0 引言

Pawlak提出的粗糙集理论^[1-2]是一种分析和处理不精确和不确定性问题的数学工具.目前，粗糙集理论已在数据挖掘、知识发现、图像处理、模式识别^[3-6]等领域得到广泛应用.

Pawlak粗糙集在等价关系对论域产生的划分(单粒空间)上给出了目标概念的近似描述.此后，许多学者将等价关系推广到容差、相似或优势关系等^[7-11]，或将目标概念推广到模糊集等^[12]研究粗糙近似.Lin提出了粒计算的概念^[13-14]，讨论了二元关系下的模糊集和粗糙集方法，并将粒计算方法引入到数据挖掘和机器学习中.钱宇华等从粒计算角度建立了等价关系族下的多粒空间，提出了多粒度粗糙集模型^[15-17]，证明了经典的粗糙集是多粒度粗糙集的特殊情况.

在实际应用中，有些概念往往不能精确定义，概念的外延也不能由实体集精确表达.现实世界中表示不确定、不精确、含糊或者部分已知概念的方法有：不确定边界概念、部分已知概念、不可定义概念和近似以及系统转换与概念近似.Yao引入了区间集^[18]来表示部分已知概念.胡宝清提出了区间集粗糙集，研究了区间集三支决策^[19].本文基于区间集粗糙集和多粒度粗糙集的思想，提出了乐观多粒度区间集粗糙集的概念，研究了其性质.建立了一族属性子集不同运算下的单粒空间和多粒空间，讨论了单粒空间下区间集粗糙集和多粒空间下乐观多粒度区间集粗糙集之间的关系.

1 乐观多粒度粗糙集

本节给出有关粗糙集和乐观多粒度粗糙集的基本概念和性质，相关内容请参考文献[1, 15-17].

1.1 Pawlak粗糙集

S=(U, AT, V, f)称为信息系统^[1](简记为S=(U, AT)), 其中：U是非空有限对象集, 称为论域；AT是非空有限属性集；$ V=\bigcup\limits_{a\in AT}{{{V}_{a}}} $, V_a是属性a的值域；f:U×AT→V是信息函数, 即∀x∈U, a∈AT, f(x, a)∈V_a.对任意A⊆AT，等价关系R_A={(x, y)∈U×U:f(x, a)=f(y, a), ∀a∈A}.由R_A诱导的划分U/R_A={[x]_A:x∈U}, 其中[x]_A={y∈U:(x, y)∈R_A}.对任意X⊆U，X的下、上近似定义为^[1]：

$ \underline A \left( X \right) = \left\{ {x \in U:{{\left[ x \right]}_A} \subseteq X} \right\};\bar A\left( X \right) = \left\{ {x \in U:{{\left[ x \right]}_A} \cap X \ne \emptyset } \right\}. $

(1)

从粒计算角度看，划分U/R_A是由等价关系R_A导出的单粒空间.Pawlak粗糙集是在单粒空间中对目标概念进行近似刻画.由于粒度世界存在不同形式、不同数量的粒空间，钱宇华等提出了多粒度粗糙集^[15]，在一族等价关系诱导的多粒空间下构建了乐观多粒度粗糙集对目标概念进行近似描述.

1.2 乐观多粒度粗糙集

设信息系统S=(U, AT), A₁, …, A_m是AT的m个属性子集.对任意X⊆U，X关于属性集A₁, …, A_m的乐观多粒度粗糙下、上近似^[15]分别定义为：

$ \begin{matrix} \underline{\sum\nolimits_{i=1}^{m}{A_{i}^{o}}}\left( X \right)=\left\{ x\in U:{{\left[ x \right]}_{{{A}_{1}}}}\subseteq X\vee \cdots \vee {{\left[ x \right]}_{{{A}_{m}}}}\subseteq X \right\}; \\ \overline{\sum\nolimits_{i=1}^{m}{A_{i}^{o}}}\left( X \right)=\tilde{\ }\underline{\sum\nolimits_{i=1}^{m}{A_{i}^{o}}}\left( \tilde{\ }X \right)=\left\{ x\in U:{{\left[ x \right]}_{{{A}_{1}}}}\cap X\ne \varnothing \wedge \cdots \wedge {{\left[ x \right]}_{{{A}_{m}}}}\cap X\ne \varnothing \right\}. \\ \end{matrix} $

