﻿ 模糊粗糙集的相似度及其生成

# 模糊粗糙集的相似度及其生成The Similarity Degree of Fuzzy Rough Sets and Its Generating

Abstract: Based on the analysis of the properties of similarity of the L-fuzzy sets, the definition and the generating method of the fuzzy rough sets similarity are given. The generation of strong similarity can also be realized by constructing special functions from different angles.

1. 引言

2. 相似度

1) 当 $\alpha \le \beta$ 时， $D\left(\beta /\alpha \right)=1$

2) 当 $\alpha \le \beta \le \gamma$ 时， $D\left(\alpha /\gamma \right)\le D\left(\alpha /\beta \right)$，则称D为L上的包含度函数。 $D\left(\beta /\alpha \right)$$\alpha$$\beta$ 中的包含度。

${U}_{1},{U}_{2},\cdots ,{U}_{n}$ 为论域， $U={U}_{1}×{U}_{2}×\cdots ×{U}_{n}$${\mathcal{F}}_{L}\left({U}_{i}\right)$${U}_{i}$ 上L-模糊集组成的集合，( $i=1,2,\cdots ,n$ )， ${H}_{n}\left(U\right)=\left\{A|A=\underset{i=1}{\overset{n}{\prod }}{A}_{i},{A}_{i}\in {\mathcal{F}}_{L}\left({U}_{i}\right),i=1,2,\cdots ,n\right\}$$A\left(u\right)=\left(\underset{i=1}{\overset{n}{\prod }}{A}_{i}\right)\left({u}_{1},{u}_{2},\cdots ,{u}_{n}\right)=\underset{i=1}{\overset{n}{\wedge }}{A}_{i}\left({u}_{i}\right)$$u\in U$

1) $SM\left(\alpha ,\alpha \right)=1$

2) $SM\left(\alpha ,\beta \right)=SM\left(\beta ,\alpha \right)$

3) 当 $\alpha \le \beta \le \gamma$ 时， $SM\left(\alpha ,\gamma \right)\le SM\left(\alpha ,\beta \right)$。则称SM为L上的相似度函数。 $SM\left(\alpha ,\beta \right)$$\alpha$$\beta$ 的相似度量。

1) 显然 $0\le SM\left(A,B\right)\le 1$

2) $SM\left(A,A\right)=S\left[S{M}_{1}\left({A}_{1},{A}_{1}\right),S{M}_{2}\left({A}_{2},{A}_{2}\right)\right]=S\left(1,1\right)=1$

3) $SM\left(A,B\right)=S\left[S{M}_{1}\left({A}_{1},{B}_{1}\right),S{M}_{2}\left({A}_{2},{B}_{2}\right)\right]=S\left[S{M}_{1}\left({B}_{1},{A}_{1}\right),S{M}_{2}\left({B}_{2},{A}_{2}\right)\right]=SM\left(B,A\right)$

4) $A\subseteq B\subseteq C⇒{A}_{i}\subseteq {B}_{i}\subseteq {C}_{i},i=1,2$

$⇒SM\left(A,C\right)=S\left[S{M}_{1}\left({A}_{1},{C}_{1}\right),S{M}_{2}\left({A}_{2},{C}_{2}\right)\right]\le S\left[S{M}_{1}\left({A}_{1},{B}_{1}\right),S{M}_{2}\left({A}_{2},{B}_{2}\right)\right]=SM\left(A,B\right)$

$SM\left(A,C\right)=S\left[S{M}_{1}\left({A}_{1},{C}_{1}\right),S{M}_{2}\left({A}_{2},{C}_{2}\right)\right]\le S\left[S{M}_{1}\left({B}_{1},{C}_{1}\right),S{M}_{2}\left({B}_{2},{C}_{2}\right)\right]=SM\left(B,C\right)$

$⇒SM\left(A,C\right)\le SM\left(A,B\right)\wedge SM\left(B,C\right)$

1) 显然 $0\le SM\left(A,B\right)\le 1$

2) $SM\left(A,A\right)=S\left(D\left(A/A\right),D\left(A/A\right)\right)=S\left(1,1\right)\ge S\left(0,1\right)=1$，故 $SM\left(A,A\right)=1$

3) 由反三角模的对称性，易证SM的对称性。

4) $A\subseteq B\subseteq C$ $⇒D\left(A/C\right)=D\left(A/B\right)\wedge D\left(B/C\right)$$D\left(C/A\right)=D\left(B/A\right)=D\left(C/B\right)=1$ $⇒SM\left(A,C\right)=S\left(1,D\left(A/C\right)\right)$$SM\left(A,B\right)=S\left(1,D\left(A/B\right)\right)$$SM\left(B,C\right)=S\left(1,D\left(B/C\right)\right)$ $⇒SM\left(A,C\right)\le SM\left(A,B\right)$$SM\left(A,C\right)\le SM\left(B,C\right)$ $⇒SM\left(A,C\right)\le SM\left(A,B\right)\wedge SM\left(B,C\right)$。从而SM为强相似度。

1) $\begin{array}{c}SM\left(A,B\right)=T\left[T\left({D}_{1}\left({B}_{1}/{A}_{1}\right),{D}_{1}\left({A}_{1}/{B}_{1}\right)\right),T\left({D}_{2}\left({B}_{2}/{A}_{2}\right),{D}_{2}\left({A}_{2}/{B}_{2}\right)\right)\right]\\ =T\left[T\left({D}_{1}\left({B}_{1}/{A}_{1}\right),{D}_{2}\left({B}_{2}/{A}_{2}\right)\right),T\left({D}_{1}\left({A}_{1}/{B}_{1}\right),{D}_{2}\left({A}_{2}/{B}_{2}\right)\right)\right]\end{array}$

