﻿ 粒协调决策形式背景的属性约简与规则融合
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 智能系统学报  2019, Vol. 14 Issue (6): 1138-1143  DOI: 10.11992/tis.201905050 0

### 引用本文

ZHANG Xiaohe, MI Jusheng, LI Meizheng. Attribute reduction and rule fusion in granular consistent formal decision contexts[J]. CAAI Transactions on Intelligent Systems, 2019, 14(6): 1138-1143. DOI: 10.11992/tis.201905050.

### 文章历史

1. 河北师范大学 数学与信息科学学院，河北 石家庄 050024;
2. 河北师范大学 计算机与网络空间安全学院，河北 石家庄 050024

Attribute reduction and rule fusion in granular consistent formal decision contexts
ZHANG Xiaohe 1, MI Jusheng 1, LI Meizheng 2
1. College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050024, China;
2. College of Computer and Cyber Security, Hebei Normal University, Shijiazhuang 050024, China
Abstract: Attribute reduction and rule acquisition based on formal decision contexts can acquire knowledge more conveniently and effectively; thus, rule acquisition and attribute reduction are two key research directions of the theory of formal concept analysis (FCA). This study investigates attribute reduction and rule acquisition based on an equivalence relation in formal granular consistent decision contexts. In this paper, the granular consistent set and granular reduction are defined, and the judgment theory of the granular consistent set is given, and by combination with the Boolean method, the granular reduction is formulated. Finally, using the inclusion degree of set-valued vectors, optimistic and pessimistic rule fusion methods in formal decision contexts are proposed.
Key words: attribute reduction    decision rules    formal context    discernibility matrix    inclusions    extracting rules    granular computing    concept lattice

2009年，Wu[6]研究了保持概念格的粒结构不变的属性约简与规则提取，并给出具体算法。Li等[7-9]提出了一种新的决策形式背景知识约简框架，给出约简算法，且在决策形式背景研究了保持决策规则不变的属性约简。Li等[10]研究了基于同余关系的不协调决策形式背景的属性约简。Li等[11]提出了基于最大规则的决策形式背景中的新型属性约简。Chen[12-13]提出了一种大数据模型下的快速属性约简模型，并结合实例分析其算法复杂度。Yang[14]基于蕴涵映射对实数集决策形式背景中的属性约简和规则提取问题进行研究。Qi[15]从多角度讨论了两类三支概念格与传统概念格之间的联系, 并给出了在经典概念格基础上构造三支概念格的算法。Zhang等[16]利用概念格提出了一种基于案例的层次化分类器。Zhang等[17]研究了模糊决策格值信息系统上的近似约简与规则提取。张[18-19]叙述了信息系统上的知识发现与知识约简, 提出了协调近似表示空间上的规则融合方法。

1 预备知识

 ${{{X}}^*} = \{ a \in {{A}}: \forall x \in {{X}}, (x,a) \in {{I}}\}$ (1)
 ${{{B}}^ \triangleleft } = \{ x \in {{U}}: \forall a \in {{B}}, (x,a) \in {{I}}\}$ (2)

 ${x^{{\rm{*\{ a\} }}}} = \left\{ {\begin{array}{*{20}{c}} {\{ a\} ,\quad (x,a) \in {{I}} }\\ { \text{Ø} , \quad (x,a) \notin {{I}} } \end{array}} \right.$ (3)

 $\begin{array}{l} {{P}}{\rm{ = \{ }}{{E}}{\rm{ = (}}{{{E}}_{\rm{1}}}{\rm{,}}{{{E}}_{\rm{2}}}{\rm{,}} \cdots {\rm{ ,}}{{{E}}_m}{\rm{) }}:{{{E}}_l} \leqslant {{{V}}_l}{\rm{(}}l \leqslant m{\rm{)\} }}\subseteq \\ \;\;\;{{P}}({{{E}}_{\rm{1}}}{\rm{)}} \times {{P}}({{{E}}_{\rm{2}}}{\rm{)}} \times \cdots \times ({{{E}}_m}{\rm{)}} \end{array}$