(2)

性质1^[15-17]  设S=(U, AT)为信息系统，A₁, …, A_m是AT的m个属性子集.对任意X, Y⊆U：

$ \begin{array}{l} ①\;\;\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \emptyset \right) = \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \emptyset \right) = \emptyset ,\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( U \right) = \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( U \right) = U;\\ ②\;\;\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( { \sim X} \right) = \sim \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( X \right),\overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( { \sim X} \right) = \sim \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( X \right);\\ ③\;\;\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( X \right) \subseteq X \subseteq \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( X \right);\\ ④\;\;\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( X \right) = \bigcup\limits_{i = 1}^m {\underline {{A_i}} \left( X \right)} ,\overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( X \right) = \bigcap\limits_{i = 1}^m {\overline {{A_i}} \left( X \right)} ;\\ ⑤\;\;X \subseteq Y \Rightarrow \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( X \right) \subseteq \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( Y \right),\overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( X \right) \subseteq \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( Y \right). \end{array} $

2 乐观多粒度区间集粗糙集

Yao利用一对集合作为下界和上界对概念进行描述，从而引入了区间集^[18].胡宝清提出了区间集粗糙集^[19].本节研究乐观多粒度区间集粗糙集.

2.1 区间集粗糙集

设U是论域，2^U为其幂集.区间集$ \mathscr{X} $定义^[18]为

$ \mathscr{X} = \left[ {{X_l},{X_u}} \right] = \left\{ {X \in {2^U}:{X_l} \subseteq X \subseteq {X_u}} \right\},{X_l} \subseteq {X_u} \subseteq U. $

所有区间集的集合用I(2^U)来表示I(2^U)＝{$ \mathscr{X} $＝[X_l, X_u]:X_l⊆X_u⊆U}, 称为U的区间集幂集.区间集上的运算^{[18, 20-23]}定义为：对任意区间集$ \mathscr{X} $=[X_l, X_u], $ \mathscr{Y} $=[Y_l, Y_u]∈I(2^U),

$ \begin{array}{*{20}{c}} {\mathscr{X} \sqcap \mathscr{Y} = \left[ {{X_l} \cap {Y_l},{X_u} \cap {Y_u}} \right] = \left\{ {X \cap Y\left| {X \in \mathscr{X},Y \in \mathscr{Y}} \right.} \right\};}\\ {\mathscr{X} \sqcup \mathscr{Y} = \left[ {{X_l} \cup {Y_l},{X_u} \cup {Y_u}} \right] = \left\{ {X \cup Y\left| {X \in \mathscr{X},Y \in \mathscr{Y}} \right.} \right\};}\\ {\mathscr{X} - \mathscr{Y} = \left[ {{X_l} - {Y_u},{X_u} - {Y_l}} \right] = \left\{ {X - Y\left| {X \in \mathscr{X},Y \in \mathscr{Y}} \right.} \right\};}\\ {\neg \mathscr{X} = \left[ {U,U} \right] - \left[ {{X_l},{X_u}} \right] = \left[ { \sim {X_u}, \sim {X_l}} \right].} \end{array} $

(3)

记$ \hat{X}=[X, X](X\subseteq U) $称为单点区间集.区间集幂集I(2^U)上的序关系“⊑”定义为

$ \mathscr{X} \sqsubseteq \mathscr{Y} \Leftrightarrow {X_l} \subseteq {Y_l},{X_u} \subseteq {Y_u}. $

(4)

于是, $ \mathscr{X} $⊑$ \mathscr{Y} $⇔$ \mathscr{X} $⊓$ \mathscr{Y} $＝$ \mathscr{X} $⇔$ \mathscr{X} $⊔$ \mathscr{Y} $＝$ \mathscr{Y} $.

设S=(U, AT)为信息系统, A⊆AT.区间集$ \mathscr{X} $=[X_l, X_u]的区间集粗糙下、上近似定义^[21]为：

$ \underline A \left( \mathscr{X} \right) = \left[ {\underline A \left( {{X_l}} \right),\underline A \left( {{X_u}} \right)} \right];\bar A\left( \mathscr{X} \right) = \left[ {\bar A\left( {{X_l}} \right),\bar A\left( {{X_u}} \right)} \right], $

(5)

其中，A(X_l)、A(X_l)与A(X_u)、A(X_u)分别为区间集$ \mathscr{X} $的下界X_l和上界X_u的粗糙下、上近似.

