﻿ 不协调区间值决策系统的最大分布约简
«上一篇
 文章快速检索 高级检索

 智能系统学报  2018, Vol. 13 Issue (3): 469-478  DOI: 10.11992/tis.201710011 0

引用本文

YIN Jiliang, ZHANG Nan, TONG Xiangrong, et al. Maximum distribution reduction in inconsistent interval-valued decision systems[J]. CAAI Transactions on Intelligent Systems, 2018, 13(3): 469-478. DOI: 10.11992/tis.201710011.

文章历史

1. 烟台大学 数据科学与智能技术山东省高校重点实验室，山东 烟台 264005;
2. 烟台大学 计算机与控制工程学院，山东 烟台 264005

Maximum distribution reduction in inconsistent interval-valued decision systems
YIN Jiliang1,2, ZHANG Nan1,2, TONG Xiangrong1,2, CHEN Manru1,2
1. Key Lab for Data Science and Intelligence Technology of Shandong Higher Education Institutes, Yantai University, Yantai 264005, China;
2. School of Computer and Control Engineering, Yantai University, Yantai 264005, China
Abstract: Distribution reduction is one of the important methods of attribute reduction as it can guarantee consistent confidence coefficients of all decision rules before and after reduction. Maximum distributed reduction keeps the unchanged rule with the highest confidence coefficient in the decision system, and extracting a rule with a high confidence coefficient has a wide application value. This paper introduces the concept of maximum confidence coefficient for inconsistent interval-valued decision systems based on compatibility relation and proposes a maximum distribution reduction algorithm based on discernibility matrix, whereby a discernibility matrix is constructed to keep the unchanged maximum distribution. The relationship between the maximum distribution reduction algorithm in inconsistent interval-valued decision systems and other reduction algorithms was analyzed. Experiments were performed using UCI standard data sets, and the proposed algorithm proved to be effective.
Key words: distributed reduction    maximum distributed reduction    confidence coefficient    compatibility relation    discernibility matrix    inharmonious    interval-valued    decision system

1 基本知识 1.1 区间值决策系统的粗糙近似

1)区间值交运算为

 ${\eta _1} \cap {\eta _2} = \left\{ {\begin{array}{*{20}{l}}{0, \quad (u_i^k < l_j^k) \vee (u_j^k < l_i^k)\;\;}\\{[{\rm{max}}(l_i^k,l_j^k),{\rm{min}}(u_i^k,u_j^k)], \quad \text{其他}}\end{array}} \right.$

2)区间值并运算为

 ${\eta _1} \cup {\eta _2} = [{\rm{min}}(l_i^k,l_j^k),{\rm{max}}(u_i^k,u_j^k)]\;$

 $\alpha _{ij}^k{\rm{ = }}\frac{{{\rm{|}}[l_i^k,u_i^k] \cap [l_j^k,u_j^k]{\rm{|}}}}{{{\rm{|}}[l_i^k,u_i^k] \cup [l_j^k,u_j^k]{\rm{|}}}}$

Jaccard相似率为两个区间数的交集与并集长度的比值，它适合度量长度相似的两个区间数。

${\eta _1} = [l_1^1,u_1^1]$ ${\eta _1} = [l_2^1,u_2^1]$ ，分别计算 ${\eta _1}$ ${\eta _2}$ 的交、并：

 ${\eta _1} \cap {\eta _2} = [0.86,3.13] \cap [{\rm{ - }}0.12,2.13] = {\rm{[0}}{\rm{.86,2}}{\rm{.13]}}$
 ${\eta _1} \cup {\eta _2} = [0.86,3.13] \cup [{\rm{ - }}0.12,2.13] = [{\rm{ - }}0.12,3.13{\rm{]}}$

 $\alpha _{12}^1 = \frac{{|[l_1^1,u_1^1] \cap [l_2^1,u_2^1]|}}{{|[l_1^1,u_1^1] \cup [l_2^1,u_2^1]|}} = 0.391$

 ${T}_{\{ {a_k}\} }^\alpha = \{ ({x_i},{x_j})|({x_i},{x_j}) \in U \times U,\alpha _{ij}^k > \alpha \}$

