[J]. Natural hazards, 2016, 84(Suppl 1): 63-76. (0) [32] XIE Xuecai, GUO Deyong. Human factors risk assessment and management: process safety in engineering[J]. Process safety and environmental protection, 2018(113): 467-482. (0) [33] WEI Chao, DAI Xiaoyan, YE Shufeng, et al. Prediction analysis model of integrated carrying capacity using set pair analysis[J]. Ocean and coastal management, 2016, 120: 39-48. (0) [34] BAO Danwen, ZHANG Xiaoling. Measurement methods and influencing mechanisms for the resilience of large airports under emergency events[J]. Transportmetrica A: transport science, 2018, 14(10): 855-880. DOI:10.1080/23249935.2018.1448016 (0) [35] GARG H, 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 (0) [36] YAN Fang, XU Kaili. A set pair analysis based layer of protection analysis and its application in quantitative risk assessment[J]. Journal of loss prevention in the process industries, 2018, 55: 313-319. DOI:10.1016/j.jlp.2018.07.007 (0) [37] LI Peiyue, QIAN Hui, WU Jianhua. Application of set pair analysis method based on entropy weight in groundwater quality assessment−a case study in Dongsheng City, Northwest China[J]. Journal of chemistry, 2010, 8(2): 851-858. (0) [38] ZENG Jiajun, HUANG Guoru. Set pair analysis for karst waterlogging risk assessment based on AHP and entropy weight[J]. Hydrology research, 2017, 49(4): 1143-1155. (0) [39] KUMAR K, GARG H. Connection number of set pair analysis based TOPSIS method on intuitionistic fuzzy sets and their application to decision making[J]. Applied intelligence, 2018, 48(8): 2112-2119. DOI:10.1007/s10489-017-1067-0 (0) [40] GARG H, KUMAR K. Some aggregation operators for linguistic intuitionistic fuzzy set and its application to group decision-making process using the set pair analysis[J]. Arabian journal for science and engineering, 2018, 43(6): 3213-3227. DOI:10.1007/s13369-017-2986-0 (0) [41] FENG Yixiong, LU Runjie, GAO Yicong, et al. Multi-objective optimization of VBHF in sheet metal deep-drawing using Kriging, MOABC, and set pair analysis[J]. The international journal of advanced manufacturing technology, 2018, 96(9/10/11/12): 3127-3138. (0) [42] 赵克勤. 偏联系数[C]//中国人工智能进展2005. 北京: 北京邮电大学出版社, 2005: 884–885. ZHAO Keqin. Partial connection number[C]//Progress of Artificial Intelligence 2005. Beijing: Progress of Beijing University of Posts and Telecommunications (BUPT) Publishing House, 2005: 884–885. (0) [43] 施志坚, 王华伟, 王祥. 基于多元联系数集对分析的航空维修风险态势评估[J]. 系统工程与电子技术, 2016, 38(3): 588-594. SHI Zhijian, WANG Huawei, WANG Xiang. Risk state evaluation of aviation maintenance based on multiple connection number set pair analysis[J]. Systems engineering and electronics, 2016, 38(3): 588-594. (0) [44] 李聪, 陈建宏, 杨珊, 等. 五元联系数在地铁施工风险综合评价中的应用[J]. 中国安全科学学报, 2013, 23(10): 21-26. LI Cong, CHEN Jianhong, YANG Shan, et al. Application of five-element connection number to comprehensive evaluation of risks involved with subway construction[J]. China safety science journal, 2013, 23(10): 21-26. DOI:10.3969/j.issn.1003-3033.2013.10.004 (0) [45] 谢红涛, 李波, 赵云胜. 基于联系数的地铁隧道施工邻近建筑物风险评价[J]. 工业安全与环保, 2014, 40(7): 16-19. XIE Hongtao, LI Bo, ZHAO Yunsheng. Risk assessment of neighboring building in metro tunneling construction based on connection number[J]. Industrial safety and environmental protection, 2014, 40(7): 16-19. DOI:10.3969/j.issn.1001-425X.2014.07.006 (0) [46] 马成正, 王洪德. 联系数在地铁车站火灾安全风险评价中的应用[J]. 辽宁工程技术大学学报(自然科学版), 2015, 37(1): 26-31. MA Chengzheng, WANG Hongde. Application of connection number to safety risk evaluation of fire accident in subway station[J]. Journal of Liaoning Technical University (Natural Science Edition), 2015, 37(1): 26-31. (0) [47] 金菊良, 沈时兴, 潘争伟, 等. 