2.2 乐观多粒度区间集粗糙集

定义1  设信息系统S=(U, AT), A₁, …, A_m⊆AT.对任意区间集$ \mathscr{X} $=[X_l, X_u], $ \mathscr{X} $的乐观多粒度区间集下、上近似分别定义为：

$ \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) = \left[ {\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{X_l}} \right),\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{X_u}} \right)} \right];\\ \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) = \left[ {\overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{X_l}} \right),\overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{X_u}} \right)} \right], $

(6)

其中，$ \underline{\sum\nolimits_{i=1}^{m}{A_{i}^{o}}}({{X}_{l}}) $, $ \overline{\sum\nolimits_{i=1}^{m}{A_{i}^{o}}}({{X}_{l}}) $, $ \underline{\sum\nolimits_{i=1}^{m}{A_{i}^{o}}}({{X}_{u}}) $, $ \overline{\sum\nolimits_{i=1}^{m}{A_{i}^{o}}}({{X}_{u}}) $分别由公式(2)给出.

性质2  设信息系统S=(U, AT), A₁, …, A_m⊆AT.对任意区间集$ \mathscr{X} $, $ \mathscr{Y} $∈I(2^U)，有下列性质：

$ \begin{array}{l} ①\;\;\neg \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) = \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {\neg \mathscr{X}} \right),\neg \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) = \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {\neg \mathscr{X}} \right);\\ ②\;\;\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {\hat \emptyset } \right) = \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {\hat \emptyset } \right)=\hat \emptyset ,\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {\hat U} \right) = \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {\hat U} \right) = \hat U;\\ ③\;\;\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) \sqsubseteq \mathscr{X} \sqsubseteq \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right);\\ ④\;\;\mathscr{X} \sqsubseteq \mathscr{Y} \Rightarrow \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) \sqsubseteq \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{Y} \right),\overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) \sqsubseteq \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{Y} \right);\\ ⑤\;\;\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {\mathscr{X} \sqcup \mathscr{Y}} \right) \sqsupseteq \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) \sqcup \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{Y} \right),\\ \;\;\;\;\overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {\mathscr{X} \sqcup \mathscr{Y}} \right) \sqsupseteq \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) \sqcup \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{Y} \right);\\ ⑥\;\;\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {\mathscr{X} \sqcap \mathscr{Y}} \right) \sqsubseteq \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) \sqcap \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{Y} \right),\\ \;\;\;\;\overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {\mathscr{X} \sqcap \mathscr{Y}} \right) \sqsubseteq \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) \sqcap \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{Y} \right);\\ ⑦\;\;\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) = \mathop \sqcup \limits_{i = 1}^m \underline {{A_i}} \left( \mathscr{X} \right),\overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) = \mathop \sqcup \limits_{i = 1}^m \overline {{A_i}} \left( \mathscr{X} \right). \end{array} $

证明  设区间集$ \mathscr{X} $＝[X_l, X_u], $ \mathscr{Y} $＝[Y_l, Y_u], 下证多粒度区间集下近似的性质，上近似类似可证.

① 由定义1、性质1②及区间集补运算的性质知

$ \begin{array}{l} \neg \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) = \left[ { \sim \left( {\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{X_u}} \right)} \right), \sim \left( {\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{X_l}} \right)} \right)} \right] = \\ \left[ {\overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( { \sim {X_u}} \right),\overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( { \sim {X_l}} \right)} \right] = \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {\neg \mathscr{X}} \right). \end{array} $

② 由性质1①及定义1易证结论成立.

