Roughness and granularity measures using Hausdorff metric: a new approach

摘要

Measures for uncertainty due to approximation of sets in rough set theory are accuracy and roughness. In determining these quantities, the cardinality of a set is always used and never the numerical values of the attributes (if they exist) of elements in the sets. Therefore, distances between the exact set and the corresponding upper and lower approximations can give a better quantitative measure of the roughness. Here, we propose a measure based on Hausdorff metric which takes into account the distance between two sets, the exact set and its two approximations (lower and upper). Using this measure, we can quantify the uncertainty of a rough set based on the values in the domain of sample points but not on the basis of number of sample points. Also, we propose a new measure for granulation which is again based on the Hausdorff metric. The effectiveness of the proposed measures is demonstrated on a synthetic data.