Generalized intuitionistic fuzzy probabilistic rough set models on two universes and their applications
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摘要: 已有的双论域直觉模糊概率粗糙集模型通过设置两个阈值
${\lambda _1}$ 、${\lambda _2} $ ，讨论了经典集合在直觉模糊二元关系下的概率粗糙下上近似。该模型不能计算直觉模糊集合在直觉模糊二元关系下的概率粗糙下上近似，这在一定程度上限制了该模型的应用。首先给出了直觉模糊条件概率的定义。在直觉模糊概率空间下构造了双论域广义直觉模糊概率粗糙集模型，讨论了模型的主要性质。最后，将模型应用到临床诊断系统中。与其他模型相比，所提出的广义直觉模糊概率粗糙集模型进一步丰富了概率粗糙集理论，更适合于实际应用。Abstract: In the existing intuitionistic fuzzy (IF) probabilistic rough set model of two universes, the probabilistic rough lower and upper approximations of a classical set under IF binary relation are discussed by setting two thresholds,${\lambda _1}\;{\rm{and}}\;{\lambda _2}$ . However, this model is unable to calculate the probabilistic rough lower and upper approximations of IF sets under IF binary relation, which limits the application of this model to a certain extent. In this study, we first define the IF conditional probability. Then, the generalized IF probability rough set models are constructed in IF probabilistic approximation space. In addition, the main properties of the model are discussed. Finally, the numerical examples of clinical diagnostic systems are applied to illustrate the validity of the proposed models. Compared with other models, the proposed generalized IF probabilistic rough set models further enrich the probabilistic rough set theory and are more suitable for practical applications. -
粗糙集最早是由Pawlak[1]于1982年提出的。经典的粗糙集模型是建立在等价关系之上，由于等价关系的要求过于严苛，为了适应各类复杂数据环境，许多学者提出了包括优势关系[2]、容差关系[3]、模糊关系[4]、邻域关系[5]等扩展的粗糙集模型。考虑集合之间的重叠关系，许多研究者给出了该理论的概率推广[6-8]。概率粗糙集模型是其中一种重要的推广[9-11]。该模型中的概率近似算子是根据条件概率和阈值（能承受不确定性或分类错误的程度）来确定的。当前，概率粗糙集模型大致分为三类：决策理论粗糙集模型、变精度粗糙集模型和贝叶斯粗糙集模型。Yao在文献[12]中，基于粗糙隶属度和粗糙包含度，重新讨论了概率粗糙集近似算子，给出了3个模型的统一框架。近年来，概率粗糙集模型在应用方面取得了很大进展[13-15]，Sun等[16]基于概率粗糙集的三支决策原理，提出了一种改进的Pawlak冲突分析模型。与原有的Pawlak冲突分析模型相比，所提出的模型不仅为处理冲突分析问题提供了新的视角和方法，而且克服了原有模型的局限性。基于模糊熵，Ma[17]给出了三支概率粗糙集模型中两种属性约简方法。Yang等[18]提出了基于模糊关系的概率粗糙集模型，并将其用于医疗诊断系统中。尽管该模型中采用了模糊关系，但真正起作用的是模糊关系的截关系。这意味着它仍然是基于经典二元关系的概率粗糙集模型。
概率理论中的事件一般指样本空间中的精确指定的元素集合[19]。然而日常生活中，人们也常常遇到模糊的、不清晰的事件[20]。这就需要讨论模糊概率近似空间框架下，模糊事件的概率粗糙近似问题。文献[21]提出了4种模糊概率近似算子，然后应用贝叶斯决策理论，研究了模糊事件的三支决策问题，指出了决策理论粗糙集与模糊概率粗糙集的关系。作为模糊集理论的重要推广，直觉模糊集理论是Atanassov[22-23]于1986年提出的。由于该理论在考虑隶属度的同时，增加了元素对集合的非隶属度和犹豫度等信息，因此比模糊集在表达模糊性上更加细腻，在处理不确定性问题方面更具灵活性和实用性。郭智莲等[24]提出了基于直觉模糊关系的概率粗糙集模型。与文献[18]类似，利用阈值将直觉模糊关系转化成经典二元关系，它依然是基于经典二元关系的概率粗糙集模型。当前，在直觉模糊概率近似空间框架下，很少文献对直觉模糊事件进行概率粗糙近似研究。因此，有必要将概率粗糙集模型进一步推广到直觉模糊概率近似空间，进而拓展模型的应用。
在现有文献的基础上，本文首先定义了直觉模糊条件概率。在直觉模糊概率空间下构造了双论域广义直觉模糊概率粗糙集模型，讨论了模型的主要性质。最后，将模型应用到临床诊断系统中。
1. 预备知识
1.1 直觉模糊集
定义1[22-23] 设
$U = \{ {x_1},{x_2}, \cdots ,{x_n}\} $ 是非空有限论域，称$A = \left\{ {\left\langle {{\mu _A}\left( x \right),{\nu _A}\left( x \right)} \right\rangle \left| {x \in U} \right.} \right\}$ 为$U$ 上的直觉模糊集合，其中${\mu }_{A}:U\to \left[0,1\right]$ ，${\nu }_{A}:U\to \left[0,1\right]$ 分别为$U$ 中元素$ a $ 关于$A$ 的隶属度和非隶属度，且对于任意$ x \in U $ 都满足$0 \leqslant {\mu _A}\left( x \right) + {\nu _A}\left( x \right) \leqslant 1$ 。称${\pi _A}\left( x \right) = 1 - {\mu _A}\left( x \right) - $ $ {\nu _A}\left( x \right)$ 为$ x $ 关于$A$ 的犹豫度或不确定度。如果对于任意$ x \in U $ 都有${\pi _A}\left( x \right) = 0$ ，则直觉模糊集合退化成模糊集合。$ U $ 上全体直觉模糊集记为${\rm{IFS}}\left( U \right)$ 。定义2[22-23] 对于任意
$A,B \in {\rm{IFS}}\left( U \right)$ ，有：1)
$A\subseteq B \iff \forall x\in U,{\mu }_{A}\left(x\right)\leqslant {\mu }_{B}\left(x\right)$ 且${\nu }_{A}\left(x\right) \geqslant {\nu }_{B}\left(x\right)$ ；2)
$A=B \iff \forall x\in U,{\mu }_{A}\left(x\right)={\mu }_{B}\left(x\right)且{\nu }_{A}\left(x\right)={\nu }_{B}\left(x\right)；$ 3)
$\sim A=\left\{\langle {\nu }_{A}\left(x\right),{\mu }_{A}\left(x\right)\rangle |x\in U\right\}；$ 4)
$ A\cap B\iff \left\{\langle {\mu }_{A}\left(x\right)\wedge {\mu }_{B}\left(x\right),{\nu }_{A}\left(x\right)\vee {\nu }_{B}\left(x\right)\rangle |x\in U\right\} $ ；5)
$ A\cup B\iff \left\{\langle {\mu }_{A}\left(x\right)\vee {\mu }_{B}\left(x\right),{\nu }_{A}\left(x\right)\wedge {\nu }_{B}\left(x\right)\rangle |x\in U\right\} $ ；6)
$A\oplus B\text{=}\{\langle {\mu }_{A}\left(x\right)+{\mu }_{B}\left(x\right)-{\mu }_{A}\left(x\right){\mu }_{B}\left(x\right),{\nu }_{A}\left(x\right){\nu }_{B}\left(x\right)\rangle $ $ |x\in U\}$ ；7)
$A\otimes B\text{=}\{\langle {\mu }_{A}\left(x\right){\mu }_{B}\left(x\right),{\nu }_{A}\left(x\right)+{\nu }_{B}\left(x\right)-{\nu }_{A}\left(x\right){\nu }_{B}\left(x\right)\rangle $ $ |x\in U\}$ 。设
$U$ 、$V $ 是两个非空有限论域，称${\rm{IR}}:U \times V \to $ $ \left[ {0,1} \right] \times \left[ {0,1} \right]$ 是$U$ 到$V$ 上的直觉模糊二元关系。对于任意$x \in U$ ，$\operatorname{IR}(x)=\left\{\left\langle\mu_{\mathrm{IR}}\left(x, y_{1}\right), \quad v_{\mathrm{IR}}\left(x, y_{1}\right)\right\rangle,\left\langle\mu_{\mathrm{IR}}\left(x, y_{2}\right)\right.\right., $ $\left.v_{\mathrm{IR}}\left(x, y_{2}\right)\right\rangle,$ $\left. {\left. { \cdots ,\left\langle {{\mu _{\rm{IR}}}\left( {x,{y_m}} \right),{\nu _{\rm{IR}}}\left( {x,{y_m}} \right)} \right\rangle } \right|{y_j} \in V,j = 1,2, \cdots ,m} \right\}$ 表示
$V$ 上的直觉模糊集合。从粒的角度来看，${\rm{IR}} \left( x \right)$ 可以看成对象$x$ 的一个直觉模糊粒，而$\left\{ {\left. {{\rm{IR}}\left( x \right)} \right|x \in U} \right\}$ 可看成$U$ 上的一个直觉模糊粒结构。1.2 粗糙集与概率粗糙集
设
$U$ 是一个非空有限论域，$ R $ 是$U \times U$ 上的一个等价关系，则称$ \left( {U,R} \right) $ 为一个近似空间。于是R产生了$U$ 上的一个划分${U \mathord{\left/ {\vphantom {U R}} \right. } R} = \left\{ {\left. {{{\left[ {{x_i}} \right]}_R}} \right|{x_i} \in U} \right\}$ ，${\left[ {{x_i}} \right]_R}$ 称为含$ {x_i} $ 的等价类。$\forall X \subseteq U$ ，$X$ 关于$ R $ 的下近似和上近似分别定义为$\underline R \left( X \right) = \left\{ {x \in U\left| {{{\left[ x \right]}_R} \subseteq X} \right.} \right\}$ ，$\bar{R}(X)=\{x \in U \mid$ $\left.[x]_{R} \cap X \neq \varnothing\right\}$ 。称序对$\left( {\underline R \left( X \right),\overline R \left( X \right)} \right)$ 为$X$ 关于等价关系$ R $ 的粗糙集[1]。设