${{P}}$ 为集合向量空间。对于 ${{P}}$ 中的两个集合向量 ${{E}}{\rm{ = (}}{{{E}}_1}{\rm{,}} \cdots {\rm{ ,}}{{{E}}_m}{\rm{) }}$ ${{S}}{\rm{ = (}}{{{S}}_{\rm{1}}}{\rm{,}} \cdots {\rm{ ,}}{{{S}}_m}{\rm{) }}$ ，如果 $\forall l \leqslant m$ ，均有 ${{{E}}_l} \leqslant {{{S}}_l}$ ，则记 ${{E}} \leqslant {{S}}$ ，且 ${\rm{(}}{{P}}{\rm{,}} \leqslant {\rm{)}}$ 为偏序集。

1) ${\rm{0}} \leqslant {{D(S/E)}} \leqslant 1$

2) ${{E}} \leqslant {{S}}$ 时， ${{D}}({{S}}/{{E}}) = 1$

3) ${{E}} \leqslant {{S}} \leqslant {{G}}$ 时， ${{D}}({{E}}/{{G}}) \leqslant {{D}}({{E}}/{{S}})$

2 基于等价关系的粒协调决策形式背景的属性约简

${{F}} = ({{U}},{{A}}, {{I}},{{D}},{{G}} )$ 为一个决策形式背景， ${{A}}$ 为条件属性集， ${{D}}$ 为决策属性集，并且 ${{A}} \cap {{D}} = \text{Ø}$ ${{I}} \subseteq {{U}} \times {{A}}$ 。定义 ${{{R}}_{{D}}} = \{ (x,y) \in {{U}} \times {{U}}: {d_l}(x) = {d_l}(y) (\forall {d_l} \in$ ${{D}})\}$ ，则由上述等价关系可产生 ${{U}}$ 上的一个划分：

 ${{U}}/{{{R}}_{{D}}} = \{ {[x]_{{D}}}: x \in {{U}}\} {\rm{ = \{ }}{{{D}}_1}{\rm{,}}{{{D}}_2}, \cdots ,{{{D}}_r}{\rm{\} }}$

 ${[x]_D}{\rm{ = \{ }}y \in {{U}}: (x,y) \in {{{R}}_D}\}$

2)对于 $x$ $y \in {{{D}}_j}$ $\forall {d_l} \in {{D}}$ ${d_l}(x) = {d_l}(y)$ ，显然决策类中所有对象决策值相同，记为 ${d_l}({{{D}}_j})$ 。称 ${{{T}}_j} =$ $\{ {d_1}({{{D}}_j}), {d_2}({{{D}}_j}),\cdots ,{d_{|{{D}}|}}({{{D}}_j})\}$ 为决策类 ${{{D}}_j}$ 的决策值。

${{F}} = ({{U}},{{A}}, {{I}},{{D}},{{G}} )$ 是一个粒协调决策形式背景， ${{B}} \subseteq {{A}}$ ，记

 ${{{R}}_{{B}}} = \{ ({x_i},{x_j}) \in {{U}} \times {{U}}:x_i^{*\{ a\} } \subseteq x_j^{*\{ a\} },\forall a \in {{B}}\}$ (4)

1) $x_i^{*B \triangleleft } = \{ {x_j} \in {{U}}:({x_i},{x_j}) \in {{{R}}_B}\}$

2) $x_i^{*B \triangleleft } \subseteq {[{x_i}]_D} \Leftrightarrow {{{R}}_B} \subseteq {{{R}}_D}$ ，即 ${{B}}$ 是粒协调集等价于 ${{{R}}_B} \subseteq {{{R}}_D}$

 ${{{D}}_d}(x,y){\rm{ = }}\left\{ {\begin{array}{*{20}{c}} {\{ a \in {{A}}:x_{}^{*\{ a\} }\not \subseteq y_{}^{*\{ a\} }\} ,}& {{d_l}(x) \ne {d_l}(y)}\\ {\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\text{Ø} ,}& {\text{其他}} \end{array}} \right.$ (5)

