云模型是李德毅院士创立的定性与定量相互转换的不确定性模型[1-2]。云模型在空间数据挖掘[3]、粒度计算[4]、图像分割[5]、控制[6]等领域有着广泛的深入应用。集合论是现代数学的基础,创立至今已有百年之久[7]。集合论的观点和方法渗透在现代数学的各个分支以及科学技术的许多领域之中,也是目前对系统进行数学描述的主要工具[8],集合论在电力系统[9]、指挥信息系统[10]和计算机科学[11]等领域均有应用。1965年Zadeh[12]提出了模糊集合理论,集合与模糊集合均是人工智能的基础理论。模糊数学在金融[13]、故障分析[14]、物资需求分析[15]、控制优化[16-17]、聚类算法[18]等方面均有应用。模糊集合中给出了交、并、补等基本运算,研究了截集及其运算,给出了分解定理,这些基本运算和定理是模糊数学进一步应用的基础。云模型表示的半定性半定量概念的运算,有赖于云模型的集合视角的理论扩展。如果能够从集合论的角度建立云模型的集合基础理论与方法,即云集合理论,并建立云集合与模糊集合、经典集合的转换桥梁,则可以进一步应用集合理论与方法拓展云模型的应用范围和领域,将来的进一步研究则有可能将云模型扩展到函数、关系、基数、有序集和序数等方面。因此构建云模型的集合论基础理论和方法具有十分重要的理论和实际意义。参照模糊集合,本文对云集合定义及其集合基础运算方法、云集合的截集与分解定理进行了研究。
1 云集合及其组成元素 1.1 云集合定义1 云集合。设
云集合和模糊集的关系:一个云集合对应多个模糊集,理论上对应无限个模糊集,因此云集合是一个无限集合;同时由于元素隶属度的随机性,云集合可看作包含多个模糊集的随机集合,所以云集合又是一个随机隶属度集合。模糊集可以看作是云集合的一次具体实现。云集合与模糊集之间的转化关系:设
$\mathop A\limits_\wp = \Bigg\{ \mathop x\limits^\wp \left| {{u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )} \right. = {{\rm{e}}^{ - \frac{{{{(\mathop x\limits^\wp - {\rm{E}}x)}^2}}}{{2{\rm{E}}n{{_j'}^2}}}}}{\rm{, }}{\rm{E}}n_j' = {\rm{Nor}}({\rm{E}}n,He)\Bigg\} $ |
式中:
对集合内的所有
$\mathop A\limits_{\sim} = \Bigg\{ \left. {\mathop x\limits^{\sim} } \right|{u_{\mathop A\limits_{\sim} }}(\mathop x\limits^{\sim} ) = {{\rm{e}}^{ - \frac{{{{(\mathop x\limits^{\sim} - {\rm{E}}x)}^2}}}{{2{{({\rm{Cer}}({\rm{E}}n_j'))}^2}}}}}\Bigg\} $ |
式中
定义2 云集合元素。云集合元素定义为云集合中具有随机隶属度的数(元素)。当
1)首先生成以
2)令
因为在1)中生成的随机数有多种可能的值,所以2)中
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$I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) = \left[ {{{\rm{e}}^{ - \frac{{{{(\mathop x\limits^\wp - {\rm{E}}{x_A})}^2}}}{{2{{({\rm{E}}{n_A}{\rm{ - }}3H{{\rm{e}}_A})}^2}}}}},\;{{\rm{e}}^{ - \frac{{{{(\mathop x\limits^\wp - {\rm{E}}{x_A})}^2}}}{{2{{({\rm{E}}{n_A} + 3H{{\rm{e}}_A})}^2}}}}}} \right],\quad \forall \mathop x\limits^\wp \in \mathop A\limits_\wp $ | (1) |
式(1)主要依据正态分布的
$I({\rm{E}}n_j') = [{\rm{E}}n - 3He,{\rm{E}}n + 3He]$ |
$\begin{array}{l} P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) \geqslant I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp ))) = P([{a^l},{b^r}] \geqslant [{a^l},{b^r}]) = \\ \left\{ {\begin{array}{*{20}{l}} 1,\quad{{a^l} \geqslant {b^r}} \\ {\dfrac{{{a^r} - {b^r}}}{{{a^r} - {b^l}}} + \dfrac{{{b^r} - {a^l}}}{{2({a^r} - {b^l})}}},\quad{{b^l} < {a^l} < {b^r} < {a^r}} \\ {\dfrac{1}{2}},\quad{{a^l} = {b^l},{a^r} = {b^r}} \\ {\dfrac{{{a^r} - {b^l}}}{{2({b^r} - {a^l})}}},\quad{{a^l} < {b^l} < {a^r} < {b^r}} \\ {\dfrac{{{a^r} - {b^r}}}{{{a^r} - {a^l}}} + \dfrac{{{b^r} - {b^l}}}{{2({a^r} - {a^l})}}},\quad{{a^l} < {b^l} < {b^r} < {a^r}} \\ {\dfrac{{{a^r} - {a^l}}}{{2({b^r} - {b^l})}}},\quad{{b^l} < {a^l} < {a^r} < {b^r}} \\ 0,\quad{{b^l} \geqslant {a^r}} \end{array}} \right. \\ \end{array} $ |
设
云集合
$\begin{array}{l} \mathop A\limits_\wp = \mathop B\limits_\wp \Leftrightarrow {C_A}({\rm{E}}{x_A},{\rm{E}}{n_A},H{e_A}) = \\ {C_B}({\rm{E}}{x_B},{\rm{E}}{n_B},H{e_B}) \Leftrightarrow \left\{ {\begin{array}{*{20}{c}} {{\rm{E}}{x_A} = {\rm{E}}{x_B}} \\ {{\rm{E}}{n_{\rm{A}}} = {\rm{E}}{n_{\rm{B}}}} \\ {H{e_A} = H{e_B}} \end{array}} \right. \end{array} $ |
从集合运算的角度,对于
$\left\{ {\begin{array}{*{20}{l}} {\mathop A\limits_\wp = \mathop B\limits_\wp \Leftrightarrow P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp ))) = \dfrac{1}{2}} \\ {{a^l} = {b^l},{a^r} = {b^r}} \end{array}} \right.$ |
云集合
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对于
$\begin{array}{c} \mathop A\limits_\wp \supset \mathop B\limits_\wp \Leftrightarrow P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp ))) = 1 \Leftrightarrow \\ P(I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )) >I( {u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))) = 0 \end{array} $ |
对于论域内