③ 由性质1③及定义1知，

$ \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) = \left[ {\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{X_l}} \right),\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{X_u}} \right)} \right] \sqsubseteq \left[ {{X_l},{X_u}} \right] = \mathscr{X}. $

④ 由公式(4)、定义1和性质1⑤可得

$ \begin{array}{l} \mathscr{X} \sqsubseteq \mathscr{Y} \Rightarrow {X_l} \subseteq {Y_l},{X_u} \subseteq {Y_u} \Rightarrow \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{X_l}} \right) \subseteq \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{Y_l}} \right),\\ \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{X_u}} \right) \subseteq \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{Y_u}} \right) \Rightarrow \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) \sqsubseteq \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{Y} \right). \end{array} $

⑤ 由公式(3)、定义1可得

$ \begin{array}{l} \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {\mathscr{X} \sqcup \mathscr{Y}} \right) = \left[ {\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{X_l} \cup {Y_l}} \right),\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{X_u} \cup {Y_u}} \right)} \right] \sqsupseteq \\ \left[ {\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{X_l}} \right) \cup \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{Y_l}} \right),\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{X_u}} \right) \cup \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{Y_u}} \right)} \right] = \\ \left[ {\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{X_l}} \right),\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{X_u}} \right)} \right] \sqcup \left[ {\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{Y_l}} \right),\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{Y_u}} \right)} \right] = \\ \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {\left[ {{X_l},{X_u}} \right]} \right) \sqcup \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {\left[ {{Y_l},{Y_u}} \right]} \right) = \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) \sqcup \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{Y} \right). \end{array} $

⑥ 类似⑤的证明可证结论成立.

⑦ 由性质1④和公式(3)可得

$ \begin{array}{l} \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) = \left[ {\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{X_l}} \right),\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{X_u}} \right)} \right] = \left[ {\mathop \sqcup \limits_{i = 1}^m \underline {{A_i}} \left( {{X_l}} \right),\mathop \sqcup \limits_{i = 1}^m \underline {{A_i}} \left( {{X_u}} \right)} \right] = \\ \mathop \sqcup \limits_{i = 1}^m \left[ {\underline {{A_i}} \left( {{X_l}} \right),\underline {{A_i}} \left( {{X_u}} \right)} \right] = \mathop \sqcup \limits_{i = 1}^m \underline {{A_i}} \left( \mathscr{X} \right). \end{array} $

由性质2可知，对任意区间集$ \mathscr{X} $∈I(2^U)，

$ \underline {{A_i}} \left( \mathscr{X} \right) \sqsubseteq \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) \sqsubseteq \mathscr{X} \sqsubseteq \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) \sqsubseteq \overline {{A_i}} \left( \mathscr{X} \right). $

(7)

性质3  设信息系统S=(U, AT), A₁, …, A_m⊆AT.对任意区间集$ \mathscr{X} $_j∈I(2^U), j=1, 2, …, n, 都有

$ \begin{array}{l} ①\;\;\underline {\sum\nolimits_{i = 1}^m {A_i^o}} \left( {\mathop \sqcap \limits_{j = 1}^n {\mathscr{X}_j}} \right) = \mathop \sqcup \limits_{i = 1}^m \left( {\mathop \sqcap \limits_{j = 1}^n \underline {{A_i}} \left( {{\mathscr{X}_j}} \right)} \right) \sqsubseteq \mathop \sqcap \limits_{j = 1}^n \left( {\mathop \sqcup \limits_{i = 1}^m \underline {{A_i}} \left( {{\mathscr{X}_j}} \right)} \right),\\ \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {\mathop \sqcup \limits_{j = 1}^n {\mathscr{X}_j}} \right) = \mathop \sqcap \limits_{i = 1}^m \left( {\mathop \sqcup \limits_{j = 1}^n \overline {{A_i}} \left( {{\mathscr{X}_j}} \right)} \right) \sqsupseteq \mathop \sqcup \limits_{j = 1}^n \left( {\mathop \sqcap \limits_{i = 1}^m \overline {{A_i}} \left( {{\mathscr{X}_j}} \right)} \right);\\ ②\;\;\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {\mathop \sqcup \limits_{j = 1}^n {\mathscr{X}_j}} \right) \sqsupseteq \mathop \sqcup \limits_{j = 1}^n \left( {\mathop \sqcup \limits_{i = 1}^m \underline {{A_i}} \left( {{\mathscr{X}_j}} \right)} \right),\overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {\mathop \sqcap \limits_{j = 1}^n {\mathscr{X}_j}} \right) \sqsubseteq \mathop \sqcap \limits_{j = 1}^n \left( {\mathop \sqcap \limits_{i = 1}^m \overline {{A_i}} \left( {{\mathscr{X}_j}} \right)} \right);\\ ③\;\;{\mathscr{X}_1} \sqsubseteq {\mathscr{X}_2} \sqsubseteq \cdots \sqsubseteq {\mathscr{X}_n} \Rightarrow \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{\mathscr{X}_1}} \right) \sqsubseteq \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{\mathscr{X}_2}} \right) \sqsubseteq \cdots \sqsubseteq \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{\mathscr{X}_n}} \right),\\ \;\;\;\;\overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{\mathscr{X}_1}} \right) \sqsubseteq \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{\mathscr{X}_2}} \right) \sqsubseteq \cdots \sqsubseteq \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{\mathscr{X}_n}} \right). \end{array} $

