﻿ 基于对象变化的邻域决策粗糙集动态更新算法
«上一篇
 文章快速检索 高级检索

 智能系统学报  2021, Vol. 16 Issue (4): 746-756  DOI: 10.11992/tis.202010028 0

### 引用本文

SUN Haixia. Dynamic updating algorithm of neighborhood decision-theoretic rough set model based on object change[J]. CAAI Transactions on Intelligent Systems, 2021, 16(4): 746-756. DOI: 10.11992/tis.202010028.

### 文章历史

Dynamic updating algorithm of neighborhood decision-theoretic rough set model based on object change
SUN Haixia
College of Computer Engineering, Anhui Sanlian University, Hefei 230601, China
Abstract: In accordance with the dynamic characteristics of a dataset in a real environment, an incremental updating algorithm of the neighborhood decision-theoretic rough set model is proposed after conducting simple to complex research. In this paper, the change law of the probability between the target approximation set and the neighborhood class is first analyzed in the context of the domain of the neighborhood information system increasing or decreasing individual objects, which is then adopted to construct the incremental updating of the upper and lower approximation sets of the neighborhood decision-theoretic rough set model. On the basis of the change of a single object and given the variety of multiple objects, an incremental updating algorithm is designed through step-by-step iteration. Experimental results show that the proposed algorithm has a high incremental updating performance, thereby making it suitable for the dynamic updating of the neighborhood decision-theoretic rough set model in a dynamic data environment.
Key words: rough set    decision-theoretic rough set    neighborhood    incremental learning    approximation set    object    iteration    dynamic

1 邻域决策粗糙集模型

Yao等[1-2]提出的决策粗糙集仅应用于离散型的信息系统，Li等将邻域关系引入传统的决策粗糙集模型中，提出了邻域决策粗糙集[10]

 $N_A^\delta = \{ (x,y) \in U \times U\left| {{\Delta _A}(x,y) \leqslant \delta } \right.\}$

 ${\Delta _A}(x,y) = {\left( {\sum_{\forall a \in A} {|a(x) - a(y){|^k}} } \right)^{{1 / k}}}$

 ${\delta _A}(x) = \{ y \in U|(x,y) \in N_A^\delta \}$

1) ${R_P} = R({a_P}|\delta (x))$ $= {\lambda _{PP}} \cdot P(X|\delta (x)) + {\lambda _{PN}} \cdot P({X^c}|\delta (x))$ ;

2) ${R_B} = R({a_B}|\delta (x))$ $= {\lambda _{BP}} \cdot P(X|\delta (x)) + {\lambda _{BN}} \cdot P({X^c}|\delta (x))$ ;

3) ${R_N} = R({a_N}|\delta (x))$ $= {\lambda _{NP}} \cdot P(X|\delta (x)) + {\lambda _{NN}} \cdot P({X^c}|\delta (x))$

1) ${R_P} \leqslant {R_B}$ ${R_P} \leqslant {R_N}$ 时， $x \in {\rm{POS}}(X)$ ;

2) ${R_B} \leqslant {R_P}$ ${R_B} \leqslant {R_N}$ 时， $x \in {\rm{BUN}}(X)$ ;

3) ${R_N} \leqslant {R_P}$ ${R_N} \leqslant {R_B}$ 时， $x \in {\rm{NEG}}(X)$

1)当 $P(X|\delta (x)) \geqslant \alpha$ $P(X|\delta (x)) \geqslant \gamma$ 时， $x \in {\rm{POS}}(X)$

2)当 $P(X|\delta (x)) \leqslant \alpha$ $P(X|\delta (x)) \geqslant \beta$ 时， $x \in {\rm{BUN}}(X)$

3)当 $P(X|\delta (x)) \leqslant \beta$ $P(X|\delta (x)) \leqslant \gamma$ 时， $x \in {\rm{NEG}}(X)$