$U$ 是一个非空有限论域，$ R $ 是$U \times U$ 上的一个等价关系，$ P $ 是定义在由$U$ 的子集构成的$\sigma $ 代数上的概率测度，则称$\left( {U,R,P} \right)$ 为一个概率近似空间。$\forall X \subseteq U$ ，$0 \leqslant \beta < \alpha \leqslant 1$ ，$X$ 关于近似空间$\left( {U,R,P} \right)$ 及阈值$ \alpha$ 、$ \beta $ 的概率粗糙下近似和上近似分别定义为$$ {\underline R _\alpha }\left( Y \right) = \left\{ {x \in U\left| {P\left( {X\left| {{{\left[ x \right]}_R}} \right.} \right) \geqslant \alpha } \right.} \right\} $$ $$ {\overline R _\beta }\left( X \right) = \left\{ {x \in U\left| {P\left( {X\left| {{{\left[ x \right]}_R}} \right.} \right) > \beta } \right.} \right\} $$ 其中
$P\left( {X\left| {{{\left[ x \right]}_R}} \right.} \right)$ 表示${\left[ x \right]_R}$ 中元素属于$X$ 的概率。称序对$\left( {{{\underline R }_\alpha }\left( X \right),{{\overline R }_\beta }\left( X \right)} \right)$ 为$X$ 关于$\left( {U,R,P} \right)$ 及阈值$ \alpha$ 、$ \beta $ 的概率粗糙集[8]。2. 双论域上的模糊概率粗糙集与直觉模糊概率粗糙集
定义3[18] 设
$U$ 、$V $ 是两个非空有限论域，称$R:U \times $ $ V \to \left[ {0,1} \right]$ 是$U$ 到$V$ 上的模糊二元关系。$ \forall \lambda \in \left[ {0,1} \right] $ ，称$ {R_\lambda } $ 为$ R $ 的$ \lambda $ 截关系，即${R_\lambda } = \left\{ {\left( {x,y} \right) \in U \times V\left| {R\left( {x,y} \right) \geqslant \lambda } \right.} \right\}$ 。$\forall x \in U$ ，令${R_\lambda }\left( x \right) = \left\{ {y \in V\left| {R\left( {x,y} \right) \geqslant \lambda } \right.} \right\}$ 。定义4[18] 设
$U$ 、$V $ 是两个非空有限论域，R是U到V上的模糊二元关系。P是定义在由V的子集构成的$\sigma $ 代数上的概率测度。称$\left( {U,V,R,P} \right)$ 为U到V上的模糊概率近似空间。$ \forall \lambda \in \left[ {0,1} \right] $ ，$ 0 \leqslant \beta < \alpha \leqslant 1 $ ，$ Y \subseteq V $ ，则$ Y $ 关于$\left( {U,V,R,P} \right)$ 及阈值$ \lambda $ 、$\alpha $ 、$\beta $ 的下、上近似分别为$$ {\underline R _{\left( {\lambda ,\alpha } \right)}}\left( Y \right) = \left\{ {x \in U\left| {P\left( {Y\left| {{R_\lambda }\left( x \right)} \right.} \right) \geqslant \alpha ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {R_\lambda }\left( x \right) \ne \varnothing } \right.} \right\} $$ $$\begin{aligned} {\overline R _{\left( {\lambda ,\beta } \right)}}\left( Y \right) =& \left\{ {x \in U\left| {P\left( {Y\left| {{R_\lambda }\left( x \right)} \right.} \right) > \beta ,{R_\lambda }\left( x \right) \ne \varnothing } \right.} \right\} \cup \\ &\left\{ {x \in U\left| {{R_\lambda }\left( x \right) = \varnothing } \right.} \right\} \end{aligned}$$ 称序对
$ \left( {{{\underline R }_{\left( {\lambda ,\alpha } \right)}}\left( Y \right),{{\overline R }_{\left( {\lambda ,\beta } \right)}}\left( Y \right)} \right) $ 为$ Y $ 关于$ \left( {U,V,R,P} \right) $ 及阈值$ \lambda$ 、$ \alpha $ 、$ \beta $ 的模糊概率粗糙集。根据上下近似，很容易计算
$ Y $ 关于$\left( {U,V,R,P} \right)$ 及阈值$ \lambda$ 、$ \alpha $ 、$\beta $ 的模糊概率正域、负域以及边界域，即$$\begin{aligned} {{\rm{POS}}_{{R_{\left( {\lambda ,\alpha } \right)}}}}\left( Y \right) = {\underline R _{\left( {\lambda ,\alpha } \right)}}\left( Y \right) = \quad\quad\quad\\ \left\{ {x \in U\left| {P\left( {Y\left| {{R_\lambda }\left( x \right)} \right.} \right) \geqslant \alpha , {R_\lambda }\left( x \right) \ne \varnothing } \right.} \right\}\end{aligned}$$ $$\begin{aligned} {{\rm{NEG}}_{{R_{\left( {\lambda ,\beta } \right)}}}}\left( Y \right) = U - {\overline R _{\left( {\lambda ,\beta } \right)}}\left( Y \right) =\quad\quad\\ \left\{ {x \in U\left| {P\left( {Y\left| {{R_\lambda }\left( x \right)} \right.} \right) \leqslant \beta ,{R_\lambda }\left( x \right) \ne \varnothing } \right.} \right\} \end{aligned}$$ $$\begin{aligned} {{\rm{BN}}_{{R_{\left( {\lambda ,\alpha ,\beta } \right)}}}}\left( Y \right) = {\overline R _{\left( {\lambda ,\alpha } \right)}}\left( Y \right) - {\underline R _{\left( {\lambda ,\alpha } \right)}}\left( Y \right) =\quad\\ \left\{ {x \in U\left| {\beta < P\left( {Y\left| {{R_\lambda }\left( x \right)} \right.} \right) < \alpha , {R_\lambda }\left( x \right) \ne \varnothing } \right.} \right\} \cup \\ \left\{x\in U|{R}_{\lambda }\left(x\right)=\varnothing \right\}\quad\quad\quad\quad \end{aligned}$$ 若
${\underline R _{\left( {\lambda ,\alpha } \right)}}\left( Y \right) = {\overline R _{\left( {\lambda ,\beta } \right)}}\left( Y \right)$ ，则称$ Y $ 是关于$\left( {U,V,R,P} \right)$ 及阈值$\lambda$ 、$ \alpha $ 、$\beta $ 的可定义集。否则，$ Y $ 是关于$\left( {U,V,R,P} \right)$ 及阈值$\lambda $ 、$\alpha $ 、$\beta $ 的不可定义集。如果
$P\left( {Y\left| {{R_\lambda }\left( x \right)} \right.} \right) = {{\left| {Y \cap {R_\lambda }\left( x \right)} \right|} \mathord{\left/ {\vphantom {{\left| {Y \cap {R_\lambda }\left( x \right)} \right|} {\left| {{R_\lambda }\left( x \right)} \right|}}} \right. } {\left| {{R_\lambda }\left( x \right)} \right|}}$ ，其中$\left| {{R_\lambda }\left( x \right)} \right|$ 表示集合的${R_\lambda }\left( x \right)$ 基数，R是$U$ 到$V$ 上的经典等价二元关系，令$\alpha = 1,\;\beta = 0$ ，则定义4中模糊概率粗糙集将退化为经典粗糙集。定义5[24] 设
$U$ 、$V $ 是两个非空有限论域，称IR是$U$ 到$V$ 上的直觉模糊二元关系。$\forall {\lambda _1},{\lambda _2} \in \left[ {0,1} \right]$ ，称${{\rm{IR}} _{\left( {{\lambda _1},{\lambda _2}} \right)}}$ 为$\rm {IR}$ 关于$\left( {{\lambda _1},{\lambda _2}} \right)$ 的截关系，即${{\rm{IR}} _{\left( {{\lambda _1},{\lambda _2}} \right)}} = \{ \left( {x,y} \right) \in $ $ U \times V\left| {{\mu_{{\rm{IR}} }}\left( {x,y} \right) \geqslant {\lambda _1},{\nu_{{\rm{IR}} }}\left( {x,y} \right) \leqslant {\lambda _2}} \right. \}$ 。$\forall x \in U$ ，${{\rm{IR}} _{\left( {{\lambda _1},{\lambda _2}} \right)}}\left( x \right)$ 为对象$ x $ 的截关系类，其中$\operatorname{IR}_{\left(\lambda_{1}, \lambda_{2}\right)}(x)=\left\{y \in V \mid \mu_{\mathrm{IR}}(x, y) \geqslant\right.$ $\left.\lambda_{1}, v_{\mathrm{IR}}(x, y) \leqslant \lambda_{2}\right\}$ 。定义6[24] 设