证明  设$ \mathscr{X} $_j=[X_l^j, X_u^j]∈I(2^U), j=1, 2, …, n.由区间集的⊓运算及性质2⑦知

$ \begin{array}{l} \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {\mathop \sqcap \limits_{j = 1}^n {\mathscr{X}_j}} \right) = \mathop \sqcup \limits_{i = 1}^m \underline {{A_i}} \left( {\left[ {\bigcap\limits_{j = 1}^n {X_l^j} ,\bigcap\limits_{j = 1}^n {X_u^j} } \right]} \right) = \mathop \sqcup \limits_{i = 1}^m \left[ {\underline {{A_i}} \left( {\bigcap\limits_{j = 1}^n {X_l^j} } \right),\underline {{A_i}} \left( {\bigcap\limits_{j = 1}^n {X_u^j} } \right)} \right] = \\ \mathop \sqcup \limits_{i = 1}^m \left[ {\bigcap\limits_{j = 1}^n {\underline {{A_i}} \left( {X_l^j} \right)} ,\bigcap\limits_{j = 1}^n {\underline {{A_i}} \left( {X_u^j} \right)} } \right] = \mathop \sqcup \limits_{i = 1}^m \left( {\mathop \sqcap \limits_{j = 1}^n \underline {{A_i}} \left( {{\mathscr{X}_j}} \right)} \right), \end{array} $

由性质2⑥和性质2⑦知

$ \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {\mathop \sqcap \limits_{j = 1}^n {\mathscr{X}_j}} \right) \sqsubseteq \mathop \sqcap \limits_{j = 1}^n \left( {\underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( {{\mathscr{X}_j}} \right)} \right) = \mathop \sqcap \limits_{j = 1}^n \left( {\mathop \sqcup \limits_{i = 1}^m \underline {{A_i}} \left( {{\mathscr{X}_j}} \right)} \right), $

即$ \underline{\sum\nolimits_{i=1}^{m}{A_{i}^{o}}}(\underset{j=1}{\overset{n}{\mathop{\sqcap }}}\, {\mathscr{X}_{j}})=\underset{i=1}{\overset{m}{\mathop{\sqcup }}}\, (\underset{j=1}{\overset{n}{\mathop{\sqcap }}}\, \underline{{{A}_{i}}}({\mathscr{X}_{j}}))\sqsubseteq \underset{j=1}{\overset{n}{\mathop{\sqcap }}}\, (\underset{i=1}{\overset{m}{\mathop{\sqcup }}}\, \underline{{{A}_{i}}}({\mathscr{X}_{j}})) $.类似可证$ \overline{\sum\nolimits_{i=1}^{m}{A_{i}^{o}}}(\underset{j=1}{\overset{n}{\mathop{\sqcup }}}\, {\mathscr{X}_{j}})=\underset{i=1}{\overset{m}{\mathop{\sqcap }}}\, (\underset{j=1}{\overset{n}{\mathop{\sqcup }}}\, \overline{{{A}_{i}}}({\mathscr{X}_{j}}))\sqsupseteq \underset{j=1}{\overset{n}{\mathop{\sqcup }}}\, (\underset{i=1}{\overset{m}{\mathop{\sqcap }}}\, \overline{{{A}_{i}}}({\mathscr{X}_{j}})) $.

由性质2④, ⑤, ⑥和⑦类似，可证性质3②与3③成立.