 $\alpha = \frac{{{\lambda _{PN}} - {\lambda _{BN}}}}{{({\lambda _{PN}} - {\lambda _{BN}}) + ({\lambda _{BP}} - {\lambda _{PP}})}}$
 $\beta = \frac{{{\lambda _{BN}} - {\lambda _{NN}}}}{{({\lambda _{BN}} - {\lambda _{NN}}) + ({\lambda _{NP}} - {\lambda _{BP}})}}$
 $\gamma = \frac{{{\lambda _{PN}} - {\lambda _{NN}}}}{{({\lambda _{PN}} - {\lambda _{NN}}) + ({\lambda _{NP}} - {\lambda _{PP}})}}$

1)当 $P(X|\delta (x)) > \alpha$ 时，那么 $x \in {\rm{POS}}(X)$

2)当 $\beta \leqslant P(X|\delta (x)) \leqslant \alpha$ 时，那么 $x \in {\rm{BUN}}(X)$

3)当 $P(X|\delta (x)) < \beta$ 时， $x \in {\rm{NEG}}(X)$

 $\underline N _A^{(\alpha ,\beta )}(X) = \{ x \in U|P(X|{\delta _A}(x)) > \alpha \}$
 $\overline N _A^{(\alpha ,\beta )}(X) = \{ x \in U|P(X|{\delta _A}(x)) \geqslant \beta \}$

 $P(X|{\delta _A}(x)) = \dfrac{{|{\delta _A}(x) \cap X|}}{{|{\delta _A}(x)|}}$
2 邻域决策粗糙集模型的增量式更新

2.1 论域中对象增加时模型的增量式更新

1) $\forall x \in \delta _A^{U{}^ + }(x{}^ + ) - \{ x{}^ + \}$ ，有

 $P({X^ + }|\delta _A^{{U^ + }}(x)) \geqslant P(X|\delta _A^U(x));$
 $P({Y^ + }|\delta _A^{{U^ + }}(x)) < P(Y|\delta _A^U(x)).$

2) $\forall x \in {U^ + } - \delta _A^{{U^ + }}({x^ + })$ ，有

 $P({X^ + }|\delta _A^{{U^ + }}(x)) = P(X|\delta _A^U(x));$
 $P({Y^ + }|\delta _A^{{U^ + }}(x)) = P(Y|\delta _A^U(x)).$

 \begin{aligned} 1)\underline N _A^{(\alpha ,\beta )}&({X^ + }) = \underline N _A^{(\alpha ,\beta )}(X) \cup \\ &\{ x \in \delta _A^{{U^ + }}({x^ + }) - \underline N _A^{(\alpha ,\beta )}(X)|P({X^ + }|\delta _A^{{U^ + }}(x)) > \alpha \} \end{aligned}
 \begin{aligned} 2)\underline N _A^{(\alpha ,\beta )}&({Y^ + }) = (\underline N _A^{(\alpha ,\beta )}(Y) - \\ &\{ x \in \underline N _A^{(\alpha ,\beta )}(Y) \cap \delta _A^{{U^ + }}({x^ + })|P({Y^ + }|\delta _A^{{U^ + }}(x)) \leqslant \alpha \} ) \cup \\ &\{ x \in \{ {x^ + }\} |P({Y^ + }|\delta _A^{{U^ + }}(x)) > \alpha \} \end{aligned}

 $P({X^ + }|\delta _A^{{U^ + }}(x)) = P(X|\delta _A^U(x)) > \alpha$

$x \in \delta _A^{{U^ + }}({x^ + }) - \{ {x^ + }\}$ ，根据定理1可得

 $P({X^ + }|\delta _A^{{U^ + }}(x)) \geqslant P(X|\delta _A^U(x))$

$P(X|\delta _A^U(x)) \leqslant \alpha$ ，因此 $P({X^ + }|\delta _A^{{U^ + }}(x))$ $\alpha$ 的大小无法确定，只有当 $P({X^ + }|\delta _A^{{U^ + }}(x)) > \alpha$ $x \in \underline N _A^{(\alpha ,\beta )}({X^ + })$ 。也就是说，只需对 $x \in \delta _A^{{U^ + }}({x^ + }) - \underline N _A^{(\alpha ,\beta )}(X)$ 中的对象进行具体的计算便可以完成最终的下近似更新，因此