${{ {{D}}}^*} = \{ {{{D}}_d}(x,y)\mid x,y \in {{U}}\}$ ${{F}}$ 的辨识矩阵。

1) ${{{D}}_d}(x,x) = \text{Ø}$

2) ${{{D}}_d}(x,y) \bigcap {{{{D}}_d}} (y,x) = \text{Ø}$

$\left( \Leftarrow \right)$ ${{{D}}_d}(x,y) \ne \text{Ø}$ ，则 $\forall {d_l} \in {{D}}$ ${d_l}(x) \ne {d_l}(y)$ ，即 $(x,y)\not \in {{{R}}_{{D}}}$ 。故 ${{B}} \bigcap {{{{D}}_d}(x,y)} \ne \text{Ø}$ ，一定存在 $b \in {{B}},$ 使得 $b \in {{{D}}_d}(x,y)$ ，则有 $x_{}^{*\{ b\} }\not \subseteq y_{}^{*\{ b\} }$ ，从而 $(x,y) \notin {{R}}_B^{}$ 。当 $(x,y)\not \in {{{R}}_{{D}}}$ $(x,y) \notin {{R}}_B^{}$ ，故 ${{{R}}_B} \subseteq {{{R}}_{{D}}}$ 。综上， ${{B}}$ ${{F}}$ 的粒协调集。

$\left( \Leftarrow \right)$ 如果 ${D_d}(x,y{\rm{) = \{ }}a{\rm{\} }}$ ，定义7可知 ${d_l}(x) \ne {d_l}(y)$ $x_{}^{*\{ a\} }\not \subseteq y_{}^{*\{ a\} }$ 并且任取 $b \in {{A}} - a,$ 必有 $x_{}^{*\{ b\} } \subseteq y_{}^{*\{ b\} }$ 。因此 $(x,y)\not \in {{{R}}_D}$ $(x,y) \in {{{R}}_{{{A}} - \{ a\} }}$ 。故可得 ${{{R}}_{{{A}} - \{ a\} }}\not \subseteq {{{R}}_D}$ ，即 ${{A}} - {\rm{\{ }} a {\rm{\} }}$ 不是 ${{F}}$ 的粒约简， $a$ ${{F}}$ 的核心属性。

 ${{M}}{\rm{ = }} \wedge ( \vee {{{D}}_d}(x,y)) = \wedge ( \vee \{ {a_l}:{a_l} \in {{{D}}_d}(x,y)\} )$ (6)

 ${{M}} = \vee _{k = 1}^p( \wedge _{t = 1}^{{q_k}}{a_{{i_t}}})$ (7)

1) for 1≤i≤|U| and 1≤s≤|A|

2) if $\left( {{x_i},{a_s}} \right) \in {{I}},{x_i}^{*\left\{ {{a_s}} \right\}} = {a_s}$

3) else ${x_i}^{*\left\{ {{a_s}} \right\}} = \text{Ø}$

4) initialize ${{{D}}_d}\left( {{x_i},{x_j}} \right) = \text{Ø}$

5) for 1≤s≤|A| and 1≤l≤|D| do

6)　if ${x_i}^{*\left\{ {{a_s}} \right\}}\not \subseteq {x_j}^{*\left\{ {{a_s}} \right\}}\;\;\;{\rm{and}}\;\;{d_l}\left( {{x_i}} \right) \ne {d_l}\left( {{x_j}} \right)$ then

7) 　　 ${{{D}}_d}\left( {{x_i},{x_j}} \right) \leftarrow {{{D}}_d}\left( {{x_i},{x_j}} \right) \vee {a_s}$

8) 　end if

9) end for

10) initialize ${{B}}= \text{Ø}$

11) for 1≤i≤|U|, 1≤j≤|U| do

12) 　 ${{B}} = {{B}} \wedge {{{D}}_d}\left( {{x_i},{x_j}} \right)$

13) end for

14) compute ${{B}} = \vee _{m = 1}^p\left( { \wedge _{q = 1}^{{s_i}}{a_q}} \right)$