2) $\begin{array}{c}SM\left(A,B\right)=S\left[S\left({D}_{1}\left({B}_{1}/{A}_{1}\right),{D}_{1}\left({A}_{1}/{B}_{1}\right)\right),S\left({D}_{2}\left({B}_{2}/{A}_{2}\right),{D}_{2}\left({A}_{2}/{B}_{2}\right)\right)\right]\\ =S\left[S\left({D}_{1}\left({B}_{1}/{A}_{1}\right),{D}_{2}\left({B}_{2}/{A}_{2}\right)\right),S\left({D}_{1}\left({A}_{1}/{B}_{1}\right),{D}_{2}\left({A}_{2}/{B}_{2}\right)\right)\right]\end{array}$

3) $\begin{array}{c}SM\left(A,B\right)=T\left[S\left({D}_{1}\left({B}_{1}/{A}_{1}\right),{D}_{1}\left({A}_{1}/{B}_{1}\right)\right),S\left({D}_{2}\left({B}_{2}/{A}_{2}\right),{D}_{2}\left({A}_{2}/{B}_{2}\right)\right)\right]\\ =T\left[S\left({D}_{1}\left({B}_{1}/{A}_{1}\right),{D}_{2}\left({B}_{2}/{A}_{2}\right)\right),S\left({D}_{1}\left({A}_{1}/{B}_{1}\right),{D}_{2}\left({A}_{2}/{B}_{2}\right)\right)\right]\end{array}$

4) $\begin{array}{c}SM\left(A,B\right)=S\left[T\left({D}_{1}\left({B}_{1}/{A}_{1}\right),{D}_{1}\left({A}_{1}/{B}_{1}\right)\right),T\left({D}_{2}\left({B}_{2}/{A}_{2}\right),{D}_{2}\left({A}_{2}/{B}_{2}\right)\right)\right]\\ =S\left[T\left({D}_{1}\left({B}_{1}/{A}_{1}\right),{D}_{2}\left({B}_{2}/{A}_{2}\right)\right),T\left({D}_{1}\left({A}_{1}/{B}_{1}\right),{D}_{2}\left({A}_{2}/{B}_{2}\right)\right)\right]\end{array}$

$\begin{array}{c}SM\left(A,B\right)=S\left[T\left(\alpha ,\beta \right),T\left(a,b\right)\right]=S\left[\alpha ,T\left(\beta ,T\left(a,b\right)\right)\right]=S\left[\alpha ,T\left(\beta ,T\left(b,a\right)\right)\right]\\ =S\left[\alpha ,T\left(T\left(\beta ,b\right),a\right)\right]=S\left[\alpha ,T\left(a,T\left(\beta ,b\right)\right)\right]=S\left[T\left(\alpha ,a\right),T\left(\beta ,b\right)\right]\end{array}$。即(4)成立。

3. 模糊粗糙集的相似度及生成

1) 当 $A=B$ 时， $S{M}_{R}\left[\left({A}_{L},{A}_{U}\right),\left({B}_{L},{B}_{U}\right)\right]=1$

2) $S{M}_{R}\left[\left({A}_{L},{A}_{U}\right),\left({B}_{L},{B}_{U}\right)\right]=S{M}_{R}\left[\left({B}_{L},{B}_{U}\right),\left({A}_{L},{A}_{U}\right)\right]$

3) 当 $A\subseteq B\subseteq C$$S{M}_{R}\left[\left({A}_{L},{A}_{U}\right),\left({C}_{L},{C}_{U}\right)\right]\le S{M}_{R}\left[\left({A}_{L},{A}_{U}\right),\left({B}_{L},{B}_{U}\right)\right]$

2) $S{M}_{R}^{2}\left(A,B\right)={\omega }_{1}S{M}_{L}\left({A}_{L},{B}_{L}\right)+{\omega }_{2}S{M}_{U}\left({A}_{U},{B}_{U}\right)$$\left(FR\left(X\right),\subseteq \right)$ 上的(强)相似度。其中 ${\omega }_{1}+{\omega }_{2}=1,{\omega }_{1}\ge 0,{\omega }_{2}\ge 0$

1) $SM\left(A,B\right)=f\left(D\left(A/B\right),D\left(B/A\right)\right)=f\left(D\left(B/A\right),D\left(A/B\right)\right)=SM\left(B,A\right)$

2) $SM\left(A,A\right)=f\left(D\left(A/A\right),D\left(A/A\right)\right)=f\left(1,1\right)=1$

3) $A\subseteq B\subseteq C$，则由f的非减性， $\begin{array}{c}SM\left(A,C\right)=f\left(D\left(A/C\right),D\left(C/A\right)\right)=f\left(D\left(A/C\right),1\right)\le f\left(D\left(A/B\right),1\right)\\ =f\left(D\left(A/B\right),D\left(B/A\right)\right)=SM\left(A,B\right)\end{array}$

$SM\left(A,B\right)=f\left(D\left(A/B\right),D\left(B/A\right)\right)$ 为A与B之间的强相似度量。

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[2] 袁修久, 张文修. 模糊粗糙集的包含度和相似度[J]. 模糊系统与数学, 2005, 19(1): 111-115.

[3] 张文修, 徐宗本, 等. 包含度理论[J]. 模糊系统与数学, 1996, 10(4): 1-9.

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