$U$ 、$V $ 是两个非空有限论域，称IR是$U$ 到$V$ 上的直觉模糊二元关系。$ P $ 是定义在由$V$ 的子集构成的$\sigma $ 代数上的概率测度。称$( U, $ $ V,{\rm{IR}},P )$ 为$U$ 到$V$ 上的直觉模糊概率近似空间。$\forall {\lambda _1}, $ $ {\lambda _2} \in \left[ {0,1} \right]$ ，$0 \leqslant \beta < \alpha \leqslant 1$ ，$ Y \subseteq V $ ，则$ Y $ 关于$\left( {U,V,{\rm{IR}},P} \right)$ 及阈值${\lambda _1} $ 、${\lambda _2}$ 、$\alpha $ 、$\beta $ 的下、上近似分别为$$ {\underline {{\rm{IR}} } _{\left( {{\lambda _1},{\lambda _2},\alpha } \right)}}\left(Y\right)=\left\{x\in U|P\left(Y|{\rm{IR}}_{\left({\lambda }_{1},{\lambda }_{2}\right)}\left(x\right)\right) \geqslant \alpha ,{\rm{IR}}_{\left({\lambda }_{1},{\lambda }_{2}\right)}\left(x\right)\ne \varnothing \right\} $$ $$\begin{aligned} {\overline {{\rm{IR}} } _{\left( {{\lambda _1},{\lambda _2},\beta } \right)}}\left( Y \right) = \left\{ {x \in U| {P\left( {Y| {{{\rm{IR} }_{\left( {{\lambda _1},{\lambda _2}} \right)}}\left( x \right)} } \right) > \beta , {{\rm{IR} }_{\left( {{\lambda _1},{\lambda _2}} \right)}}\left( x \right) \ne \varnothing } } \right\} \cup\\ \left\{x\in U|{\rm{IR}}_{\left({\lambda }_{1},{\lambda }_{2}\right)}\left(x\right)=\varnothing \right\}\quad\quad\quad\quad\quad\quad\quad \end{aligned}$$ 称序对
$\left( {{{\underline {\rm{IR}} }_{\left( {{\lambda _1},{\lambda _2},\alpha } \right)}}\left( Y \right),{{\overline {{\rm{IR}}} }_{\left( {{\lambda _1},{\lambda _2},\beta } \right)}}\left( Y \right)} \right)$ 为$ Y $ 关于$( U,V, {\rm{IR}}, $ $ P )$ 及阈值${\lambda _1} $ 、${\lambda _2} $ 、$\alpha $ 、$ \beta $ 的直觉模糊概率粗糙集。根据上、下近似，很容易计算
$ Y $ 关于$( U,V,{\rm{IR}}, $ $ P )$ 及阈值${\lambda _1} $ 、${\lambda _2} $ 、$\alpha$ 、$ \beta $ 的直觉模糊概率正域、负域、以及边界域，即$$\begin{aligned} {{\rm{POS}}_{{{\rm{IR} }_{\left( {{\lambda _1},{\lambda _2},\alpha } \right)}}}}\left( Y \right) = {\underline {\rm{IR} } _{\left( {{\lambda _1},{\lambda _2},\alpha } \right)}}\left( Y \right)=\quad\quad\quad \\ \left\{x\in U|P\left(Y|{\rm{IR}}_{\left({\lambda }_{1},{\lambda }_{2}\right)}\left(x\right)\right)\geqslant \alpha ,{\rm{IR}}_{\left({\lambda }_{1},{\lambda }_{2}\right)}\left(x\right)\ne \varnothing \right\}\end{aligned}$$ $$ \begin{aligned}{{\rm{NEG}}_{{{\rm{IR} }_{\left( {{\lambda _1},{\lambda _2},\beta } \right)}}}}\left( Y \right) = U - {\overline {\rm{IR} } _{\left( {{\lambda _1},{\lambda _2},\beta } \right)}}\left( Y \right)=\quad\quad\\ \left\{x\in U|P\left(Y|{\rm{IR}}_{\left({\lambda }_{1},{\lambda }_{2}\right)}\left(x\right)\right)\leqslant \beta ,{\rm{IR}}_{\left({\lambda }_{1},{\lambda }_{2}\right)}\left(x\right)\ne \varnothing \right\}\end{aligned}$$ $$\begin{aligned} {{\rm{BN}}_{{{\rm{IR}} _{\left( {{\lambda _1},{\lambda _2},\alpha ,\beta } \right)}}}}\left( Y \right) = {\overline {\rm{IR} } _{\left( {{\lambda _1},{\lambda _2},\beta } \right)}}\left( Y \right) - {\underline {\rm{IR} } _{\left( {{\lambda _1},{\lambda _2},\alpha } \right)}}\left( Y \right)= \quad\\ \left\{ {x \in U\left| {\beta < P\left( {Y\left| {{{\rm{IR} }_{\left( {{\lambda _1},{\lambda _2}} \right)}}\left( x \right)} \right.} \right) < \alpha ,{{\rm{IR} }_{\left( {{\lambda _1},{\lambda _2}} \right)}}\left( x \right) \ne \varnothing } \right.} \right\}\cup \\ \left\{x\in U|{\rm{IR}}_{\left({\lambda }_{1},{\lambda }_{2}\right)}\left(x\right)=\varnothing \right\}\quad\quad\quad\quad\quad\quad\end{aligned}$$ 若
${\underline {\rm{IR}} _{\left( {{\lambda _1},{\lambda _2},\alpha } \right)}}\left( Y \right) = {\overline {{\rm{IR}}} _{\left( {{\lambda _1},{\lambda _2},\beta } \right)}}\left( Y \right)$ ，则称$ Y $ 是关于$( U,V, $ $ {\rm{IR}},P )$ 及阈值${\lambda _1} $ 、${\lambda _2}$ 、$\alpha $ 、$ \beta $ 的可定义集。否则，$ Y $ 是关于$\left( {U,V,{\rm{IR}},P} \right)$ 及阈值${\lambda _1} $ 、${\lambda _2}$ 、$\alpha $ 、$\beta $ 的不可定义集。虽然上述直觉模糊概率粗糙集采用了直觉模糊关系，但真正起作用的是直觉模糊关系的
$\left( {{\lambda _1},{\lambda _2}} \right)$ 截关系，即将直觉模糊关系转化成经典二元关系后的经典概率粗糙集模型。若$Y \in {\rm{IFS}}\left( V \right)$ 为$V$ 上的直觉模糊集合，条件概率公式$ P\left(Y|I{R}_{\left({\lambda }_{1}，{\lambda }_{2}\right)}\left(x\right)\right) $ 将无法计算，直觉模糊集合$ Y $ 的概率粗糙下、上近似也将无法计算。其次，令$V = \{ {y_1},{y_2},{y_3},{y_4}\} $ ，${\rm{IR}} \in {\rm{IFS}}\left( V \right)$ ，其中${\rm{IR}} = \left\{ \left\langle {0.7,0.21} \right\rangle ,\left\langle {0.8,0.1} \right\rangle ,\left\langle {0.69,0.2} \right\rangle ,\left\langle {0.71,0.21} \right\rangle \right\}$ 。若取$ \left( {{\lambda _1},{\lambda _2}} \right) = \left( {0.7,0.2} \right) $ ，则根据定义5有${{\rm{IR}} _{\left( {0.7,0.2} \right)}} = \left\{ {{y_2}} \right\}$ 。若取$ \left( {{\lambda _1},{\lambda _2}} \right) = \left( {0.7,0.21} \right) $ ，则${{\rm{IR}} _{\left( {0.7,0.21} \right)}} = \left\{ {{y_1},{y_2},{y_4}} \right\}$ 。隶属度和非隶属的极小改变，会导致截关系的结果出现很大差异，即直觉模糊关系的$\left( {{\lambda _1},{\lambda _2}} \right)$ 截关系不具有鲁棒性。文献[24]中指出，同一疾病的不同症状表现强度是不同的，因此选取一组$\left( {{\lambda _1},{\lambda _2}} \right)$ 参数所得的结果也不近合理。为了解决这一问题，下面我们给出一种新的双论域上的广义直觉模糊概率粗糙集模型。
3. 双论域上的广义直觉模糊概率粗糙集
一个直觉模糊集合可以看作一个直觉模糊事件，根据文献[25]可以定义直觉模糊事件的概率。
定义7[25] 设
$U$ 是非空有限论域，$ P $ 是定义在由$U$ 的子集构成的$\sigma $ 代数上的概率测度。对于任意直觉模糊集（直觉模糊事件）$A \in {\rm{IFS}}\left( U \right)$ ，$A$ 的直觉模糊概率${P^*}\left( A \right)$ 定义如下：$${P}^{*}\left(A\right)={\displaystyle\sum_{x\in U}\left({\left(1-{\nu }_{A}\left(x\right)\right)}^{2}-{\left(1-{\mu }_{A}\left(x\right)-{\nu }_{A}\left(x\right)\right)}^{2}\right)}P\left(x\right)$$ 显然
$0 \leqslant {P^*}\left( A \right) \leqslant 1$ ，且对于任意$A,B \in {\rm{IFS}}\left( U \right)$ ，若$A \subseteq B$ ，则${P^*}\left( A \right) \leqslant {P^*}\left( B \right)$ 。若$\forall x \in U$ ，$P\left( x \right){\text{ = }}{{\text{1}} \mathord{\left/ {\vphantom {{\text{1}} {\left| U \right|}}} \right. } {\left| U \right|}}$ ，当$A$ 是一个经典集合时，则${P^*}\left( A \right){\text{ = }}P\left( A \right){\text{ = }}{{\left| A \right|} \mathord{\left/ {\vphantom {{\left| A \right|} {\left| U \right|}}} \right. } {\left| U \right|}}$ 。例1 设
