本节我们将继续介绍粗糙集有关的概念。
上节我们介绍了知识粒度的度量，本节将介绍知识粒度的矩阵表示形式。
我们先简单介绍矩阵的相关概念。
矩阵
先看矩阵的和，差。
矩阵的和：
若\(A=(a_{ij})_{m \times n}\)，\(B=(b_{ij})_{m \times n}\)是两个\(m \times n\)的矩阵，则两个矩阵的和\(C=(c_{ij})_{m \times n}\)为
\[ C = A+B \quad \Longrightarrow \quad c_{ij}=a_{ij}+b_{ij} \]
\[ =\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} & \cdots & b_{1n} \\ b_{21} & b_{22} & \cdots & b_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ b_{m1} & b_{m2} & \cdots & b_{mn} \\ \end{bmatrix} \]
\[ =\begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} & \cdots & a_{1n}+b_{1n} \\ a_{21}+b_{21} & a_{22}+b_{22} & \cdots & a_{2n}+b_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1}+b_{m1} & a_{m2}+b_{m2} & \cdots & a_{mn}+b_{mn} \\ \end{bmatrix} \]
类似的，两个矩阵的差：
\[ C = A-B \quad \Longrightarrow \quad c_{ij}=a_{ij}-b_{ij} \]
\[ = \begin{bmatrix} a_{11}-b_{11} & a_{12}-b_{12} & \cdots & a_{1n}-b_{1n} \\ a_{21}-b_{21} & a_{22}-b_{22} & \cdots & a_{2n}-b_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1}-b_{m1} & a_{m2}-b_{m2} & \cdots & a_{mn}-b_{mn} \\ \end{bmatrix} \]
矩阵的转置：
\[ A= \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \\ \end{bmatrix} \]
则矩阵\(A\)的转置矩阵\(A^T\)为：
\[ A^T= \begin{bmatrix} a_{11} & a_{21} & \cdots & a_{n1} \\ a_{12} & a_{22} & \cdots & a_{n2} \\ \vdots & \vdots & \ddots & \vdots \\ a_{1n} & a_{2n} & \cdots & a_{nn} \\ \end{bmatrix} \]
最后来看矩阵的乘积：
若\(A=(a_{ij})_{m \times n}\)，\(B=(b_{ij})_{n \times p}\)是两个矩阵
则两个矩阵的乘积\(A \times B =C=(c_{ij})_{m \times p}\) 为：
\[ C = A \times B \quad \Longrightarrow \quad (c_{ij})_{m \times p}=(\sum_{k=1}^{n} a_{ik}\cdot b_{kj})_{m \times p} \]
\[ = \begin{bmatrix} \sum_{k=1}^{n} a_{1k}b_{k1} & \sum_{k=1}^{n}a_{1k}b_{k2} & \cdots & \sum_{k=1}^{n} a_{1k}b_{kp} \\ \sum_{k=1}^{n} a_{2k}b_{k1} & \sum_{k=1}^{n}a_{2k}b_{k2} & \cdots & \sum_{k=1}^{n} a_{2k}b_{kp} \\ \vdots & \vdots & \ddots & \vdots \\ \sum_{k=1}^{n} a_{mk}b_{k1} & \sum_{k=1}^{n}a_{mk}b_{k2} & \cdots & \sum_{k=1}^{n} a_{mk}b_{kp} \\ \end{bmatrix} \]
知识粒度的矩阵表现形式
我们依旧使用该表
| \(U\) | \(a\) | \(b\) | \(c\) | \(e\) | \(f\) | \(d\) |
|---|---|---|---|---|---|---|
| 1 | 0 | 1 | 1 | 1 | 0 | 1 |
| 2 | 1 | 1 | 0 | 1 | 0 | 1 |
| 3 | 1 | 0 | 0 | 0 | 1 | 0 |
| 4 | 1 | 1 | 0 | 1 | 0 | 1 |