由等价关系的定义知，A₁, …, A_m⊆AT, X∈2^U,

$ \left\{ \begin{array}{l} {R_{\bigcup\nolimits_{i = 1}^m {{A_i}} }} = \bigcap\nolimits_{i = 1}^m {{R_{{A_i}}}} \subseteq {R_{{A_i}}} \subseteq {R_{\bigcap\nolimits_{i = 1}^m {{A_i}} }},\\ \underline {\bigcap\nolimits_{i = 1}^m {{A_i}} } \left( X \right) \subseteq \underline {{A_i}} \left( X \right) \subseteq \underline {\bigcup\nolimits_{i = 1}^m {{A_i}} } \left( X \right) \subseteq X,\\ X \subseteq \overline {\bigcup\nolimits_{i = 1}^m {{A_i}} } \left( X \right) \subseteq \overline {{A_i}} \left( X \right) \subseteq \overline {\bigcap\nolimits_{i = 1}^m {{A_i}} } \left( X \right). \end{array} \right. $

(8)

性质4  设S=(U, AT)为信息系统, A₁, …, A_m⊆AT.对任意区间集$ \mathscr{X} $∈I(2^U)，

$ \underline {\bigcap\nolimits_{i = 1}^m {{A_i}} } \left( \mathscr{X} \right) \sqsubseteq \underline {{A_i}} \left( \mathscr{X} \right) \sqsubseteq \underline {\bigcup\nolimits_{i = 1}^m {{A_i}} } \left( \mathscr{X} \right) \sqsubseteq \mathscr{X} \sqsubseteq \overline {\bigcup\nolimits_{i = 1}^m {{A_i}} } \left( \mathscr{X} \right) \sqsubseteq \overline {{A_i}} \left( \mathscr{X} \right) \sqsubseteq \overline {\bigcap\nolimits_{i = 1}^m {{A_i}} } \left( \mathscr{X} \right). $

证明  由公式(8)知，$ \mathscr{X} $的边界X_l、X_u满足

$ \underline {\bigcap\nolimits_{i = 1}^m {{A_i}} } \left( {{X_l}} \right) \subseteq \underline {{A_i}} \left( {{X_l}} \right) \subseteq \underline {\bigcup\nolimits_{i = 1}^m {{A_i}} } \left( {{X_l}} \right) \subseteq {X_l},\underline {\bigcap\nolimits_{i = 1}^m {{A_i}} } \left( {{X_u}} \right) \subseteq \underline {{A_i}}\\ \left( {{X_u}} \right) \subseteq \underline {\bigcup\nolimits_{i = 1}^m {{A_i}} } \left( {{X_u}} \right) \subseteq {X_u}. $

于是由区间集粗糙下、上近似的定义知，

$ \begin{array}{l} \underline {\bigcap\nolimits_{i = 1}^m {{A_i}} } \left( \mathscr{X} \right) = \left[ {\underline {\bigcap\nolimits_{i = 1}^m {{A_i}} } \left( {{X_l}} \right),\underline {\bigcap\nolimits_{i = 1}^m {{A_i}} } \left( {{X_u}} \right)} \right] \sqsubseteq \left[ {\underline {{A_i}} \left( {{X_l}} \right),\underline {{A_i}} \left( {{X_u}} \right)} \right] = \underline {{A_i}} \left( \mathscr{X} \right) \sqsubseteq \\ \left[ {\underline {\bigcup\nolimits_{i = 1}^m {{A_i}} } \left( {{X_l}} \right),\underline {\bigcup\nolimits_{i = 1}^m {{A_i}} } \left( {{X_u}} \right)} \right] = \underline {\bigcup\nolimits_{i = 1}^m {{A_i}} } \left( \mathscr{X} \right) \sqsubseteq \mathscr{X}. \end{array} $

由区间集粗糙集的对偶性即证$ \mathscr{X}\sqsubseteq \overline{\bigcup\nolimits_{i=1}^{m}{{{A}_{i}}}}\left( \mathscr{X} \right)\sqsubseteq \overline{{{A}_{i}}}\left( \mathscr{X} \right)\sqsubseteq \overline{\bigcap\nolimits_{i=1}^{m}{{{A}_{i}}}}\left( \mathscr{X} \right) $成立.