$U = \{ {x_1},{x_2},{x_3},{x_4}\} $ ，若对于任意$x \in U$ ，$P\left( x \right) = {{\text{1}} \mathord{\left/ {\vphantom {{\text{1}} {\left| U \right|}}} \right. } {\left| U \right|}}$ 。$A$ 是$U$ 上的直觉模糊集合且$A=\{\langle 0.7,0.1\rangle,\langle 0.2,0.6\rangle,\langle 0.7,0.0\rangle,\langle 0.1,0.8\rangle\}$ ，则根据定义7计算可知：$$\begin{aligned} {P^*}\left( A \right)&= \frac{1}{4}\sum_{i = 1}^4 {\left[ {{{\left( {1 - {\nu _A}\left( {{x_i}} \right)} \right)}^2} - {{\left( {1 - {\mu _A}\left( {{x_i}} \right) - {\nu _A}\left( {{x_i}} \right)} \right)}^2}} \right]}\\ &\quad\quad\quad\quad\quad\quad\quad= 0.46\end{aligned}$$ 如果两个直觉模糊事件A、
$B \in {\rm{IFS}}\left( U \right)$ 是相互独立的，则直觉模糊条件概率定义如定义8。定义8 设
$U$ 是非空有限论域，P是定义在由$U$ 的子集构成的$\sigma $ 代数上的概率测度。对于任意直觉模糊集（直觉模糊事件）A、$B \in {\rm{IFS}}\left( U \right)$ ，若${P^*}\left( B \right) \ne 0$ ，$A$ 关于$B$ 的直觉模糊条件概率${P^*}\left( {A\left| B \right.} \right)$ 定义为$$ {P^*}\left( {A\left| B \right.} \right) = \frac{{{P^*}\left( {A \otimes B} \right)}}{{{P^*}\left( B \right)}} $$ 例2 设
$A,B \in {\rm{IFS}}\left( U \right)$ ，其中$A=\{\langle 0.7,0.1\rangle,\langle 0.2,0.6\rangle,\langle 0.7,0.0\rangle,\langle 0.1,0.8\rangle\}$ ，$B=\{\langle 0.8,0.1\rangle,\langle 0.6,0.1\rangle,\langle 0.3,0.6\rangle,\langle 0.3,0.6\rangle\}$ 若$\;\forall x \in U,\;P\left( x \right){\text{ = }}{{\text{1}} / {\vphantom {{\text{1}} {\left| U \right|}} } {\left| U \right|}}$ ，则$$ {P^*}\left( {A\left| B \right.} \right) = \frac{{{P^*}\left( {A \otimes B} \right)}}{{{P^*}\left( B \right)}} = \frac{{0.204}}{{0.458}} = 0.45 $$ 如果
$A$ 和$B$ 退化成经典集合，那么上述直觉模糊条件概率则退化成经典集合的条件概率，即$$ {P^*}\left( {A\left| B \right.} \right) = P\left( {A\left| B \right.} \right){\text{ = }}\frac{{P\left( {A \cap B} \right)}}{{P\left( B \right)}} $$ 性质1
$\forall A,B,C \in {\rm{IFS}}\left( U \right)$ ，如果${P^*}\left( A \right) \ne 0$ ，则：1）
${P^*}\left( {\varnothing\left| A \right.} \right) = 0$ ，${P^*}\left( {U\left| A \right.} \right) = 1$ ；2）如果
$B \subseteq C$ ，则${P^*}\left( {B\left| A \right.} \right) \leqslant {P^*}\left( {C\left| A \right.} \right)$ 。证明 根据定义2、7，结论1)和2)显然成立。
定义9 设
$\left( {U,V,{\rm{IR}},P} \right)$ 为$U$ 到$V$ 上的直觉模糊概率近似空间。$A \in {\rm{IFS}}\left( V \right)$ ，令$0 \leqslant \beta < \alpha \leqslant 1$ ，则$A$ 关于$\left( {U,V,{\rm{IR}},P} \right)$ 及阈值$ \alpha $ 、$\beta $ 的下、上近似分别为$$ {\underline {\rm{IR}} _\alpha }\left( A \right) = \left\{ {x \in U\left| {{P^*}\left( {A\left| {{\rm{IR}}\left( x \right)} \right.} \right) \geqslant \alpha } \right.} \right\} $$ $$ {\overline {\rm{IR}} _\beta }\left( A \right) = \left\{ {x \in U\left| {{P^*}\left( {A\left| {{\rm{IR}}\left( x \right)} \right.} \right) > \beta } \right.} \right\} $$ 称序对
$\left( {{{\underline {\rm{IR}} }_\alpha }\left( A \right),{{\overline {\rm{IR}} }_\beta }\left( A \right)} \right)$ 为$A$ 关于$\left( {U,V,{\rm{IR}},P} \right)$ 的$\left( {\alpha ,\beta } \right)$ –广义直觉模糊概率粗糙集。根据上、下近似，可以计算A关于
$( U,V, {\rm{IR}}, $ $ P)$ 及阈值$\alpha $ 、$ \beta $ 的直觉模糊概率正域、负域，以及边界域，即$$ {{\rm{POS}}_{{{\rm{IR}}_\alpha }}}\left( A \right) = {\underline {\rm{IR}} _\alpha }\left( A \right) = \left\{ {\left. {x \in U} \right|P\left( {A\left| {{\rm{IR}}\left( x \right)} \right.} \right) \geqslant \alpha } \right\} $$ $$ {\rm{NEG}}_{{\rm{IR}}_{\beta }}\left(A\right)=U-{\overline{\rm{IR}}}_{\beta }\left(A\right)=\left\{x\in U|{P}^{*}\left(A|\rm{IR}\left(x\right)\right) \leqslant \beta \right\} $$ $$\begin{aligned}{{\rm{BN}}_{{{\rm{IR} }_{\left( {\alpha ,\beta } \right)}}}}\left( A \right) = {\overline {\rm{IR} } _\beta }\left( A \right) - {\underline {\rm{IR} } _\alpha }\left( A \right) =\\ \left\{x\in U|\beta < P\left(A|\mathrm{IR}\left(x\right)\right) < \alpha \right\}\quad \end{aligned}$$ 若
${\underline {\rm{IR}} _\alpha }\left( A \right) = {\overline {\rm{IR}} _\beta }\left( A \right)$ ，则称A是关于$\left( {U,V,{\rm{IR}},P} \right)$ 及阈值$\alpha $ 、$\beta $ 的可定义集。否则，$A$ 是关于$\left( {U,V,{\rm{IR}},P} \right)$ 及阈值$ \alpha$ 、$ \beta $ 的不可定义集。如果直觉模糊关系
${\rm{IR}}$ 和直觉模糊集$A$ 分别退化成模糊关系和模糊集合，则$ \left( {\alpha ,\beta } \right) $ –广义直觉模糊概率粗糙集退化成文献[21]中$ \left( {\alpha ,\beta } \right)$ –模糊概率粗糙集；如果直觉模糊关系${\rm{IR}}$ 和直觉模糊集$A$ 分别退化成经典等价关系和经典集合，则$ \left( {\alpha ,\beta } \right) $ –广义直觉模糊概率粗糙集退化成文献[8]中$ \left( {\alpha ,\beta } \right) $ –概率粗糙集。性质2 设
$\left( {U,V,{\rm{IR}},P} \right)$ 为$U$ 到$V$ 上的直觉模糊概率近似空间。对于$A,B \in {\rm{IFS}}\left( V \right)$ ，$ 0 \leqslant \beta < \alpha \leqslant 1 $ ，有下列结论成立：1)
${\underline {\rm{IR} } _\alpha }\left( A \right) \subseteq {\overline {\rm{IR} } _\beta }\left( A \right)$ ；2)
${\underline {\rm{IR} } _\alpha }(\varnothing )={\overline{\rm{IR}}}_{\alpha }(\varnothing )=\varnothing ，\underline {\rm{IR} }_{\beta }\left(V\right)={\overline{\rm{IR}}}_{\beta }\left(V\right)=U$ ；3)若
$A \subseteq B$ ，则${\underline {\rm{IR}} _\alpha }\left( A \right) \subseteq {\underline {\rm{IR}} _\alpha }\left( B \right)$ ，${\overline {\rm{IR}} _\beta }\left( A \right) \subseteq {\overline {\rm{IR}} _\beta }\left( B \right)$ ；4)
${\underline {\rm{IR} } _\alpha }\left( {A \cap B} \right) \;\subseteq\; {\underline {\rm{IR} } _\alpha }\left( A \right) \;\cap\; {\underline {\rm{IR} } _\alpha }\left( B \right)$ ，${\overline {\rm{IR} } _\beta }\left( {A \cap B} \right) \subseteq $ $ {\overline {\rm{IR} } _\beta }\left( A \right) \cap {\overline {\rm{IR} } _\beta }\left( B \right)$ ；5)
${\underline {\rm{IR} } _\alpha }\left( {A \cup B} \right) \;\supseteq\; {\underline {\rm{IR} } _\alpha }\left( A \right)\; \cup \;{\underline {\rm{IR} } _\alpha }\left( B \right)$ ，${\overline {\rm{IR} } _\beta }\left( {A \cup B} \right)\; \supseteq $ $ {\overline {\rm{IR} } _\beta }\left( A \right) \cup {\overline {\rm{IR} } _\beta }\left( B \right)$ ；6)若
$0 \leqslant {\beta _1} \leqslant {\beta _2} < {\alpha _1} \leqslant {\alpha _2} \leqslant 1$ ，则${\underline{\rm{IR}}}_{{\alpha }_{2}}\left(A\right)\subseteq {\underline{\rm{IR}}}_{{\alpha }_{1}}\left(A\right)， $ $ {\overline{\rm{IR}}}_{{\beta }_{2}}\left(A\right)\subseteq {\overline{\rm{IR}}}_{{\beta }_{1}}\left(A\right)$ 。证明 根据定义8、9易证结论1）、2）。下面来证明结论3）~6）。