15) return B

 \qquad\begin{aligned} & x_1^{* \triangleleft } = \{ {x_1},{x_2},{x_3}\} = {[{x_1}]_D}, x_2^{* \triangleleft } = \{ {x_2},{x_3}\} \subseteq {[{x_2}]_D} = {[{x_1}]_D}\\ & x_3^{* \triangleleft } = \{ {x_3}\} \subseteq {[{x_3}]_D} = {[{x_1}]_D}, x_4^{* \triangleleft } = \{ {x_4}\} \subseteq \{ {x_4},{x_7}\} = {[{x_4}]_D}\\ & x_5^{* \triangleleft } = \{ {x_5},{x_6}\} \subseteq \{ {x_5},{x_6}\} = {[{x_5}]_D}\\ & x_6^{* \triangleleft } = \{ {x_6}\} \subseteq {[{x_6}]_D} = {[{x_5}]_D},x_7^{* \triangleleft } = \{ {x_7}\} \subseteq {[{x_7}]_D} = {[{x_4}]_D} \end{aligned}

 $\begin{array}{l} {{M}} = \wedge ( \vee {D_d}(x,y)) \\ \;\;\;\;{\rm{ = }}{a_1} \wedge {a_4} \wedge {a_6} \wedge {a_8} \wedge ({a_3} \vee {a_5}) \\ \;\;\;\;{\rm{ = }}({a_1} \wedge {a_4} \wedge {a_6} \wedge {a_8} \wedge {a_3}) \vee ({a_1} \wedge {a_4} \wedge {a_6} \wedge {a_8} \wedge {a_5})\\ \end{array}$

3 基于等价关系的粒协调决策形式背景上的规则融合

${{F}} = ({{U}},{{A}}, {{I}},{{D}},{{G}} )$ 是一个粒协调决策形式背景， $\forall x \in {{U}}$ ${a_l} \in {{A}}$ ，定义

 ${f_l}({x_i}) = \left\{ {\begin{array}{*{20}{c}} {1,\quad ({x_i},{a_l}) \in {{I}}} \\ { 0,\quad ({x_i},{a_l}) \notin {{I}}} \end{array}} \right.$

$\forall {{B}} \in {{A}}$ ${a_l} \in {{A}}$ ，定义 ${v_l}({{B}}) = \left\{ {\begin{array}{*{20}{c}} {1,\quad {a_l} \in {{B}}\;\;\;\; } \\ { 0,\quad {a_l}\not \in {{B}}{\text{。}}} \end{array}} \right.$

 ${{D}}({{S}}/{{E}}) = \sum\limits_{i = 1}^m {\frac{{\left| {{{{S}}_l}} \right|}}{{\sum\limits_{i = 1}^m {\left| {{{{S}}_i}} \right|} }}} {\chi _{{{{F}}_l}}}({{{E}}_l})$ (8)

3.1 乐观规则融合

1)由 ${{{R}}_D}$ ${{U}}$ 进行划分，得到

 ${{U}}/{{{R}}_{{D}}} = \{ {{{D}}_1},{{{D}}_2}, \cdots ,{{{D}}_r}\}$

2)对于决策类 ${{{D}}_j}(j \leqslant r)$ ，记

 ${{{M}}_j} = \{ (\{ {f_1}({x_i})\} ,\{ {f_2}({x_i})\} , \cdots ,\{ {f_m}({x_i})\} )\left| {x_i^{* \triangleleft } \subseteq {{{D}}_j}} \right.\} (j \leqslant r)$

3)将 ${{{M}}_j}$ 中的向量取并运算，即每个分量取并运算，得到 ${{{S}}_j} = ({{S}}_1^j,{{S}}_2^j, \cdots ,{{S}}_m^j) (j \leqslant r)$

4)对于概念 $({{X}},{{B}}) \in {{L}}({{U}},{{A}},{{I}})$ ，记

 ${{E}} = (\{ {v_1}\} ,\{ {v_2}\} , \cdots ,\{ {v_m}\} )$

 ${\rm{If}}\;{{B}}, {\rm{then}} \;{{{T}}_{{j_o}}} \;\;\;({{D}}({{{S}}_{{j_o}}}/{{E}}))$

