﻿ 双论域下多粒度模糊粗糙集上下近似的包含关系
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 智能系统学报  2019, Vol. 14 Issue (1): 115-120  DOI: 10.11992/tis.201804046 0

引用本文

HU Zhiyong, MI Jusheng, FENG Tao, et al. Inclusion relation of upper and lower approximations of multigranularity fuzzy rough set in two universes[J]. CAAI Transactions on Intelligent Systems, 2019, 14(1): 115-120. DOI: 10.11992/tis.201804046.

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. 河北科技大学 理学院, 河北 石家庄 050024

Inclusion relation of upper and lower approximations of multigranularity fuzzy rough set in two universes
HU Zhiyong 1, MI Jusheng 1, FENG Tao 2, YAO Aimeng 1
1. College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050024, China;
2. School of Sciences, Hebei University of Science and Technology, Shijiazhuang 050024, China
Abstract: To solve the problem that the upper and lower approximations of multigranularity rough set in two universe may no longer have an inclusion relation, this paper will present a sufficient condition for the inclusion relation of the upper and lower approximations in two universes. Furthermore, we use the standardized method to transform upper and lower approximations with no inclusion relation into upper and lower approximations with an inclusion relation. It is verified by an example that this method can effectively solve the problem that the upper and lower approximations of the multigranularity fuzzy rough set in two universes have inclusion relation.
Key words: fuzzy set    rough set    dual domain    multi-granularity    upper approximation    lower approximation    standardized method    inclusion relation    sufficient condition

1 多粒度模糊粗糙集

 $\underline {\sum\limits_{t = 1}^m {R_t^O} } (X){\rm{( }}x{\rm{)}} = \vee _{t = 1}^m{ \wedge _{z \in U}}[(1 - {R_t}(x, z)) \vee X(z)], \forall x, z \in U$
 $\overline {\sum\limits_{t = 1}^m {R_t^O} } (X) = {\rm{\sim }}\underline {\sum\limits_{t = 1}^m {R_t^O} } {\rm{(\sim }}X)$

$X$ 的悲观多粒度下上近似分别为 $\underline {\sum\limits_{t = 1}^m {R_t^P} } (X)$ $\overline {\sum\limits_{t = 1}^m {R_t^P} (} X)$ ，定义如下：

 $\underline {\sum\limits_{t = 1}^m {R_t^P} } (X){\rm{(}}x{\rm{)}} = \wedge _{t = 1}^m{ \wedge _{z \in U}}[(1 - {R_t}(x, z)) \vee X(z)], \forall x, z \in U$
 $\overline {\sum\limits_{t = 1}^m {R_t^P} (} X) = \sim\underline {\sum\limits_{t = 1}^m {R_t^P} } \left( {\sim X} \right)$

 $\underline {\Re _{\sum\limits_{t = 1}^m {{R_i}} }^O} (A)(x) = \vee _{t = 1}^m{ \wedge _{y \in V}}[(1 - {R_t}(x, y)) \vee A(y)], \forall x \in U, y \in V$
 $\overline {\Re _{\sum\limits_{t = 1}^m {{R_t}} }^O} (A)(x) = \wedge _{t = 1}^m{ \vee _{y \in V}}{\rm{[}}{R_t}(x, y) \wedge A(y){\rm{]}}, \forall x \in U, y \in V$

$A$ $(U, V, \Re )$ 中的悲观下近似 $\underline {\Re _{\sum\limits_{t = 1}^m {{R_t}} }^P} (A)$ 和悲观上近似 $\overline {\Re _{\sum\limits_{t = 1}^m {{R_t}} }^P} (A)$ 分别定义为：

 $\underline {\Re _{\sum\limits_{t = 1}^m {{R_t}} }^P} (A)(x) = \wedge _{t = 1}^m{ \wedge _{y \in V}}[(1 - {R_t}(x, y)) \vee A(y)], \forall x \in U, y \in V$
 $\overline {\Re _{\sum\limits_{t = 1}^m {{R_t}} }^P} (A)(x) = \vee _{t = 1}^m{ \vee _{y \in V}}{\rm{[}}{R_t}(x, y) \wedge A(y){\rm{]}}, \forall x \in U, y \in V$