 $\begin{split} \underline N _A^{(\alpha ,\beta )}({X^ + }) = \underline N _A^{(\alpha ,\beta )}(X) \cup \quad\quad\quad\\ \{ x \in \delta _A^{{U^ + }}({x^ + }) - \underline N _A^{(\alpha ,\beta )}(X)|P({X^ + }|\delta _A^{{U^ + }}(x)) > \alpha \} \\ \end{split}$

2)对于 $\forall x \in U - \underline N _A^{(\alpha ,\beta )}(Y)$ ，都有 $P(Y|\delta _A^U(x)) \leqslant$ $\alpha$ 。那么根据定理1，当 $x \in \delta _A^{{U^ + }}({x^ + }) - \{ {x^ + }\}$ 时，有

 $P({Y^ + }|\delta _A^{{U^ + }}(x)) < P(Y|\delta _A^U(x)) \leqslant \alpha$

$x \in U{}^ + - \delta _A^{U{}^ + }(x{}^ + )$ 时，有

 $P({Y^ + }|\delta _A^{{U^ + }}(x)) = P(Y|\delta _A^U(x)) \leqslant \alpha$

$P({Y^ + }|\delta _A^{{U^ + }}(x)) < P(Y|\delta _A^U(x))$ $P(Y|\delta _A^U(x)) > \alpha$

$x \in U{}^ + - \delta _A^{U{}^ + }(x{}^ + )$ 时，有

 $P({Y^ + }|\delta _A^{{U^ + }}(x)) = P(Y|\delta _A^U(x)) > \alpha$

$x \in \underline N _A^{(\alpha ,\beta )}({Y^ + })$ ，则需要对 $x \in \delta _A^{{U^ + }}({x^ + }) - \{ {x^ + }\}$ 中的对象计算 $P({Y^ + }|\delta _A^{{U^ + }}(x))$ 可得到新的下近似结果，即

 $\begin{split} \underline N _A^{(\alpha ,\beta )}({Y^ + }) = (\underline N _A^{(\alpha ,\beta )}(Y) - \quad\quad\quad\quad \\ \{ x \in \underline N _A^{(\alpha ,\beta )}(Y) \cap \delta _A^{{U^ + }}({x^ + })|P({Y^ + }|\delta _A^{{U^ + }}(x)) \leqslant \alpha \} ) \cup \\ \{ x \in \{ {x^ + }\} |P({Y^ + }|\delta _A^{{U^ + }}(x)) > \alpha \} \quad\quad \quad \\ \end{split}$

 $\begin{split} 1)\overline N _A^{(\alpha ,\beta )}&({X^ + }) = \overline N _A^{(\alpha ,\beta )}(X) \cup \\ &\{ x \in \delta _A^{{U^ + }}({x^ + }) - \overline N _A^{(\alpha ,\beta )}(X)|P({X^ + }|\delta _A^{{U^ + }}(x)) \geqslant \beta \} \\ \end{split}$
 $\begin{split} 2)\overline N _A^{(\alpha ,\beta )}&({Y^ + }) = (\overline N _A^{(\alpha ,\beta )}(Y) - \\ &\{ x \in \overline N _A^{(\alpha ,\beta )}(Y) \cap \delta _A^{{U^ + }}({x^ + })|P({Y^ + }|\delta _A^{{U^ + }}(x)) < \beta \} ) \cup \\ &\{ x \in \{ {x^ + }\} |P({Y^ + }|\delta _A^{{U^ + }}(x)) \geqslant \beta \} \\ \end{split}$