推论1  设信息系统S=(U, AT), A₁, …, A_m是AT的属性子集.对任意区间集$ \mathscr{X} $∈I(2^U)，

$ \begin{array}{l} ①\;\;\underline {\bigcap\nolimits_{i = 1}^m {{A_i}} } \left( \mathscr{X} \right) \sqsubseteq \underline {{A_i}} \left( \mathscr{X} \right) \sqsubseteq \mathop \sqcup \limits_{i = 1}^m \underline {{A_i}} \left( \mathscr{X} \right) = \underline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) \sqsubseteq \underline {\bigcup\nolimits_{i = 1}^m {{A_i}} } \left( \mathscr{X} \right) \sqsubseteq \mathscr{X},\\ ②\;\;\mathscr{X} \sqsubseteq \overline {\bigcup\nolimits_{i = 1}^m {{A_i}} } \left( \mathscr{X} \right) \sqsubseteq \mathop \sqcap \limits_{i = 1}^m \overline {{A_i}} \left( \mathscr{X} \right) = \overline {\sum\nolimits_{i = 1}^m {A_i^o} } \left( \mathscr{X} \right) \sqsubseteq \overline {{A_i}} \left( \mathscr{X} \right) \sqsubseteq \overline {\bigcap\nolimits_{i = 1}^m {{A_i}} } \left( \mathscr{X} \right). \end{array} $

2.3 风险投资分析中的应用

考虑风险投资公司的风险投资问题^[24].公司给出了20个投资方案x_i(i=1, 2, …, 20)，投资方案的风险水平由5个专家给出，风险水平值1、2、3分别表示风险低、中和高.风险水平值越大，投资计划的风险越高，收益也就越高，反之亦然.表 1是5个专家给出的投资方案的风险水平评估表.

记U={x₁, …, x₂₀}是投资方案集合, AT={a₁, …, a₅}是专家集合.根据经验，3个以上专家打分超过2的投资方案的获益较高，风险也高，而3个以上专家打分低于2的投资方案风险较低，但获益也低.故投资风险一定高的方案集合X={x₁, x₆, x₈, x₉, x₁₉}，而投资风险一定低，同时收益也较低的方案集合Y={x₃, x₄, x₁₁, x₁₃, x₁₄, x₁₅, x₁₈, x₂₀}, Y^c是风险不低的投资方案.记X_l=X, X_u=Y^c, 则$ \mathscr{X} $=[X_l, X_u]给出了风险一定高和风险可能高作为下界和上界的方案集合，即$ \mathscr{X} $中的集合是投资风险相对高而获益也相对高的方案集合.

在专家组A₁={a₁, a₂}诱导的单粒空间U/R_A1上, 区间集$ \mathscr{X} $=[X_l, X_u]的边界X_l, X_u的近似刻画为：

$ \begin{array}{*{20}{c}} {\underline {{A_1}} \left( {{X_l}} \right) = \left\{ {{x_9},{x_{19}}} \right\},\underline {{A_1}} \left( {{X_u}} \right) = \left\{ {{x_1},{x_5},{x_6},{x_7},{x_8},{x_9},{x_{10}},{x_{12}},{x_{17}},{x_{19}}} \right\},}\\ {\overline {{A_1}} \left( {{X_l}} \right) = \left\{ {{x_1},{x_6},{x_7},{x_8},{x_9},{x_{17}},{x_{19}}} \right\},\overline {{A_1}} \left( {{X_u}} \right) = \\ \left\{ {{x_1},{x_2},{x_5},{x_6},{x_7},{x_8},{x_9},{x_{10}},{x_{12}},{x_{13}},{x_{15}},{x_{16}},{x_{17}},{x_{19}}} \right\}.} \end{array} $

在专家组A₂={a₃}诱导的单粒空间U/R_A2上，区间集$ \mathscr{X} $=[X_l, X_u]的边界X_l, X_u的近似刻画为：

$ \begin{array}{*{20}{c}} {\underline {{A_2}} \left( {{X_l}} \right) = \emptyset ,\underline {{A_2}} \left( {{X_u}} \right) = \left\{ {{x_1},{x_2},{x_8},{x_9},{x_{12}}} \right\},}\\ {\overline {{A_2}} \left( {{X_l}} \right) = U - \left\{ {{x_3},{x_4},{x_{11}},{x_{13}},{x_{16}},{x_{17}},{x_{18}}} \right\},\overline {{A_2}} \left( {{X_u}} \right) = U.} \end{array} $