$\mathop A\limits_\wp \mathop \supseteq \limits^p \mathop B\limits_\wp \Leftrightarrow \frac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )))}}{{P(I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} \geqslant 1$ |
云集合
${u_{\mathop A\limits_\wp \cup \mathop B\limits_\wp }}\left( {\mathop x\limits^\wp } \right) = {u_{\mathop A\limits_\wp }}\left( {\mathop x\limits^\wp } \right) \vee {u_{\mathop B\limits_\wp }}\left( {\mathop x\limits^\wp } \right)$ |
符号“
$\begin{array}{c} {u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ) \vee {u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp ) = \\ \left\{ {\begin{array}{*{20}{l}} {{u_{\mathop A\limits_\wp }}\left( {\mathop x\limits^\wp } \right)},\quad{\dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )))}}{{P(I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} > 1}\\ {{u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ),{u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )},\quad{\dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )))}}{{P(I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} = 1}\\ {{u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )},\quad{\dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )))}}{{P(I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} <1}\\ {{u_{\mathop A\limits_\wp }}\left( {\mathop x\limits^\wp } \right)},\quad{P(I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))) = 0} \end{array}} \right. \end{array}$ | (2) |
对于在论域
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云集合
${u_{\mathop A\limits_\wp \cap \mathop B\limits_\wp }}(\mathop x\limits^\wp ) = {u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ) \wedge {u_{\mathop B\limits_\wp }}\left( {\mathop x\limits^\wp } \right)$ |
符号“
$\begin{array}{c} {u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ) \wedge {u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp ) = \\ \left\{ {\begin{array}{*{20}{l}} {{u_{\mathop A\limits_\wp }}\left( {\mathop x\limits^\wp } \right)},\quad{\dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )))}}{{P(I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} <1}\\ {{u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ),{u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )},\quad{\dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )))}}{{P(I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} = 1}\\ {{u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )},\quad{\dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )))}}{{P(I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} > 1}\\ {{u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )},\quad{P(I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))) = 0} \end{array}} \right. \end{array}$ | (3) |
对于在论域
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云集合
${u_{{{\mathop A\limits_\wp }^c}}}(\mathop x\limits^\wp ) = 1 - {u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )$ |
云集合的余(补)集隶属函数如图5所示。
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参照经典集合与模糊集合中直积的定义[20],设云集合
${u_{\mathop A\limits_\wp {\rm{(}}X{\rm{)}} \times \mathop B\limits_\wp {\rm{(}}Y{\rm{)}}}} = {u_{\mathop A\limits_\wp {\rm{(}}X{\rm{)}}}} \wedge {u_{\mathop B\limits_\wp {\rm{(}}Y{\rm{)}}}}$ |
设
$({{\mathop A\limits_\wp} _\lambda })(\mathop x\limits^\wp ) = \left\{ {\begin{array}{*{20}{l}} {{u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ),\quad I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))\mathop \geqslant \limits^{\rm{L}} \lambda } \\ {0,\quad I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))\neg {\mathop \geqslant \limits^{\rm{L}}} \lambda } \end{array}} \right.$ |
云集合的