结论3)证明。若
$A \subseteq B$ ，由定义2可知$\forall y \in V$ ，${\mu }_{A}\left(y\right)\leqslant {\mu }_{B}\left(y\right) \;\; 且\;\;{\nu }_{A}\left(y\right)\geqslant {\nu }_{B}\left(y\right)$ 。而$A \otimes {\rm{IR}}\left( x \right) =$ $\{ \left\langle {\mu _A}\left( y \right) {\mu _{\rm{IR}}}\left( {x,y} \right),\right.$ ${\nu _A}\left( y \right) + \left.{\nu _{\rm{IR}}}\left( {x,y} \right) - {\nu _A}\left( y \right){\nu _{\rm{IR}}}\left( {x,y} \right) \right\rangle \left| {y \in V} \right. \}$ $\leqslant \{ \left\langle {\mu _B}\!\left( y \right) {\mu _{\rm{IR}}}\!\left( {x,y} \right), \!{\nu _B}\left( y \right) \!+\! {\nu _{\rm{IR}}}\left( {x,y} \right) \!-\! {\nu _B}\left( y \right){\nu _{\rm{IR}}}\left( {x,y} \right) \right\rangle \left| {y \!\in\! V\!} \right. \}$ $= B \otimes {\rm{IR}}\left( x \right)$ ，从而${P^*}( A \otimes {\rm{IR}} \left( x \right) ) \leqslant {P^*}\left( {B \otimes {\rm{IR}} \left( x \right)} \right)$ 。$\forall x \in {\underline {\rm{IR}} _\alpha }\left( A \right)$ ，根据定义9有，${P^*}\left( {B\left| {{\rm{IR}}\left( x \right)} \right.} \right) = \dfrac{{{P^*}\left( {B \otimes {\rm{IR}}\left( x \right)} \right)}}{{{P^*}\left( {{\rm{IR}}\left( x \right)} \right)}} \geqslant \dfrac{{{P^*}\left( {A \otimes {\rm{IR}}\left( x \right)} \right)}}{{{P^*}\left( {{\rm{IR}}\left( x \right)} \right)}} \geqslant \alpha$ ，则$x \in $ $ {\underline {\rm{IR}} _\alpha }\left( B \right)$ ，由此可知${\underline {\rm{IR}} _\alpha }\left( A \right) \subseteq {\underline {\rm{IR}} _\alpha }\left( B \right)$ 。类似的可以证明
${\overline {\rm{IR}} _\beta }\left( A \right) \subseteq {\overline {\rm{IR}} _\beta }\left( B \right)$ 。结论4)证明。由于
${\kern 1pt} A \cap B \subseteq A$ ，${\kern 1pt} A \cap B \subseteq B$ ，根据结论3)有${\underline {\rm{IR}} _\alpha }\left( {A \cap B} \right)\subseteq {\underline {\rm{IR}} _\alpha }\left( A \right)$ ，${\underline {\rm{IR}} _\alpha }\left( {A \cap B} \right) \subseteq {\underline {\rm{IR}} _\alpha }\left( B \right)$ ，从而${\underline {\rm{IR} } _\alpha }\left( {A \cap B} \right) \subseteq {\underline {\rm{IR} } _\alpha }\left( A \right) \cap {\underline {\rm{IR} } _\alpha }\left( B \right)$ 。类似的可以证明${\overline {\rm{IR}} _\beta }\left( {A \cap B} \right) \subseteq {\overline {\rm{IR}} _\beta }\left( A \right) \cap {\overline {\rm{IR}} _\beta }\left( B \right)$ 。结论5)证明过程和结论4)相同。
结论6)证明。
$\forall x \in {\underline {\rm{IR}} _{{\alpha _2}}}\left( A \right)$ ，根据定义9有，${P^*}\left( A\left| \right.\right. $ $ \left.{\rm{IR}} \left( x \right) \right) = \dfrac{{{P^*}\left( {A \otimes {\rm{IR}} \left( x \right)} \right)}}{{{P^*}\left( {{\rm{IR}} \left( x \right)} \right)}} \geqslant {\alpha _2} \geqslant {\alpha _1}$ ，则$x \;\in \;{\underline {\rm{IR} } _{{\alpha _1}}}\left( A \right)$ ，即${\underline {\rm{IR} } _{{\alpha _2}}}\left( A \right) \;\subseteq\; {\underline {\rm{IR} } _{{\alpha _1}}}\left( A \right)$ 。同理${\overline {\rm{IR} } _{{\beta _2}}}\left( A \right) \subseteq {\overline {\rm{IR} } _{{\beta _1}}}\left( A \right)$ 。若
$\beta = 1 - \alpha$ ，就得到了一种特殊的$\left( {\alpha ,\beta } \right)$ –广义直觉模糊概率粗糙集，这时只需要确定参数$ \alpha $ 一个阈值。定义10 设
$\left( {U,V,{\rm{IR}},P} \right)$ 为$U$ 到$V$ 上的直觉模糊概率近似空间。$\forall A \in {\rm{IFS}}\left( V \right)$ ，令$0.5 < \alpha \leqslant 1$ ，则$A$ 关于$\left( {U,V,{\rm{IR}},P} \right)$ 及阈值$ \alpha $ 的下上近似分别为$$ {\underline {\rm{IR}} _\alpha }\left( A \right) = \left\{ {x \in U\left| {{P^*}\left( {A\left| {{\rm{IR}}\left( x \right)} \right.} \right) \geqslant \alpha } \right.} \right\} $$ $$ {\overline {\rm{IR}} _\alpha }\left( A \right) = \left\{ {x \in U\left| {{P^*}\left( {A\left| {{\rm{IR}}\left( x \right)} \right.} \right) > 1 - \alpha } \right.} \right\} $$ 称序对
$\left( {{{\underline {\rm{IR}} }_\alpha }\left( A \right),{{\overline {\rm{IR}} }_\alpha }\left( A \right)} \right)$ 为$A$ 关于$\left( {U,V,{\rm{IR}},P} \right)$ 的$\alpha - $ 广义直觉模糊概率粗糙集。根据上、下近似，可以计算A关于
$\left( {U,V,{\rm{IR}},P} \right)$ 及阈值$\alpha $ 的直觉模糊概率正域、负域，以及边界域，即$$ {{\rm{POS}}_{{{\rm{IR}}_\alpha }}}\left( A \right) = {\underline {\rm{IR}} _\alpha }\left( A \right) = \left\{ {\left. {x \in U} \right|P\left( {A\left| {{\rm{IR}}\left( x \right)} \right.} \right) \geqslant \alpha } \right\} $$ $$ {{\rm{NEG}}_{{{\rm{IR} }_\alpha }}}\left( A \right) = U - {\overline {\rm{IR} } _\alpha }\left( A \right) =\left\{x\in U|{P}^{*}\left(A|\rm{IR}\left(x\right)\right)\leqslant 1-\alpha \right\}$$ $$\begin{aligned} {{\rm{BN}}_{{{\rm{IR} }_{\left( {\alpha ,\beta } \right)}}}}\left( A \right) = {\overline {\rm{IR} } _\alpha }\left( A \right) - {\underline {\rm{IR} } _\alpha }\left( A \right)=\quad\\ \left\{x\in U|1-\alpha < P\left(A|{\rm{IR}}\left(x\right)\right) < \alpha \right\}\quad\end{aligned} $$ 若
${\underline {\rm{IR}} _\alpha }\left( A \right) = {\overline {\rm{IR}} _\alpha }\left( A \right)$ ，则称$A$ 是关于$( U,V,{\rm{IR}}, P )$ 及阈值$\alpha $ 的可定义集。否则，$A$ 是关于$( U,V,{\rm{IR}}, P)$ 及阈值$\alpha $ 的不可定义集。如果直觉模糊关系
${\rm{IR}}$ 和直觉模糊集$A$ 分别退化成模糊关系和模糊集合，则$\alpha - $ 广义直觉模糊概率粗糙集退化成文献[21]中$\alpha - $ 模糊概率粗糙集；如果直觉模糊关系${\rm{IR}}$ 和直觉模糊集$A$ 分别退化成经典等价关系和经典集合，则$\alpha - $ 广义直觉模糊概率粗糙集退化成经典的$\alpha - $ 概率粗糙集。当只关心那些在一定程度上支持直觉模糊事件的对象时，可以使用
$\alpha - $ 广义直觉模糊概率粗糙集。这时只需要确定一个阈值$\alpha $ 。类似的，也能得到性质3。性质3 设
$\left( {U,V,{\rm{IR}},P} \right)$ 为$U$ 到$V$ 上的直觉模糊概率近似空间。对于$A,B \in {\rm{IFS}}\left( V \right)$ ，$0.5 < \alpha \leqslant 1$ ，有下列结论成立：1)
${\underline {\rm{IR} } _\alpha }\left( A \right) \subseteq {\overline {\rm{IR} } _\alpha }\left( A \right)$ ；2)
${\underline{\mathrm{IR}}}_{\alpha }(\varnothing )={\overline{\mathrm{IR}}}_{\alpha }(\varnothing )=\varnothing ，$ ${\underline {\rm{IR} } _\alpha }\left( V \right) = {\overline {\rm{IR} } _\alpha }\left( V \right) = U$ ；3) 若