 ${{ R}_1} = \left[ {\begin{array}{*{20}{c}} {0.2}&{0.5}&{0.2} \\ {0.1}&{0.3}&{0.7} \\ {0.6}&{0.3}&{0.4} \end{array}} \right]$
 ${{ R}_2} = \left[ {\begin{array}{*{20}{c}} {0.2}&{0.5}&{0.3} \\ {0.4}&{0.4}&{0.3} \\ {0.3}&{0.6}&{0.2} \end{array}} \right]$
 ${{ R}_3} = \left[ {\begin{array}{*{20}{c}} {0.4}&{0.3}&{0.3} \\ {0.5}&{0.5}&{0.4} \\ {0.3}&{0.8}&{0.4} \end{array}} \right]$

 \begin{aligned} \overline {\Re _{\sum\limits_{t = 1}^m {{R_t}} }^P} (A)({x_1}) = & [(0.2 \wedge 0.4) \vee (0.5 \wedge 0.6) \vee\\[-10pt] & (0.2 \wedge 0.9)] \vee[(0.2 \wedge 0.4) \vee\\ & (0.5 \wedge 0.6) \vee(0.3 \wedge 0.9)] \vee\\ & [(0.4 \wedge 0.4) \vee (0.3 \wedge 0.6) \vee\\ & (0.3 \wedge 0.9)] = 0.5 \end{aligned}
 \begin{aligned} \underline {\Re _{\sum\limits_{t = 1}^m {{R_t}} }^P} (A)({x_1}) = & ([(1 - 0.2) \vee 0.4] \wedge [(1 - 0.5) \vee 0.6] \wedge\\[-10pt] & [(1 - 0.2) \vee 0.9]) \wedge([(1 - 0.2) \vee 0.4] \wedge\\ & [(1 - 0.5) \vee 0.6] \wedge [(1 - 0.3) \vee 0.9]) \wedge\\ & ([(1 - 0.4) \vee 0.4] \wedge [(1 - 0.3) \vee 0.6] \wedge\\ & [(1 - 0.3) \vee 0.9]) = 0.6 \end{aligned}

 $\overline {\Re _{\sum\limits_{t = 1}^m {{R_t}} }^P} (A){\rm{ = }}\frac{{0.5}}{{{x_1}}} + \frac{{0.7}}{{{x_2}}} + \frac{{0.6}}{{{x_3}}}$
 $\underline {\Re _{\sum\limits_{t = 1}^m {{R_t}} }^P} (A){\rm{ = }}\frac{{0.6}}{{{x_1}}} + \frac{{0.5}}{{{x_2}}} + \frac{{0.4}}{{{x_3}}}$

$\overline {\Re _{\sum\limits_{t = 1}^m {{R_t}} }^P} (A)$ 表示悲观情况下病人患上述几种病的乐观估计， $\underline {\Re _{\sum\limits_{t = 1}^m {{R_t}} }^P} (A)$ 表示悲观情况下病人患几种病的保守估计。

 \begin{aligned} \overline {\Re _{\sum\limits_{t = 1}^m {{R_t}} }^O} (A)({x_1}) = & [(0.2 \wedge 0.4) \vee (0.5 \wedge 0.6) \vee\\[-10pt] & (0.2 \wedge 0.9)] \wedge[(0.2 \wedge 0.4) \vee\\ & (0.5 \wedge 0.6) \vee(0.3 \wedge 0.9)]\wedge\\ & [(0.4 \wedge 0.4) \vee (0.3 \wedge 0.6)\vee\\ & (0.3 \wedge 0.9)] = 0.4 \end{aligned}
 \begin{aligned} \underline {\Re _{\sum\limits_{t = 1}^m {{R_t}} }^O} (A)({x_1}) = & ([(1 - 0.2) \vee 0.4] \wedge [(1 - 0.5) \vee 0.6] \hfill \wedge\\[-12pt] & [(1 - 0.2) \vee 0.9]) \vee ([(1 - 0.2) \vee 0.4] \wedge \\ & [(1 - 0.5) \vee 0.6] \wedge[(1 - 0.3) \vee 0.9]) \vee\\ & ([(1 - 0.4) \vee 0.4] \wedge [(1 - 0.3) \vee 0.6] \wedge \hfill \\ & [(1 - 0.3) \vee 0.9]) = 0.6 \end{aligned}

 $\overline {\Re _{\sum\limits_{t = 1}^m {{R_t}} }^O} (A){\rm{ = }}\frac{{0.4}}{{{x_1}}} + \frac{{0.4}}{{{x_2}}} + \frac{{0.4}}{{{x_3}}}$
 $\underline {\Re _{\sum\limits_{t = 1}^m {{R_t}} }^O} (A){\rm{ = }}\frac{{0.6}}{{{x_1}}} + \frac{{0.7}}{{{x_2}}} + \frac{{0.6}}{{{x_3}}}$