$({A_\lambda })(\mathop x\limits^\wp ) = \left\{ {\begin{array}{*{20}{l}} {1,\quad I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))\mathop \geqslant \limits^{\rm{L}} \lambda } \\ {0,\quad I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))\neg \mathop \geqslant \limits^{\rm{L}} \lambda } \end{array}} \right.$ |
云集合的
$({{\mathop A\limits_{\sim}} _\lambda })(\mathop x\limits^{\sim} ) = \left\{ {\begin{array}{*{20}{l}} {{u_{\mathop A\limits_{\sim} }}(\mathop x\limits^{\sim} ),\quad I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))\mathop \geqslant \limits^{\rm{L}} \lambda } \\ {0,\quad I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))\neg \mathop \geqslant \limits^{\rm{L}} \lambda } \end{array}} \right.$ |
定义区间I “左大于”
对于正态云集合,根据隶属度函数有
${{\rm{e}}^{ - \frac{{{{\left( {x - {\rm{E}}x} \right)}^2}}}{{2{{({\rm{E}}n')}^2}}}}} \geqslant \lambda $ |
因此,对于正态云模型,云集合
${A_\lambda } = \left\{ {\left. x \right|{\rm{E}}x{\rm{ - }}\sqrt {{\rm{ - 2}}{\rm{E}}{{n'}^2}\ln \lambda } \leqslant x \leqslant {\rm{E}}x + \sqrt {{\rm{ - 2}}{\rm{E}}{{n'}^2}\ln \lambda } } \right\}$ |
式中:
设
$ ({{\mathop A\limits_\wp} _{{{\mathop x\limits^\wp }_i}}})(\mathop x\limits^\wp ) = \left\{ {\begin{array}{*{20}{c}} {{u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ),\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop {{x_i}}\limits^\wp )))}}{{P(I({u_{\mathop A\limits_\wp }}(\mathop {{x_i}}\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} \geqslant 1} \\ \begin{array}{l} 0,\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})))}}{{P(I({u_{\mathop A\limits_\wp }}({x_i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))}} < 1 \\ {u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ),\quad P(I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp} _i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))) = 0 \end{array} \end{array}} \right. $ |
$ ({A_{{{\mathop {\rm{x}}\limits^\wp }_i}}})(x) = \left\{ {\begin{array}{*{20}{l}} {1,\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})))}}{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} \geqslant 1} \\ \begin{array}{l} 0,\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})))}}{{P(I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} < 1 \\ 1,\quad P(I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp} _i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))) = 0 \end{array} \end{array}} \right. $ | (4) |
称集合
设
${{\mathop A\limits_\wp} _{[a,b]}}\left( {\mathop x\limits^\wp } \right) = \left\{ {\begin{array}{*{20}{c}} {{u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ),\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I([a,b]))}}{{P(I([a,b])) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} \geqslant 1}\\ \begin{array}{l} 0,\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I([a,b]))}}{{P(I([a,b])) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} < 1\\ {u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ),\quad P(I([a,b]) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))) = 0 \end{array} \end{array}} \right.$ |
${A_{[a,b]}}\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {1,\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I([a,b]))}}{{P(I([a,b])) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} \geqslant 1}\\ \begin{array}{l} 0,\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I([a,b])}}{{P(I([a,b])) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} < 1\\ 1,\quad P(I([a,b]) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))) = 0 \end{array} \end{array}} \right.$ |
称集合
设