$A \subseteq B$ ，则${\underline {\rm{IR} } _\alpha }\left( A \right) \subseteq {\underline {\rm{IR} } _\alpha }\left( B \right)$ ，${\overline {\rm{IR} } _\alpha }\left( A \right) \subseteq {\overline {\rm{IR} } _\alpha }\left( B \right)$ ；4)
${\underline {\rm{IR} } _\alpha }\left( {A \cap B} \right) \;\subseteq\; {\underline {\rm{IR} } _\alpha }\left( A \right) \;\cap\; {\underline {\rm{IR} } _\alpha }\left( B \right)$ ，${\overline {\rm{IR} } _\alpha }\left( {A \cup B} \right) \;\supseteq \; $ $ {\overline {\rm{IR} } _\alpha } \left( A \right) \;\cup \; {\overline {\rm{IR} } _\alpha }\left( B \right)$ ；5)
${\underline {\rm{IR} } _\alpha }\left( {A \cup B} \right) \;\supseteq\; {\underline {\rm{IR} } _\alpha }\left( A \right) \;\cup\; {\underline {\rm{IR} } _\alpha }\left( B \right)$ ，${\overline {\rm{IR} } _\alpha }\left( {A \cap B} \right) \;\subseteq $ $ {\overline {\rm{IR} } _\alpha } \left( A \right) \;\cap\; {\overline {\rm{IR} } _\alpha }\left( B \right)$ ；6) 若
$0.5 \; < \; {\alpha _1} \;\leqslant\; {\alpha _2} \;\leqslant \;1$ ，则${\underline {\rm{IR} } _{{\alpha _2}}}\left( A \right) \;\;\subseteq\;\; {\underline {\rm{IR} } _{{\alpha _1}}}\left( A \right)$ ，${\overline {\rm{IR} } _{{\alpha _2}}} \left( A \right)\;\; \subseteq\;\; {\overline {\rm{IR} } _{{\alpha _1}}}\left( A \right)$ 。根据定义8、10，类似性质2易证性质3，这里就不再重复了。
4. 实例应用
本节将讨论双论域上的广义直觉模糊概率粗糙集在医疗诊断上的应用。
例3 有一个医疗诊断实例，其中
$U = \{ {x_1}, $ $ {x_2}, \cdots ,{x_9}\}$ 为一组患者集合，$V{\text{ = }}\left\{ {{y_1},{y_2}, \cdots ,{y_5}} \right\}$ 为一组症状集合。每个患者$x \in U$ 关于症状$y \in V$ 的隶属度和非隶属度如表1所示。假设
$A$ 为某种疾病，它的临床诊断表现为$A = $ $ \left\{ {\left\langle {0.7,0} \right\rangle ,\left\langle {0.2,0.6} \right\rangle ,\left\langle {0,0.9} \right\rangle ,\left\langle {0.7,0} \right\rangle ,\left\langle {0.1,0.8} \right\rangle } \right\}$ 设症状集合上的概率分布函数为$P\left( y \right) = {1 \mathord{\left/ {\vphantom {1 {\left| V \right|}}} \right. } {\left| V \right|}} $ $ \left( {\forall y \in V} \right)$ ，根据定义8，疾病A关于每个患者${x_i} \in U$ （即信息粒度${\rm{IR}} \left( {{x_i}} \right),i = 1,2, \cdots ,9$ ）的直觉模糊条件概率分别为${P^*}\left( {A\left| {{\rm{IR}} \left( {{x_1}} \right)} \right.} \right) = 0.57,\;{P^*}\left( {A\left| {{\rm{IR}} \left( {{x_2}} \right)} \right.} \right) = 0.06$ ,${P^*}\left( {A\left| {{\rm{IR}} \left( {{x_3}} \right)} \right.} \right) \;=\; $ $ 0.51,\;{P^*}\left( {A\left| {{\rm{IR}} \left( {{x_4}} \right)} \right.} \right)\; =\; 0.52$ ，${P^*}\left( {A\left| {{\rm{IR}} \left( {{x_5}} \right)} \right.} \right) \;=\; 0.80,\;\; {P^*}( A| $ $ {{\rm{IR}} \left( {{x_6}} \right)} ) = 0.74$ ,${P^*}\left( {A\left| {{\rm{IR}} \left( {{x_7}} \right)} \right.} \right) = 0.33,{P^*}\left( {A\left| {{\rm{IR}} \left( {{x_8}} \right)} \right.} \right) = 0.09$ ,${P^*}\left( {A\left| {{\rm{IR}} \left( {{x_9}} \right)} \right.} \right) = 0.28 $ 。表 1 患者与症状之间的直觉模糊关系Table 1 Intuitionistic fuzzy relationship between patients and symptoms$U$ $V$ ${y_1}$ ${y_2}$ ${y_3}$ ${y_4}$ ${y_5}$ ${x_1}$ ${\left\langle {0.8,0.1} \right\rangle } $ ${\left\langle {0.6,0.1} \right\rangle } $ ${\left\langle {0.2,0.8} \right\rangle } $ ${\left\langle {0.6,0.1} \right\rangle } $ ${\left\langle {0.1,0.6} \right\rangle } $ ${x_2}$ ${\left\langle {0.0,0.8} \right\rangle } $ ${\left\langle {0.4,0.4} \right\rangle } $ ${\left\langle {0.6,0.1} \right\rangle } $ ${\left\langle {0.1,0.7} \right\rangle } $ ${\left\langle {0.1,0.8} \right\rangle } $ ${x_3}$ ${\left\langle {0.8,0.1} \right\rangle } $ ${\left\langle {0.8,0.1} \right\rangle } $ ${\left\langle {0.0,0.6} \right\rangle } $ ${\left\langle {0.2,0.7} \right\rangle } $ ${\left\langle {0.0,0.5} \right\rangle } $ ${x_4}$ ${\left\langle {0.6,0.1} \right\rangle } $ ${\left\langle {0.5,0.4} \right\rangle } $ ${\left\langle {0.3,0.4} \right\rangle } $ ${\left\langle {0.7,0.2} \right\rangle } $ ${\left\langle {0.3,0.4} \right\rangle } $ ${x_5}$ ${\left\langle {0.7,0.2} \right\rangle } $ ${\left\langle {0.1,0.7} \right\rangle } $ ${\left\langle {0.1,0.9} \right\rangle } $ 〈 ${\left\langle {0.6,0.1} \right\rangle } $ ${\left\langle {0.0,0.9} \right\rangle } $ ${x_6}$ ${\left\langle {0.6,0.0} \right\rangle } $ ${\left\langle {0.2,0.7} \right\rangle } $ ${\left\langle {0.0,0.8} \right\rangle } $ ${\left\langle {0.6,0.4} \right\rangle } $ ${\left\langle {0.2,0.7} \right\rangle } $ ${x_7}$ ${\left\langle {0.3,0.3} \right\rangle } $ ${\left\langle {0.6,0.1} \right\rangle } $ ${\left\langle {0.2,0.7} \right\rangle } $ ${\left\langle {0.2,0.6} \right\rangle } $ ${\left\langle {0.1,0.9} \right\rangle } $ ${x_8}$ ${\left\langle {0.1,0.7} \right\rangle } $ ${\left\langle {0.2,0.4} \right\rangle } $ ${\left\langle {0.8,0.0} \right\rangle } $ ${\left\langle {0.2,0.7} \right\rangle } $ ${\left\langle {0.2,0.7} \right\rangle } $ ${x_9}$ ${\left\langle {0.1,0.8} \right\rangle } $ ${\left\langle {0.0,0.8} \right\rangle } $ ${\left\langle {0.2,0.8} \right\rangle } $ ${\left\langle {0.4,0.3} \right\rangle } $ ${\left\langle {0.8,0.1} \right\rangle } $ 令
$\alpha = 0.7,\beta = 0.5$ ，根据定义9有如下结果：$$ {{\rm{POS}}_{{{\rm{IR} }_{0.7}}}}\left( A \right) = {\underline {\rm{IR} } _{0.7}}\left( A \right) = \left\{ {{x_5},{x_6}} \right\}$$ $$ {{\rm{NEG}}_{{{\rm{IR} }_{0.5}}}}\left( A \right) = U - {\overline {\rm{IR} } _{0.5}}\left( A \right) = \left\{ {{x_2},{x_7},{x_8},{x_9}} \right\} $$ $$ {\rm{BN}}_{{\rm{IR}}_{\left(0.7,0.5\right)}}\left(A\right)=\left\{{x}_{1},{x}_{3},{x}_{4},{x}_{5},{x}_{6}\right\}-\left\{{x}_{5},{x}_{6}\right\}=\left\{{x}_{1},{x}_{3},{x}_{4}\right\}$$ 因此可以得到以下结论：在给定阈值
$\alpha = 0.7, $ $ \beta = 0.5$ 的情况下，患者${x_5}、{x_6}$ 一定感染了疾病$A$ ，需要立即进行治疗；患者${x_2}、{x_7}、{x_8}、{x_9}$ 一定没有感染疾病A，暂时不需要进行治疗；患者${x_1}、 {x_3}、 $ $ {x_4}$ 可能感染了疾病A，需要进一步检查确诊。若令