1) for 1≤t≤|D| do

2) 　 ${{U}}/{{{R}}_{{D}}} = \left\{ {{{{D}}_1},{{{D}}_2}, \cdots ,{{{D}}_r}} \right\} = \left\{ {{{\left[ x \right]}_{{D}}}:x \in {{U}}} \right\}$

3) end for

4) for ${{{D}}_j}\left( {j \leqslant r} \right),\;\;{\rm{do}}$

5) ${{{M}}_j} = \left\{ {\left( {\left\{ {{f_1}\left( {{x_i}} \right)} \right\},\left\{ {{f_2}\left( {{x_i}} \right)} \right\}, \cdots ,\left. {\left\{ {{f_{|{{A}}|}}\left( {{x_i}} \right)} \right\}} \right)|} \right.x_i^{* \triangleleft } \subseteq {{{D}}_j}} \right\}$

6) compute ${{{S}}_j} \!=\! \left\{ {\left( { \vee \left\{ {{f_1}\left( {{x_i}} \right)} \right\}, \!\cdots\! , \vee \left\{ {{f_{|{{A}}|}}\left( {{x_i}} \right)} \right\}} \right)} \right.$ $\left. {| x_i^{* \triangleleft } \subseteq {{{D}}_j}} \right\}$

7) for $\left( {{{X}},{{B}}} \right) \in {{L}}\left( {{{U}},{{A}},{{I}}} \right)$ do

8) 　 ${{E}} = \left( {\left\{ {{v_1}\left( {{B}} \right)} \right\},\left\{ {{v_2}\left( {{B}} \right)} \right\}, \cdots ,\left\{ {{v_{|{{A}}|}}\left( {{B}} \right)} \right\}} \right)$

9) end for

10) compute D(Sj/E)

11) if D(Sjo/E)=maxD(Sj/E) ，then

12) 　 ${{{{R}}:}}\;{{B}} \Rightarrow {{{T}}_{{j_o}}}\left( {{{D}}\left( {{{{S}}_{{j_o}}}/{{E}}} \right)} \right)$

13) end if

14) return R

 $\!\!\!\!\begin{array}{l} ({\rm{\{ }}{x_1},{x_2},{x_3}\},\!\{ {a_1},{a_6}{\rm{\} }});({\rm{\{ }}{x_2},\!{x_3}\} ,\!\{ {a_1},\!{a_6},{a_7}\} );(\{ {x_3}\} ,\{ {a_1},\!{a_2},\!{a_6},\!{a_7}\} )\\ ({\rm{\{ }}{x_4}{\rm{\} }},\{ {a_2},{a_6},{a_7},{a_8}\} );(\{ {x_5},{x_6}\} ,\{ {a_1},{a_3},{a_5}\} )\\ ({\rm{\{ }}{x_6}{\rm{\} }},{\rm{\{ }}{a_1}{\rm{,}}{a_2},{a_3},{a_5}{\rm{\} }});(\{ {x_7}\} ,\{ {a_2},{a_3},{a_4}\} )\\ (\{ {x_1},{x_2},{x_3},{x_4}\} ,\{ {a_6}\} );(\{ {x_1},{x_2},{x_3},{x_5},{x_6}\} ,\{ a { _1}\} )\\ \{ {x_2},{x_3},{x_4}\} ,\{ {a_6},{a_7}\} );(\{ {x_3},{x_4}\} ,{\rm{\{ }}{a_2},{a_6},{a_7}{\rm{\} }})\\ (\{ {x_3},{x_6}\} ,{\rm{\{ }}{a_1},{a_2}{\rm{\} }});(\{ {x_3},{x_4},{x_6},{x_7}\} ,{\rm{\{ }}{a_2}{\rm{\} }});({x_{567}},\{ {a_3}\} )\\ (\{ {x_6},{x_7}\} ,{\rm{\{ }}{a_2},{a_3}{\rm{\} }});(U,\text{Ø} );(\text{Ø} ,{{A}}) \end{array}$

 ${{U}}/{{{R}}_{{D}}} = \{ \{ {x_1},{x_2},{x_3}\} ,\{ {x_4},{x_7}\} ,\{ {x_5},{x_6}\} \}$

