基于矩阵的粗糙集上下近似的计算方法

Matrix-Based Computational Method for Upper and Lower Approximations of Rough Sets

借助于论域子集的布尔列矩阵表示的思想,引入等价关系矩阵的诱导矩阵和矩阵的λ-截矩阵等概念,提出Pawlak粗糙集模型中概念上、下近似计算的矩阵方法,即利用论域子集的布尔列矩阵、论域上的等价关系矩阵和诱导矩阵三个矩阵间的运算来计算该子集的上、下近似集,并从理论上证明该方法的正确性.然后给出运用该方法计算论域子集上、下近似的计算步骤.进一步将该方法和计算步骤推广至变精度粗糙集模型中上、下近似的计算.该方法填补用矩阵方法描述粗糙集时下近似计算方面的空缺,从而统一Pawlak粗糙集模型和变精度粗糙集模型中的上、下近似的矩阵计算方法.

The concepts of an induced matrix of equivalence relation matrix and a λ-cat matrix of matrix are introduced on the basis of Boolean column matrix representation of subsets in the universe. Then, a matrix method for computing upper and lower approximations of a concept in Pawlak rough set model is proposed, which means the upper and lower approximations of a subset can be derived from the operation among the Boolean column matrix of the subset, the equivalence relation matrix of the universe and the induced matrix. The correctness of the proposed method is proved. In addition, an algorithm for computing upper and lower approximations of a subset by the proposed method is given. Furthermore, the proposed method and its algorithm are generalized to compute upper and lower approximations of a concept in variable precision rough set model. The vacancy on the computation of lower approximation is filled by the proposed method. Therefore, it unifies the two matrix-based methods for computing upper and lower approximations of a concept in Pawlak rough set model and variable precision rough set model.