 ${T}_A^\alpha = \{ ({x_i},{x_j})|({x_i},{x_j}) \in U \times U,\alpha _{ij}^k > \alpha ,{a_k} \in A\}$

 ${T}_A^\alpha = \mathop \cap \limits_{{a_k} \in A} {T}_{\{ {a_{k\} }}}^\alpha$

1)自反性：任意xiU，则 $({x_i},{x_j}) \in {T}_A^\alpha$

2)对称性：任意xixjU，若 $({x_i},{x_j}) \in {T}_A^\alpha$ ，则 $({x_j},{x_i}) \in {T}_A^\alpha$

3)非传递性：任意 ${x_i},{x_j},{x_k} \in U$ ，若满足 $({x_i},{x_k}) \in$ ${T}_A^\alpha$ $({x_k},{x_j}) \in {T}_A^\alpha$ ，则 $({x_i},{x_j}) \in {T}_A^\alpha$ 不一定成立。

 $S_A^\alpha ({x_i}) = \{ {x_j}|{x_j} \in U,({x_i},\,{x_j}) \in {T}_A^\alpha \}$

 $S_A^\alpha (U) = \{ S_A^\alpha ({x_1}),S_A^\alpha ({x_2}), \cdots ,S_A^\alpha ({x_n})\}$

 $\overline {{\rm{apr}}_A^\alpha } (X) = \{ {x_i}|{x_i} \in U,S_A^\alpha ({x_i}) \cap X \ne \text{Ø} \}$
 $\underline {{\rm{apr}}_A^\alpha } (X) = \{ {x_i}|{x_i} \in U,S_A^\alpha ({x_i}) \subseteq X\}$

 ${\rm{POS}}_A^\alpha (X) = \underline {{\rm{apr}}_A^\alpha } (X)$

 $\mu _A^\alpha (X) = \frac{{|\underline {{\rm{apr}}_A^\alpha } (X)|}}{{|\overline {{\rm{apr}}_A^\alpha } (X)|}}$

 $\overline {{\rm{apr}}_A^\alpha } (D) = \{ {x_i}|{x_i} \in U,{D_j} \in U/D,S_A^\alpha ({x_i}) \cap {D_j} \ne \text{Ø} \}$
 $\underline {{\rm{apr}}_A^\alpha } (D) = \{ {x_i}|{x_i} \in U,{D_j} \in U/D,S_A^\alpha ({x_i}) \subseteq {D_j}\}$

 ${\rm{POS}}_A^\alpha (D) = \overline {{\rm{apr}}_A^\alpha } (D)$

 $\mu _A^\alpha (D) = \frac{{|\underline {{\rm{apr}}_A^\alpha } (D)|}}{{|\overline {{\rm{apr}}_A^\alpha } (D)|}}$

 ${{T}}_C^{0.6}{\rm{ = }}\left[ {\begin{array}{*{20}{c}} 1&0&0&0&0&0 \\ 0&1&0&0&0&0 \\ 0&0&1&0&1&1 \\ 0&0&0&1&0&0 \\ 0&0&1&0&1&1 \\ 0&0&1&0&1&1 \end{array}} \right]$

 $S_C^{0.6}(U) = \{ S_C^{0.6}({x_1}),S_C^{0.6}({x_2}), \cdots ,S_C^{0.6}({x_6})\}$

$U/D = \left\{ {\left\{ {{x_1},{x_5},{x_6}} \right\},\left\{ {{x_2},{x_3},{x_4}} \right\}} \right\}$ 为决策属性DU的划分，计算决策属性D关于相容关系 ${{T}}_C^{0.6}$ 的上、下近似：

$\overline {{\rm{apr}}_C^{0.6}} (D) = U$ $\underline {{\rm{apr}}_C^{0.6}} (D) = \{ {x_1},{x_2},{x_4}\}$

 $\mu _C^{0.6}(D) = \frac{{|\underline {{\rm{apr}}_C^\alpha } (D)|}}{{|\overline {{\rm{apr}}_C^\alpha } (D)|}} = \frac{3}{6} = 0.5$
1.2 区间值决策系统的分布约简