 $({\rm{\{ }}{x_1},{x_2},{x_3}{\rm{\} }},{\rm{\{ }}1{\rm{\} }});({\rm{\{ }}{x_4},{x_7}{\rm{\} }},{\rm{\{ }}3{\rm{\} }});({\rm{\{ }}{x_5},{x_6}{\rm{\} }},{\rm{\{ }}2{\rm{\} }})$

 ${{{D}}_1} = \{ {x_1},{x_2},{x_3}\} ,\;{{{D}}_2} = \{ {x_4},{x_7}\} ,\;{{{D}}_3} = \{ {x_5},{x_6}\}$

 $\begin{array}{l}{{{M}}_1} = \{ ({\rm{\{ }}1{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}1{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}0{\rm{\} }}),\\[-1pt] \;\;\;\;\;\;\;\;\;({\rm{\{ }}1{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}1{\rm{\} }},{\rm{\{ }}1{\rm{\} }},{\rm{\{ }}0{\rm{\} }}),\\[-1pt] \;\;\;\;\;\;\;\;\;({\rm{\{ }}1{\rm{\} }},{\rm{\{ }}1{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}1{\rm{\} }},{\rm{\{ }}1{\rm{\} }},{\rm{\{ }}0{\rm{\} }}){\rm{\} }} \end{array}$
 $\begin{array}{l} {{{M}}_2} = \{ ({\rm{\{ }}0{\rm{\} }},{\rm{\{ }}1{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}1{\rm{\} }},{\rm{\{ }}1{\rm{\} }},{\rm{\{ }}1{\rm{\} }}),\\[-1pt] \;\;\;\;\;\;\;\;\;({\rm{\{ }}0{\rm{\} }},{\rm{\{ }}1{\rm{\} }},{\rm{\{ }}1{\rm{\} }},{\rm{\{ }}1{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}0{\rm{\} }})\} \end{array}$
 $\begin{array}{l} {{{M}}_3} = \{ ({\rm{\{ }}1{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}1{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}1{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}0{\rm{\} }}),\\[-1pt] \;\;\;\;\;\;\;\;\;({\rm{\{ }}1{\rm{\} }},{\rm{\{ }}1{\rm{\} }},{\rm{\{ }}1{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}1{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}0{\rm{\} }},{\rm{\{ }}0{\rm{\} }})\} \end{array}$

 ${{{S}}_1} = (\{ 1\} ,\{ 0,1\} ,\{ 0\} ,\{ 0\} ,\{ 0\} ,\{ 1\} ,\{ 0,1\} ,\{ 0\} )$
 ${{{S}}_2} = (\{ 0\} ,\{ 1\} ,\{ 0,1\} ,\{ 0,1\} ,\{ 0\} ,\{ 0 ,1\} ,\{ 0 ,1\} ,\{ 0 ,1\} )$
 ${{{S}}_3} = (\{ 1\} ,\{ 0,1\} ,\{ 1\} ,\{ 0\} ,\{ 1\} ,\{ 0\} ,\{ 0{\rm{\} }},\{ 0\} )$

 $\begin{array}{l} \;\;{{D}}({{{S}}_1}/{{E}}) = 0.9\\ {{D}}({{{S}}_2}/{{E}}) \approx 0.92\\ \;{{D}}({{{S}}_3}/{{E}}) \approx 0.56 \end{array}$

$\{ {a_1},{a_6}{\rm{\} }} \Rightarrow d = 1 (1);$ $\{ {a_1},{a_6},{a_7}{\rm{\} }} \Rightarrow d = 1 (1)$

$\{ {a_1},{a_2},{a_6},{a_7}\} \Rightarrow d = 1 (1);$ $\{ {a_2},{a_6},{a_7},{a_8}\} \Rightarrow d = 2(1)$

$\{ {a_1},{a_3},{a_5}\} \Rightarrow d = 3 (1);$ ${\rm{\{ }}{a_1}{\rm{,}}{a_2},{a_3},{a_5}{\rm{\} }} \Rightarrow d = 3(1)$