$\alpha = 0.55$ ，根据定义10有如下结果：$$ {{\rm{POS}}_{{{\rm{IR} }_{0.55}}}}\left( A \right) = {\underline {\rm{IR} } _{0.55}}\left( A \right) = \left\{ {{x_1},{x_5},{x_6}} \right\} $$ $$ {{\rm{NEG}}_{{{\rm{IR} }_{0.55}}}}\left( A \right) = U - {\overline {\rm{IR} } _{0.55}}\left( A \right) = \left\{ {{x_2},{x_7},{x_8},{x_9}} \right\} $$ $$ {{\rm{BN}}_{{{\rm{IR} }_{0.55}}}}\left( A \right) = \left\{ {{x_1},{x_3},{x_4},{x_5},{x_6}} \right\} - \left\{ {{x_1},{x_5},{x_6}} \right\} = \left\{ {{x_3},{x_4}} \right\} $$ 此时，可以得到以下结论：在给定阈值
$\alpha = 0.55$ 的情况下，患者${x_1}、{x_5}、{x_6}$ 一定感染了疾病$A$ ，需要立即进行治疗；患者${x_2}、{x_7}、{x_8}、{x_9}$ 一定没有感染了疾病${{A}}$ ，暂时不需要进行治疗；患者${x_3}、{x_4}$ 可能感染了疾病${{A}}$ ，需要进一步检查确诊。另外，还考虑下面一个医疗诊断问题，数据来源于文献[26]。
例4 假设有四名患者，记为
$U= \{ {\rm{Al}},{\rm{Bob}}, $ $ {\rm{Joe}},{\rm{Ted}} \}$ ，症状集合$V{\text{ = }}\left\{ {{\text{T,H,S,C,CP}}} \right\}$ ，其中T表示温度，H表示头痛，S表示胃痛，C表示咳嗽，CP表示胸痛。诊断结果集合$D{\text{ = }}\left\{ {{\text{Vf,Ma,Ty,St,Ch}}} \right\}$ ，其中Vf表示病毒性感冒，Ma表示疟疾，Ty表示伤寒，St表示胃病，Ch表示胸肺病。表2为患者与症状之间的直觉模糊关系，表3为疾病与症状之间的直觉模糊关系。下面我们来确定每位患者的诊断结果。表 2 患者与症状之间的直觉模糊关系Table 2 Intuitionistic fuzzy relationship between patients and symptoms患者 T H S C CP Al ${\left\langle {0.8,0.1} \right\rangle } $ ${\left\langle {0.6,0.1} \right\rangle }$ ${\left\langle {0.2,0.8} \right\rangle } $ ${\left\langle {0.6,0.1} \right\rangle } $ ${\left\langle {0.1,0.6} \right\rangle } $ Bob ${\left\langle {0.0,0.8} \right\rangle } $ ${\left\langle {0.4,0.4}\right\rangle } $ ${\left\langle {0.6,0.1} \right\rangle } $ ${\left\langle {0.1,0.7}\right\rangle } $ ${\left\langle {0.1,0.8} \right\rangle } $ Joe ${\left\langle {0.8,0.1} \right\rangle } $ ${\left\langle {0.8,0.1} \right\rangle } $ ${\left\langle {0.0,0.6}\right\rangle } $ ${\left\langle {0.2,0.7}\right\rangle } $ ${\left\langle {0.0,0.5} \right\rangle } $ Ted ${\left\langle {0.6,0.1} \right\rangle } $ ${\left\langle {0.5,0.4} \right\rangle } $ ${\left\langle {0.3,0.4} \right\rangle } $ ${\left\langle {0.7,0.2} \right\rangle } $ ${\left\langle {0.3,0.4} \right\rangle } $ 表 3 疾病与症状之间的直觉模糊关系Table 3 Intuitionistic fuzzy relationship between diseases and symptoms疾病 T H S C CP Vf ${\left\langle {0.4,0.0} \right\rangle } $ ${\left\langle {0.3,0.5}\right\rangle } $ ${\left\langle {0.1,0.7} \right\rangle } $ ${\left\langle {0.4,0.3}\right\rangle } $ ${\left\langle {0.1,0.7}\right\rangle } $ Ma ${\left\langle {0.7,0.0}\right\rangle } $ ${\left\langle {0.2,0.6}\right\rangle } $ ${\left\langle {0.0,0.9}\right\rangle } $ ${\left\langle {0.7,0.0}\right\rangle } $ ${\left\langle {0.1,0.8}\right\rangle } $ Ty ${\left\langle {0.3,0.3}\right\rangle } $ ${\left\langle {0.6,0.1}\right\rangle } $ ${\left\langle {0.2,0.7}\right\rangle } $ ${\left\langle {0.2,0.6}\right\rangle } $ ${\left\langle {0.1,0.9} \right\rangle } $ St ${\left\langle {0.1,0.7} \right\rangle } $ ${\left\langle {0.2,0.4} \right\rangle } $ ${\left\langle {0.8,0.0}\right\rangle } $ ${\left\langle {0.2,0.7}\right\rangle } $ ${\left\langle {0.2,0.7}\right\rangle } $ Ch ${\left\langle {0.1,0.8}\right\rangle } $ ${\left\langle {0.0,0.8} \right\rangle } $ ${\left\langle {0.2,0.8} \right\rangle } $ ${\left\langle {0.2,0.8}\right\rangle } $ ${\left\langle {0.8,0.1} \right\rangle } $ 由于每种疾病的主要表现症状各不相同，根据这一特点，令
$P\left( y \right) = {{{\mu _V}\left( y \right)} \mathord{/ {\vphantom {{{\mu _V}\left( y \right)} {\sum\nolimits_{y \in V} {{\mu _V}\left( y \right)} }}} } {\sum\nolimits_{y \in V} {{\mu _V}\left( y \right)} }}{\kern 1pt} {\kern 1pt}$ 。表2中，用A1 =$ \left\{ {\left\langle {0.8,0.1} \right\rangle ,\left\langle {0.6,0.1} \right\rangle ,\left\langle {0.2,0.8} \right\rangle ,\left\langle {0.6,0.1} \right\rangle ,\left\langle {0.1,0.6} \right\rangle } \right\}$ 表示患者Al在每种症状下的直觉模糊集合；表3中，用Vf =$\left\{ \left\langle {0.4,0.0} \right\rangle ,\left\langle {0.3,0.5} \right\rangle ,\left\langle {0.1,0.7} \right\rangle ,\left\langle {0.4,0.3} \right\rangle ,\right. $ $ \left.\left\langle {0.1,0.7} \right\rangle \right\}$ 表示疾病Vf在每种症状下的直觉模糊集合，其他以此类推。根据定义8，计算诊断结果中病毒性感冒关于每位患者的直觉模糊条件概率：$$ {P^*}\left( {{\text{Vf}}\left| {{\text{Al}}} \right.} \right){\text{ = }}\frac{{{P^*}\left( {{\text{Vf}} \otimes {\text{Al}}} \right)}}{{{P^*}\left( {{\text{Al}}} \right)}}{\text{ = }}0.39 $$ $$ {P^*}\left( {{\text{Vf}}\left| {{\text{Bob}}} \right.} \right){\text{ = }}\frac{{{P^*}\left( {{\text{Vf}} \otimes {\text{Bob}}} \right)}}{{{P^*}\left( {{\text{Bob}}} \right)}}{\text{ = }}0.14 $$ $$ {P^*}\left( {{\text{Vf}}\left| {{\text{Joe}}} \right.} \right){\text{ = }}\frac{{{P^*}\left( {{\text{Vf}} \otimes {\text{Joe}}} \right)}}{{{P^*}\left( {{\text{Joe}}} \right)}}{\text{ = }}0.42 $$ $$ {P^*}\left( {{\text{Vf}}\left| {{\text{Ted}}} \right.} \right){\text{ = }}\frac{{{P^*}\left( {{\text{Vf}} \otimes {\text{Ted}}} \right)}}{{{P^*}\left( {{\text{Ted}}} \right)}}{\text{ = }}0.38 $$ 计算其他疾病关于每位患者的直觉模糊条件概率，结果如表4所示。
表 4 疾病关于患者的直觉模糊条件概率Table 4 Intuitionistic fuzzy conditional probability of diseases about patients患者 Vf Ma Ty St Ch Al 0.39 0.75 0.43 0.17 0.17 Bob 0.14 0.32 0.38 0.79 0.14 Joe 0.42 0.71 0.55 0.13 0.03 Ted 0.38 0.76 0.34 0.40 0.36 设阈值
$\alpha = 0.7,\beta = 0.5$ 。 据定义9可知，Vf关于阈值$\alpha 、\beta$ 的下、上近似为${\underline {\rm{IR}} _{0.7}}\left( {{\text{Vf}}} \right) = \varnothing$ ，${\overline {\rm{IR}} _{0.5}}\left( {{\text{Vf}}} \right) = \varnothing$ 。 根据上、下近似，Vf关于阈值$\alpha 、\beta$ 的直觉模糊概率正域、负域，以及边界域分别为${\rm{POS}}_{0.7}\left(\text{Vf}\right)=\varnothing ， $ $ {\rm{NEG}}_{0.5}\left({\rm{Vf}}\right)={\rm{U}}， {\rm{BN}}_{\left(0.7,0.5\right)}\left({\rm{Vf}}\right)=\varnothing$ 。也就是说，所有患者均都没有患有疾病Vf（病毒性感冒）。类似的，计算其他疾病关于阈值$\alpha、\beta$ 的直觉模糊概率正域、负域，以及边界域如下：${\rm{POS}}_{0.7}\left({\rm{Ma}}\right)=\left\{{\rm{Al}},\rm{Joe,Ted}\right\}$ ，${\rm{NEG}}_{0.5}\left({\rm{Ma}}\right)=\left\{{\rm{Bob}}\right\}$ ，${{\rm{BN}}_{\left( {0.7,0.5} \right)}}\left( {{\rm{Ma}}} \right) = \varnothing$ ；${\rm{POS}}_{0.7}\left({\rm{Ty}}\right)\;=\;\varnothing$ ，${\rm{NEG}}_{0.5}\left({\rm{Ty}}\right)\;=\;\{{\rm{Al}}, $ $ {\rm{Bob}}, {\rm{Ted}}\}$ ，${{\rm{BN}}_{\left( {0.7,0.5} \right)}}\left( {{\rm{Ty}}} \right) = \left\{ {{\rm{Joe}}} \right\}$ ；${\rm{POS}}_{0.7}\left({\rm{St}}\right)\;=\;\left\{{\rm{Bob}}\right\}$ ，${\rm{NEG}}_{0.5}\left({\rm{St}}\right) \;=\;\left\{{\rm{Al}},{\rm{Joe}},{\rm{Ted}}\right\}$ ，${{\rm{BN}}_{\left( {0.7,0.5} \right)}} \left( {{\rm{St}}} \right) = \varnothing$ ；${\rm{POS}}_{0.7}\left({\rm{Ch}}\right) = \varnothing$ ，${\rm{NEG}}_{0.5}\left({\rm{Ch}}\right)={\rm{U}}$ ，${{\rm{BN}}_{\left( {0.7,0.5} \right)}}\left( {{\rm{Ch}}} \right) = \varnothing$ 由上面的计算可知，Al一定患有疾病Ma（疟疾）；Bob一定患有疾病St（胃病）；Joe一定患有疾病Ma（疟疾），可能患有疾病Ty（伤寒）；Ted一定患有疾病Ty（伤寒）。