2.2 论域中对象减少时模型的增量式更新

1) $\forall x \in \delta _A^U(x{}^ - ) - \{ x{}^ - \}$ ,有

 $P({X^ - }|\delta _A^{{U^ - }}(x)) \leqslant P(X|\delta _A^U(x))$
 $P({Y^ - }|\delta _A^{{U^ - }}(x)) > P(Y|\delta _A^U(x))$

2) $\forall x \in U - \delta _A^U({x^ - })$ ,有

 $P({X^ - }|\delta _A^{{U^ - }}(x)) = P(X|\delta _A^U(x))$
 $P({Y^ - }|\delta _A^{{U^ - }}(x)) = P(Y|\delta _A^U(x))$

 $\begin{split} 1)\underline N _A^{(\alpha ,\beta )}&({X^ - }) = \underline N _A^{(\alpha ,\beta )}(X) - \{ x \in \underline N _A^{(\alpha ,\beta )}(X) \cap \\ &(\delta _A^U({x^ - }) - \{ {x^ - }\} )|P({X^ - }|\delta _A^{{U^ - }}(x)) \leqslant \alpha \} \quad\quad\quad\quad\quad \\ \end{split}$
 $\begin{split} 2)\underline N _A^{(\alpha ,\beta )}&({Y^ - }) = \underline N _A^{(\alpha ,\beta )}(Y) \cup \\ &\{ x \in \delta _A^U({x^ - }) - \{ {x^ - }\} - \underline N _A^{(\alpha ,\beta )}(Y)|P({Y^ - }|\delta _A^{{U^ - }}(x) > \alpha \} \\ \end{split}$

 $P({X^ - }|\delta _A^{{U^ - }}(x)) \leqslant P(X|\delta _A^U(x))$

 $P({X^ - }|\delta _A^{{U^ - }}(x)) \leqslant P(X|\delta _A^U(x)) \leqslant \alpha .$

2)对于 $\forall x \in \underline N _A^{(\alpha ,\beta )}(Y)$ ，若对象满足 $x \in \delta _A^U(x{}^ - ) -$ $\{ x{}^ - \}$ ，那么由定理4可以得到

 $P({Y^ - }|\delta _A^{{U^ - }}(x)) > P(Y|\delta _A^U(x)) > \alpha$

 $P({Y^ - }|\delta _A^{{U^ - }}(x)) = P(Y|\delta _A^U(x)) > \alpha$

 $P({Y^ - }|\delta _A^{{U^ - }}(x)) > P(Y|\delta _A^U(x)),P(Y|\delta _A^U(x)) \leqslant \alpha$

 $P({Y^ - }|\delta _A^{{U^ - }}(x)) = P(Y|\delta _A^U(x)) \leqslant \alpha$

$x \notin \underline N _A^{(\alpha ,\beta )}({Y^ - })$

 $\begin{split} 1)\overline N _A^{(\alpha ,\beta )}({X^ - }) = &\overline N _A^{(\alpha ,\beta )}(X) - \{ x \in \overline N _A^{(\alpha ,\beta )}(X) \cap \\ &(\delta _A^U({x^ - }) - \{ {x^ - }\} )|P({X^ - }|\delta _A^{{U^ - }}(x)) < \beta \} \quad\quad \\ \end{split}$
 $\begin{split} 2)\overline N _A^{(\alpha ,\beta )}&({Y^ - }) = \overline N _A^{(\alpha ,\beta )}(Y) \cup \\ &\{ x \in \delta _A^U({x^ - }) - \{ {x^ - }\} - \overline N _A^{(\alpha ,\beta )}(Y)|P({Y^ - }|\delta _A^{{U^ - }}(x) \geqslant \beta \} \\ \end{split}$

2)根据定理5的证明结果，对于 $\forall x \in \overline N _A^{(\alpha ,\beta )}(Y)$ 都有 $x \in \overline N _A^{(\alpha ,\beta )}({Y^ - })$ 。对于 $\forall x \in U - \overline N _A^{(\alpha ,\beta )}(Y)$ $x \in U - \delta _A^U({x^ - })$ 时，都有 $x \notin \overline N _A^{(\alpha ,\beta )}({Y^ - })$ ，只有当 $\forall x \in U - \overline N _A^{(\alpha ,\beta )}(Y)$ $x \in \delta _A^U(x{}^ - ) - \{ x{}^ - \}$ 时，不能直接确定 $P({Y^ - }|\delta _A^{{U^ - }}(x))$ $\alpha$ 之间的大小关系，因此定理6中的2)得到证明。