2 双论域多粒度模糊粗糙集上下近似包含关系的充分条件

 ${{ R}_1} = \left[ {\begin{array}{*{20}{c}} {a_{11}^1}&{a_{12}^1}&{a_{13}^1} \\ {a_{21}^1}&{a_{22}^1}&{a_{23}^1} \\ {a_{31}^1}&{a_{32}^1}&{a_{33}^1} \end{array}} \right]$
 ${{ R}_2} = \left[ {\begin{array}{*{20}{c}} {a_{11}^2}&{a_{12}^2}&{a_{13}^2} \\ {a_{21}^2}&{a_{22}^2}&{a_{23}^2} \\ {a_{31}^2}&{a_{32}^2}&{a_{33}^2} \end{array}} \right]$
 ${{ R}_3} = \left[ {\begin{array}{*{20}{c}} {a_{11}^3}&{a_{12}^3}&{a_{13}^3} \\ {a_{21}^3}&{a_{22}^3}&{a_{23}^3} \\ {a_{31}^3}&{a_{32}^3}&{a_{33}^3} \end{array}} \right]$

 \begin{aligned} \underline {\Re _{\sum\limits_{t = 1}^m {{R_t}} }^P} (A)({x_1}) = & ([(1 - a_{11}^1) \vee {d_1}] \wedge [(1 - a_{12}^1) \vee {d_2}] \wedge\\[-10pt] & [(1 - a_{1{\rm{3}}}^1) \vee {d_{\rm{3}}}])\wedge([(1 - a_{11}^{\rm{2}}) \vee {d_1}] \wedge\\ & [(1 - a_{12}^{\rm{2}}) \vee {d_2}] \wedge[(1 - a_{1{\rm{3}}}^{\rm{2}}) \vee {d_{\rm{3}}}])\wedge \\ & ([(1 - a_{11}^{\rm{3}}) \vee {d_1}] \wedge [(1 - a_{12}^{\rm{3}}) \vee {d_2}] \wedge\\ & [(1 - a_{1{\rm{3}}}^{\rm{3}}) \vee {d_{\rm{3}}}]) \end{aligned}
 \begin{aligned} \overline {\Re _{\sum\limits_{t = 1}^m {{R_t}} }^P} (A)({x_1}) = & [(a_{11}^1 \wedge {d_1}) \vee (a_{12}^1 \wedge {d_2}) \vee (a_{1{\rm{3}}}^1 \wedge {d_{\rm{3}}})] \vee \\ & [(a_{11}^{\rm{2}} \wedge {d_1}) \vee (a_{12}^{\rm{2}} \wedge {d_2}) \vee (a_{1{\rm{3}}}^{\rm{2}} \wedge {d_{\rm{3}}})] \vee \\ & [(a_{11}^{\rm{3}} \wedge {d_1}) \vee (a_{12}^{\rm{3}} \wedge {d_2}) \vee (a_{1{\rm{3}}}^{\rm{3}} \wedge {d_{\rm{3}}})]& \end{aligned}

 $\begin{gathered} p \vee q = \frac{{p + q}}{2} + \frac{{|p - q|}}{2} \hfill \\ p \wedge q = \frac{{p + q}}{2} - \frac{{|p - q|}}{2} \hfill \\ \end{gathered}$