| 5 | 1 | 0 | 0 | 0 | 1 | 0 |
| 6 | 0 | 1 | 1 | 1 | 1 | 0 |
| 7 | 0 | 1 | 1 | 1 | 1 | 0 |
| 8 | 1 | 0 | 0 | 1 | 0 | 1 |
| 9 | 1 | 0 | 0 | 1 | 0 | 0 |
等价关系矩阵的定义如下：
设\(S=(U,A=C \bigcup D,V,f)\)是一个决策信息系统，论域\(U=\{u_{1},u_{2},...,u_{n} \}\)，\(n\)是论域内元素个数，\(U/C=\{X_{1},X_{2},...,X_{m}\}\)，\(R_{C}\)是论域\(U\)的等价关系。则等价关系矩阵\(U_{U}^{R_{C}} = (m_{ij})_{n \times n}\)定义如下：
\[ m_{ij} =\begin{cases} 1 & (u_{i},u_{j}) \in R_{C} \\ 0 & (u_{i},u_{j}) \notin R_{C} \end{cases} \]
其中，\({1 \leq i,j \leq n}\)。
基于矩阵的知识粒度如下：
设\(S=(U,A=C \bigcup D,V,f)\)是一个决策信息系统，\(U_{U}^{R_{C}} = (m_{ij})_{n \times n}\)是等价关系矩阵，条件属性\(C\)基于矩阵的知识粒度定义如下：
\[ GP_{U}(C)=\frac{sum\left(M_{U}^{R_{C}}\right)}{|U|^{2}}=\overline{M_{U}^{R_{C}}} \]
其中，\(sum\left(M_{U}^{R_{C}}\right)\)是等价矩阵内\(1\)的个数总和，\(\overline{M_{U}^{R_{C}}}\)是矩阵内所有元素的均值。
依旧上表，我们可以计算\(GP_{U}(C)\):
\[ GP_{U}(C)=\overline{M_{U}^{R_{C}}}=\frac{1}{81}\times\operatorname{sum}(\left[\begin{array}{ccccccccc} {1} & {0} & {0} & {0} & {0} & {0} & {0} & {0} & {0} \\ {0} & {1} & {0} & {1} & {0} & {0} & {0} & {0} & {0} \\ {0} & {0} & {1} & {0} & {1} & {0} & {0} & {0} & {0} \\ {0} & {1} & {0} & {1} & {0} & {0} & {0} & {0} & {0} \\ {0} & {1} & {0} & {1} & {0} & {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} & {0} & {1} & {1} & {0} & {0} \\ {0} & {0} & {0} & {0} & {0} & {1} & {1} & {0} & {0} \\ {0} & {0} & {0} & {0} & {0} & {0} & {0} & {1} & {1} \\ {0} & {0} & {0} & {0} & {0} & {0} & {0} & {1} & {1} \end{array}\right])=\frac{17}{81} \]
这和我们在上节计算得到的结果是一致的。
类似的，相对知识粒度的定义如下：
若\(S=(U,A=C \bigcup D,V,f)\)是一个决策信息系统，\(U_{U}^{R_{C}}\)，\(U_{U}^{R_{C \bigcup D}}\)是等价关系矩阵，则决策属性\(D\)关于条件属性\(C\)基于矩阵的相对知识粒度定义如下：
\[ G P_{U}(D\mid C)=\overline{U_{U}^{R_{C}}}-\overline{U_{U}^{R_{C \bigcup D}}} \]
根据上表，我们可以计算\(GP_{U}(D \mid C)\):
\[ GP_{U}(D \mid C)=\overline{U_{U}^{R_{C}}}-\overline{U_{U}^{R_{C \bigcup D}}} \]
\[ =\frac{1}{81}\times\operatorname{sum}(\left[\begin{array}{ccccccccc} {1} & {0} & {0} & {0} & {0} & {0} & {0} & {0} & {0} \\ {0} & {1} & {0} & {1} & {0} & {0} & {0} & {0} & {0} \\ {0} & {0} & {1} & {0} & {1} & {0} & {0} & {0} & {0} \\ {0} & {1} & {0} & {1} & {0} & {0} & {0} & {0} & {0} \\ {0} & {1} & {0} & {1} & {0} & {0} & {0} & {0} & {0} \\ {0} & {0} & {1} & {0} & {1} & {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} & {0} & {1} & {1} & {0} & {0} \\ {0} & {0} & {0} & {0} & {0} & {1} & {1} & {0} & {0} \\ {0} & {0} & {0} & {0} & {0} & {0} & {0} & {1} & {1} \\ {0} & {0} & {0} & {0} & {0} & {0} & {0} & {1} & {1} \end{array}\right] - \left[\begin{array}{ccccccccc} {1} & {0} & {0} & {0} & {0} & {0} & {0} & {0} & {0} \\ {0} & {1} & {0} & {1} & {0} & {0} & {0} & {0} & {0} \\ {0} & {0} & {1} & {0} & {1} & {0} & {0} & {0} & {0} \\ {0} & {1} & {0} & {1} & {0} & {0} & {0} & {0} & {0} \\ {0} & {1} & {0} & {1} & {0} & {0} & {0} & {0} & {0} \\ {0} & {0} & {1} & {0} & {1} & {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} & {0} & {1} & {1} & {0} & {0} \\ {0} & {0} & {0} & {0} & {0} & {1} & {1} & {0} & {0} \\ {0} & {0} & {0} & {0} & {0} & {0} & {0} & {1} & {0} \\ {0} & {0} & {0} & {0} & {0} & {0} & {0} & {0} & {1} \end{array}\right]) =\frac{2}{81} \]
这与我们之前计算的结果是一致的。
类似的，基于矩阵的内外部属性重要度的定义如下：
内部属性重要度:
若\(S=(U,A=C \bigcup D,V,f)\)是一个决策信息系统，\(B\subseteq C\)，且\(U_{U}^{R_{B}}\)，\(U_{U}^{R_{B-\{a\} }}\),\(U_{U}^{R_{B \bigcup D}}\),\(U_{U}^{R_{(B -\{a\}) \bigcup D}}\)都是等价关系矩阵，\(\forall a \in B\)，则属性\(a\)关于条件属性\(B\)相对于决策属性集\(D\)的基于矩阵的相对知识粒度定义如下：
\[ \operatorname{Sig}_{U}^{inner }(a, B, D)=GP_{U}(D \mid B-\{a\})-GP_{U}(D \mid B) \]
\[ =\{ GP_{U}(B-\{a\})-GP_{U}((B-\{a\}) \bigcup D) \}-\{GP_{U}(B)-GP_{U}(B \bigcup D) \} \]
\[ =\overline{M_{U}^{R_{B-\{a \}}}}-\overline{M_{U}^{R_{(B -\{a\}) \bigcup D}}}-\overline{M_{U}^{R_{B}}}+\overline{M_{U}^{R_{B \bigcup D}}} \]
外部属性重要度:
若\(S=(U,A=C \bigcup D,V,f)\)是一个决策信息系统，\(B\subseteq C\)，且\(U_{U}^{R_{B}}\)，\(U_{U}^{R_{B \bigcup D}}\),\(U_{U}^{R_{B \bigcup \{a\} }}\),\(U_{U}^{R_{(B \bigcup \{a\}) \bigcup D}}\)都是等价关系矩阵，\(\forall a \in (C-B)\)，则属性\(a\)关于条件属性\(B\)相对于决策属性集\(D\)的基于矩阵的相对知识粒度定义如下：
\[ \operatorname{Sig}_{U}^{outer }(a, B, D)=GP_{U}(D \mid B)-GP_{U}(D \mid B \bigcup \{a\}) \]
\[ =\{ GP_{U}(B)-GP_{U}(B\bigcup D)\} - \{ GP_{U}(B \bigcup \{a\})-GP_{U}((B\bigcup \{a\}) \bigcup D) \} \]
\[ =\overline{M_{U}^{R_{B}}}-\overline{M_{U}^{R_{B \bigcup D}}}-\overline{M_{U}^{R_{B \bigcup \{a \} }}}+\overline{M_{U}^{R_{(B \bigcup \{a\}) \bigcup D}}} \]
参考上节的案例，如果使用矩阵表示的话，结果是一样的，但是基于矩阵的方式在面对大规模数据集是可能不是好的选择。
本文参考了：
- 景运革. 基于知识粒度的动态属性约简算法研究[D].西南交通大学,2017.