$\{ {a_2},{a_3},{a_4}\} \Rightarrow d = 2 (1);$ $\{ {a_6}\} \Rightarrow d = 2 (0.92)$

$\{ {a_1}\} \Rightarrow d = 1 (0.9);$ $\{ {a_6},{a_7}\} \Rightarrow d = 2 (0.92)$

${\rm{\{ }}{a_2},{a_6},{a_7}{\rm{\} }} \Rightarrow d = 2 (1);$ ${\rm{\{ }}{a_1},{a_2}{\rm{\} }} \Rightarrow d = 2 \;(0.92)$

${\rm{\{ }}{a_2}{\rm{\} }} \Rightarrow d = 2 (1);$ ${\rm{\{ }}{a_3}{\rm{\} }} \Rightarrow d = 2 (0.92)$

${\rm{\{ }}{a_2},{a_3}{\rm{\} }} \Rightarrow d = 2 (1)$

$\{ {a_1},{a_6}{\rm{\} }} \Rightarrow d = 1 (1);$ $\{ {a_1},{a_6},{a_7}{\rm{\} }} \Rightarrow d = 1 (1)$

$\{ {a_1},{a_2},{a_6},{a_7}\} \Rightarrow d = 1(1);$ $\{ {a_2},{a_6},{a_7},{a_8}\} \Rightarrow d = 2(1)$

$\{ {a_1},{a_3},{a_5}\} \Rightarrow d = 3 (1)$

3.2 悲观规则融合

1)由 ${{{R}}_{{D}}}$ ${{U}}$ 进行划分，得到

 ${{U}}/{{{R}}_{{D}}} = \{ {{{D}}_1},{{{D}}_2}, \cdots ,{{{D}}_r}\}$

2)对于决策类 ${{{D}}_j}(j \leqslant r)$ ，记

 ${{{M}}_j} = \{ (\{ {f_1}({x_i})\} ,\{ {f_2}({x_i})\} , \cdots ,\{ {f_m}({x_i})\} )\left| {x_i^{* \triangleleft } \subseteq {{{D}}_j}} \right.\} (j \leqslant r)$

3)将 ${{{M}}_j}$ 中的向量取最大值，即每个分量取最大值，得到 ${{{L}}_j} = ({{L}}_1^j,{{L}}_2^j, \cdots ,{{L}}_m^j) (j \leqslant r)$

4)对于概念 $({{X}},{{B}}) \in {{L}}({{U}},{{A}},{{I}})$ ，记

 ${{E}} = (\{ {v_1}\} ,\{ {v_2}\} , \cdots ,\{ {v_m}\} )$

 ${\rm{If}} \;{{B}}, {\rm{then}} \;{{{T}}_{{j_o}}} ({{D}}({{{L}}_{{j_o}}}/{{E}}))$

1) for 1≤t≤|D|

2) ${{U}}/{{{R}}_{{D}}} = \left\{ {{{{D}}_1},{{{D}}_2}, \cdots ,{{{D}}_r}} \right\} = \left\{ {{{\left[ x \right]}_{{D}}}:x \in {{U}}} \right\}$

3) for ${{{D}}_j}\left( {j \leqslant r} \right),\;\;{\rm{do}}$

4) ${{{M}}_j} = \left\{ {\left( {\left\{ {{f_1}\left( {{x_i}} \right)} \right\},\left\{ {{f_2}\left( {{x_i}} \right)} \right\}, \cdots ,\left. {\left\{ {{f_{|{{A}}|}}\left( {{x_i}} \right)} \right\}} \right)|} \right.x_i^{* \triangleleft } \subseteq {{{D}}_j}} \right\}$

5) ${{{L}}_j} \!=\! \Big\{ {\left( {\max \left\{ {{f_1}\left( {{x_i}} \right)} \right\}{\rm{ }},\max \left\{ {{f_2}\left( {{x_i}} \right)} \right\} \cdots ,\max \left. {\left\{ {{f_{|{{A}}|}}\left( {{x_i}} \right)} \right\}} \right)|{\rm{ }}} \right.} \Big.$ $\Big. {x_i^{* \triangleleft } \subseteq {{{D}}_j}} \Big\}$