 $\begin{array}{c}\mu _A^\alpha ({x_i}) = (D({D_1}/S_A^\alpha ({x_i})),D({D_2}/S_A^\alpha ({x_i})), \cdots ,\\D({D_j}/S_A^\alpha ({x_i}), \cdots ,D({D_{|U|}}/S_A^\alpha ({x_i})))\end{array}$

 ${\rm{DM}}_D^\alpha (i,j) = \left\{ {\begin{array}{*{20}{l}} {\{ {a_k}|{a_k} \in C \wedge \alpha _{ij}^k < \alpha \} ,\quad \mu _A^\alpha ({x_i}) \ne \mu _A^\alpha ({x_j})} \\ { \text{Ø} ,\quad \mu _A^\alpha ({x_i}) = \mu _A^\alpha ({x_j})} \end{array}} \right.$

 $f_D^\alpha (C)({\bar a_1},{\bar a_2}, \cdots ,{\bar a_{|C|}}) = \wedge \{ \vee {\rm{DM}}_D^\alpha (i,j):{\rm{DM}}_D^\alpha (i,j) \ne \text{Ø} \}$

1)计算区间值决策系统 ${\rm{DS}}$ 的在阈值 $\alpha$ 下的相容类集合 $S_C^\alpha (U)$

2)根据每个对象对应的相容类，计算每个对象相对于每一个决策类的概率分布 $\mu _C^\alpha ({x_i})$

3)根据每个对象的可信度不同构造分布约简可辨识矩阵 ${\bf{DM}}_D^\alpha$

4)由可辨识矩阵 ${\bf{DM}}_D^\alpha$ 计算分布约简可辨识函数 $f_D^\alpha \left( C \right)$

5)利用分配率和吸收率将 $f_D^\alpha (C)$ 转化为 $h_D^\alpha (C)$ $h_D^\alpha (C)$ 中每一个蕴含项为一个分布保持的约简。

$\mu _C^{0.6}({x_1}) = \{ 1,0\}$ $\mu _C^{0.6}({x_2}) = \{ 0,1\}$ ，　　 $\mu _C^{0.6}({x_3}) = \{ 2/3,1/3\}$ $\mu _C^{0.6}({x_4}) = \{ 0,1\}$ ，　　 $\mu _C^{0.6}({x_5}) = \{ 2/3,1/3\}$ $\mu _C^{0.6}({x_6}) = \{ 2/3,1/3\}$

 ${{\bf{DM}}}_D^{0.6}(6,6) = \left[ {\begin{array}{*{20}{c}} \text{Ø} &{}&{}&{}&{}&{} \\ {{a_1},{a_2},{a_3},{a_4}}&\text{Ø} &{}&{}&{}&{} \\ {{a_1},{a_2}}&{{a_3},{a_4}}&\text{Ø} &{}&{}&{} \\ {{a_1},{a_2},{a_3}}&\text{Ø} &{{a_3}}&\text{Ø} &{}&{} \\ {{a_1},{a_2}}&{{a_3},{a_4}}&\text{Ø} &{{a_3}}&\text{Ø} &{} \\ {{a_1},{a_2}}&{{a_3},{a_4}}&\text{Ø} &{{a_3}}&\text{Ø} &\text{Ø} \end{array}} \right]$

 $f_D^{0.6}(C)({\bar a_1},{\bar a_2},{\bar a_3},{\bar a_4}) = {a_3} \wedge ({a_2} \vee {a_1})$

 $h_D^{0.6}(C)({\bar a_1},{\bar a_2},{\bar a_3},{\bar a_4}) = ({a_1} \wedge {a_3}) \vee ({a_2} \wedge {a_3})$

2 区间值决策系统的最大分布约简

 $m_A^\alpha ({x_i}) = D({D_{{j_0}}}/S_A^\alpha ({x_i}))= \mathop {{\rm{max}}}\limits_{j \leqslant q} D({D_j}/S_A^\alpha ({x_i}))$

${x_i} \in U$ 对应的最大分布为

 $\gamma _A^\alpha ({x_i}) = \{ {D_j}|D({D_j}/S_A^\alpha ({x_i})) = D({D_{{j_0}}}/S_A^\alpha ({x_i})\}$