3 邻域决策粗糙集更新算法

2)信息系统增加对象集 $\Delta {U^ + } = \{ x_1^ + ,x_2^ + , \cdots ,x_s^ + \}$ ，新的邻域型信息系统为 ${{\rm{IS}}^ + } = ({U^ + } = U \cup \Delta {U^ + },{\rm{AT}})$ ，新的近似对象集为 ${X^ + }$

1)将初始时的信息系统 ${\rm{IS}} = (U,{\rm{AT}})$ 记为 ${{\rm{IS}}^{(0)}} = ({U^{(0)}},{\rm{AT}})$ $X$ 记为 ${X^{(0)}}$ ，近似集 $\underline N _A^{(\alpha ,\beta )}(X)$ $\overline N _A^{(\alpha ,\beta )}(X)$ 分别记为 $\underline N _A^{(\alpha ,\beta )}({X^{(0)}})$ $\overline N _A^{(\alpha ,\beta )}({X^{(0)}})$

2)对于 $x_i^ + \in \Delta {U^ + }(1 \leqslant i \leqslant s)$ ，将 $x_i^ +$ 加入邻域型信息系统 ${{\rm{IS}}^{(i - 1)}} = ({U^{(i - 1)}},{\rm{AT}})$ 中，新的信息系统表示为 ${{\rm{IS}}^{(i)}} = ({U^{(i)}},{\rm{AT}})$ ，其中 ${U^{(i)}} = {U^{(i - 1)}} \cup \{ x_i^ + \}$ ，此时新的近似对象集为 ${X^{(i)}}$

3)计算对象 $x_i^ +$ 的邻域类 $\delta _A^{{U^{(i)}}}(x_i^ + )$ ，然后根据定理2和定理3在 $\underline N _A^{(\alpha ,\beta )}({X^{(i - 1)}})$ $\overline N _A^{(\alpha ,\beta )}({X^{(i - 1)}})$ 的基础上增量式计算 $\underline N _A^{(\alpha ,\beta )}({X^{(i)}})$ $\overline N _A^{(\alpha ,\beta )}({X^{(i)}})$

4)重复迭代计算步骤2）~3）。

5)令

 $\begin{array}{l} \underline N _A^{(\alpha ,\beta )}({X^ + }) = \underline N _A^{(\alpha ,\beta )}({X^{(s)}});\\ \overline N _A^{(\alpha ,\beta )}({X^ + }) = \overline N _A^{(\alpha ,\beta )}({X^{(s)}}). \end{array}$

2)信息系统减少对象集 $\Delta {U^ - } = \{ x_1^ - ,x_2^ - , \cdots ,x_t^ - \}$ ，新的邻域型信息系统为 ${{\rm{IS}}^ - } = ({U^ - } = U - \Delta {U^ - },{\rm{AT}})$ ，新的近似对象集为 ${X^ - }$

1)将初始时的信息系统 ${\rm{IS}} = (U,{\rm{AT}})$ 记为 ${{\rm{IS}}^{(0)}} = ({U^{(0)}},{\rm{AT}})$ $X$ 记为 ${X^{(0)}}$ ，近似集 $\underline N _A^{(\alpha ,\beta )}(X)$ $\overline N _A^{(\alpha ,\beta )}(X)$ 分别记为 $\underline N _A^{(\alpha ,\beta )}({X^{(0)}})$ $\overline N _A^{(\alpha ,\beta )}({X^{(0)}})$

