聊一聊粗糙集
Posted gedanke
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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 imes n}),(B=(b_{ij})_{m imes n})是两个(m imes n)的矩阵,则两个矩阵的和(C=(c_{ij})_{m imes n})为
[
C = A+B quad Longrightarrow quad c_{ij}=a_{ij}+b_{ij}
]
[ =egin{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} + egin{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} ]
[ =egin{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}
]
[
= egin{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= egin{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= egin{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 imes n}),(B=(b_{ij})_{n imes p})是两个矩阵
则两个矩阵的乘积(A imes B =C=(c_{ij})_{m imes p}) 为:
[
C = A imes B quad Longrightarrow quad (c_{ij})_{m imes p}=(sum_{k=1}^{n} a_{ik}cdot b_{kj})_{m imes p}
]
[
= egin{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 igcup 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 imes n})定义如下:
[
m_{ij}
=egin{cases}
1 & (u_{i},u_{j}) in R_{C} & (u_{i},u_{j})
otin 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 igcup D,V,f))是一个决策信息系统,(U_{U}^{R_{C}} = (m_{ij})_{n imes 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{sumleft(M_{U}^{R_{C}}
ight)}{|U|^{2}}=overline{M_{U}^{R_{C}}}
]
其中,(sumleft(M_{U}^{R_{C}}
ight))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} imesoperatorname{sum}(left[egin{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}
ight])=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 igcup D,V,f))是一个决策信息系统,(U_{U}^{R_{C}}),(U_{U}^{R_{C igcup 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}(Dmid C)=overline{U_{U}^{R_{C}}}-overline{U_{U}^{R_{C igcup 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 igcup D}}}
]
[
=frac{1}{81} imesoperatorname{sum}(left[egin{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}
ight] - left[egin{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}
ight]) =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 igcup D,V,f))是一个决策信息系统,(Bsubseteq C),且(U_{U}^{R_{B}}),(U_{U}^{R_{B-{a} }}),(U_{U}^{R_{B igcup D}}),(U_{U}^{R_{(B -{a}) igcup 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}) igcup D) }-{GP_{U}(B)-GP_{U}(B igcup D) } ]
[
=overline{M_{U}^{R_{B-{a }}}}-overline{M_{U}^{R_{(B -{a}) igcup D}}}-overline{M_{U}^{R_{B}}}+overline{M_{U}^{R_{B igcup D}}}
]
外部属性重要度:
若(S=(U,A=C igcup D,V,f))是一个决策信息系统,(Bsubseteq C),且(U_{U}^{R_{B}}),(U_{U}^{R_{B igcup D}}),(U_{U}^{R_{B igcup {a} }}),(U_{U}^{R_{(B igcup {a}) igcup 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 igcup {a})
]
[ ={ GP_{U}(B)-GP_{U}(Bigcup D)} - { GP_{U}(B igcup {a})-GP_{U}((Bigcup {a}) igcup D) } ]
[
=overline{M_{U}^{R_{B}}}-overline{M_{U}^{R_{B igcup D}}}-overline{M_{U}^{R_{B igcup {a } }}}+overline{M_{U}^{R_{(B igcup {a}) igcup 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.
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