水资源集对分析理论与应用研究进展[J]. 华北水利水电大学学报(自然科学版), 2017, 38(4): 54-66. JIN Juliang, SHEN Shixing, PAN Zhengwei, et al. Advances in theoretical and applied research on set pair analysis method for water resources system[J]. Journal of North China University of Water Resources and Electric Power (Natural Science Edition), 2017, 38(4): 54-66. DOI:10.3969/j.issn.1002-5634.2017.04.008 (0) [48] 李子彪, 张静, 李林琼. 区域创新极创新态势测度方法研究: 对北京的集对分析[J]. 科技进步与对策, 2016, 33(15): 111-117. LI Zibiao, ZHANG Jing, LI Linqiong. Study on the measure method of regional innovation poles innovation trend based on set pair analysis[J]. Science and technology progress and policy, 2016, 33(15): 111-117. DOI:10.6049/kjjbydc.2016010013 (0) [49] 李柏洲, 李新. 基于集对分析的企业技术依赖预警及其演化趋势测度[J]. 运筹与管理, 2015, 24(2): 262-271. LI Baizhou, LI Xin. Technology dependence early warning and evolution tendency evaluation based on set pair analysis[J]. Operations research and management science, 2015, 24(2): 262-271. DOI:10.12005/orms.2015.0073 (0) [50] 吴亭. 五元联系数在学生成绩发展趋势分析中的应用[J]. 数学的实践与认识, 2009, 39(5): 53-59. WU Ting. Application on the analysis of developmental trend of the student mark with five-element partial connection number[J]. Mathematics in practice and theory, 2009, 39(5): 53-59. (0) [51] 赵金楼, 高宏玉, 成俊会. 基于三元联系数的网络舆情传播中主体参与意愿演化评价方法[J]. 情报科学, 2017, 35(8): 118-120, 140. ZHAO Jinlou, GAO Hongyu, CHENG Junhui. Research on evolution of participation willingness in network public opinion: based on three-element connection number[J]. Information science, 2017, 35(8): 118-120, 140. (0) [52] 杨斯玲, 蒋根谋. 基于约束理论和集对分析的EPC建筑供应链风险管理[J]. 技术经济, 2016, 35(8): 111-117. YANG Siling, JIANG Genmou. Risk management of EPC construction supply chain based on theory of constraints and set pair analysis[J]. Technology economics, 2016, 35(8): 111-117. DOI:10.3969/j.issn.1002-980X.2016.08.016 (0) [53] 高晓辉. 基于联系数的老年人健康状态潜在发展趋势分析[J]. 中国卫生统计, 2012, 29(2): 265-266. GAO Xiaohui. Analysis of medical quality development trend based on the concomitant function of contact number[J]. Chinese journal of health statistics, 2012, 29(2): 265-266. DOI:10.3969/j.issn.1002-3674.2012.02.039 (0) [54] 周兴慧, 张吉军. 基于五元联系数的风险综合评价方法及其应用[J]. 系统工程理论与实践, 2013, 33(8): 2169-2176. ZHOU Xinghui, ZHANG Jijun. Risk comprehensive evaluation method and its application based on the five-element connection number[J]. Systems engineering -theory and practice, 2013, 33(8): 2169-2176. DOI:10.3969/j.issn.1000-6788.2013.08.035 (0) [55] 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. YAN Yan, 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 (0) [56] 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): 062604. ZHANG Mengmeng, LIU Yian, SONG Ping. Applications of partial connection clustering algorithm and random forest algorithm in radar signal sorting[J]. Laser & optoelectronics progress, 2019, 56(6): 062604. (0) [57] 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, 中国, 2007: 66−67. (0) [58] 金菊良, 张浩宇, 宁少尉, 等. 效应全偏联系数及其在区域水资源承载力评价中的应用[J]. 华北水利水电大学学报(自然科学版), 2019, 40(1): 1-8. JIN Juliang, ZHANG Haoyu, NING Shaowei, et al. Effect full partial connection number and its application in evaluation of regional water resources carrying capacity[J]. Journal of North China University of Water Resources and Electric Power (Natural Science Edition), 2019, 40(1): 1-8. (0) [59] 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): 16-23, 72. ZHAO Keqin, XUAN Aili. Set pair theory: A new theory method of non-define and its applications[J]. Systems engineering, 1996, 14(1): 16-23, 72. (0) [60] 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. JIANG Yunliang, 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. (0)