 $\begin{array}{c} [(1 - a_{ij}^t) \vee {d_j}] - [a_{ij}^t \wedge {d_j}] = \\ \displaystyle\frac{{1 - a_{ij}^t + {d_j}}}{2} + \frac{{|1 - a_{ij}^t - {d_j}|}}{2} - \left[\frac{{a_{ij}^t + {d_j}}}{2} - \frac{{|a_{ij}^t - {d_j}|}}{2}\right] = \\ \displaystyle\frac{{1 - a_{ij}^t + {d_j}}}{2} + \frac{{|1 - a_{ij}^t - {d_j}|}}{2} - \frac{{(a_{ij}^t + {d_j})}}{2} + \frac{{|a_{ij}^t - {d_j}|}}{2} = \\ \displaystyle\frac{{1 - 2a_{ij}^t}}{2} + \frac{{|1 - a_{ij}^t - {d_j}| + |a_{ij}^t - {d_j}|}}{2} \geqslant \\ \displaystyle\frac{{1 - 2a_{ij}^t}}{2} + \frac{{|1 - a_{ij}^t - {d_j} + a_{ij}^t - {d_j}|}}{2} =\frac{{1 - 2a_{ij}^t}}{2} + \frac{{|1 - 2{d_j}|}}{2}\geqslant \\ 1 - (a_{ij}^t + {d_j}) \end{array}$
 $\begin{array}{c} [(1 - a_{ij}^t) \vee {d_j}] - [a_{ij}^{t'} \wedge {d_j}] =\\ \displaystyle \frac{{1 - a_{ij}^t + {d_j}}}{2} + \frac{{|1 - a_{ij}^t - {d_j}|}}{2} - \left[\frac{{a_{ij}^{t'} + {d_j}}}{2} - \frac{{|a_{ij}^{t'} - {d_j}|}}{2}\right] = \\ \displaystyle \frac{{1 - a_{ij}^t + {d_j}}}{2} + \frac{{|1 - a_{ij}^t - {d_j}|}}{2} - \frac{{(a_{ij}^{t'} + {d_j})}}{2} + \frac{{|a_{ij}^{t'} - {d_j}|}}{2} = \\ \displaystyle\frac{{1 - (a_{ij}^t + a_{ij}^{t'})}}{2} + \frac{{|1 - a_{ij}^t - {d_j}| + |a_{ij}^{t'} - {d_j}|}}{2} \geqslant \\ \displaystyle\frac{{1 - (a_{ij}^t + a_{ij}^{t'})}}{2} + \frac{{|1 - a_{ij}^t - {d_j}| + a_{ij}^{t'} - {d_j}}}{2} = \\ \displaystyle\frac{{1 - a_{ij}^t - {d_j}}}{2} + \frac{{|1 - a_{ij}^t - {d_j}|}}{2} \geqslant 0 \end{array}$
 $\begin{array}{c} [(1 - a_{ij}^t) \vee {d_j}] - [a_{hk}^t \wedge {d_k}] = \\ \displaystyle\frac{{1 - a_{ij}^t + {d_j}}}{2} + \frac{{|1 - a_{ij}^t - {d_j}|}}{2} - \left[\frac{{a_{hk}^t + {d_k}}}{2} - \frac{{|a_{hk}^t - {d_k}|}}{2}\right] = \\ \displaystyle\frac{{1 - a_{ij}^t + {d_j}}}{2} + \frac{{|1 - a_{ij}^t - {d_j}|}}{2} - \frac{{(a_{hk}^t + {d_k})}}{2} + \frac{{|a_{hk}^t - {d_k}|}}{2} = \\ \displaystyle \frac{{1 - a_{ij}^t - a_{hk}^t + {d_j} - {d_k}}}{2} + \frac{{|1 - a_{ij}^t - {d_j}| + |a_{hk}^t - {d_k}|}}{2} \geqslant \\ \displaystyle\frac{{1 - a_{ij}^t - a_{hk}^t + {d_j} - {d_k}}}{2} + \frac{{|1 - a_{ij}^t| - {d_j} + |a_{hk}^t - {d_k}|}}{2} = \\ \displaystyle\frac{{1 - a_{ij}^t - a_{hk}^t - {d_k}}}{2} + \frac{{|1 - a_{ij}^t| + |a_{hk}^t - {d_k}|}}{2} \geqslant \\ \displaystyle\frac{{1 - a_{ij}^t - a_{hk}^t - {d_k}}}{2} + \frac{{|1 - a_{ij}^t| + a_{hk}^t - {d_k}}}{2} = \\ \displaystyle \frac{{1 - a_{ij}^t - {d_k}}}{2} + \frac{{|1 - a_{ij}^t| - {d_k}}}{2} = 1 - (a_{ij}^t + {d_k}) \end{array}$

 ${{ R}_t} = \left[ {\begin{array}{*{20}{c}} {a_{11}^t}& {a_{12}^t}\cdots &{a_{1l}^t} \\ {a_{21}^t}&{a_{22}^t}\cdots &{a_{2l}^t} \\ \vdots & \vdots& \vdots \\ {a_{n1}^t}& {a_{n2}^t} &{a_{nl}^t} \end{array}} \right]$

 \begin{aligned} \underline {\Re _{\sum\limits_{t = 1}^m {{R_t}} }^P} (A)({x_1}) = & ([(1 - a_{11}^1) \vee {d_1}] \wedge [(1 - a_{12}^1) \vee {d_2}] \cdots \wedge \\[-10pt] & [(1 - a_{1l}^1) \vee {d_l}]) \cdots \wedge([(1 - a_{11}^m) \vee {d_1}] \wedge\\ & [(1 - a_{12}^m) \vee {d_2}] \cdots \wedge [(1 - a_{1l}^m) \vee {d_l}]) \end{aligned}
 \begin{aligned} \overline {\Re _{\sum\limits_{t = 1}^m {{R_t}} }^P} (A)({x_1}) = & [(a_{11}^1 \wedge {d_1}) \vee (a_{12}^1 \wedge {d_2}) \cdots \vee \\[-10pt] & (a_{1l}^1 \wedge {d_l})] \cdots \vee [(a_{11}^m \wedge {d_1}) \vee\\ & (a_{12}^m \wedge {d_2}) \cdots \vee (a_{1l}^m \wedge {d_l})] \end{aligned}