1) $\gamma _A^\alpha ({x_i}) = \gamma _C^\alpha ({x_i})$

2)任意 $B \subset A$ ，满足 $\gamma _B^\alpha ({x_i}) \ne \gamma _C^\alpha ({x_i})$

$\Leftarrow$ ”：对任意 ${x_i},{x_j} \in U$ ，当 $S_A^\alpha ({x_i}) = S_A^\alpha ({x_j})$ ，有 $\gamma _C^\alpha ({x_i}) = \gamma _C^\alpha ({x_j})$ ，对于任意的 ${D_{{j_0}}} \in \gamma _C^\alpha ({x_i})$ ，有 ${D_{{j_0}}} \in \gamma _C^\alpha ({x_j})$ 。由于 $S_A^\alpha ({x_i}) = \cup \{ S_C^\alpha ({x_j})|S_C^\alpha ({x_j}) \in$ $J(S_A^\alpha ({x_i})){\rm{\} }}$ ，于是对任意的 $k \leqslant q$ ，有

 $\begin{gathered} D({D_k}/S_A^\alpha ({x_i})){\rm{ = }} \hfill \\ \frac{{\sum {\{ |{D_k} \cap S_C^\alpha ({x_j})|:S_C^\alpha ({x_j}) \in J(S_A^\alpha ({x_i}))\} } }}{{|S_A^\alpha ({x_i})|}}{\rm{ = }} \hfill \\ \sum {\left\{ {\frac{{|{D_k} \cap S_C^\alpha ({x_j})|}}{{|S_C^\alpha ({x_j})|}}} \right.} \times \left. {\frac{{|S_C^\alpha ({x_j})|}}{{|S_A^\alpha ({x_i})|}}:S_C^\alpha ({x_j}) \in J(S_A^\alpha ({x_i}))} \right\} \leqslant \hfill \\ \sum {\left\{ {\frac{{|{D_{{j_0}}} \cap S_C^\alpha ({x_j})|}}{{|S_C^\alpha ({x_i})|}}} \right.} \times \left. {\frac{{|S_C^\alpha ({x_j})|}}{{|S_A^\alpha ({x_i})|}}:S_C^\alpha ({x_j}) \in J(S_A^\alpha ({x_i}))} \right\}{\rm{ = }} \hfill \\ \frac{{|{D_{{j_0}}} \cap S_A^\alpha ({x_i})|}}{{|S_A^\alpha ({x_i})|}}{\rm{ = }}D({D_{{j_0}}}/S_A^\alpha ({x_i})) \hfill \\ \end{gathered}$

${D_{{j_0}}} \in \gamma _C^\alpha ({x_i})$ ，从而 $\gamma _A^\alpha ({x_i}) = \gamma _C^\alpha ({x_i})$

 $\begin{gathered} D({D_{{k_0}}}/S_A^\alpha ({x_i})){\rm{ = }} \\ \sum {\left\{ {\frac{{|{D_{{k_0}}} \cap S_C^\alpha ({x_j})|}}{{|S_C^\alpha ({x_j})|}}} \right.} \times \left. {\frac{{|S_C^\alpha ({x_j})|}}{{S_A^\alpha ({x_i})}}:S_C^\alpha ({x_j}) \in J(S_A^\alpha ({x_i}))} \right\}{\rm{ = }} \\ \sum {\left\{ {m_C^\alpha ({x_j}) \times \frac{{|S_C^\alpha ({x_j})|}}{{|S_A^\alpha ({x_i})|}}:S_C^\alpha ({x_j}) \in J(S_A^\alpha ({x_i}))} \right\} > } \\ \sum {\left\{ {D({D_{{j_0}}}/S_C^\alpha ({x_j})) \times \frac{{|S_{\rm{AT}}^\alpha ({x_j})|}}{{|S_A^\alpha ({x_i})|}}:S_C^\alpha ({x_j}) \in J(S_A^\alpha ({x_i})) } \right\}} = \\ \sum {\left\{ {\frac{{|{D_{{j_0}}} \cap S_C^\alpha ({x_j})|}}{{|S_C^\alpha ({x_j})|}}} \right.} \times \left. {\frac{{|S_C^\alpha ({x_j})|}}{{|S_A^\alpha ({x_i})|}}:S_C^\alpha ({x_j}) \in J(S_A^\alpha ({x_i}))} \right\} = \\ \frac{{|{D_{{j_0}}} \cap S_A^\alpha ({x_i})|}}{{|S_A^\alpha ({x_i})|}} = D({D_{{j_0}}}/S_A^\alpha ({x_i})) \end{gathered}$