6) for $\left( {{{X}},{{B}}} \right) \in {{L}}\left( {{{U}},{{A}},{{I}}} \right)$ do

7) 　 ${{E}} = \left( {\left\{ {{v_1}\left( {{B}} \right)} \right\},\left\{ {{v_2}\left( {{B}} \right)} \right\}, \cdots ,\left\{ {{v_{|A|}}\left( {{B}} \right)} \right\}} \right)$

8) 　compute D(Lj/E)

9) 　if D(Ljo/E)=maxD(Lj/E)

10) 　then ${{R}}:\;{{B}} \Rightarrow {{{T}}_{{j_o}}}\left( {{{D}}\left( {{{{L}}_{{j_o}}}/{{E}}} \right)} \right)$

11) 　return R

12) end for

 ${{{L}}_1} = (\{ 1\} ,\{ 1\} ,\{ 0\} ,\{ 0\} ,\{ 0\} ,\{ 1\} ,\{ 1\} ,\{ 0\} )$
 ${{{L}}_2} = (\{ 0\} ,\{ 1\} ,\{ 1\} ,\{ 1\} ,\{ 0\} ,\{ 1\} ,\{ 1\} ,\{ 1\} )$
 ${{{L}}_3} = (\{ 1\} ,\{ 1\} ,\{ 1\} ,\{ 0\} ,\{ 1\} ,\{ 0\} ,\{ 0{\rm{\} }},\{ 0\} )$

 $\begin{array}{l} {{D}}({{{L}}_1}/{{E}}) = 0.625\\ {{D}}({{{L}}_2}/{{E}}){\rm{ = }}0.375\\ {{D}}({{{L}}_3}/{{E}}){\rm{ = }}0.375 \end{array}$

$\{ {a_1},{a_6}{\rm{\} }} \Rightarrow d = 1 (0.75);$ $\{ {a_1},{a_6},{a_7}{\rm{\} }} \Rightarrow d = 1 (0.875)$

$\{ {a_1},{a_2},{a_6},{a_7}\} \Rightarrow d \!=\! 1 (1);$ $\{ {a_2},{a_6},{a_7},{a_8}\} \Rightarrow d \!=\! 1(0.75)$

$\{ {a_2},{a_6},{a_7},{a_8}\} \Rightarrow d \!=\! 2 (0.75);$ $\{ {a_1},{a_3},{a_5}\} \Rightarrow d \!=\! 3 (0.75)$

${\rm{\{ }}{a_1}{\rm{,}}{a_2},{a_3},{a_5}{\rm{\} }} \Rightarrow d = 3 (1);$ $\{ {a_2},{a_3},{a_4}\} \Rightarrow d = 2 (0.625)$

$\{ {a_2},{a_3},{a_4}\} \Rightarrow d = 3 (0.625);$ $\{ {a_6}\} \Rightarrow d = 1 (0.625)$

$\{ {a_1}\} \Rightarrow d = 1 (0.625);$ ${\rm{\{ }}{a_1}{\rm{\} }} \Rightarrow d = 3 (0.625)$

$\{ {a_6},{a_7}\} \Rightarrow d = 1 (0.75);$ ${\rm{\{ }}{a_2},{a_6},{a_7}{\rm{\} }} \Rightarrow d = 1 (0.875)$

${\rm{\{ }}{a_1},{a_2}{\rm{\} }} \Rightarrow d = 1 (0.75);$ ${\rm{\{ }}{a_1},{a_2}{\rm{\} }} \Rightarrow d = 3 (0.75)$

${\rm{\{ }}{a_2}{\rm{\} }} \Rightarrow d = 1 (0.75);$ ${\rm{\{ }}{a_2}{\rm{\} }} \Rightarrow d = 3 (0.625)$

${\rm{\{ }}{a_3}{\rm{\} }} \Rightarrow d = 3 (0.625);$ ${\rm{\{ }}{a_2},{a_3}{\rm{\} }} \Rightarrow d = 3 (0.75)$

4 结束语

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