3 标准化方法

 ${l_{11}} = 0.2 + 0.1 + 0.6 = 0.9$
 $I_{11}^{1 + } = \frac{{I_{{\rm{11}}}^{\rm{1}}}}{{{l_{11}}}} = \frac{{0.2}}{{2 \times 0.9}} = \frac{1}{9}$

${{ R}_1}{\text{、}} {{ R}_2}{\text{、}} {{ R}_3}$ 进行标准化，结果如下：

 ${ R}_1^ + = \left[ {\begin{array}{*{20}{c}} {\displaystyle\frac{1}{9}}&{\displaystyle\frac{5}{{22}}}&{\displaystyle\frac{1}{{13}}} \\ {\displaystyle\frac{1}{{18}}}&{\displaystyle\frac{3}{{22}}}&{\displaystyle\frac{7}{{26}}} \\ {\displaystyle\frac{1}{3}}&{\displaystyle\frac{3}{{22}}}&{\displaystyle\frac{2}{{13}}} \end{array}} \right]$
 ${ R}_2^ + = \left[ {\begin{array}{*{20}{c}} {\displaystyle\frac{1}{9}}&{\displaystyle\frac{1}{6}}&{\displaystyle\frac{3}{{16}}} \\ {\displaystyle\frac{2}{9}}&{\displaystyle\frac{2}{{15}}}&{\displaystyle\frac{3}{{16}}} \\ {\displaystyle\frac{1}{6}}&{\displaystyle\frac{1}{5}}&{\displaystyle\frac{1}{8}} \end{array}} \right]$
 ${ R}_3^ + = \left[ {\begin{array}{*{20}{c}} {\displaystyle\frac{1}{6}}&{\displaystyle\frac{3}{{32}}}&{\displaystyle\frac{3}{{22}}} \\ {\displaystyle\frac{5}{{24}}}&{\displaystyle\frac{5}{{32}}}&{\displaystyle\frac{2}{{11}}} \\ {\displaystyle\frac{1}{8}}&{\displaystyle\frac{1}{4}}&{\displaystyle\frac{2}{{11}}} \end{array}} \right]$

A进行标准化后为 ${A^ + }$ ，则

 ${A^ + } = \displaystyle\frac{{\displaystyle\frac{2}{{19}}}}{{{x_1}}} + \displaystyle\frac{{\displaystyle\frac{3}{{19}}}}{{{x_2}}} + \displaystyle\frac{{\displaystyle\frac{9}{{38}}}}{{{x_3}}}$

 $\underline {\Re _{\sum\limits_{t = 1}^m {R_t^ + } }^P} ({A^ + }){\rm{ = }}\displaystyle\frac{{\displaystyle\frac{{17}}{{22}}}}{{{x_1}}} + \displaystyle\frac{{\displaystyle\frac{{19}}{{26}}}}{{{x_2}}} + \displaystyle\frac{{\displaystyle\frac{2}{3}}}{{{x_3}}}$
 $\overline {\Re _{\sum\limits_{t = 1}^m {R_t^ + } }^P} ({A^ + }){\rm{ = }}\displaystyle\frac{{\displaystyle\frac{3}{{16}}}}{{{x_1}}} +\displaystyle \frac{{\displaystyle\frac{9}{{38}}}}{{{x_2}}} + \displaystyle\frac{{\displaystyle\frac{2}{{11}}}}{{{x_3}}}$
 $\underline {\Re _{\sum\limits_{t = 1}^m {R_t^ + } }^O} ({A^ + }){\rm{ = }}\displaystyle\frac{{\displaystyle\frac{5}{6}}}{{{x_1}}} + \displaystyle\frac{{\displaystyle\frac{{19}}{{24}}}}{{{x_2}}} + \displaystyle\frac{{\displaystyle\frac{4}{5}}}{{{x_3}}}$
 $\overline {\Re _{\sum\limits_{t = 1}^m {R_t^ + } }^O} ({A^ + }){\rm{ = }}\displaystyle\frac{{\displaystyle\frac{3}{{22}}}}{{{x_1}}} + \displaystyle\frac{{\displaystyle\frac{2}{{11}}}}{{{x_2}}} + \displaystyle\frac{{\displaystyle\frac{2}{{13}}}}{{{x_3}}}$

4 结束语

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