${D_{{j_0}}} \in \gamma _A^\alpha ({x_i})$ 矛盾，因此 ${D_{{j_0}}} \in \gamma _C^\alpha ({x_i})$ ，于是有 $\gamma _A^\alpha ({x_i}) \subseteq$ $\gamma _C^\alpha ({x_i})$

 ${\rm{DM}}_{D{\rm{Max}}}^\alpha(i,j) = \left\{ {\begin{array}{*{20}{l}}{\{ {a_k}|{a_k} \in C \wedge \alpha _{ij}^k < \alpha \} ,\quad\gamma _A^\alpha ({x_i}) \ne \gamma _A^\alpha ({x_j})}\\{\text{Ø} ,\quad \gamma _A^\alpha ({x_i}) = \gamma _A^\alpha ({x_j})}\end{array}} \right.$

${\rm{DM}}_{D{\rm{Max}}}^\alpha(i,j)$ 为基于α-相容类的最大分布约简可辨识矩阵 ${\bf{DM}}_{D{\rm{Max}}}^\alpha$ $i$ $j$ 列的元素， ${\bf{DM}}_{D{\rm{Max}}}^\alpha$ 简称为最大分布可辨识矩阵，其中 $i,j = 1,2, \cdots ,|U|$ ，Ø表示空集。

$\Leftarrow$ ”：假设存在xixjU，满足 $\gamma _C^\alpha ({x_i}) \ne$ $\gamma _C^\alpha ({x_j})$ ，且 ${\rm{DM}}_{D{\rm{Max}}}^\alpha(i,j) \cap A \ne \text{Ø}$ ，则对任意akA，有 ${a_k} \notin$ ${\rm{DM}}_{D{\rm{Max}}}^\alpha(i,j)$ $\alpha _{ij}^k > \alpha$ ，因此 $({x_i},{x_j}) \in {T}_A^\alpha$ 。假设 ${x_i}\text{、}{x_j}$ 对应的 $\alpha$ -相容类分别为 $S_A^\alpha ({x_i})$ $S_A^\alpha ({x_j})$ ，则有 $S_A^\alpha ({x_i}) = S_A^\alpha ({x_j})$ ，由定理1得 $A$ 不是最大分布协调集。定理得证。

${\rm{DM}}_{D{\rm{Max}}}^\alpha(i,j) \ne \text{Ø} \}$ ，为基于 $\alpha$ -相容类的最大分布约简可辨识函数，简称最大分布可辨识函数。 $\vee {\rm{DM}}_{D{\rm{Max}}}^\alpha(i,j)$ 表示满足 $a \in {\rm{DM}}_{D{\rm{Max}}}^\alpha(i,j)$ 的全体布尔变量 $\bar a$ 的析取式。

$\Leftarrow$ ”：根据定义16可得 $h_D^\alpha {(C)_{{\rm{Max}}}}({\bar a_1},{\bar a_2}, \cdots ,{\bar a_m}) =$ $( \wedge {\theta _1}) \vee \cdots \vee ({\theta _l}),{\theta _k} \subseteq C,k = 1,2, \cdots ,l$ ，若在θ中去掉一个元素形成θ′，则存在 $S_C^\alpha ({x_i})$ $S_C^\alpha ({x_j})$ 满足 $\gamma _C^\alpha ({x_i}) \ne \gamma _C^\alpha ({x_j})$ ，使得 ${\rm{DM}}_{D{\rm{Max}}}^\alpha(i,j) \cap \theta ' \ne \text{Ø}$ ，故θ′ 不是最大分布约简，从而θ是其中一个最大分布约简。定理得证。