$ {{\mathop A\limits_\wp} _{{u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_{\rm{i}}})}}\left( {\mathop x\limits^\wp } \right) = \left\{ {\begin{array}{*{20}{l}} {{u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ),\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_i})))}}{{P(I({u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} \geqslant 1}\\ \begin{array}{l} 0,\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_i})))}}{{P(I({u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} < 1\\ {u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ),\quad P(I({u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp} _i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))) = 0 \end{array} \end{array}} \right. $ |
$ {{\mathop A\limits_{}} _{{u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_i})}}\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {1,\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_i})))}}{{P(I({u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} \geqslant 1}\\ \begin{array}{l} 0,\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_i})))}}{{P(I({u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} < 1\\ 1,\quad P(I({u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp} _i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))) = 0 \end{array} \end{array}} \right. $ | (5) |
称
定理1 设
$\mathop A\limits_\wp = \bigcup\limits_{x \in U}^{} {{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i}){A_{{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})}}} \left( x \right)$ | (6) |
式中:
${u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp} _i})A = \left\{ {\begin{array}{*{20}{c}} {{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i}),\quad x \in A} \\ {0,\quad x \notin A} \end{array}} \right.$ |
证明 根据随机隶属度
$\begin{array}{c} \left( {{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i}){A_{_{\mathop A\limits_\wp }({{\mathop x\limits^\wp }_i})}}\left( x \right)} \right)\left( {\mathop x\limits^\wp } \right) = \\ \left\{ {\begin{array}{*{20}{c}} {{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i}),\quad\mathop x\limits^\wp \in {A_{{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_{\rm{i}}})}},\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})))}}{{P(I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} \geqslant 1}\\ \begin{array}{l} 0,\quad \mathop x\limits^\wp \notin {A_{{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_{\rm{i}}})}},\dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})))}}{{P(I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} < 1\\ {u_{\mathop A\limits_\wp }}({\mathop x\limits^\wp _i}),\quad \mathop x\limits^\wp \in {A_{{u_{\mathop A\limits_\wp }}\left( {{{\mathop x\limits^\wp }_i}} \right)}},P(I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp} _i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))) = 0 \end{array} \end{array}} \right. \end{array}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!$ | (7) |
由式(7)可进一步得
定理2 设
$\mathop A\limits_\wp = \bigcup\limits_{{{\mathop x\limits^\wp }_i} \in U} {{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i}){A_{{{\mathop x\limits^\wp }_i}}}} (x)$ | (8) |
证明 根据式(4)和式(5),易知:
${A_{{{\mathop x\limits^\wp }_i}}}(x){\rm{ = }}{A_{{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})}}\left( x \right)$ |
于是有
$\bigcup\limits_{{{\mathop x\limits^\wp }_i} \in U} {{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i}){A_{{{\mathop x\limits^\wp }_i}}}} (x){\rm{ = }}\bigcup\limits_{{{\mathop x\limits^\wp }_i} \in U} {{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i}){A_{{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})}}\left( x \right)} $ |
根据云分定理1得到:
$\bigcup\limits_{{{\mathop x\limits^\wp }_i} \in U} {{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i}){A_{{{\mathop x\limits^\wp }_i}}}} (x) = \mathop A\limits_\wp $ |
本文参照模糊集,提出一种云模型的集合理论与方法。提出
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