2)对于 $x_i^ - \in \Delta {U^ - }(1 \leqslant i \leqslant t)$ ，将 $x_i^ -$ 从邻域型信息系统 ${ {\rm{IS}}^{(i - 1)}} = ({U^{(i - 1)}},{\rm{AT}})$ 中移除，新的信息系统表示为 ${{\rm{IS}}^{(i)}} = ({U^{(i)}},{\rm{AT}})$ ，其中 ${U^{(i)}} = {U^{(i - 1)}} - \{ x_i^ - \}$ ，此时新的近似对象集为 ${X^{(i)}}$

3)计算对象 $x_i^ -$ 的邻域类 $\delta _A^{{U^{(i)}}}(x_i^ - )$ ，然后根据定理5和定理6在 $\underline N _A^{(\alpha ,\beta )}({X^{(i - 1)}})$ $\overline N _A^{(\alpha ,\beta )}({X^{(i - 1)}})$ 的基础上增量式计算 $\underline N _A^{(\alpha ,\beta )}({X^{(i)}})$ $\overline N _A^{(\alpha ,\beta )}({X^{(i)}})$

4)重复迭代计算步骤2)~3)

5)令

 $\begin{array}{l} \underline N _A^{(\alpha ,\beta )}({X^ - }) = \underline N _A^{(\alpha ,\beta )}({X^{(t)}});\\ \overline N _A^{(\alpha ,\beta )}({X^ - }) = \overline N _A^{(\alpha ,\beta )}({X^{(t)}}). \end{array}$

4 实验分析

 Download: 图 1 论域增加时两类算法的更新用时比较 Fig. 1 Comparison of update time of two algorithms when universe is added

 Download: 图 2 论域减少时两类算法的更新用时比较 Fig. 2 Comparison of update time of two algorithms when universe is reduced

 Download: 图 3 数据集pima在不同邻域半径下算法更新用时比较 Fig. 3 Comparison of algorithm updating time of pima data set under different neighborhood radius
 Download: 图 4 数据集wdbc在不同邻域半径下算法更新用时比较 Fig. 4 Comparison of algorithm updating time of wdbc data set under different neighborhood radius
 Download: 图 5 数据集biodeg在不同邻域半径下算法更新用时比较 Fig. 5 Comparison of algorithm updating time of biodeg data set under different neighborhood radius
 Download: 图 6 数据集musk在不同邻域半径下算法更新用时比较 Fig. 6 Comparison of algorithm updating time of musk data set under different neighborhood radius
5 结束语