1)计算区间值决策系统DS在阈值 $\alpha$ 下的相容类集合 $S_C^\alpha (U)$

2)根据每个对象对应的相容类，计算每个对象相对于每一个决策类的概率分布 $\, \mu _C^\alpha ({x_i})$

3)根据每个对象的概率分布，计算所对应的最大分布 $\gamma _C^\alpha ({x_i})$

4)根据每个对象的可信度不同构造最大分布约简可辨识矩阵 ${\bf{DM}}_{D{\rm{Max}}}^\alpha$

5)由可辨识矩阵 ${\bf{DM}}_{D{\rm{Max}}}^\alpha$ 计算最大分布约简可辨识函数 $f_D^\alpha {(C)_{{\rm{Max}}}}$

6)利用分配率和吸收率将 $U$ 转化为 $h_D^\alpha {(C)_{{\rm{Max}}}}$ $h_D^\alpha {(C)_{{\rm{Max}}}}$ 中每一个蕴含项为一个最大分布保持的约简。

$U/D = \{ {D_1},{D_2}\} = \Bigr\{ \{ {x_1},{x_5},{x_6}\} ,\{ {x_2},{x_3},{x_4}\} \Bigr\}$

$\mu _C^{0.6}({x_1}) = \{ 1,0\}$ $\mu _C^{0.6}({x_2}) = \{ 0,1\}$

$\mu _C^{0.6}({x_3}) = \{ 2/3,1/3\}$ $\mu _C^{0.6}({x_4}) = \{ 0,1\}$

$\mu _C^{0.6}({x_5}) = \{ 2/3,1/3\}$ $\mu _C^{0.6}({x_6}) = \{ 2/3,1/3\}$

$\gamma _C^{0.6}({x_1}) = \{ {D_1}\}$ $\gamma _C^{0.6}({x_2}) = \{ {D_2}\}$

$\gamma _C^{0.6}({x_3}) = \{ {D_1}\}$ $\gamma _C^{0.6}({x_4}) = \{ {D_2}\}$

$\gamma _C^{0.6}({x_5}) = \{ {D_1}\}$ $\gamma _C^{0.6}({x_6}) = \{ {D_1}\}$

 ${\bf{DM}}_{D{\rm{Max}}}^{0.6}(6,6) = \left[ {\begin{array}{*{20}{c}} \text{Ø} &{}&{}&{}&{}&{}\\{{a_1},{a_2},{a_3},{a_4}}& \text{Ø} &{}&{}&{}&{}\\ \text{Ø} &{{a_3},{a_4}}& \text{Ø} &{}&{}&{}\\{{a_1},{a_2},{a_3}}& \text{Ø} &{{a_3}}& \text{Ø} &{}&{}\\ \text{Ø} &{{a_3},{a_4}}& \text{Ø} &{{a_3}}& \text{Ø} &{}\\ \text{Ø} &{{a_3},{a_4}}& \text{Ø} &{{a_3}}& \text{Ø} & \text{Ø} \end{array}} \right]$

 $f_D^{0.6}{(C)_{{\rm{Max}}}}({\bar a_1},{\bar a_2},{\bar a_3},{\bar a_4}) = {a_3}$

 $h_D^{0.4}(C)({\bar a_1},{\bar a_2},{\bar a_3},{\bar a_4}) = {a_1} \vee {a_4}$
 $h_D^{0.5}(C)({\bar a_1},{\bar a_2},{\bar a_3},{\bar a_4}) = ({a_1} \wedge {a_3}) \vee ({a_2} \wedge {a_3})$
 $h_D^{0.6}(C)({\bar a_1},{\bar a_2},{\bar a_3},{\bar a_4}) = ({a_1} \wedge {a_3}) \vee ({a_2} \wedge {a_3})$
 $h_D^{0.7}(C)({\bar a_1},{\bar a_2},{\bar a_3},{\bar a_4}) = ({a_1} \wedge {a_3}) \vee ({a_2} \wedge {a_3})$