将计算结果与其他文献进行比较，结果如表5所示。与其他方法不同的是，由于条件概率的加入，根据不同的阈值，患者Joe一定患有疾病Ma（疟疾），可能患有疾病Ty（伤寒）。这样的诊断结果更加符合实际，因为临床症状表现可能不是由单一疾病引起的。因此，需进一步检查，从而对症治疗。
5. 结束语
已有的双论域直觉模糊概率粗糙集模型通过设置两个阈值
${\lambda _1}、{\lambda _2}$ ，讨论了经典集合的概率粗糙下、上近似。从概率角度出发，一个直觉模糊集合就是一个直觉模糊事件。本文首先给出了直觉模糊条件概率的定义。随后，在直觉模糊概率空间下构造了双论域广义直觉模糊概率粗糙集模型，讨论了模型的主要性质。最后，将模型应用到临床诊断系统中，并与已有文献中的结果进行比较。结果表明，所提出的概率粗糙集模型进一步丰富了概率粗糙集理论，更加符合实际应用。下一步，考虑直觉模糊条件概率具有单调性，将讨论广义直觉模糊概率粗糙集中的属性约简问题。另外，由于直觉模糊集具有接受、拒绝和犹豫的语义，将该模型与三支决策模型相结合，继续拓展该模型在其他领域的应用也是我们未来的研究内容。 -
表 1 患者与症状之间的直觉模糊关系
Table 1 Intuitionistic fuzzy relationship between patients and symptoms
$U$ $V$ ${y_1}$ ${y_2}$ ${y_3}$ ${y_4}$ ${y_5}$ ${x_1}$ ${\left\langle {0.8,0.1} \right\rangle } $ ${\left\langle {0.6,0.1} \right\rangle } $ ${\left\langle {0.2,0.8} \right\rangle } $ ${\left\langle {0.6,0.1} \right\rangle } $ ${\left\langle {0.1,0.6} \right\rangle } $ ${x_2}$ ${\left\langle {0.0,0.8} \right\rangle } $ ${\left\langle {0.4,0.4} \right\rangle } $ ${\left\langle {0.6,0.1} \right\rangle } $ ${\left\langle {0.1,0.7} \right\rangle } $ ${\left\langle {0.1,0.8} \right\rangle } $ ${x_3}$ ${\left\langle {0.8,0.1} \right\rangle } $ ${\left\langle {0.8,0.1} \right\rangle } $ ${\left\langle {0.0,0.6} \right\rangle } $ ${\left\langle {0.2,0.7} \right\rangle } $ ${\left\langle {0.0,0.5} \right\rangle } $ ${x_4}$ ${\left\langle {0.6,0.1} \right\rangle } $ ${\left\langle {0.5,0.4} \right\rangle } $ ${\left\langle {0.3,0.4} \right\rangle } $ ${\left\langle {0.7,0.2} \right\rangle } $ ${\left\langle {0.3,0.4} \right\rangle } $ ${x_5}$ ${\left\langle {0.7,0.2} \right\rangle } $ ${\left\langle {0.1,0.7} \right\rangle } $ ${\left\langle {0.1,0.9} \right\rangle } $ 〈 ${\left\langle {0.6,0.1} \right\rangle } $ ${\left\langle {0.0,0.9} \right\rangle } $ ${x_6}$ ${\left\langle {0.6,0.0} \right\rangle } $ ${\left\langle {0.2,0.7} \right\rangle } $ ${\left\langle {0.0,0.8} \right\rangle } $ ${\left\langle {0.6,0.4} \right\rangle } $ ${\left\langle {0.2,0.7} \right\rangle } $ ${x_7}$ ${\left\langle {0.3,0.3} \right\rangle } $ ${\left\langle {0.6,0.1} \right\rangle } $ ${\left\langle {0.2,0.7} \right\rangle } $ ${\left\langle {0.2,0.6} \right\rangle } $ ${\left\langle {0.1,0.9} \right\rangle } $ ${x_8}$ ${\left\langle {0.1,0.7} \right\rangle } $ ${\left\langle {0.2,0.4} \right\rangle } $ ${\left\langle {0.8,0.0} \right\rangle } $ ${\left\langle {0.2,0.7} \right\rangle } $ ${\left\langle {0.2,0.7} \right\rangle } $ ${x_9}$ ${\left\langle {0.1,0.8} \right\rangle } $ ${\left\langle {0.0,0.8} \right\rangle } $ ${\left\langle {0.2,0.8} \right\rangle } $ ${\left\langle {0.4,0.3} \right\rangle } $ ${\left\langle {0.8,0.1} \right\rangle } $ 表 2 患者与症状之间的直觉模糊关系
Table 2 Intuitionistic fuzzy relationship between patients and symptoms
患者 T H S C CP Al ${\left\langle {0.8,0.1} \right\rangle } $ ${\left\langle {0.6,0.1} \right\rangle }$ ${\left\langle {0.2,0.8} \right\rangle } $ ${\left\langle {0.6,0.1} \right\rangle } $ ${\left\langle {0.1,0.6} \right\rangle } $ Bob ${\left\langle {0.0,0.8} \right\rangle } $ ${\left\langle {0.4,0.4}\right\rangle } $ ${\left\langle {0.6,0.1} \right\rangle } $ ${\left\langle {0.1,0.7}\right\rangle } $ ${\left\langle {0.1,0.8} \right\rangle } $ Joe ${\left\langle {0.8,0.1} \right\rangle } $ ${\left\langle {0.8,0.1} \right\rangle } $ ${\left\langle {0.0,0.6}\right\rangle } $ ${\left\langle {0.2,0.7}\right\rangle } $ ${\left\langle {0.0,0.5} \right\rangle } $ Ted ${\left\langle {0.6,0.1} \right\rangle } $ ${\left\langle {0.5,0.4} \right\rangle } $ ${\left\langle {0.3,0.4} \right\rangle } $ ${\left\langle {0.7,0.2} \right\rangle } $ ${\left\langle {0.3,0.4} \right\rangle } $ 表 3 疾病与症状之间的直觉模糊关系
Table 3 Intuitionistic fuzzy relationship between diseases and symptoms
疾病 T H S C CP Vf ${\left\langle {0.4,0.0} \right\rangle } $ ${\left\langle {0.3,0.5}\right\rangle } $ ${\left\langle {0.1,0.7} \right\rangle } $ ${\left\langle {0.4,0.3}\right\rangle } $ ${\left\langle {0.1,0.7}\right\rangle } $ Ma ${\left\langle {0.7,0.0}\right\rangle } $ ${\left\langle {0.2,0.6}\right\rangle } $ ${\left\langle {0.0,0.9}\right\rangle } $ ${\left\langle {0.7,0.0}\right\rangle } $ ${\left\langle {0.1,0.8}\right\rangle } $ Ty ${\left\langle {0.3,0.3}\right\rangle } $ ${\left\langle {0.6,0.1}\right\rangle } $ ${\left\langle {0.2,0.7}\right\rangle } $ ${\left\langle {0.2,0.6}\right\rangle } $ ${\left\langle {0.1,0.9} \right\rangle } $ St ${\left\langle {0.1,0.7} \right\rangle } $ ${\left\langle {0.2,0.4} \right\rangle } $ ${\left\langle {0.8,0.0}\right\rangle } $ ${\left\langle {0.2,0.7}\right\rangle } $ ${\left\langle {0.2,0.7}\right\rangle } $ Ch ${\left\langle {0.1,0.8}\right\rangle } $ ${\left\langle {0.0,0.8} \right\rangle } $ ${\left\langle {0.2,0.8} \right\rangle } $ ${\left\langle {0.2,0.8}\right\rangle } $ ${\left\langle {0.8,0.1} \right\rangle } $ 表 4 疾病关于患者的直觉模糊条件概率
Table 4 Intuitionistic fuzzy conditional probability of diseases about patients
患者 Vf Ma Ty St Ch Al 0.39 0.75 0.43 0.17 0.17 Bob 0.14 0.32 0.38 0.79 0.14 Joe 0.42 0.71 0.55 0.13 0.03 Ted 0.38 0.76 0.34 0.40 0.36 -
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