在专家组A₃={a₄, a₅}诱导的单粒空间U/R_A3上，区间集$ \mathscr{X} $=[X_l, X_u]的边界X_l, X_u的近似刻画为：

$ \begin{array}{*{20}{c}} {\underline {{A_3}} \left( {{X_l}} \right) = \left\{ {{x_1},{x_6},{x_8},{x_9},{x_{19}}} \right\},\underline {{A_3}} \left( {{X_u}} \right) = \left\{ {{x_1},{x_2},{x_5},{x_6},{x_7},{x_8},{x_9},{x_{16}},{x_{17}},{x_{19}}} \right\},}\\ {\overline {{A_3}} \left( {{X_l}} \right) = \left\{ {{x_1},{x_6},{x_8},{x_9},{x_{19}}} \right\},\overline {{A_3}} \left( {{X_u}} \right) = U - \left\{ {{x_3},{x_{15}}} \right\}.} \end{array} $

3个专家组A₁, A₂, A₃诱导的多粒空间{U/R_Ai:i=1, 2, 3}上边界X_l, X_u的近似刻画为：

$ \begin{array}{*{20}{c}} {\underline {\sum\nolimits_{i = 1}^3 {A_i^o} } \left( {{X_l}} \right) = \left\{ {{x_1},{x_6},{x_8},{x_9},{x_{19}}} \right\},\underline {\sum\nolimits_{i = 1}^3 {A_i^o} } \left( {{X_u}} \right) = \\ \left\{ {{x_1},{x_2},{x_5},{x_6},{x_7},{x_8},{x_9},{x_{10}},{x_{12}},{x_{16}},{x_{17}},{x_{19}}} \right\},}\\ {\overline {\sum\nolimits_{i = 1}^3 {A_i^o} } \left( {{X_l}} \right) = \left\{ {{x_1},{x_6},{x_8},{x_9},{x_{19}}} \right\},\overline {\sum\nolimits_{i = 1}^3 {A_i^o} } \left( {{X_u}} \right) =\\ U - \left\{ {{x_3},{x_4},{x_{11}},{x_{14}},{x_{15}},{x_{18}},{x_{20}}} \right\}.} \end{array} $

由上面的计算可知, 对任意i=1, 2, 3, $ \underline{{{A}_{i}}}\left( \mathscr{X} \right)\sqsubseteq \underline{\sum\nolimits_{i=1}^{3}{A_{i}^{o}}}\left( \mathscr{X} \right)\sqsubseteq \mathscr{X} \sqsubseteq \overline{\sum\nolimits_{i=1}^{3}{A_{i}^{o}}}\left( \mathscr{X} \right)\sqsubseteq \overline{{{A}_{i}}}\left( \mathscr{X} \right) $.故乐观多粒度区间集粗糙集($ \underline{\sum\nolimits_{i=1}^{3}{A_{i}^{o}}}\left( \mathscr{X} \right)$, $\overline{\sum\nolimits_{i=1}^{3}{A_{i}^{o}}}\left( \mathscr{X} \right) $)所确定的投资方案较区间集粗糙集($ \underline{{{A}_{i}}}\left( \mathscr{X} \right) $, $ \overline{{{A}_{i}}}\left( \mathscr{X} \right) $)(i=1, 2, 3)确定的投资方案获益要高.

3 结论

Pawlak粗糙集利用一个等价关系对论域的划分(单粒空间)给出了目标概念的近似刻画.由于对实际问题的研究存在不同的观点、不同的粒度，一族等价关系对论域的划分产生了多粒空间，从而产生了对目标概念的乐观多粒度粗糙近似描述.由于解决问题时存在的不精确、不确定性等原因产生了区间集，进而产生了区间集粗糙集.基于区间集粗糙集和乐观多粒度粗糙集的思想，本文提出了乐观多粒度区间集粗糙集，讨论了它们的性质，并给出了不同属性产生的单个和多个粒空间下几种区间集粗糙集和乐观多粒度区间集粗糙集之间的关系，最后将乐观多粒度区间集粗糙集应用于风险投资分析中，分析了乐观多粒度区间集粗糙集的逼近效果优于区间集粗糙集.

下一步将研究多粒度空间上的悲观多粒度区间集粗糙集，以及乐观、悲观多粒度区间集粗糙集与不同粒度空间上的区间集粗糙集之间的关系.