 $h_D^{0.4}{(C)_{{\rm{Max}}}}({\bar a_1},{\bar a_2},{\bar a_3},{\bar a_4}) = {a_1} \vee {a_4}$
 $h_D^{0.5}{(C)_{{\rm{Max}}}}({\bar a_1},{\bar a_2},{\bar a_3},{\bar a_4}) = {a_3}$
 $h_D^{0.6}{(C)_{{\rm{Max}}}}({\bar a_1},{\bar a_2},{\bar a_3},{\bar a_4}) = {a_3}$
 $h_D^{0.7}{(C)_{{\rm{Max}}}}({\bar a_1},{\bar a_2},{\bar a_3},{\bar a_4}) = {a_3}$

3 实验验证与分析

 $\sigma _t^k = \sqrt {\frac{1}{{|{D_k}| - 1}}\sum\limits_{{x_i} \in {D_k}} {{{({a_t}({x_j}) - \bar a_t^k)}^2}} }$

 $l_i^t = {a_t}({x_i}) - \lambda \bar a_t^k$
 $u_i^t = {a_t}({x_i}) + \lambda \bar a_t^k$

3.1 约简结果对比

3.2 约简效率对比

 Download: 图 1 约简效率对比 ( $\alpha {\rm{ = 0}}{\rm{.4}}$ ) Fig. 1 Comparison of reduction efficiency ( $\alpha {\rm{ = 0}}{\rm{.4}}$ )
 Download: 图 2 约简效率对比 ( $\alpha {\rm{ = 0}}{\rm{.5}}$ ) Fig. 2 Comparison of reduction efficiency ( $\alpha {\rm{ = 0}}{\rm{.5}}$ )
 Download: 图 3 约简效率对比 ( $\alpha {\rm{ = 0}}{\rm{.6}}$ ) Fig. 3 Comparison of reduction efficiency ( $\alpha {\rm{ = 0}}{\rm{.6}}$ )
 Download: 图 4 约简效率对比 ( $\alpha {\rm{ = 0}}{\rm{.7}}$ ) Fig. 4 Comparison of reduction efficiency ( $\alpha {\rm{ = 0}}{\rm{.7}}$ )
 Download: 图 5 约简效率对比 ( $\alpha {\rm{ = 0}}{\rm{.8}}$ ) Fig. 5 Comparison of reduction efficiency ( $\alpha {\rm{ = 0}}{\rm{.8}}$ )