 [1] YAO Yiyu. Three-way decisions with probabilistic rough sets[J]. Information sciences, 2010, 180(3): 341-353. DOI:10.1016/j.ins.2009.09.021 (0) [2] YAO Yiyu. The superiority of three-way decisions in probabilistic rough set models[J]. Information sciences, 2011, 181(6): 1080-1096. DOI:10.1016/j.ins.2010.11.019 (0) [3] 陈家俊, 徐华丽, 魏赟. 多重代价多粒度决策粗糙集模型研究[J]. 计算机科学与探索, 2018, 12(5): 839-850. CHEN Jiajun, XU Huali, WEI Yun. Multi-cost based multi-granulation decision-theoretic rough set model[J]. Journal of frontiers of computer science and technology, 2018, 12(5): 839-850. DOI:10.3778/j.issn.1673-9418.1705019 (0) [4] JIA Xiuyi, LI Weiwei, SHANG Lin. A multiphase cost-sensitive learning method based on the multiclass three-way decision-theoretic rough set model[J]. Information sciences, 2019, 485: 248-262. DOI:10.1016/j.ins.2019.01.067 (0) [5] 张婷, 张红云, 王真. 基于三支决策粗糙集的迭代量化的图像检索算法[J]. 南京大学学报(自然科学版), 2018, 54(4): 714-724. ZHANG Ting, ZHANG Hongyun, WANG Zhen. image retrieval: Iterative quantization based on saliency detection and three-way decision based rough sets[J]. Journal of Nanjing University (natural sciences edition), 2018, 54(4): 714-724. (0) [6] ZHAO Xuerong, HU Baoqing. Three-way decisions with decision-theoretic rough sets in multiset-valued information tables[J]. Information sciences, 2020, 507: 684-699. DOI:10.1016/j.ins.2018.08.024 (0) [7] LIU Dun, LIANG Decui, WANG Changchun. A novel three-way decision model based on incomplete information system[J]. Knowledge-based systems, 2016, 91: 32-45. DOI:10.1016/j.knosys.2015.07.036 (0) [8] SUN Bingzhen, MA Weimin, ZHAO Haiyan. Decision-theoretic rough fuzzy set model and application[J]. Information sciences, 2014, 283: 180-196. DOI:10.1016/j.ins.2014.06.045 (0) [9] DOU Huili, YANG Xibei, SONG Xiaoning, et al. Decision-theoretic rough set: a multicost strategy[J]. Knowledge-based systems, 2016, 91: 71-83. DOI:10.1016/j.knosys.2015.09.011 (0) [10] LI Weiwei, HUANG Zhiqiu, JIA Xiuyi, et al. Neighborhood based decision-theoretic rough set models[J]. International journal of approximate reasoning, 2016, 69: 1-17. DOI:10.1016/j.ijar.2015.11.005 (0) [11] SHU Wenhao, QIAN Wenbin, XIE Yonghong. Incremental approaches for feature selection from dynamic data with the variation of multiple objects[J]. Knowledge-based systems, 2019, 163: 320-331. DOI:10.1016/j.knosys.2018.08.028 (0) [12] 段海玲, 王光琼. 一种高效的复杂信息系统增量式属性约简[J]. 华南理工大学学报(自然科学版), 2019, 47(6): 18-30. DUAN Hailing, WANG Guangqiong. An efficient incremental attribute reduction for complex information systems[J]. Journal of South China University of Technology (natural science edition), 2019, 47(6): 18-30. (0) [13] HU Chengxiang, ZHANG Li, WANG Bangjun, et al. Incremental updating knowledge in neighborhood multigranulation rough sets under dynamic granular structures[J]. Knowledge-based systems, 2019, 163: 811-829. DOI:10.1016/j.knosys.2018.10.010 (0) [14] ZHANG Qinghua, LV Gongxun, CHEN Yuhong, et al. A dynamic three-way decision model based on the updating of attribute values[J]. Knowledge-based systems, 2018, 142: 71-84. DOI:10.1016/j.knosys.2017.11.026 (0) [15] XU Jianfeng, MIAO Duoqian, ZHANG Yuanjian, et al. A three-way decisions model with probabilistic rough sets for stream computing[J]. International journal of approximate reasoning, 2017, 88: 1-22. DOI:10.1016/j.ijar.2017.05.001 (0) [16] CHEN Hongmei, LI Tianrui, LUO Chuan, et al. A decision-theoretic rough set approach for dynamic data mining[J]. IEEE transactions on fuzzy systems, 2015, 23(6): 1958-1970. DOI:10.1109/TFUZZ.2014.2387877 (0) [17] 赵小龙, 杨燕. 基于邻域粒化条件熵的增量式属性约简算法[J]. 控制与决策, 2019, 34(10): 2061-2072. ZHAO Xiaolong, YANG Yan. Incremental attribute reduction algorithm based on neighborhood granulation conditional entropy[J]. Control and decision, 2019, 34(10): 2061-2072. (0) [18] 杨臻, 邱保志. 混合信息系统的动态变精度粗糙集模型[J]. 控制与决策, 2020, 35(2): 297-308. YANG Zhen, QIU Baozhi. Dynamic variable precision rough set model of mixed information system[J]. Control and decision, 2020, 35(2): 297-308. (0) [19] LUO Chuan, LI Tianrui, ZHANG Yi, et al. Matrix approach to decision-theoretic rough sets for evolving data[J]. Knowledge-based systems, 2016, 99: 123-134. DOI:10.1016/j.knosys.2016.01.042 (0)