4 结束语

 [1] PAWLAK Z. Rough sets[J]. International journal of computer & information sciences, 1982, 11(5): 341-356. (0) [2] PAWLAK Z. Rough sets: theoretical aspects of reasoning about data[M]. Boston: Kluwer Academic Publishers, 1992. (0) [3] 王国胤, 姚一豫, 于洪. 粗糙集理论与应用研究综述[J]. 计算机学报, 2009, 32(7): 1229-1246. WANG Guoyin, YAO Yiyu, YU Hong. A survey on rough set theory and applications[J]. Chinese journal of computers, 2009, 32(7): 1229-1246. (0) [4] QIAN Yuhua, LIANG Jiye, PEDRYCZ W, et al. Positive approximation: an accelerator for attribute reduction in rough set theory[J]. Artificial intelligence, 2010, 174(9/10): 597-618. (0) [5] WANG Feng, LIANG Jiye, QIAN Yuhua. Attribute reduction: A dimension incremental strategy[J]. Knowledge-based systems, 2013, 39: 95-108. DOI:10.1016/j.knosys.2012.10.010 (0) [6] CHEN Hongmei, LI Tianrui, RUAN Da, et al. A rough-set based incremental approach for updating approximations under dynamic maintenance environments[J]. IEEE transactions on knowledge and data engineering, 2013, 25(2): 274-284. DOI:10.1109/TKDE.2011.220 (0) [7] HU Qinghua, YU Daren, XIE Zongxia. Information-preserving hybrid data reduction based on fuzzy-rough techniques[J]. Pattern recognition letters, 2006, 27(5): 414-423. DOI:10.1016/j.patrec.2005.09.004 (0) [8] SKOWRON A, RAUSZER C. The discernibility matrices and functions in information systems[M]//SŁOWIŃSKI R. Intelligent Decision Support. Dordrecht: Springer, 1992, 11: 331–362. (0) [9] KRYSZKIEWICZ M. Rough set approach to incomplete information systems[J]. Information sciences, 1998, 112(1/2/3/4): 39-49. (0) [10] 邓大勇, 黄厚宽, 李向军. 不一致决策系统中约简之间的比较[J]. 电子学报, 2007, 35(2): 252-255. DENG Dayong, HUANG Houkuan, LI Xiangjun. Comparison of various types of reductions in inconsistent systems[J]. Acta electronica sinica, 2007, 35(2): 252-255. (0) [11] MIAO Duoqian, ZHAO Yan, YAO Yiyu, et al. Relative reducts in consistent and inconsistent decision tables of the Pawlak rough set model[J]. Information sciences, 2009, 179(24): 4140-4150. DOI:10.1016/j.ins.2009.08.020 (0) [12] ZHOU Jie, MIAO Duoqian, PEDRYCZ W, et al. Analysis of alternative objective functions for attribute reduction in complete decision tables[J]. Soft computing, 2011, 15(8): 1601-1616. DOI:10.1007/s00500-011-0690-7 (0) [13] 张文修, 米据生, 吴伟志. 不协调目标信息系统的知识约简[J]. 计算机学报, 2003, 26(1): 12-18. ZHANG Wenxiu, MI Jusheng, WU Weizhi. Knowledge reductions in inconsistent information systems[J]. Chinese journal of computers, 2003, 26(1): 12-18. (0) [14] 徐伟华, 张文修. 基于优势关系下不协调目标信息系统的分布约简[J]. 模糊系统与数学, 2007, 21(4): 124-131. XU Weihua, ZHANG Wenxiu. Distribution reduction in inconsistent information systems based on dominance relations[J]. Fuzzy systems and mathematics, 2007, 21(4): 124-131. (0) [15] 张楠, 苗夺谦, 岳晓冬. 区间值信息系统的知识约简[J]. 计算机研究与发展, 2010, 47(8): 1362-1371. ZHANG Nan, MIAO Duoqian, YUE Xiaodong. Approaches to knowledge reduction in interval-valued information systems[J]. Journal of computer research and development, 2010, 47(8): 1362-1371. (0) [16] 张楠, 许鑫, 童向荣, 等. 不协调区间值决策系统的知识约简[J]. 小型微型计算机系统, 2017, 38(7): 1585-1589. ZHANG Nan, XU Xin, TONG Xiangrong, et al. Knowledge reduction in inconsistent interval-valued decision systems[J]. Journal of Chinese computer systems, 2017, 38(7): 1585-1589. (0) [17] 张楠, 许鑫, 童向荣, 等. 不协调区间值决策系统的分布约简[J]. 计算机科学, 2017, 44(9): 78-82, 104. ZHANG Nan, XU Xin, TONG Xiangrong, et al. Distribution reduction in inconsistent interval-valued decision systems[J]. Computer science, 2017, 44(9): 78-82, 104. DOI:10.11896/j.issn.1002-137X.2017.09.016 (0) [18] 刘鹏惠, 陈子春, 秦克云. 区间值信息系统的决策属性约简[J]. 计算机工程与应用, 2009, 45(28): 148-150, 229. LIU Penghui, CHEN Zichun, QIN Keyun. Decision attribute reduction of interval-valued information system[J]. Computer engineering and applications, 2009, 45(28): 148-150, 229. DOI:10.3778/j.issn.1002-8331.2009.28.044 (0) [19] ZHANG Xiao, MEI Changlin, CHEN Degang, et al. Multi-confidence rule acquisition and confidence-preserved attribute reduction in interval-valued decision systems[J]. International journal of approximate reasoning, 2014, 55(8): 1787-1804. DOI:10.1016/j.ijar.2014.05.007 (0) [20] 史德容, 徐伟华. 区间值模糊决策序信息系统的分布约简[J]. 计算机科学与探索, 2017, 11(4): 652-658. SHI Derong, XU Weihua. Distribution reduction in interval-valued fuzzy decision ordered information systems[J]. Journal of frontiers of computer science and technology, 2017, 11(4): 652-658. DOI:10.3778/j.issn.1673-9418.1602002 (0)